Press release [Feb. 19, 2015]

Richard Woods, Georgia's School Superintendent Educating Georgia's Future
State Board of Education approves traditional/discrete math options
MEDIA CONTACT: Matt Cardoza, 404-651-7358, mcardoza@gadoe.org or Meghan Frick, 404-656-5594, mfrick@doe.k12.ga.us Follow us on Twitter and Facebook
February 19, 2015 State School Superintendent Richard Woods announced today that the State Board of Education has approved new math courses that offer a traditional/discrete option for Georgia high school students.
Starting with the 2015-16 school year, high schools will be able to offer either integrated or traditional/discrete math courses.
"I have heard many times from students, teachers, and parents, that the lack of a traditional/discrete math option is an enormously troubling issue," Superintendent Woods said. "Today, thanks to the collaboration of the Governor's office and

State Board of Education, we have been able to resolve that issue in a way that offers a choice at the local level."
Surveys conducted by the State Board of Education and Regional Educational Service Agencies found that many educators desired a choice between integrated and traditional/discrete math. Public comment on the courses posted today was largely favorable.
"This is good news for Georgia's educators, parents, and students," State Board Chair Helen Rice said. "These courses are a response to feedback from all of those groups they are specific to the needs and responsive to the voices of Georgia's students."
More Information and Supporting Documents

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Geometry Standards Survey Rating and Comments

Please indicate your level of agreement with the following statements about the Geometry Standards.

The DRAFT Geometry Standards:

are clear and understandable

Strongly Agree 26.1% 43

Agree
58.8% 97

Disagree
12.1% 20

Strongly Disagree
1.8% 3

Don't Know 1.2%
2

# of Responses
165

define what students should

know and be able to do to prepare them for success in required high

26.1% 60.0% 10.3%

43

99

17

2.4% 4

1.2% 2

165

school mathematics courses

are appropriate for the next level or grade in preparation for college and career readiness
are rigorous enough to challenge learners
provide sufficient opportunity to revisit and enhance student understanding of foundational algebra concepts
are consistent with postsecondary and business/industry standards

28.0% 53.7% 11.0%

46

88

18

2.4% 4

4.9% 8

164

37.8% 56.1% 3.7%

62

92

6

0.0% 0

2.4% 4

164

31.1% 50.0% 11.0%

51

82

18

3.7% 6

4.3% 7

164

25.0% 55.5% 7.3%

41

91

12

3.7% 6

8.5% 14

164

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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1

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Please enter any additional comments you would like to make concerning the DRAFT Geometry Standards.
Comments:
Appropriate for 3rd year emotionally and academically prepared Math students. As a teacher and a parent in the state of Georgia, the integrated curriculum is not working for these students. We have to touch on so many different things during the year that the students never fully grasp the concept of the material that is being taught. Going back to traditional coverage will allow the students to have a better understanding of the material before moving on to the next course.
As a teacher who has taught QCC, GPS, and now CCGPS, I feel that GPE1 and GPE2 belong in a future course. In fact, I think that Conic Sections period belong in Advanced Algebra and not in Pre-Calculus. Things are too scattered. It will be hard for kids to make connections with conic sections if you teach two of the conics in Geometry and the other two in Pre-Calculus. Why not consider teaching all 4 in Advanced Algebra so that you can go deep and compare and contrast the conics? Conics fit perfect when we had that in Algebra II in QCC Land and Math III in GPS Land. Students had the proper background. If you are going to do this, I would suggest moving the completing the square method of solving quadratics from your proposed Algebra I standards to Advanced Algebra where you could teach this first before students could start creating equations in standard form for the Conics.
As long as these standards match up to what the students are expected to do in college, then they are fine. Being both an experienced teacher and parent, I have witnessed the stress and confusion of many students, teachers, and parents over the last several years of constantly changing math courses, names, and sequences. Local school boards should be given the option to choose the traditional course sequence.
Conic sections used to be part of Advanced Algebra and should still be taught to juniors rather than sophomores. Conics with the parabolas should be in Advanced Algebra instead of Geometry. Hope we go back to the more traditional Geometry. The integrated experiment has been a disservice to the students and teachers. Go to Geometry, get textbooks and stay with it so we can get rigor back into the Math class in a reasonable way. Mile wide inch deep is a failure.
Courses in Georgia continue to be drafted and implemented for remediation purposes. Has anyone studied the number of remedial courses offered at 2-year colleges? This semester I'm teaching Geometry A to a class of 32 students, more than half have failed this course at least once. Two seniors are enrolled. Students taking the course for the first time failed the prerequisite course at least once. What's wrong with this picture? I don't see the math difficulties that were addressed years ago improving...is anyone taking my concerns and comments seriously?
Draft Geometry look good. This new path will give teacher enough time to cover all units without rushing. Finally.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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2

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Finally...standards that are clear! The old standards were ambiguous (and had room for lots of interpretation). If these standards are adopted, teachers will now know exactly what they should be teaching!! Geometry has been taught for decades. I hope these standards are the most logical in sequence and someone compared them to many other schools' Geometry curriculum! Why reinvent a course that should be standardized? Better yet, find a good text book about Geometry and copy the book's (and math experts) sequence of learning objectives and teach that! Please provide these books for our students in Fulton County!
Geometry is a course that is learned and developed with time as students discover the relationship between early and later content. I believe this will allow them the opportunity to better grasp spatial concepts than in some of the previous courses.
GPE.1, GPE.2: These standards don't seem to fit in Geometry. I think they belong with the rest of the conics sections in a later course. MG.1, MG.2: seem to fit better in a later course as well. I am pleased with the way the standards are worded. They seem clearer. I hope I will be more aware of what they mean and what to expect from my students.
How many jobs would Geometry classes prepare students for in the "real" world? Wouldn't it be better to just integrate Geometry with one of the Algebra classes? I always felt like placing Geometry between the two algebra courses was odd, but it did make more sense when I was in school because of the varying tracks students could take (which was not a bad way to go especially with all the differentiation teachers are supposed to do and would be helpful if students/parents could differentiate their course selection). Now, though, with all students being expected to have specific math courses, why keep geometry in the middle. Wouldn't math flow better if the two algebra courses followed each other?
I am glad this course will be an option. I am not sure the standards prepare students for anything except college level courses and college level problem solving.
I am very pleased to see these three options being offered. Many 9th grade students need more support to build number sense and an understanding of operations in algebra, and Algebra I should focus on this. Specifically, I am glad to see the simple calculation topics like transformations moved to geometry. Algebra I should have a focus on the operations and their role in simplifying expressions and solving equations.
I believe focusing on a discrete area of mathematics allows for a deeper understanding of that content, rather than skimming many domains for coverage. I believe the standards for the course is about the same and the students will be able to accomplish completing these standards. I believe we should remove Standards GPE.1 and GPE.2, to be placed with the other conics standards. MG.1 and MG.2 should also be removed from the Geometry course and placed in more appropriate modeling courses. I truly believe that to fully understand the standards, students may need a year and a half to study the current standards.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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3

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
I DO NOT agree to have districts choosing between discrete or integrated. We need consistency! This is going to be a huge mess when students transfer between districts. The vote was 85% in favor of discrete what happened? I do not understand why there are statistics standards in a discrete Geometry course. I think they should be shifted over to the course called statistics. They do not belong in Geometry. Period. I feel that there are too many standards to cover at an in-depth level in one year. I feel the following standards should be moved to being taught with other conic sections: GPE.1 and GPE.2. I do not feel that MG.1 and MG.2 fit in with this course, and would be more appropriate in the modeling unit in Advances Algebra.
I feel the trigonometry should be removed so that teachers could do more in-depth study of relationships among the different functions and graphs. Leave the unit circle, trig graphs, and identities for a pre-calculus class. I like Option 1 for the Math Courses to take in High School. This is what I took back in the 70's and it has served me well. Traditional math and teaching students how to think and solve problems is very important! It is also a lot easier to understand when the courses are not Integrated.
I like this. I think putting all geometry standards together and all algebra standards together will be helpful in developing the needed skills for students to be successful in Advanced Algebra.
I think that these standards represent a much more reasonable amount of content to cover thoroughly in a school year. It is also good to see that the standards are now more related to each other rather than having multiple areas of mathematics crammed into a single course.
I think that those of us who understand the standards and how they are connected to each other can teach them in just about any order. What we want is for these courses/standards to STOP CHANGING! We really cannot truly tell how students are progressing because we are having to compare apples to oranges every other year so it seems. Teachers have not adjusted to the new courses because they are still trying to use old, outdated methodologies that are not meeting the needs of our students. Instead of changing the courses, let's add some rigorous professional development. If we change the expectation of the way we teach mathematics, it will definitely have a greater impact on the way students learn and progress.
I think the Geometry standards ask too much of our regular ed students and there is too much content to teach in the amount of time given to teach the material.
I think the traditional way of teaching is more effective. Not including the algebra and probability in the standards will make the impact of the geometry taught more meaningful. I think these standards work.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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4

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
I would like the standards to address how to deal with students who have deficits in basic math skills such as multiplication and division. Also, if the whole class fails a unit, what happens? Currently, the teacher offers a recovery process after the unit test. She teaches nothing. She just let's them learn how to cheat by taking and retaking an online assessment up to five times. Their grade can be raised in her grade book by learning the process of elimination in the recovery process, but none of the class learned the concepts in the unit.
I would like to see an example provided for Geometry MCC9-12.G.GPE.6. This cannot be called Discrete Geometry considering that we are still including Statistics standards. However, I understand that it is not integrated either. The subtraction of a few standards from the integrated tenth grade course does not make this course discrete. I am glad that they removed S.CP.8 as well as G.SRT.9-11 as recommended in the previous survey. The Foundations of Algebra looks great for those that cannot handle AMDM or Pre-Calculus senior year. However, how does this affect graduation and subsequent post-secondary admittance? We need clear guidelines on this. The creation of the EOC tests needs to be completed in a timely enough manner so that teachers get study guides and sample questions--well before the test is administered. I would like to see the conics section in the third year of math, not with Geometry. If possible I would like to see the following standard moved to a later course to teach all of the conics together. MCC9 - 12.G.GPE.2 Derive the equation of a parabola given a focus and directrix. Also, you may want to move the quadratic unit back into this course and take it out of the Algebra 1 course. The Algebra 1 course has too many standards. Thank you for all that you do.
If the class is Geometry, then why are students studying typically algebra concepts such as complex numbers, quadratics, and statistics? Too much content for support children to learn in a short amount of time.
If this course is designed for students who were struggling and are more of the applied/remedial type of student who will not be going to college, I believe that there could be less proofs required, similar to the Informal Geometry class that schools used to offer.
If we are making a wholesale return to traditional rather than integrated math courses, I think this course is reasonable, appropriate, and desirable. As I expressed at length in my comments regarding the Algebra I course, my larger concern is the coherence of our high school mathematics curriculum and the provision for different levels of performance. Just changing course names and restacking standards isn't going to full address the underlying issues.
Keep Geometry in one class and stop throwing it into all classes. The scattering of topics through out the other classes is not helping the students understand Geometry. They cannot relate one topic to another with things scattered into different classes. Go back to just teaching Geometry.
Let geometry be geometry and put algebra standards in algebra!!! Make sure that all Special Educations students have a clear path for graduation at every level. MG #1 and MG #2 do not meet standards and should be removed.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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5

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Most of the Geometry standards ask students to prove. Because Geometry is so scattered before students get to Geometry, they do not have a strong enough basis of the properties they need to know before they have to prove. Students need to be able to develop the concepts before having to prove them. I do like that the standards are more clear and easier to read. There is also not enough time to properly teach the content to where students understand the content fully. I also feel that GPE.1 and GPE.2 should be moved to a later course to be taught with all other conics. MG.1 and MG.2 should be moved to the modeling unit in Advanced Algebra.
Most of the standards are very clear. A few on circles could use additional definition as to what exactly is expected. Move the conic sections of circles and parabolas to Advanced Algebra. This information should not be a separated or chunked standard! We strongly believe these standards should be together representing all the ways to slice a cone.
Move the conic sections of circles and parabolas to Advanced Algebra. We strongly believe these standards should be together representing all the ways to slice a cone. Move the conic sections of circles and parabolas to Advanced Algebra. We strongly believe these standards should be together representing all the ways to slice a cone. One or more job descriptions for each chapter! Please continue to move backwards towards the old math standards (you know the one from 10 years ago that worked and made sense.... when subjects weren't disjointed facts).
Please decide traditional vs integrated for the entire state! One-in-six students transfer schools each year! Having different schools on different math tracks with continue to make things difficult. Sure, some people will be angry, but we must do what is best for kids!
PLEASE let us teach using the Discrete Model. Please move the conic sections of circles and parabolas to Advanced Algebra. We believe that this information should not be separated. We believe these standards should be together representing all the ways to slice a cone.
Please remove GPE.1, GPE.2, MG.1, and MG.2. Remove the following standards. These standards should be grouped with other conics. MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.
Some standards seem unnecessary, specifically, MCC9-12.G.GPE.6 (partitioning a line), MCC9-12.G.GMD.1, 2 (Cavalieri's principle). Concept of radians should be introduced at the appropriate time that it will be used...with advanced trig topics. Higher-level statistics and probability concepts should be included in a separate course.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

6

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Standards are not always clear. For example, one or more standards say "Prove theorems about triangles. Theorems include....." Why not state exactly which theorems to prove? "Theorems include" leaves the reader/teacher believing there could be others assessed on the EOC. Second issue is that there is no need for Statistics and Probability to be included in a Geometry course. Tell the public that the course is still integrated and quit trying to trick them into believing that we've gone back to the days of Alg I, Geom, Alg II. Or take out the Statistics and Probability. Ridiculous. Period.
Take out the statistics and probability. Thank you for putting geometry together again. Students will be able to make stronger connections to the information because there is once again a systematic approach to the presentation of the material.
Thank you, GADOE, for a focus on providing students in GA with different paths to success. Let's keep this going!!! The accelerated courses are too long for one semester. The following standards should be combined with the other conic sections (hyperbola and ellipse), taught in the 3rd year of math (algebra II or advanced algebra): MCC912.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.
The following standards should be combined with the other conic sections (hyperbola and ellipse), taught in the 3rd year of math (algebra II or advanced algebra): MCC912.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix
The following standards should be combined with the similar standards with the other conic sections (hyperbola and ellipses) in the third year course: Translate between the geometric description and the equation for a conic section: MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.
The following two standards NEED to be combined with a FULL conic representation with hyperbolas and ellipses. Whether it is combined in Geometry or the third level where hyperbolas and ellipses are now. It doesn't matter. Translate between the geometric description and the equation for a conic section: MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix. As for the amount of content in each class. It is overwhelming and does not allow for a very rich or deep learning of important content. That is disappointing since we have been told for years this transition would lead to more depth and less width of content.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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7

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
The geometry needs to be taught in one full year unlike presently taught over three years. The students need the consistency of these standards in one year to fully grasp the concept involved in geometry. Most of the time we have to reteach the previous years geometry because the students don't remember or the teacher didn't cover it well.
The Geometry standards seem appropriate. If these are passed, resources will need to be developed to include algebraic application of these geometric standards so students do not loose their algebraic skills between Algebra I and Advanced Algebra. This (students not applying algebra skills to Geometric problems) was a frequent problem when we taught QCC Algebra I, QCC Geometry, and QCC Algebra II.
The majority is old Geometry, which is good. The previous number of standards for 10th grade was excessive. This is much more appropriate and more logical for the student and the teacher. The stand-alone Geometry course (the word "discrete" has a specific meaning in mathematics, so its use here is confusing) is certainly preferable to the integrated courses. It would be even better if the entire year was devoted to Geometry topics, but I supposed probability had to be squeezed in somewhere. As far as clarity, again, it is better than the current standards, but the CCGPS standards are still not logically organized from a teacher's perspective. The old QCC standards were more logical as they were organized by skill progression, so teachers could actually plan their calendars based on the standards.
The standards G.GPE.1, G.GPE.2, and G.GPE.3 should be moved to the Advanced Algebra/Algebra 2 course where the remainder of conics will be learned. This course is FULL already and does not need these standards. Also, G.MG.1, G.MG.2 and G.MG.3 should be moved to the Advanced Algebra/Algebra 2 course as well for age appropriateness. The course is FULL it will have plenty in it with all of the Geometry plus Data. Please move these standards where they belong. They don't belong in this course for tenth graders.
The standards have been simply pulled from the current common core standards. We currently teach these standards along with about a semester of algebra standards. How is it any better to just cut the standards in half? Why not remove the algebra standards AND add enriching standards to geometry that include all points of concurrency, area of polygons, a unit for surface area and volume, a unit for parallel and perpendicular lines, a basic geometry unit and a unit for reasoning and proofs all by itself? Students are not able to connect the current standards because we are not teaching basics because they are not in the standards
There are far too many standards for Geometry to be covered in one year. They are very in depth standards that the students need more time to work through in order to succeed.
There seems to be too many standards in this course. Take out the Statistics and Probability content.... it does not relate to Geometry. Let those concepts be covered in the later math course or in college.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

8

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
These are the skills my higher-level students are missing as a result of the GPS and CCGPS curricula. Thank you!!! I would leave out the "modeling with Geometry" standards. That can be covered in a higher-level math course. What about "honors" versions of these courses instead of another "accelerated" track? Due to the structure of the accelerated track, students are not able to move into "honors" math courses unless they were there from day 1. Some students end up in regular math classes due to poor registration practices and then are later recommended for honors math. With the accelerated tracking, they are left out. A kid should be able to move into Honors Geometry after excelling in an Algebra 1 course. These standards are much more suitable to a year long course. The students will have much more time to understand the material thoroughly without feeling rushed. These standards need to be combined with the other conic sections i.e., hyperbola and ellipses. Translate between the geometric description and the equation for a conic section: MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.
These standards need to be taught with the other conic sections such as hyperbola: MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.
These standards should be combined with the conics section. Translate between the geometric description and the equation for a conic section: MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix. These standards should be combined with the conics section. Translate between the geometric description and the equation for a conic section: MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.
This is a general comment not specific to the math standards. Georgia middle and high schools need to be reinvented from the ground up. The vast majority of students leave school unprepared for work or higher education. First, many (20-30%) drop out with no skills. Next, many actually graduate with a diploma, but really have no marketable skills. A large percentage of those who go on to college or tech school are unprepared and never complete a degree. Even those who complete a degree are often lacking the hard and soft skills employers demand. All in all, just look at the examples of your friends, neighbors, and relatives and you will see my point here. Georgia needs to reinvent education grades 6-16 into a comprehensive, sequenced 6, 8, and 10-year program to prepare young people to enter the workforce. Extensive use of practical, hands-on skill development, internships, and apprenticeships should be integral to this approach. Joe Strickland

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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9

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
This is a step in the right direction. What also needs to change is the state's approach with the Accelerated track. We should go back to the model with the honors level course meant a meatier course instead of a track meet trying to finish the standards. Especially in the 10th grade accelerated, it's nearly impossible to complete the course in the time provided and give students what they need to be successful in PreCalculus and beyond.
This is very similar to what we teach now. He is still lacking the influence of proofs and the addition to probability is a little difficult to integrate in Geometry (I like it in Algebra better). Over all very pleased to see the Quadratic unit removed!! :) This should help students who are in need of extra help. Too many concepts to teach and it is hard for students to retain all concepts. Transformations are already done in 8th grade. It should not be done in 10th grade. Modeling Geometry should move to 11th grade. We have lost the beauty and structure of Euclidean Geometry. A great loss. We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!! We need textbooks!!!!!!
Why don't we teach standard deviation instead of just mean absolute deviation? In every other statistics course you are required to use standard deviation, so why isn't it in the standards? Why is graphing circles from a given equation considered a geometry skill? Conic sections should be moved to Algebra 2. Glad there is more focus on actual geometry topics. With this course being "lighter", how heavy will advanced algebra become?

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
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10

Geometry Standards

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Algebra I Standards Survey Rating and Comments
Please indicate your level of agreement with the following statements about the Algebra I Standards.

The DRAFT Algebra I Standards:

are clear and understandable

Strongly Agree 25.7% 47

Agree
59.6% 109

Disagree
10.4% 19

Strongly Disagree
2.2% 4

Don't Know 2.2%
4

# of Responses
183

define what students should

know and be able to do to prepare them for success in required high

24.6% 57.9% 13.1%

45

106

24

2.2% 4

2.2% 4

183

school mathematics courses

are appropriate for the next level or grade in preparation for college and career readiness
are rigorous enough to challenge learners
provide sufficient opportunity to revisit and enhance student understanding of foundational algebra concepts
are consistent with postsecondary and business/industry standards

28.0% 53.3% 9.3%

51

97

17

3.8% 7

5.5% 10

182

36.8% 56.0% 1.6%

67

102

3

1.6% 3

3.8% 7

182

26.4% 51.6% 8.2%

48

94

15

8.8% 16

4.9% 9

182

28.6% 58.2% 4.4%

52

106

8

1.6% 3

7.1% 13

182

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

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Algebra I Standards

1

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"

Please enter any additional comments you would like to make concerning the DRAFT Algebra I Standards.
Comments: A heavy course for 9th graders. Better than the Coordinate Algebra Course. Adding all quadratics, graphing, factoring, solving and completing the square is too much for 9th grade. Some standards were removed but things that have been added will take more time that what was taken out. Completing the square is not needed until Conics and should be taught in Advanced Algebra. Graphing quadratics and describing max, min, vertex, symmetry should not be taught in Algebra 1. We can teach factoring so that it can be used in the Geometry course.
Algebra has been taught for decades. I hope these standards are the most logical in sequence and someone compared them to many other schools' Algebra curriculum! Why reinvent a course that should be standardized? Better yet, find a good text book about Algebra and copy the book's (and math experts) sequence of learning objectives and teach that! Please provide these books for our students in Fulton County!
Algebra needs to be taught in the traditional way. Very important!!! I use what I learned in Algebra almost every day.
As a professional who has used statistics for the past 45 years, I have a quibble over the omission of 'standard deviation' in describing the variability of data. Mean average deviation (MAD) is a fine thing to talk about; however, omitting the standard deviation would disadvantage anyone going on to more advanced statistics topics as it enters into much of what is involved in looking at errors of prediction (especially in linear models) and is even directly involved in computing things like correlation coefficients and the coefficients that go into linear prediction models. MAD doesn't enter into any of that. So, ADD 'standard deviation' to the set of things to be covered dealing with statistics. To be a little more helpful with statistics that often show up in professional publications, also cover '5 number summaries' (min, 25th %ile, 50th %ile, 75th %ile and max)!
As long as the standards match what the students are expected to do at the college level, they are fine. Clarification is needed in some standards. In some standards functions are defined as linear, exponential and quadratic, but not in all standards. Can it be any other functions? Clear College Prep curriculum. Completing the square and exponential functions should not be in this course. Ninth grade students will not be successful in completing the square. It needs to be moved to advanced algebra. Also, why are these standards listed as Algebra I and the course map still has it has Coordinate Algebra? Coming full circle.
Consider the ramifications of allowing districts to choose between offering a discrete and integrated math curriculum. It is quite challenging to fit transfer students from one curriculum into the other when they change schools. Regardless of which they come from and go to, there will be significant gaps in their learning.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

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Algebra I Standards

2

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Courses in Georgia continue to be drafted and implemented for remediation purposes. Has anyone studied the number of remedial courses offered at 2-year colleges? This semester I'm teaching Geometry A to a class of 32 students, more than half have failed this course at least once. Two seniors are enrolled. Students taking the course for the first time failed the prerequisite course at least once. What's wrong with this picture? I don't see the math difficulties that were addressed years ago improving...is anyone taking my concerns and comments seriously?
Finally. FINALLY - quadratics back where it should be - in the Algebra course!!! First and foremost. We need Textbooks!!!!!!!!!!!!!!!!!!!!!!!!!! Go back to QCC's. Be specific. Line item standards. This document is a waste. Teachers do not have the time to read the extensive rhetoric of the document. Go back to the old courses. There is not enough reinforcement in these courses and the students are forgetting the material. The old courses retaught and reinforced the material on different levels. They also separated the courses so that the topics were not jumbled...they flowed together. Right now they are just scrambled and nothing flows to help students relate one topic to the next.
Graphing exponential functions used to illustrate geometric sequences and their rates of change is fine, but further study of exponential functions should not be included with freshman level mathematics. Exponential functions should be studied fully in a 2nd or 3rd year math course along with logarithms. Ideally, rates of change could be covered well using the quadratics vs linear, so there is no real need for exponential functions in the 9th grade course. Also, we should not be asking students to memorize a bunch of useless information, but provide them with any formula resource they might need and expect them to illustrate that they understand how to USE the formula effectively. Great draft and makes logical mathematical sense for both educators and students
Hope we go back to the more traditional Algebra I. The integrated experiment has been a disservice to the students and teachers. Go to Algebra I, get textbooks and stay with it so we can get rigor back into the Math class in a reasonable way. Mile wide inch deep is a failure.
How does this differ from the CCGPS course? Why are the topics the same in both courses? I am a supporter of discrete math -- Algebra I, Geometry, Algebra II, Pre-Calc, Calc. GA Milestones EOC exams should be realigned to match. In Fulton County, we teach many students at an accelerated rate -- 9th and half of 10th in 1 year. They should take an EOC for this course, and not for just 9th grade. ALSO, if they are in the 8th grade taking this course, they should not be required to take the 8th grade End of Grade Math test -- they make 100's anyway and taking this exam and the day or 2 to review for it removes precious days away from learning.
I am concerned with including exponential functions in this course. I believe that linear and quadratic functions are enough to tackle in one course. I am glad this course will be an option.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

An Equal Opportunity Employer

Algebra I Standards

3

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
I am still concerned about the time allotted for these standards. The time allocated to the standards that were removed and the standards that were added are not the same. Some of the material included in this course used to be taught in the Algebra 2 course. I have found that students do not have enough time to fully grasp a concept before moving on to another concept. I have seen the results of this in my upper level courses. If this means that some of the topics need to be moved to Advanced Algebra or Pre-Calculus, then that is what needs to happen.
I appreciate that schools will be given the option to teach a "discrete" algebra and geometry curriculum. I wish the standards were written with the same clarity and specificity as the old GPS standards were. I do not like having to read and interpret an entire paragraph to figure out how it applies to my students, and it still does not say how in-depth or how high of a standard to which the students will be held.
I appreciate the work that has gone into making our high school math courses more manageable for students and teachers. I am, however, concerned about districts being able to offer discrete or integrated options. With the transient nature of our society and knowing how many issues our system faced when we had students transfer in from the GPS Algebra track versus the Math I track, it would have been nice to keep everyone in our state following one path.
I believe that allowing students to learn in more isolated domains will allow them to go more in depth with that area of content, rather than "skimming" the surface trying to cover many domains. I think this will allow for better understanding of the concepts in the subject area. I believe that the Algebra 1 standards are decent. I believe that the Algebra 1 standards are advanced for the students. I believe the changes are too difficult for ninth graders. They struggle with the standards as is and will fall further behind with these new standards. It is very unrealistic! I DO NOT agree to have districts choosing between discrete or integrated. We need consistency! This is going to be a huge mess when students transfer between districts. The vote was 85% in favor of discrete what happened? I do not feel it is realistic to expect freshman to be able to do quadratics and exponential functions during the ninth grade year. I do not feel like the time is available. I feel that if one must stay, then quadratics should stay; however, I feel quadratics and exponential both fit better in advanced algebra where polynomials are covered. If something must fill the void in Algebra 1 and that's all you are trying to do, then move more stats/probability to it.
I don't, and have never understood why you must have standards sound like they were written by Math Lawyers they trying to come across as super smart. Why can't they be written in a way that a parent could actually understand. As a Math Teacher I understand but know no one else probably does.
I feel like this course should focus on linear and quadratic. Including exponential might be too much especially since we will be doing so much with quadratics.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

An Equal Opportunity Employer

Algebra I Standards

4

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"

I feel that AREI. 4a and A.SSE.3b need to moved up to one of the other two courses. You have already got the kids solving quadratic equations by using the quadratic formula, taking square roots, and factoring in this course. I am afraid adding one more method will cause kids to get even more confused.

I feel there are too many standards in the Algebra I course. It should only consist of linear and quadratics topics. Students are not mentally able to really make deep connections with exponential functions until later.

I haven't had an opportunity to review the algebra I standards. I like the approach of granting students the opportunity to study algebra and only algebra. I believe this approach gives them a chance to master the algebra standards alone and equip them with the knowledge to master subsequent courses that will follow such as geometry, as opposed to bombarding them with a vast variety, attempting to make them jacks of all (mathematical) trades and a master of few to none.

I noticed that when the standards for solving a system of equations were stated only

elimination and graphing are emphasized...I think substitution method should be

stated if we are to teach it because graphing and elimination alone would fit the

description of estimate and exact answers. Also...when we teach graphing of linear

systems we should also teach systems of inequalities because students need to

distinguish between the two - it is very slight when seeing a system of equations only

but it is very big in amount of solution choices when looking at it graphically. If the

students don't learn these at the same time they will not see clearly their differences.

The standard as it is now is just to graph one inequality but not a system. I think it

should be but under systems. (We are having that trouble now with only teaching

linear and exponential without the quadratic to contrast the two because students are

making the false conclusion that if the function has a variable rate of change it must

be an exponential because that is the only one they have to deal with that has a

variable rate of change.) I am afraid there will be a false assumption about only

graphing the inequalities one at a time and not as a system. They may think that

systems can only be a set of equations. Also it should be added because if you have

to teach them to graph one linear inequality and a system of linear equations that is

so close to a system of linear inequalities so why not do it here instead of some other

class?I read the new Algebra Standards when they were released, but have now re-

read them in conjunction with the Foundations of Algebra standards. This has given

me a better picture of the flow of the curriculum. The obvious difference is the creation

of discrete courses. And, while I can make an argument in favor of integrated math, I

do believe that discrete math will better serve the needs of the majority of our

students. Below are Questions and Comments about the Algebra Standards. Most of

my questions ask for clarification of the rigor. In my (very humble opinion), I do not

think any standards should be removed, nor any added. The curriculum has a strong

flow that ties all the standards together. MCC9-12.N.Q.1 Use units of measure (linear,

area, capacity, rates, and time) as a way to understand problems. Giving the

conversion factor for Standard to Metric is wonderful! It would also be beneficial if

teachers knew which standard conversions the students would be required to know.

Inches to feet to miles are all very useful, basic conversions. But, what about bushels

to pecks, acres to square feet? Specifying the exact units would allow educators to

target teaching, and therefore, student's learning. I would also very much like

guidance on level of rigor: On page 22 of the EOCT review guide August 2013, I read
2th0e66nTewwiAnlgTeobwraerSstEanadstard2s05whJeesnsethHeyillwJerr.eDrreilveeaseAdt,labnuttah, aGveeonrgoiwa 3re0-3re3a4d twhewmwi.ngadoe.org

conjunction with the FoundationsAonfEAqlugael bOrpaposrttaunidtyarEdmsp. loTyheirs has given me a better

picture of the flow of the curriculum. The obvious difference is the creation of discrete
courses. And, while I can make an argument 5in favor of integrated math, I do believe
that discrete math will better serve the needs of the majority of our students. Below

Algebra I Standards Survey Summary Report

are Questions and Comments about the Algebra Standards. Most of my questions

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
target teaching, and therefore, student's learning. I would also very much like guidance on level of rigor: On page 22 of the EOCT review guide August 2013,
problem #3 is a very complex unit conversion question (gram-centimeters per second squared). Key Idea #4 on page 19 (miles per hour to feet per minute) and Review Example #1 on page 21 (kg per m^3) are not. MCC9-12.A.APR.1 Add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations. To what level should the multiplication be taken? A binomial by a trinomial? A trinomial by a trinomial? MCC9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. Does this include equations that have to be factored first? Example: Solve y=2ac + 3bc for c. MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Generally, the scope of high school math defines this subset as the set of natural numbers 1,2,3,4...) By graphing or calculating terms, students should be able to show how the recursive sequence a1=7, an=an-1 +2; the sequence sn = 2(n1) + 7; and the function f(x) = 2x + 5 (when x is a natural number) all define the same sequence. Absolutely LOVE how this is written! It ties together the standards quite well. I started seeing (with clarity) all the connections this past year, but this clearly spells out the purpose and flow of placing these specific standards within one course. MCC9-12.F.IF.7e Graph exponential functions, showing intercepts and end behavior. Should we use the language of "Horizontal Asymptotes" within the classroom? MCC9-12.F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with $15 and earns $2 a day, the explicit expression "2x+15" can be described recursively (either in writing or verbally) "to find out how much money Jimmy will have tomorrow, you add $2 to his total today." This example is using "J sub 0". The standard MCC9-12.F.IF.3 says `Generally, the scope of high school math defines this subset as the set of natural numbers 1,2,3,4.....' These two standards are somewhat contradictory. Students struggle immensely with domain. Connecting linear functions with a domain of all real numbers to linear functions with a restricted domain of natural numbers to an arithmetic sequence with a domain of all natural numbers and then changing the domain of the arithmetic sequence to include zero within context is a lot to ask. I understand that the standard MCC9-12.F.IF.3 states `Generally,' but if that is the generally accepted scope, I find it odd that the example is a specific exception. Maybe some further clarification would help me to better present this material to my students. Statistics/Data Analysis. It appears that residuals have been removed from the Statistics/Data Analysis unit. I firmly believe that this is a great change! There is much confusion between the residual plot and the scatter plot, as well as the correlation coefficient to the residual. I believe students will master (and need to master) the given standards if residuals are not thrown in the bucket as well. The addition of quadratic models into the statistics/data analysis standard connects the curriculum throughout the course. I find this to be the `final touch' that pulls the course together. Well done! Again, I thank you for all of your hard work and effort. I am confident that this revised curriculum will help fill the needs of many of our students.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

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Algebra I Standards

6

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
I see exponential functions are still present in the ninth grade course despite the fact that students will not learn logarithms until eleventh grade. It makes more sense to go back to teaching function families like we did under GPS with Math I. The students learned linear, absolute value, quadratic, cubic, square root, and rational functions. We had time to teach all the functions to a depth of knowledge that prepared the students for the next level. Exponential functions belong in eleventh grade.
I think the Algebra I standards are going back to what they used to be - ONLY COVERING ALGEBRA. Which is awesome. Integrating so many things into one subject confuses the kids so so much! I love the new standards. Keep it simple for the kids and the teachers. Also this allows us more opportunities as a teacher to do fun activities in the classroom without stressing about covering a thousand standards that don't flow together.
I think the standard for rational exponents should be moved to a later course (Advanced Algebra).
I wish we could see the DRAFT standards for accelerated math as well. When will they be available? I would like the standards to address how to deal with students who have deficits in basic math skills such as multiplication and division. Also, if the whole class fails a unit, what happens? Currently, the teacher offers a recovery process after the unit test. She teaches nothing. She just let's them learn how to cheat by taking and retaking an online assessment up yo five times. Their grade can be raised in her grade book by learning the process of elimination in the recovery process, but none of the class learned the concepts in the unit.
If solving quadratics is in the standards, I'd like to see complex numbers added into the standards as well. it will be helpful to allow students to focus on completely understanding algebra concepts. It will good to see the old Algebra I slowly coming back. It would be great to provide sample problems for each standard to clarify what each standard means. Keep moving backwards towards what was right in the first place. Let's go back to the more traditional Algebra 1.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

An Equal Opportunity Employer

Algebra I Standards

7

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Looking at this draft as a stand-alone document, I find it reasonable and appropriate. However, I am extremely concerned about the introduction of yet another structural change to our highly unstable mathematics curriculum. As a state, we need to decide whether we wish to implement a traditional or integrated curriculum, as well as how we wish to address different levels of performance and preparation for different types of college and career choices. I fear that the restacking of these standards back into traditional courses is just another political and semantic tactic; the public may like it and like many teachers, I see the familiar course names and feel hope that sanity might be returning to our subject area. However, I do not think offering school districts within Georgia the choice between the two types of courses is reasonable, practical, fair, or wise. Recall a similar move a few years ago when GPS Algebra and GPS Geometry were introduced. Even a quick perusal of the DOE documentation from that time will show a maze of flow charts and remediation modules just to facilitate the transfer of credit from one Georgia county to another. Another concern is the cost of creating another set of assessments for a different course configuration. There are persuasive arguments for and against the Algebra I course itself, strongly connected to the student population being considered. Pro: I spoke with a colleague from a high performing school concerned about its college bound students. Even before the public review period ends, this high school is making plans to implement the new Algebra I and Geometry courses. They are eager to get back to the traditional sequence to prepare their graduates for success in traditional college courses. They are also eager to see quadratic functions and factoring addressed in the 9th grade, so that their better students are adequately prepared for a truly advanced study of algebra in the 11th grade. Their student population is relatively stable, so their concerns about difficulties for transfer students take second place. I understand their point of view but I work in a school that serves a different student population and I have quite a different perspective. Con: I work in a charter school that serves at-risk students and focuses on drop-out recovery. Despite all the flexibility, creativity, and accommodations we can provide our students, a majority of our students struggle with math. For many of them, math is THE stumbling block that makes the goal of high school graduation seem further out of reach. For our students, the move from Math I to Coordinate Algebra was huge improvement. We were able to shorten our self-paced course from 26 to 20 modules. We were also so thankful to see the study of quadratics removed from the 9th grade. Focusing in depth on linear v. exponential growth and making the connection to arithmetic v. geometric sequences was far more appropriate, attainable, and concrete than also mixing in the study of quadratics. As you might imagine, our student population has a high rate of poverty and the

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

An Equal Opportunity Employer

Algebra I Standards

8

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"

corresponding high rate of students transferring in and out of our program because of family/economic instability. These are the students most likely to be hurt by allowing school districts to choose between integrated or traditional courses. These are the students who will miss information, test poorly, and end up hating math because of the hodgepodge of mathematics they have been exposed to in different schools. A major selling point of Common Core was the standardization among states. We are fooling ourselves if we think that offering high schools in neighboring counties a choice of different curriculums is going to be a workable option. So, which perspective is right? Valid arguments can be made for and against all the issues at stake with this curriculum change: 1) needs of college bound students v. students bound for technical school or work 2) integrated v. traditional math courses 3) local choice or concern for continuity throughout the state. In my opinion, the bottom line is that Georgia does not need another mathematics course change until we have a realistic, well-thought-out master plan. We are approaching the 8th year of implementing curriculum changes by the seat of our pants without adequate planning, preparation, or stakeholder buy-in. We need to decide as a state whether we want an integrated curriculum or a traditional one at the high school level. We need to decide how we will serve students at all performance levels in a meaningful way, perhaps with a return to different diploma seals. We need to decide this BEFORE we roll out any changes, BEFORE we design yet another set of tests, BEFORE we tie in teacher evaluations and pay to an untested assessment system. The principles of a well-designed instructional unit should be applied to the design of our state's mathematics curriculum. Love this! Coordinate Algebra has no student textbook, and includes units that shorten the time needed for in-depth understanding of algebraic manipulations, functions, and polynomials.
Make sure there are clear paths for special education students of very level. Making the curriculum easier will not make the problem go away. The kids will still fail. Only they will be failing easier material. Your "NEW" Foundations and "NEW" Algebra courses are exactly what was taught 30 years ago. Kids failed then and they will fail now. And now they will not be able to succeed in a technology rich world and to compete in our global economy. DON'T GO BACKWARDS.
More real - life - experiences would be better! Moving Exponential Functions out of Algebra I made this course much more manageable.
Our biggest problem now in the CCGPS Coord. Alg. is that there is way too much material to get through when teaching it on a block schedule. We literally have to cover more than one topic per day and still don't have enough time to cover everything we're supposed to. Hopefully a discrete course in alg. will cut down on the amount of material to be covered.
Please decide traditional vs integrated for the entire state! One-in-six students transfer schools each year! Having different schools on different math tracks with continue to make things difficult. Sure, some people will be angry, but we must do what is best for kids!
Please enter any additional comments you would like to make concerning the DRAFT Algebra I Standards.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

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Algebra I Standards

9

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"

PLEASE let us teach using the Discrete Model. Radicals... where are they? Since they still use the Common Core-aligned standards, the wording of the standards continue to be wordy without saying much. The GPS standards were much better written and understandable.
Some Standards in Algebra I need to be moved to the 11th grade math standards. These include the standards dealing with exponential functions, arithmetic and geometric sequences and series, and recursive formulas. Most 9th are not at the developmental stage to handle these concepts---the math notation that is inherent in these concepts are difficult for them to process. Students in the 11th grade, however, are quick to be able to understand the notation and therefore have no trouble with the concepts. Ninth graders get bogged down in the notation and can't get past it. It is too abstract for them at their developmental level. There are WAY too many standards at this grade level. A boatload of standards have been added about quadratics and hardly anything has been removed. The quadratic standards take a good while to teach and understand and it doesn't look like that was taken into consideration when the course standards were chosen.
Standards are clear so teachers know exactly what they should be teaching. With this curriculum, I believe we will see an increase in student achievement.
Stop changing the standards every other year. Stick to a plan so teachers do not have to figure out what they should teach each year. Provide textbooks so the students have a resource in hand without going on line.
Thank goodness for traditional Algebra standards! In my 14 years of teaching, I've taught everything from remedial Algebra 1 to AP Calculus. Most of my experience was in teaching ninth graders. The biggest struggle for my AP and IB students is their lack of a strong Algebra and Geometry foundation. While these are the BEST standards I've seen since QCC's, there are still too many for a one-year course. Students need to MASTER skills like simplifying expressions, solving equations, factoring polynomials, etc. Those are the skills they will need in order to be successful in higher math courses. My suggestion would be to choose the most important units to cover and then provide detailed sub-standards. For example: solving equations (one-step, two-step, multi-step, variables on both sides, etc.) Or for the linear functions unit: graphing with a table of values, solving for y, graphing with slope and y-intercept, etc. New teachers might not know the best way to cover all of these standards. It would be much better if these were defined sequentially. More emphasis should be placed on "students should be able to" instead of "students will explore, understand, etc.". They will build conceptual understanding through working problems. IF MY SCHOOL DISTRICT CHOOSES NOT TO ADOPT THESE NEW COURSES, THEN I WILL SEEK A POSITION IN A NEARBY DISTRICT THAT DOES ADOPT THESE STANDARDS. The Algebra standards are too expansive. There are too many concepts for a ninth grader to grasp including factoring, quadratics, etc. The present curriculum map has too many standards already and there are plans to add more. It is not feasible, even for the accelerated group. I think this will set the students up for failure and not allow the teacher to teach the concepts with any depth.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

An Equal Opportunity Employer

Algebra I Standards

10

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
The changes need to stop so we can really see where the problems are. I think some of the topics that were moved to Algebra are too much for freshmen. They are now learning- linear, exponential, and quadratics as freshmen. That is too much and they will not be able to master the standards like we want them to.
The courses absolutely need to be separated and give the students an opportunity to be successful and retain more concepts for the future. Rather than compiling algebra and geometry together, it is too much for them and they are not learning this way. The integrated math curriculum has been a disaster. Return to the basics that have worked for centuries.
The quadratic unit is one of the most difficult units to complete in Algebra II material. Moving all of it into the 9th grade curriculum is standard suicide. I teach accelerated geometry and that was their lowest scoring unit of the year this year. If they struggle, the on level students will not be able to get it to the same level at which I taught it. This should be limited to the form of "a = 1" and no more. The quadratic formula should be moved to Algebra II to join the complex number unit. They will use all of this again with the polynomial unit anyway! Also the full section on sequences have traditionally been in Algebra II, and if that is also moved out of Algebra I they may actually have time to finish the curriculum. Not only that series must include what they know from sequences and it only makes sense that they are together in Algebra II. So this would make all exponential growth and decay go into Algebra II which is where logs are taught and are the inverse of growth and decay so those type problems can be solved. If these items are not removed from the Algebra I curriculum, I'm not sure how many students will be able to pass the already difficult curriculum. I hate to see kids drop out of school after spending 3 years in the first math class not able to pass!!!! The rigor is there with these standards however there seems to be an overload of material being shipped into the 9th Grade Math classroom without any consideration for the current hardships the students already face with Coordinate Algebra alone. We already teach a lot of the Algebra as is, and with the exception of those accelerated students, we are already having issues with moving our students along as is. I'm not for changing the standards.
The standards are not clear in terms of depth and breadth to be addressed in the classroom. The standards are well defined. I appreciate taking out the redundant standards from the previous math course. CCGPS Coordinate algebra repeats almost 2 full units from 8th grade math. I am curious to the structure. Is the plan to teach exponential functions before or after quadratics? Or concurrently?
The state should provide a recommended pacing guideline based on the milestone test percentages. We should definitely have discrete math classes as an option. The traditional/discrete path has always been the best option

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

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Algebra I Standards

11

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"

There are a lot of standards crammed into the course, it is very difficult to give time for students to gain a thorough understanding of the material. However, the standards are much more clearly written than previous standards we have seen come and go. I'm really hoping this next round of standards is more long lasting. Math teachers are exhausted and overwhelmed. I'm having a hard time "learning" how to teach. We have dealt with too many changes and new courses over the last few years. I'm to the point where I don't care much WHAT I am instructed to teach as long as I'm given time to do it well and improve with experience.
There are A LOT of standards. Some may be repeated in other grades (higher or lower) that may not take as much time to cover, but it seems like a lot. There are entirely too many rigorous standards for students to master during the first year of high school. There are far too many standards in this course to be taught effectively in the time teachers are allotted. Some standards need to be taken out and placed elsewhere if student mastery at the end of the course is expected for all leathers.
There are still too many standards for Algebra I and content included that should be in 11th grade with Advanced Algebra. Pull N.RN.3, A.SSE.2, F.IF.8, F.IF.8a There are too many standards being added at one time. There are too many standards for 9th graders There seem to be way too many standards in this course. They need to be reduced. I don't think all this can be covered in the time allowed, to the depth and mastery that is wanted.
There seems to be a lot of material for a one-year class.

There should be year by year curricular objectives--not multi year standards, such as 9-12. These are immensely better, especially on more clearly defining some of the areas of confusion. I still think there needs to be more clarity with NQ1.c and NQ3. Some areas where there may still be some lack of clarity include: when are students taught square roots, do solving inequalities include compound inequalities, do we only perform linear regressions, or should students be able to perform exponential and quadratic regressions with calculator, and what should students be able to do with and without technology per assessments.
These standards are too advanced for freshman. There are too many of them and some of them are too complex This is a step in the right direction. The Coordinate Algebra course is full of fluff and needs to be beefed up. I wish the state would abandon the Accelerated model and go back to the honors model from before.
This should help students that are struggling in math, I hope. These standards look very similar to those in Algebra right now. I think most of the standards are fine. I still do not like the first few standards about conversions and unit measures. It is very hard to interpret and convey to the students.
Too many standards for a typical 9th grade student! We are pushing too many standards instead of real content knowledge through depth!!!!!!!!!!! Too many standards for one subject. Too many standards to cover in a year.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

An Equal Opportunity Employer

Algebra I Standards

12

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Too much change in mathematics standards over the past 10 years. There has been NO (ZERO) continuity for students and/or teachers. Leave the standards alone!!! It is not the standards. If you allow the teachers time to adapt, we will. If you change the standards every 3 years, we will not understand what builds and how it builds. Too much content in Algebra I. Your EOCT pass rates will go even lower (if that's possible). Unsure that a cadre of new teachers will incorporate the statistics unit into the pacing before state testing in April or whether their skill level has improved from trainings or college preparation programs. We should not be teaching linear, exponential and quadratic functions in one subject. It has been hard enough for ninth grade students to understand linear and exponential and now we have to add in quadratic. It's too much! Will there be a separate set of standards for Accelerated Algebra 1 or will it just be an Honors Algebra 1? Would students taking this course need Foundations of Algebra too? This seems to be more condensed and possible for the struggling learner.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org

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Algebra I Standards

13

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Foundations of Algebra Standards Survey Rating and Comments
Please indicate your level of agreement with the following statements about the Foundations of Algebra Standards.

The DRAFT Foundations of Algebra Standards:

are clear and understandable

Strongly Agree 29.6% 76

Agree
59.5% 153

Disagree
7.4% 19

Strongly Disagree
1.9% 5

Don't Know 1.6%
4

# of Responses
257

define what students should

know and be able to do to prepare them for success in required high

32.0% 55.9% 6.6%

82

143

17

3.1% 8

2.3% 6

256

school mathematics courses

are appropriate for the next level or grade in preparation for college and career readiness
are rigorous enough to challenge learners
provide sufficient opportunity to revisit and enhance student understanding of foundational algebra concepts
are consistent with postsecondary and business/industry standards

31.6% 52.0% 6.6%

81

133

17

5.5% 14

4.3% 11

256

33.6% 54.3% 5.9%

86

139

15

3.9% 10

2.3% 6

256

36.2% 50.4% 6.7%

92

128

17

3.9% 10

2.8% 7

254

26.7% 54.1% 8.2%

68

138

21

4.3% 11

6.7% 17

255

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

1

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Please enter any additional comments you would like to make concerning the DRAFT Foundations of Algebra Standards.
Comments: Algebra has been taught for decades. I hope these standards are the most logical in sequence and someone compared them to many other schools' Algebra curriculum! Why reinvent a course that should be standardized? Better yet, find a good text book about Algebra and copy the book's (and math experts) sequence of learning objectives and teach that! Please provide these books for our students in Fulton County! Are these standards actually aligned with college standards? As an educator, who graduated in 2007, I would have to question these standards as not really aligning with college and too rigorous for preparing students.
As a HS Math teacher, this change is MUCH needed. Students need a stronger base on which to build their HS math skills. I believe this is a good start to changing what has been so wrong with HS mathematics. I have been extremely discouraged as a mathematics educator. Since I began teaching in 2009, the math curriculum in HS has changed EVERY year and is continuing to do so. This is completely unacceptable for our students and extremely frustrating to teachers!
As a parent of a struggling learner, I am very concerned that this class would lead to him being tracked onto a lower academic track with the potential to impact his ability to complete the required four years of mathematics courses. This could have an impact on his ability to attend certain colleges in the state. I understand he might be able to double up on math courses during his senior year, but as a struggling learned I do not feel that may be a viable option for him with the large number of required courses in high school. It seems unusual to me that the Foundations course contains so many elementary school mathematics standards from grades 3,4, & 5. It would seem to make more sense to identify the areas in which students are struggling in the lower grades. Remediation plans should be developed each year. I would also think that if a large number of students are struggling in a known number of standards that increasing teacher professional development opportunities for teachers would be essential. Teachers need access to the best practices needed to teach and remediate standards. My recommendation is to look more closely at why students aren't mastering the standards in the lower grades instead of enrolling kids in a course that could impact their college enrollment. Tracking kids into a class with a focus on lower expectations does NOT seem like a good choice if we are trying to give kids the skills of being successful in their future. As a parent of high school students, I have seen them struggling in Math since they started high school. Whatever changes that are trying to be made it need to be where it is taught with understanding so the students can grasp it and be successful in passing all of the Math classes they need while in high school.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

2

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
As a SPED teacher of 6th graders, I am glad to see some attention being paid to the needs/support of students who are truly learning below grade level. Much attention has been given to students who can advance in the new "Continuous Achievement" model, yet it seemed that no one wanted to admit that many other students are, in fact, below grade level in their abilities and need instruction to meet them where they are, in order to help them move forward. Thank you!
As long as the standards match what the students will be asked in college, then they are fine.
Can students fail this course?
Courses in Georgia continue to be drafted and implemented for remediation purposes. Has anyone studied the number of remedial courses offered at 2-year colleges? This semester I'm teaching Geometry A to a class of 32 students, more than half have failed this course at least once. Two seniors are enrolled. Students taking the course for the first time failed the prerequisite course at least once. What's wrong with this picture? I don't see the math difficulties that were addressed years ago improving...is anyone taking my concerns and comments seriously?
Courses in Georgia continue to be drafted and implemented for remediation purposes. Has anyone studied the number of remedial courses offered at 2-year colleges? This semester I'm teaching Geometry A to a class of 32 students, more than half have failed this course at least once. Two seniors are enrolled. Students taking the course for the first time failed the prerequisite course at least once. What's wrong with this picture? I don't see the math difficulties that were addressed years ago improving...is anyone taking my concerns and comments seriously?
Currently, I feel okay with the standard for Foundations of Algebra. My further understanding of the standards and my ability to teach them in a realistic time frame will be determined in the years following their implementation.
Does this course grant credit to a College Prep diploma? Excellent plan for students who come to high school who are not prepared for high school math.
Factoring quadratics is a high level skill! I teach Accelerated Analytic Geometry and that test had the lowest average of the year! This is also an area of concern for the Accelerated pre-calculus teacher at our school and she included factoring as warm ups on a regular basis. For us to include this in Algebra I, they need to be factorable and "a" must equal 1 for this level. Reserve the a not = to 1 for Algebra II and using the quadratic formula for Algebra II since that is when they will be covering complex numbers anyway. With that said, the sequence section leads to growth and decay problems that have always been a part of the Algebra II curriculum, and should be moved back in order to give Algebra I teachers the time to tackle graphing parabolas. I started teaching 25 years ago. And I can't believe the failure rate we already have with the standards that are twisted and broken into such pieces and scattered among the grades. There are reasons that higher order thinking skills were reserved for algebra II and not included in algebra I. The brain must mature enough for these topics. PLEASE consider the above standard changes and lets encourage our students to not give up on school in their 9th grade year because of the outrageous math curriculum!

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

3

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Focus should be placed on connecting the concept that fractions (rational numbers) are division problems and vice-versa. In addition, the curriculum should include an intensive reading and vocabulary component to assist secondary and post secondary teachers with grade level instruction and students with the reading levels of resources and materials.
For senior year, I would like to see an option to take Money Management (personal and business aspects) in lieu of Advanced Algebra. Students who are not college bound will benefit from the money management more than advanced algebra. Foundations of Algebra is a much needed course. If our students obtain strong foundational skills they will be more successful in Algebra and Geometry. Please consider Foundations of Algebra as 1 full math credit instead of an elective credit. I believe that a student that starts with Foundations of Algebra and ends with Adv. Algebra (giving him or her 4 full math credits) will have sufficient mathematical knowledge to be successful in a four-year college or university.
Foundations of algebra would be great for our incoming 9th grade students with low test scores. These students come to use from middle school lacking basic skills. We have to spend so much time reviewing prior knowledge it takes time away from us teaching the content. The foundations of algebra would be a nice addition to our 9th grade academy, to help guide students in the direction that they need to be. Functions are an unnecessary component of the course. If functions were to be included the standard should be focused on exposing the students the different kinds of functions, than asking them to be write functions in function notation, etc. Get back to the BASICS and everything else will fall into place!
Great idea. Wish we would have introduced this course prior to now. . . High schools are in DESPERATE need of a 9th grade "pre-algebra" course for all middle school students who did not pass middle school math, or the CRCT in math, or took the modified CRCT. These students do NOT have the skills to be successful in 9th Algebra!! I am concerned that for the ability level of the students enrolled in this course, based on the state recommendations, that students will not be able to master the amount of content proposed in the standards for Foundations of Algebra and be ready to successfully move onto Algebra I/Coordinate Algebra the next year.
I am concerned that High School teachers will be teaching standards from grades 3-8, and they should be teaching it concretely, pictorially as well as algorithmically. Are the HS teachers qualified to teach these concepts in the manner that they need to be taught? What actions will be taken to make sure they are prepared?
I am glad this option will be available. I am thrilled to see a beginning high school math course that provides an opportunity for students to learn arithmetic with fractions and decimals. The pre-algebra concepts and skills are essential to success in college algebra and any math course that follows. There are too many standards, and I would recommend cutting the more advanced algebra
topics. Solving one-step equations should be the most advanced thing covered in this course. Leave the linear equations and functions standards to Algebra 1. I believe this course is absolutely necessary to ensure that our lower level students are set up for success.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

4

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
I believe this is an excellent opportunity for students who struggled with algebraic concepts from 5th grade up through 8th grade to have an opportunity to catch up and fill in any gaps that may be causing them to struggle mathematically before pushing harder math in high school courses.
I do believe this is the beginning of taking the right step to bridge the knowledge deficit that some of our students possess with respect to mathematics.
I don't recall seeing a statement in the draft the Foundations of Algebra Standards prepare students for college/career readiness.
I feel like this course needs to be offered in the 8th grade prior to entering high school. If students have not done well on mathematics standardized tests in 5th, 6th, and 7th grade then 8th grade should be used to work on remediation not sending students to high school to remediate middle school standards.
I feel that this class would be a great fit for struggling math students. So many of the students that I teach are not college prep students so this would be a much better fit for their educational needs. I feel the qualifications for the course are too restrictive.
I firmly believe that this course is needed for our 9th grades who struggle with gaps in their math reasoning skills. This course would give the appropriate students a muchneeded remediation and a chance at success. I firmly believe the Foundations of Algebra course should not be offered for the following reasons: - Limits students' opportunities for post-secondary education - high school teachers are not equipped to teach elementary and middle school math standards conceptually - this course does not provide just-in-time remediation; rather, attempts to remediate all in one course rather than throughout elementary and middle grades which should be done - There will be a very negative stigma associated with placing students in this course - course standards communicate low expectations and WILL NOT prepare students for Coordinate Algebra - students will be one year removed from 8th grade math standards which are truly foundational to the Coordinate Algebra course - creating a course designed for a homogeneous group of low achievers is NEVER a good idea -the course encourages low self-efficacy. In conclusion, we need to focus our efforts on training teachers to effectively understand and teach our standards. Additionally, rather than creating this course, we should build a curriculum for Coordinate Algebra and Analytic Geometry support so that teachers know what should be done in those courses to support the on-grade level content. I haven't had an opportunity to review the algebra standards. I'm currently a seventh grade special education teacher. I like the idea of having a foundational course. However, the standards should be connect more to Coordinate Algebra. Too many middle school standards are in this course. I like the mix of 6th, 7th and 8th grade standards. It reminds me of Pre-Algebra.
I like this course to prepare students for Coordinate Algebra. I love this addition. I believe that the class will benefit all the students who are struggling with the basics.
I LOVE this class! I think this is exactly what we've needed for years to "fill the gaps" with our learners!! I'd LOVE to teach this course!

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

5

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
I seems we agreement has been reached that middle and elementary math teachers are unable to do their jobs I think that these standards are exactly what struggling students need to be successful in the upcoming Math Sequence Courses.
I think that this course will help many students in my school district.
I think the qualifications for the course should be broadened to include students that are "bubble students".
I think the qualifications for this course need to be broader and to include the lower 10% of all students. I think the standards for this course are appropriate for the lowest achieving students. I think the standards need to be a bit more specific. For example, the section on functions, which functions are you going to include? I wouldn't include this section with the foundations class. MCC9 -12.F.IF.4.7 Analyze graphs of functions for key features-Intercepts, Intervals for Increase/decrease, maximum/minimum, symmetries, end behavior
I think there needs to be just a little bit more in this course to prepare for Algebra, but completely in favor of the course.
I think this is an awesome addition to the curriculum especially for the students who are placed and ESP students or the ones who struggle with the foundations. The students who need this I do not believe will be able to make it through a senior mathematics class nonetheless. This is the BEST option! I think we need to stick with what has worked in the past. I truly believe that we are limiting this foundations of algebra class to students who obviously desperately need it (special needs students), but there are many more grade 9 students who need this remedial style course as well, maybe even as high as 50% of the grade 9 class at any particular school.
I understand that this course is a precursor to Coordinate Algebra, but there are still a few standards that I feel should be held off until they take Coordinate Algebra. A couple of examples are rearranging formulas from the AEI standards and from the AQR standards, key features of functions. This class would be a great opportunity to make sure that students are truly comfortable with algebra. I teach students now that still cannot remember how to solve a two-step equation and cannot subtract integers.
I would just like to say that I support the effort to provide a class to incoming freshmen that will strengthen their math understanding prior to them taking Coordinate Algebra if they struggle in math.
I would like the standards to address how to deal with students who have deficits in basic math skills such as multiplication and division. Also, if the whole class fails a unit, what happens? Currently, the teacher offers a recovery process after the unit test. She teaches nothing. She just let's them learn how to cheat by taking and retaking an online assessment up to five times. Their grade can be raised in her grade book by learning the process of elimination in the recovery process, but none of the class learned the concepts in the unit.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

6

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
I would love to have this course added to the high school curriculum. So many of my students have aged out of middle school and come to high school never passing a single math class. This gives students a fighting chance to pass math and finally see some positive feedback in math. I really am excited that you have heard our concerns and are trying to implement this course! I would suggest there be opportunity for an additional pathway for the same students who are described in this draft, as the student ideal who would benefit from this course. We need students to leave us and post high school, be able to manage their own life in as many aspects as possible.
I'm only filling this out in one of the surveys because my comments apply to all. This is not what math teachers want, it is not what students want. This is just some other thing that someone has come up with to say that they are "trying to fix the problem." Go back to Algebra 1, Geometry, Algebra II, and Adv/Alg Trig. Use those course names as a starting point and put the standards where they should go. If you want to make people happy give them what they want. This version of discrete still isn't discrete and removes a complete level of rigor if a school system chooses. Taking all 4 of one is not equivalent of taking all 4 of the other, l and that is a problem.
If I read the criteria for Eligibility for the Foundations of Algebra Standards, students will have to be at a FOURTH grade mathematics level to be eligible for the course. One standard says "Use properties of integer exponents to find equivalent numerical expressions." Integers are not introduced until 6th grade under CCGPS. How is a teacher supposed to take a student with 4th grade math skills and move them to 9th grade (Algebra I) in one course? The student who meets the requirements for the Foundations course will most likely be a student with disabilities, an English Learner or on Tier 3 of RTI. A classroom of such struggling learners should have class limits and additional support required. Those students will have no one to have as an example and what recipe for disaster. A group of all remedial students is incredibly difficult to instruct much less all remedial, SWD and ELL. The ominous language of the tracking of these students makes a teacher very hesitant to volunteer to teach this course. How can we expect our best teachers to work with our lowest students if we are going to hold them to undescribed standards of progress? This group of students would need the most innovative and experienced teacher in a school.
In my opinion, it's very important to get back to math basics and learning how to compute with a calculator. In the standard for Pythagorean Theorem, remove the stipulation to focus only on solving for the hypotenuse; with emphasis on solving in the Foundations class, students should be expected to solve for missing side lengths as well.
Irrational numbers do not belong here to me. This is a foundations class and I think if this class is designed to use manipulatives and real world learning, irrational numbers can wait until Algebra I. Let's lay the foundation. Is this going to count for one of their math courses?
It is about time we accommodate all learners.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

7

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
It is important that these standards stress practice with 1) word problems, 2) percentages, and 3) application problems. The difference between rational and irrational numbers is not crucial to their understanding of mathematics, and should not be included. All standards should be based on concrete application, not abstract knowledge. They should reinforce how to use a concept, not why they are using it. Graphing systems of inequalities should not be included because they have not learned how to graph inequalities yet. It would be nice to have a class like this at the 6th grade level as well to close in on some of the major computation and number sense gaps from elementary school. If this is only a semester long course, I fear that students may not get enough time to cover all the necessary standards. Make sure that all the options for special education are clear for all levels.
Make sure that Special Education students at every high school level have options. Many of my on level 9th grade students do not have the basic math skills to be successful in Coordinate Algebra. I am thrilled that these students will have an alternative class that allows them time to solidify prerequisite math skills. Many of the standards a very elementary for high school students so the pacing guide for these should group them together so there is not a lot of time spent on the 3rd, 4th, and 5th grade standards, but more on the standards that will benefit them being successful in algebra. Mastery of topic must be met by practice. Once mastery is met then explanations follow. More hands - on - Activities are better.
My comment that would continue in all four of the options simply questions, movement from students between the foundations-alg-geo-advancedAlge... once they begin in this direction, I assume that they will remain in this 'style grouping' their entire 4 yrs, or is the advanced algebra 'mixed' (junior/seniors) Need clear directives as to how students qualify to take this course. Need more time to discuss elements with fellow teachers. There are some questions remaining. No! No! No! No! No! This course does NOT need to exist. Standards were raised significantly when Math 1-4 rolled in, and this is a huge step backwards. The bottom 10% of students today know a *LOT* more than the bottom 10% did back in the "Fundamentals of Algebra" and "Concepts of Algebra" days.
Opportunities for post-secondary education will be limited since many if not most of the students will not have a course beyond the junior course. - 9th grade struggling students will not be motivated by the elementary standards. - Placing students in a class with elementary standards communicates low expectations leading to low self-efficacy. - "Just in time" remediation should be provided beginning in 3rd grade where some of the standards originate so students won't fall behind and the necessary skills can be developed in the appropriate grade level. - Waiting 3 6 years to develop the conceptual understanding of grade-level content is not appropriate.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

8

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Overall, the standards look great. We, as teachers, however, need more direction. We need to know what the Power Standards are so we can better aim our focus when teaching the standards to students. I think it is phenomenal to see application problems in this set. We definitely need an emphasis, if not more than are listed in these standards on word problems, percent problems, and application problems. We should remove the more abstract standards, such as the one that has students understand irrational numbers approximated with rational ones. Instead, focus more on standards that students will need to know in the real world rather than random abstract ideas. A good understanding of numbers is good, but focus on the more important application of numbers, such as percentages, rather than one they'll never have to use outside of school. We need a standard that has students writing equations of lines and more graphing lines. This is taught in the 8th grade, but students are so weak in this area that it definitely needs reinforcement in this class. Remove the standard about graphing systems of inequalities. Systems of lines is fine, but we need to have them graphing one two-variable inequality first, so have that be the standard in Foundations and leave systems to 9th grade Algebra. Characteristics of functions is great, as long as the standard remains as IN CONTEXT. This is a real-world application and a much more tangible concept for students, so it is spectacular to be the lesson taught in this course. Lastly, I hope that this course will soon be offered to more than just a select few 9th graders. There are MANY students who could benefit from a solid foundation in Algebra!!!
Please continue to send examples and samples of expectations.
Please do not allow choice for districts to offer discrete or integrated math. School systems deal with many transient students. How will they adapt from a system that offers discrete to a system that offers integrated or vice versa? We dealt with this challenge at my HS where I worked with students coming in from FL. It is a nightmare!! Please ensure that this class size will be limited. The teacher or teachers who teach this class will need to have background on teaching basic concepts. Please include this course Please just put it back to the "old" math ways.
PLEASE let us teach using the Discrete Model. Please return to traditional math WITH a traditional math assessment so that we can purchase books for our kids!!!!
Please set the requirements based on failure of middle school math, failing only one middle school math can create a deficit large enough to merit enrollment in this class. Do not make the requirement too restrictive. If a student has to fail more math classes than
that then we are not creating a course that is needed, and someone has their head in the sand. Problems we are facing now is not "curriculum,'' or "standards." Teachers are... Radicals should be included so that when solving they Pythagorean theorem students can be give more challenging questions Resembles what I taught as a Pre-Algebra course. We have students that can definitely benefit from this course.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

9

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"

School systems need to go back to traditional math classes such as algebra in 9th gr, geometry in 10th grade, algebra 2 in 11th grade....NOT Math 1, 2, 3 where these areas are combined and TOO advanced and not developmentally appropriate...at this time, my middle school child, 6th gr is doing algebra!!!!!! He is doing skills I did in 10th grade math...this is NOT satisfactory because he feels like a failure and we can't help him with math homework...we need ALL standards appropriate in K-5 to begin with!!!! There is no foundation for fluency or mastery. My child is an above average student, but now he doesn't care for math because he is being taught skills his brain isn't developed for....
Should be mandatory for those recommended. Should have never went away from offering a third option like this. We're coming full circle now.
Standards MFAQR1 and MFAQR2 discuss characteristics and graphs of functions and some non-linear. Which non-linear functions? Quadratic, exponential, absolute value, piecewise....?
Standards seem much clearer and easier to implement than the current Common Corealigned standards. Nice to have an option for the weaker math students.
Students have studies these standards since sixth grade (or earlier) and should have them mastered by ninth grade. This course should not be offered for credit, only to prepare students for the high school credit algebra class.
Take us back to just Algebra I, Algebra II, Algebra III, Trig, Calculus, and Statistics. The material is not being reinforced for the students to understand it. It is being taught and then forgotten. The old curriculum reinforced it in all courses just on different levels of difficulty. Please stop trying to mix all this together in a hodgepodge and confusing them more.
THANK YOU for finally creating a pre-algebra class for high school. This has been missing for the past six years since the switch from QCC to Math 1 etc. We will have many students who now have the opportunity to be successful in math by having one more year of algebra skills before beginning Algebra I.
Thank you for recognizing that not all students learn at the same rate. Many students enter high school and are not prepared to begin the rigor of Algebra. This course will help bridge the gap and pave the way for greater student achievement. This is something that we have needed for a long time!!
Thank you, thank you, thank you for developing this course!! Our high school students DESPERATELY NEED this. We have been trying to go through the motions of teaching Coordinate Algebra's abstract and rigorous concepts to a large subgroup of students who are severely lacking in background skills; it has been an exercise in futility for everyone. We have lived out the statements on the last page of the standards explaining the pitfalls of a lack of understanding; this experience also causes students to further dislike mathematics and to become cynical about the value of a high school education in general. The addition of a course like this would go a long way toward making high school graduation within the realm of possibility for some students who advanced to high school despite being "left behind" in terms of conceptual understanding of and procedural fluency with key ideas from elementary and middle school. Please do not let the Foundations Course be sidelined by any disagreements about the potential restacking of the algebra and geometry standards. Regardless of what changes or stays the same with the other high school courses, many of our students NEED this foundations course. Overall, I

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

10

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
agree with the choice of the standards included. I would like to see operations with signed numbers also addressed. I would like some clarity on the depth/breadth of the function analysis (symmetry, max/min, etc.) as this could be interpreted at many different levels. Also, careful thought should be given to the criteria by which students are assigned to the Foundations Course. Please look at research, real data from 9th grade math courses over the past few years, & especially input from teachers who have been on the front lines of trying to help struggling 9th grade students.
Thank you! Too many students are entering high school that are totally lacking in a basic understanding of mathematical concepts. This course would help them get off to a much better start. The course in general is an excellent idea. I think there is a great need for this type of instruction.
The integrated math program is a disaster. Return to the basics that have worked for centuries. There are many students who are in need of this course. These appear to be a step backwards in preparing students for college and career. This is essentially dropping the math requirement to 3 years of HS math. If that is the intention, why not drop the requirement to 3 years of math. If that is not the goal, why would we offer credit for a MS level course?
These are great! Question: Where will complex numbers be taught? I see that we will teach solving quadratics by completing the square, taking square roots, factoring and the quadratic formula but what happens when you get a complex solution? Are we only looking at quadratics with solutions and then addressing complex/imaginary solutions in advanced algebra? I did not see this in the advanced algebra standards.
These courses should be used for students that are being placed in high school. This class is needed for our remedial population of students who struggle to be successful with the current curriculum.
This course could be very beneficial to students regardless of whether standards are realigned or not. This course is a great addition! We have a group of students who may still struggle to pass Advanced Algebra, but could benefit more from the Mathematics of Finance.
This course is definitely needed to ensure success of our students in mathematics. This course is exactly what some students need. There are students that struggle with basic foundations of Algebra - these are the students that will struggle to complete high school - especially Pre-Calculus. We need this course to help prepare this group of students for the work force and there is nothing wrong with that concept!
This course is necessary but needs to be taught as a co-teaching unit. I envision a high school teacher and a middle school special education teacher creating an atmosphere needed for learning to take place. This course is not appropriate for high school and should not be included as a core math option.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

11

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"

This course is only useful if those with gaps in their algebra knowledge are allowed to take it. As it stands, a student must fail the CRCT for 3 years in a row to enroll. Students with algebra deficiencies might fail portions of the test but still need this course to be successful at the next level. With such strict criteria for enrollment, you are tying the hands of educators and making this course useless.
This course is too remedial to be considered a core course. It should serve as a support course.

This course should help students by providing the prerequisite knowledge and skill for success the subsequent courses upon successful completion of the course. What options will be available if the students are not successful? This course steps back 30 years and will ensure that students can graduate from Georgia High Schools. But they will not be able to compete with the weakest graduates from any country in the world.
This course will be a great opportunity for SpEd student who struggle with math to get a stronger skill base.
This course will provide the students who are often overlooked in mathematics an opportunity to "catch up" perhaps. It could also be used as a summer course for 8th graders who struggled or failed mathematics.
This course would be useful for those students who really struggle with math and need extra time to understand the material. The support model has not been as effective as it could be. Not all students are going to college, and the inclusion of the Foundations of Algebra course shows that this fact is being acknowledged.
This is a course that has been needed in mathematics ever since we changed to the Math I curriculum. I have taught 9th grade math since Math I started. We have several students who can never get out of the first level because there prereq. skills are so weak. This is a realistic course to address the deficits students have arriving in high school mathematics pathways. This is an essential tool that will greatly enhance our students' achievement. This is exactly what are students who struggle with mathematics need to be successful. This gives them the chance to master the standards before moving in to higher-level math classes. Kudos to GADOE!!!! This is good for pre-algebra.
This is long overdue. Kudos to the team who made this possible. I truly believe this change will have a positive impact on student outcomes.
This is needed! Most of my 9th graders are not ready for Algebra.
This needs to be co taught to utilize sped skills
This option will give the chance to many students that are weak in math background to succeed in high school and provide them with future opportunities to their level.
This would have been really helpful to have as an option for my daughter who struggled in middle school and has failed the first semester of 9th grade integrated concepts. There needs to be a focus on ensuring that kids understand the basics before moving on to more complex topics.
Very basic standards. Seems to me these are more designed for 8th graders or struggling high school students, in which case, I feel these students would be non-college bound students.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

12

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Very excited about this course and its intent. Would like to see similar standards for a course for students already struggling leaving elementary school to get caught up during middle school, before reaching 9th grade. We desperately need a class for students who are not prepared for Algebra 1. Students should be required to take this class based on their CRCT 8th grade math scores.
We desperately need a course like this for students who are not ready for 9th grade math. We have many low level math students who enter high school needing this course. This course definitely needs to count as a core credit for these kids and not be overly restricted. We need to make sure we have a co-teacher in the class this class to have more teacher student interaction. We also should have a lower level teacher such as a middle school teacher to help teach this course. Middle school teachers come with more ways to teach lessons than high school math teachers.
Welcome back to math students can use in the workforce
Will high school teachers be equipped with appropriate strategies in teaching these students concretely and bridging to abstract thinking? One concern is the learning and retention of certain skills. If a 9th grade student has not learned certain elementary level standards in the past five years, why would they learn those skills in this course?... some being one of 27 standards that are addressed taught. If these standards are identified at each grade level as high need for 9th grade, then why doesn't the state go back to those grade level standards and stress importance (beef up) those skills making sure they are revisited throughout all grades leading up to 9th. Will students receive high school credit for Foundations of Algebra? Does the standards of this course reflect what is being taught in middle school now? Will the middle school standards change? Will this be a complete math credit for those who struggle in math? Why is this called prealgebra?
Will this count as a math credit for the students who struggle in Math?

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

13

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
with respect to the Foundations of Algebra Standards: As I understand it, this course is being offered exclusively to freshmen who struggle with math. For whatever reason, it is not their strong suit. In my (very humble opinion), everything I do in my classroom should pass my Goals Test: 1. Preparing students for the use of math in their everyday life as adults 2. Preparing students for the next class MFAPR2. Students will recognize and represent proportional relationships between quantities. It is an AWESOME standard! I am thrilled to see it listed, as the benefits to mastering this standard are endless. Kids need to learn percents. They will use them in daily life as adults. They will have to figure out sales tax, income tax, sale prices...the list is endless. Moreover, learning about percents will help set the stage for the exponential standards that they will learn in the Algebra course. This one topic works for both of goals. On the other hand, I see: MFANSQ3. Students will recognize that there are numbers that are not rational, and approximate them with rational numbers. I very much understand the need for deep conceptual understanding. I have taught this standard (or one like it) many times in the 14 years I have been teaching. But even though I have done it before, I must ask myself: How does this prepare kids for adult life? Will this help them be successful in the next class? Well, yes and no. Having a deep concept of the structure of numbers will always help students. But to what degree? This leads me to Materials: From what I have been reading, the state is planning on providing a lot of the materials for this course. The idea of having materials (both for students and teachers), in forms of examples for teachers to teach, problems for the students to solve, applications and tasks provided...it is phenomenal. A task is not enough; the kids need practice. Mindless repetition is not enough; the kids need applications. We need both. And while I have written most of my own materials for years, and am sure I will continue writing and creating for my classroom, this is yet another new class for which I have few resources. The learning curve for teachers to achieve a deep mastery of a set of new standards is always high. We can use all the help we can get! Having solid, balanced materials would also solve the question of emphasis that I raised earlier. The amount of examples, work, time, and applications devoted to a standard would very much help me guide my students toward success. For example, on page 106 of the 2013 Coordinate Algebra EOCT study guide, there are questions on even and odd functions. I find question 2 to be rather difficult. To me, a student would have to understand function notation, and that the function given is a horizontal line, and that would mean it is symmetric with respect to the y-axis, which would mean it is even, and then still understand the more formal function notation of even and odd functions. I saw this question as it relates to IF.4. However, when speaking with a coworker, she thought it came from BF.3! Taking the guess work out of new standards through solid, balanced materials would help teachers collaborate, would help alleviate the time involved in writing new materials and would put us back where we belong...teaching! Ultimately, that is the best service we can provide to the students. I am very excited to see a scaffolded set of standards being implemented to help our students achieve success! MFAEI3. Students will create algebraic models in two variables b.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

14

Survey Summary Report

Richard Woods, Georgia's School Superintendent
"Educating Georgia's Future"
Represent solutions to systems of equations and inequalities graphically or by using a table of values. (MCC6.EE.5, MCC7.EE3; MCC9-12.A.CED.2) I would very much like to see this standard changed to graphing two variable inequalities on a coordinate plane (not systems of inequalities). This could then be extended into context with applications (passes a goal test- required for success in Algebra). MFAQR2. Students will compare and graph functions e. Analyze graphs of functions for key features (intercepts, intervals of increase/decrease, maximums/minimums, symmetries, and end behavior) based on context. (MCC9-12.F.IF.4, 7) I very much like that this is added IN CONTEXT. Again, the students struggle with seeing this in an abstract manner in Algebra, and will benefit greatly from seeing it in context the year prior (passes a goal test- required for success in Algebra). I do however wonder how symmetry would be applied within context? MFANSQ2. Students will conceptualize positive and negative numbers (including decimals and fractions). FABULOUS! Meets BOTH of my "goal test" requirements! Other ideas: More emphasis on linear equations. Writing the equation of a line (given a graph, given the slope and a point). Putting lines into slope-intercept form and then graphing (multi-step). I think this is a middle school standard, but it is one that the students tend to struggle with greatly. And, it easily passes one of my "goal tests", as it is required for success in the Algebra class. Again, I thank you for all of your hard work and effort. I am confident that this new curriculum will help fill the needs of many of our students. Would like to see a math pathway that prepares students for math skills that they will use in real life instead of pushing all students toward the same track and covering the same standards.
Would like to see less cross over between algebra and geometry in courses. Let the course focus solely on algebra or geometry.
You should definitely have middle school or upper school elementary students co teaching or even teaching the Foundations of Algebra course, I am certified in both but it wouldn't be fair to high school teachers who are not trained and current with scaffolding and or interventions for remedial students.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Foundations of Algebra Standards

15

Survey Summary Report

February 9, 2015

MEMORANDUM

TO:

Members of the State Board of Education

FROM:

Pamela Smith Director of Curriculum and Instruction

RE:

Survey Results for High School Mathematics Courses

The survey period for public review and comment for the three Mathematics courses (Foundations of Algebra, Algebra I, and Geometry) that were posted on January 15, 2015, closes on Friday, February 13, 2015, at 5:00 p.m. EST. A summary of results, as well as actual comments, will be forthcoming and posted to eBoard by Monday, February 16, 2015.

2066 Twin Towers East 205 Jesse Hill Jr. Drive Atlanta, Georgia 30334 www.gadoe.org
An Equal Opportunity Employer

Grade 9th 10th

HS Mathematics Course Sequence Options

Option 1

Option 2

Option 3

Coordinate Algebra

Accelerated Coordinate Algebra/Analytic Geometry A

Foundations of Algebra

Analytic Geometry

Accelerated Analytic Geometry B/ Advanced
Algebra

Coordinate Algebra

11th

Advanced Algebra

Accelerated Pre-Calculus

Analytic Geometry

Pre-Calculus or Other 4th AP Calculus or Other 4th

12th

Advanced Algebra

Course Option

Course Option

Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry
Mathematics Georgia Performance Standards
K-12 Mathematics Introduction
Georgia Mathematics focuses on actively engaging the student in the development of mathematical understanding by working independently and cooperatively to solve problems, estimating and computing efficiently, using appropriate tools, concrete models, and a variety of representations, and conducting investigations and recording findings. There is a shift toward applying mathematical concepts and skills in the context of authentic problems and student understanding of concepts rather than merely following a sequence of procedures. In mathematics classrooms, students will learn to think critically in a mathematical way with an understanding that there are many different solution pathways and sometimes more than one right answer in applied mathematics. Mathematics is the economy of information. The central idea of all mathematics is to discover how knowing some things leads, via reasoning, to knowing more--without having to commit the information to memory as a separate fact. It is the reasoned, logical connections that make mathematics manageable. The implementation of Georgia Performance Standards in Mathematics places the expected emphasis on sense-making, problem solving, reasoning, modeling, representation, connections, and communication.
Mathematics Georgia Performance Standards Geometry
Geometry is the second course in a sequence of three required high school courses designed to ensure career and college readiness. The course represents a discrete study of geometry with correlated statistics applications.
The standards in the three-course high school sequence specify the mathematics that all students should study in order to be college and career ready. Additional mathematics content is provided in fourth credit courses and advanced courses including pre-calculus, calculus, advanced statistics, discrete mathematics, and mathematics of finance courses. High school course content standards are listed by conceptual categories including Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability. Conceptual categories portray a coherent view of high school mathematics content; a student's work with functions, for example, crosses a number of traditional course boundaries, potentially up through and including calculus. Standards for Mathematical Practice provide the foundation for instruction and assessment.
Mathematics | Standards for Mathematical Practice
Mathematical Practices are listed with each grade's mathematical content standards to reflect the need to connect the mathematical practices to mathematical content in instruction.
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the
Georgia Department of Education December 2014 Page 1 of 11 All Rights Reserved

Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry
strands of mathematical proficiency specified in the National Research Council's report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy).
1 Make sense of problems and persevere in solving them. High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
2 Reason abstractly and quantitatively. High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects.
3 Construct viable arguments and critique the reasoning of others. High school students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. High school students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and--if there is a flaw in an argument--explain what it is. High school students learn to determine domains to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
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Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry
4 Model with mathematics. High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. High school students making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
5 Use appropriate tools strategically. High school students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
6 Attend to precision. High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure. By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose functions, and transform figures.
8 Look for and express regularity in repeated reasoning. High school students notice if calculations are repeated, and look both for general methods and for shortcuts.
Georgia Department of Education December 2014 Page 3 of 11 All Rights Reserved

Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry
Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x2 + x + 1), and (x 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics should engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who do not have an understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
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Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry
Mathematics | High School--Geometry
An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts-- interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material. Although there are many types of geometry, school mathematics is devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates). Euclidean geometry is characterized most importantly by the Parallel Postulate that states that through a point not on a given line there is exactly one parallel line. (Spherical geometry, in contrast, has no parallel lines.) During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms. The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shape in general). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes--as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent. In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. For triangles, congruence means the equality of all corresponding pairs of sides and all corresponding pairs of angles. During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures. Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of "same shape" and "scale factor" developed in the middle grades. These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent. The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean Theorem, are fundamental in many real-world and theoretical situations. The Pythagorean Theorem is generalized to non-right triangles by the Law of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion. Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Just as the number line associates numbers with locations in one dimension, a pair of perpendicular axes associates pairs of numbers with locations in two dimensions. This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof. Geometric transformations of the graphs of equations correspond to algebraic changes in their equations. Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.
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Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry

Connections to Equations The correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.

Congruence

G.CO

Experiment with transformations in the plane

MCC9-12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

MCC9-12.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

MCC9-12.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

MCC9-12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

MCC9-12.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions

MCC9-12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

MCC9-12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

MCC9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS.)

Prove geometric theorems

MCC9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and

Georgia Department of Education December 2014 Page 6 of 11 All Rights Reserved

Georgia Department of Education

Georgia Performance Standards in Mathematics Geometry

corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

MCC9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

MCC9-12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Make geometric constructions

MCC9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

MCC9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.

Similarity, Right Triangles, and Trigonometry

G.SRT

Understand similarity in terms of similarity transformations

MCC9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor. a. The dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter according to the ratio given by the scale factor.

MCC9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

MCC9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

MCC9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, (and its converse); the Pythagorean Theorem using triangle similarity. MCC9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Georgia Department of Education December 2014 Page 7 of 11 All Rights Reserved

Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry

Define trigonometric ratios and solve problems involving right triangles

MCC9-12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

MCC9-12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

MCC9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Circles

G.C

Understand and apply theorems about circles

MCC9-12.G.C.1 Understand that all circles are similar.

MCC9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

MCC9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

MCC9-12.G.C.4 Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

MCC9-12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Expressing Geometric Properties with Equations

G.GPE

Translate between the geometric description and the equation for a conic section

MCC9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

MCC9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.

Georgia Department of Education December 2014 Page 8 of 11 All Rights Reserved

Georgia Department of Education

Georgia Performance Standards in Mathematics Geometry

Use coordinates to prove simple geometric theorems algebraically

MCC9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0,2). (Focus on quadrilaterals, circles, right triangles, and parabolas.)

MCC9-12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

MCC9-12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

MCC9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Geometric Measurement and Dimension

G.GMD

Explain volume formulas and use them to solve problems

MCC9-12.G.GMD.1 Give informal arguments for geometric formulas. a. Give informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments. b. Give informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri's principle.

MCC9-12.G.GMD.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

MCC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects

MCC9-12.G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Modeling with Geometry

G.MG

Apply geometric concepts in modeling situations

MCC9-12.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

MCC9-12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

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Georgia Department of Education
Georgia Performance Standards in Mathematics Geometry
MCC9-12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Mathematics | High School--Statistics and Probability
Decisions or predictions are often based on data--numbers in context. These decisions or predictions would be easy if the data always sent a clear message, but the message is often obscured by variability. Statistics provides tools for describing variability in data and for making informed decisions that take it into account. Data are gathered, displayed, summarized, examined, and interpreted to discover patterns and deviations from patterns. Quantitative data can be described in terms of key characteristics: measures of shape, center, and spread. The shape of a data distribution might be described as symmetric, skewed, flat, or bell shaped, and it might be summarized by a statistic measuring center (such as mean or median) and a statistic measuring spread (such as standard deviation or interquartile range). Different distributions can be compared numerically using these statistics or compared visually using plots. Knowledge of center and spread are not enough to describe a distribution. Which statistics to compare, which plots to use, and what the results of a comparison might mean, depend on the question to be investigated and the real-life actions to be taken. Randomization has two important uses in drawing statistical conclusions. First, collecting data from a random sample of a population makes it possible to draw valid conclusions about the whole population, taking variability into account. Second, randomly assigning individuals to different treatments allows a fair comparison of the effectiveness of those treatments. A statistically significant outcome is one that is unlikely to be due to chance alone, and this can be evaluated only under the condition of randomness. The conditions under which data are collected are important in drawing conclusions from the data; in critically reviewing uses of statistics in public media and other reports, it is important to consider the study design, how the data were gathered, and the analyses employed as well as the data summaries and the conclusions drawn. Random processes can be described mathematically by using a probability model: a list or description of the possible outcomes (the sample space), each of which is assigned a probability. In situations such as flipping a coin, rolling a number cube, or drawing a card, it might be reasonable to assume various outcomes are equally likely. In a probability model, sample points represent outcomes and combine to make up events; probabilities of events can be computed by applying the Addition and Multiplication Rules. Interpreting these probabilities relies on an understanding of independence and conditional probability, which can be approached through the analysis of two way tables. Technology plays an important role in statistics and probability by making it possible to generate plots, regression functions, and correlation coefficients, and to simulate many possible outcomes in a short amount of time.
Connections to Functions and Modeling Functions may be used to describe data; if the data suggest a linear relationship, the relationship can be modeled with a regression line, and its strength and direction can be expressed through a correlation coefficient.
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Georgia Performance Standards in Mathematics Geometry

Conditional Probability and the Rules of Probability

S.CP

Understand independence and conditional probability and use them to interpret data
MCC9-12.S.CP.1 Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not).
MCC9-12.S.CP.2 Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent.
MCC9-12.S.CP.3 Understand the conditional probability of A given B as P (A and B)/P(B). Interpret independence of A and B in terms of conditional probability; that is the conditional probability of A given B is the same as the probability of A and the conditional probability of B given A is the same as the probability of B.
MCC9-12.S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, use collected data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
MCC9-12.S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model
MCC9-12.S.CP.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in context.
MCC9-12.S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answers in context.

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Georgia Performance Standards in Mathematics Algebra I
Mathematics Georgia Performance Standards
K-12 Mathematics Introduction
Georgia Mathematics focuses on actively engaging the student in the development of mathematical understanding by working independently and cooperatively to solve problems, estimating and computing efficiently, using appropriate tools, concrete models, and a variety of representations, and conducting investigations and recording findings. There is a shift toward applying mathematical concepts and skills in the context of authentic problems and student understanding of concepts rather than merely following a sequence of procedures. In mathematics classrooms, students will learn to think critically in a mathematical way with an understanding that there are many different solution pathways and sometimes more than one right answer in applied mathematics. Mathematics is the economy of information. The central idea of all mathematics is to discover how knowing some things leads, via reasoning, to knowing more--without having to commit the information to memory as a separate fact. It is the reasoned, logical connections that make mathematics manageable. The implementation of Georgia Performance Standards in Mathematics places the expected emphasis on sense-making, problem solving, reasoning, modeling, representation, connections, and communication.
Mathematics Georgia Performance Standards Algebra I
Algebra I is the first course in a sequence of three required high school courses designed to ensure career and college readiness. The course represents a discrete study of algebra with correlated statistics applications.
The standards in the three-course high school sequence specify the mathematics that all students should study in order to be college and career ready. Additional mathematics content is provided in fourth credit courses and advanced courses including pre-calculus, calculus, advanced statistics, discrete mathematics, and mathematics of finance courses. High school course content standards are listed by conceptual categories including Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability. Conceptual categories portray a coherent view of high school mathematics content; a student's work with functions, for example, crosses a number of traditional course boundaries, potentially up through and including calculus. Standards for Mathematical Practice provide the foundation for instruction and assessment.
Mathematics | Standards for Mathematical Practice
Mathematical Practices are listed with each grade's mathematical content standards to reflect the need to connect the mathematical practices to mathematical content in instruction.
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of
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Georgia Performance Standards in Mathematics Algebra I
problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council's report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy).
1 Make sense of problems and persevere in solving them. High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
2 Reason abstractly and quantitatively. High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects.
3 Construct viable arguments and critique the reasoning of others. High school students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. High school students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and--if there is a flaw in an argument--explain what it is. High school students learn to determine domains to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
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Georgia Performance Standards in Mathematics Algebra I
4 Model with mathematics. High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. High school students making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
5 Use appropriate tools strategically. High school students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
6 Attend to precision. High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure. By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose functions, and transform figures.
8 Look for and express regularity in repeated reasoning. High school students notice if calculations are repeated, and look both for general methods and for shortcuts.
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Georgia Performance Standards in Mathematics Algebra I
Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x2 + x + 1), and (x 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics should engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who do not have an understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
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Georgia Performance Standards in Mathematics Algebra I

Mathematics | High School--Number and Quantity

Numbers and Number Systems During the years from kindergarten to eighth grade, students must repeatedly extend their conception of number. At first, "number" means "counting number": 1, 2, 3... Soon after that, 0 is used to represent "none" and the whole numbers are formed by the counting numbers together with zero. The next extension is fractions. At first, fractions are barely numbers and tied strongly to pictorial representations. Yet by the time students understand division of fractions, they have a strong concept of fractions as numbers and have connected them, via their decimal representations, with the base-ten system used to represent the whole numbers. During middle school, fractions are augmented by negative fractions to form the rational numbers. In Grade 8, students extend this system once more, augmenting the rational numbers with the irrational numbers to form the real numbers. In high school, students will be exposed to yet another extension of number, when the real numbers are augmented by the imaginary numbers to form the complex numbers. With each extension of number, the meanings of addition, subtraction, multiplication, and division are extended. In each new number system--integers, rational numbers, real numbers, and complex numbers--the four operations stay the same in two important ways: They have the commutative, associative, and distributive properties and their new meanings are consistent with their previous meanings. Extending the properties of whole-number exponents leads to new and productive notation. For example, properties of whole-number exponents suggest that (51/3)3 should be 5(1/3)3 = 51 = 5 and that 51/3 should be the cube root of 5. Calculators, spreadsheets, and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. They can be used to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with non-integer exponents.

Quantities In real world problems, the answers are usually not numbers but quantities: numbers with units, which involves measurement. In their work in measurement up through Grade 8, students primarily measure commonly used attributes such as length, area, and volume. In high school, students encounter a wider variety of units in modeling, e.g., acceleration, currency conversions, derived quantities such as person-hours and heating degree days, social science rates such as per-capita income, and rates in everyday life such as points scored per game or batting averages. They also encounter novel situations in which they themselves must conceive the attributes of interest. For example, to find a good measure of overall highway safety, they might propose measures such as fatalities per year, fatalities per year per driver, or fatalities per vehicle-mile traveled. Such a conceptual process is sometimes called quantification. Quantification is important for science, as when surface area suddenly "stands out" as an important variable in evaporation. Quantification is also important for companies, which must conceptualize relevant attributes and create or choose suitable measures for them.

The Real Number System

N.RN

Use properties of rational and irrational numbers.
MCC9-12.N.RN.2 Rewrite expressions involving radicals.
MCC9-12.N.RN.3 Explain why the sum or product of rational numbers is rational; why the sum of a rational number and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational.

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Georgia Performance Standards in Mathematics Algebra I

Quantities

N.Q

Reason quantitatively and use units to solve problems.

MCC9-12.N.Q.1 Use units of measure (linear, area, capacity, rates, and time) as a way to understand problems:
a. Identify, use, and record appropriate units of measure within context, within data displays, and on graphs;
b. Convert units and rates using dimensional analysis (English-to-English and Metric-to-Metric without conversion factor provided and between English and Metric with conversion factor);
c. Use units within multi-step problems and formulas; interpret units of input and resulting units of output.

MCC9-12.N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. Given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation.

MCC9-12.N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. For example, money situations are generally reported to the nearest cent (hundredth). Also, an answers' precision is limited to the precision of the data given.

Mathematics | High School--Algebra
Expressions An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances. Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor. Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05p is the sum of the simpler expressions p and 0.05p. Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure. A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.
Equations and Inequalities An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the

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Georgia Performance Standards in Mathematics Algebra I

equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form. The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system. An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions. Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2x + 1 = 0 is a rational number, not an integer; the solutions of x2 2 = 0 are real numbers, not rational numbers; and the solutions of x2 + 2 = 0 are complex numbers, not real numbers. The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b1+b2)/2)h, can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.

Connections to Functions and Modeling Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.

Seeing Structure in Expressions

A.SSE

Interpret the structure of expressions

MCC9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.

MCC9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients, in context.

MCC9-12.A.SSE.1b Given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors.

MCC9-12.A.SSE.2 Use the structure of an expression to rewrite it in different equivalent forms. For example, see x4 y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2) (x2 + y2).
Write expressions in equivalent forms to solve problems
MCC9-12.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

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Georgia Performance Standards in Mathematics Algebra I
MCC9-12.A.SSE.3a Factor any quadratic expression to reveal the zeros of the function defined by the expression.
MCC9-12.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression.

Arithmetic with Polynomials and Rational Expressions

A.APR

Perform arithmetic operations on polynomials

MCC9-12.A.APR.1 Add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations.

Creating Equations

A.CED

Create equations that describe numbers or relationships

MCC9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions (integer inputs only).

MCC9-12.A.CED.2 Create linear and exponential equations in two or more variables to represent
relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase
"in two or more variables" refers to formulas like the compound interest formula, in which A = P(1 + r/n)nt has multiple variables.)

MCC9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equation and/or inequalities, and interpret data points as possible (i.e. a solution) or not possible (i.e. a non-solution) under the established constraints.

MCC9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as in
solving equations. Examples: Rearrange Ohm's law V = IR to highlight resistance R; Rearrange area of a circle formula A = r2 to highlight the radius r.

Reasoning with Equations and Inequalities

A.REI

Understand solving equations as a process of reasoning and explain the reasoning

MCC9-12.A.REI.1 Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.

Solve equations and inequalities in one variable

MCC9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.

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Georgia Performance Standards in Mathematics Algebra I
MCC9-12.A.REI.4 Solve quadratic equations in one variable.
MCC9-12.A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from ax2 + bx + c = 0. MCC9-12.A.REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions).
Solve systems of equations
MCC9-12.A.REI.5 Show and explain why the elimination method works to solve a system of two-variable equations.
MCC9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Represent and solve equations and inequalities graphically
MCC9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane
MCC9-12.A.REI.11 Using graphs, tables, or successive approximations, show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.
MCC9-12.A.REI.12 Graph the solution set to a linear inequality in two variables.
Mathematics | High School--Functions
Functions describe situations where one quantity determines another. For example, the return on $10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time the money is invested. Because we continually make theories about dependencies between quantities in nature and society, functions are important tools in the construction of mathematical models. In school mathematics, functions usually have numerical inputs and outputs and are often defined by an algebraic expression. For example, the time in hours it takes for a car to drive 100 miles is a function of the car's speed in miles per hour, v; the rule T(v) = 100/v expresses this relationship algebraically and defines a function whose name is T. The set of inputs to a function is called its domain. We often infer the domain to be all inputs for which the expression defining a function has a value, or for which the function makes sense in a given context. A function can be described in various ways, such as by a graph (e.g., the trace of a seismograph); by a verbal rule, as in, "I'll give you a state, you give me the capital city;" by an algebraic expression like f(x) = a + bx; or by a recursive rule. The graph of a function is often a useful way of visualizing the relationship of the function models, and manipulating a mathematical expression for a function can throw light on the function's properties. Functions presented as expressions can model many important phenomena. Two important families of functions characterized by laws of growth are linear functions, which grow at a constant rate, and exponential functions, which grow at a constant percent rate. Linear functions with a constant term of zero describe proportional
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Georgia Performance Standards in Mathematics Algebra I

relationships. A graphing utility or a computer algebra system can be used to experiment with properties of these functions and their graphs and to build computational models of functions, including recursively defined functions.

Connections to Expressions, Equations, Modeling, and Coordinates Determining an output value for a particular input involves evaluating an expression; finding inputs that yield a given output involves solving an equation. Questions about when two functions have the same value for the same input lead to equations, whose solutions can be visualized from the intersection of their graphs. Because functions describe relationships between quantities, they are frequently used in modeling. Sometimes functions are defined by a recursive process, which can be displayed effectively using a spreadsheet or other technology.

Interpreting Functions

F.IF

Understand the concept of a function and use function notation

MCC9-12.F.IF.1 Understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range, i.e. each input value maps to exactly one output value. If f is a function, x is the input (an element of the domain), and f(x) is the output (an element of the range). Graphically, the graph is y = f(x).

MCC9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MCC9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Generally, the scope of high school math defines this subset as the set of natural numbers 1,2,3,4...) By graphing or calculating terms, students should be able to show how the recursive sequence a1=7, an=an-1 +2; the sequence sn = 2(n-1) + 7; and the function f(x) = 2x + 5 (when x is a natural number) all define the same sequence.

Interpret functions that arise in applications in terms of the context

MCC9-12.F.IF.4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.

MCC9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

MCC9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Analyze functions using different representations

MCC9-12.F.IF.7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.

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Georgia Performance Standards in Mathematics Algebra I

MCC9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context).

MCC9-12.F.IF.7e Graph exponential functions, showing intercepts and end behavior.

MCC9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

MCC9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms.

MCC9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one function and an algebraic expression for another, say which has the larger maximum.

Building Functions

F-BF

Build a function that models a relationship between two quantities

MCC9-12.F.BF.1 Write a function that describes a relationship between two quantities.

MCC9-12.F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with $15 and earns $2 a day, the explicit expression "2x+15" can be described recursively (either in writing or verbally) as "to find out how much money Jimmy will have tomorrow, you add $2 to his total today." Jn Jn1 2, J0 15

MCC9-12.F.BF.2 Write arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.

Build new functions from existing functions
MCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Linear, Quadratic, and Exponential Models

F.LE

Construct and compare linear, quadratic, and exponential models and solve problems

MCC9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

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Georgia Performance Standards in Mathematics Algebra I
MCC9-12.F.LE.1a Show that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. (This can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals).
MCC9-12.F.LE.1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
MCC9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
MCC9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
MCC9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Interpret expressions for functions in terms of the situation they model
MCC9-12.F.LE.5 Interpret the parameters in a linear (f(x) = mx + b) and exponential (f(x) = adx) function in terms of context. (In the functions above, "m" and "b" are the parameters of the linear function, and "a" and "d" are the parameters of the exponential function.) In context, students should describe what these parameters mean in terms of change and starting value.
Mathematics | High School--Statistics and Probability
Decisions or predictions are often based on data--numbers in context. These decisions or predictions would be easy if the data always sent a clear message, but the message is often obscured by variability. Statistics provides tools for describing variability in data and for making informed decisions that take it into account. Data are gathered, displayed, summarized, examined, and interpreted to discover patterns and deviations from patterns. Quantitative data can be described in terms of key characteristics: measures of shape, center, and spread. The shape of a data distribution might be described as symmetric, skewed, flat, or bell shaped, and it might be summarized by a statistic measuring center (such as mean or median) and a statistic measuring spread (such as standard deviation or interquartile range). Different distributions can be compared numerically using these statistics or compared visually using plots. Knowledge of center and spread are not enough to describe a distribution. Which statistics to compare, which plots to use, and what the results of a comparison might mean, depend on the question to be investigated and the real-life actions to be taken. Randomization has two important uses in drawing statistical conclusions. First, collecting data from a random sample of a population makes it possible to draw valid conclusions about the whole population, taking variability into account. Second, randomly assigning individuals to different treatments allows a fair comparison of the effectiveness of those treatments. A statistically significant outcome is one that is unlikely to be due to chance alone, and this can be evaluated only under the condition of randomness. The conditions under which data are collected are important in drawing conclusions from the data; in critically reviewing uses of statistics in public media and other reports, it is important to consider the study design, how the data were gathered, and the analyses employed as well as the data summaries and the conclusions drawn.
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Georgia Performance Standards in Mathematics Algebra I

Random processes can be described mathematically by using a probability model: a list or description of the possible outcomes (the sample space), each of which is assigned a probability. In situations such as flipping a coin, rolling a number cube, or drawing a card, it might be reasonable to assume various outcomes are equally likely. In a probability model, sample points represent outcomes and combine to make up events; probabilities of events can be computed by applying the Addition and Multiplication Rules. Interpreting these probabilities relies on an understanding of independence and conditional probability, which can be approached through the analysis of two way tables. Technology plays an important role in statistics and probability by making it possible to generate plots, regression functions, and correlation coefficients, and to simulate many possible outcomes in a short amount of time.

Connections to Functions and Modeling Functions may be used to describe data; if the data suggest a linear relationship, the relationship can be modeled with a regression line, and its strength and direction can be expressed through a correlation coefficient.

Interpreting Categorical and Quantitative Data

S.ID

Summarize, represent, and interpret data on a single count or measurement variable

MCC9-12.S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

MCC9-12.S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, mean absolute deviation) of two or more different data sets.

MCC9-12.S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Summarize, represent, and interpret data on two categorical and quantitative variables
MCC9-12.S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
MCC9-12.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
MCC9-12.S.ID.6a Decide which type of function is most appropriate by observing graphed data, charted data, or by analysis of context to generate a viable (rough) function of best fit. Use this function to solve problems in context. Emphasize linear, quadratic and exponential models.

MCC9-12.S.ID.6c Using given or collected bivariate data, fit a linear function for a scatter plot that suggests a linear association.

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Georgia Performance Standards in Mathematics Algebra I
Interpret linear models MCC9-12.S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. MCC9-12.S.ID.8 Compute (using technology) and interpret the correlation coefficient "r" of a linear fit. (For instance, by looking at a scatterplot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the "r" value.) After calculating the line of best fit using technology, students should be able to describe how strong the goodness of fit of the regression is, using "r". MCC9-12.S.ID.9 Distinguish between correlation and causation.
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Georgia Department of Education
Georgia Performance Standards in Mathematics Foundations of Algebra
Mathematics Georgia Performance Standards
K-12 Mathematics Introduction
Georgia Mathematics focuses on actively engaging the student in the development of mathematical understanding by working independently and cooperatively to solve problems, estimating and computing efficiently, using appropriate tools, concrete models and a variety of representations, and conducting investigations and recording findings. There is a shift toward applying mathematical concepts and skills in the context of authentic problems and student understanding of concepts rather than merely following a sequence of procedures. In mathematics classrooms, students will learn to think critically in a mathematical way with an understanding that there are many different solution pathways and sometimes more than one right answer in applied mathematics. Mathematics is the economy of information. The central idea of all mathematics is to discover how knowing some things leads, via reasoning, to knowing more--without having to commit the information to memory as a separate fact. It is the reasoned, logical connections that make mathematics manageable. The implementation of Georgia Performance Standards in Mathematics places the expected emphasis on sense-making, problem solving, reasoning, representation, modeling, representation, connections, and communication.
Mathematics Georgia Performance Standards Foundations of Algebra
Foundations of Algebra is a first year high school mathematics course option for students who have completed mathematics in grades 6 8 yet will need substantial support to bolster success in high school mathematics. The course is aimed at students who have reported low standardized test performance in prior grades and/or have demonstrated significant difficulties in previous mathematics classes.
Foundations of Algebra will provide many opportunities to revisit and expand the understanding of foundational algebra concepts, will employ diagnostic means to offer focused interventions, and will incorporate varied instructional strategies to prepare students for required high school mathematics courses. The course will emphasize both algebra and numeracy in a variety of contexts including number sense, proportional reasoning, quantitative reasoning with functions, and solving equations and inequalities.
Instruction and assessment should include the appropriate use of manipulatives and technology. Mathematics concepts should be represented in multiple ways, such as concrete/pictorial, verbal/written, numeric/data-based, graphical, and symbolic. Concepts should be introduced and used, where appropriate, in the context of realistic experiences.
The Standards for Mathematical Practice will provide the foundation for instruction and assessment. The content standards are an amalgamation of mathematical standards addressed in grades 3 through high school. The standards from which the course standards are drawn are identified for reference.
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Georgia Department of Education
Georgia Performance Standards in Mathematics Foundations of Algebra
Number Sense and Quantity
Students will compare different representations of numbers (i.e. fractions, decimals, radicals, etc.) and perform basic operations using these different representations.
MFANSQ1. Students will analyze number relationships. a. Solve multi-step real world problems, analyzing the relationships between all four operations. For example, understand division as an unknown-factor problem in order to solve problems. Knowing that 50 x 40 = 2000 helps students determine how many boxes of cupcakes they will need in order to ship 2000 cupcakes in boxes that hold 40 cupcakes each. (MCC3.OA.6, MCC4.OA.3) b. Understand a fraction a/b as a multiple of 1/b. (MCC4.NF.4) c. Explain patterns in the placement of decimal points when multiplying or dividing by powers of ten. (MCC5.NBT.2) d. Compare fractions and decimals to the thousandths place. For fractions, use strategies other than cross multiplication. For example, locating the fractions on a number line or using benchmark fractions to reason about relative size. For decimals, use place value. (MCC4.NF.2;MCC5.NBT.3,4)
MFANSQ2. Students will conceptualize positive and negative numbers (including decimals and fractions). a. Explain the meaning of zero. (MCC6.NS.5) b. Represent numbers on a number line. (MCC6.NS.5,6) c. Explain meanings of real numbers in a real world context. (MCC6.NS.5)
MFANSQ3. Students will recognize that there are numbers that are not rational, and approximate them with rational numbers.
a. Find an estimated decimal expansion of an irrational number locating the approximations on a number line. For example, for , show that is between 1 and 2, then between 1.4 and 1.5, and explain how to continue this pattern in order to obtain better approximations. (MCC8.NS.1,2)
b. Explain the results of adding and multiplying with rational and irrational numbers. (MCC912.N.RN.3)
MFANSQ4. Students will apply and extend previous understanding of addition, subtraction, multiplication, and division.
a. Find sums, differences, products, and quotients of multi-digit decimals using strategies based on place value, the properties of operations, and/or relationships between operations. (MCC5.NBT.7; MCC6.NS.3)
b. Find sums, differences, products, and quotients of all forms of rational numbers, stressing the conceptual understanding of these operations. (MCC7.NS.1,2)
c. Interpret and solve contextual problems involving division of fractions by fractions. For example, how many 3/4-cup servings are in 2/3 of a cup of yogurt? (MCC6.NS.1)
d. Illustrate and explain calculations using models and line diagrams. ( MCC7.NS.1,2)
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Georgia Performance Standards in Mathematics Foundations of Algebra
e. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using estimation strategies and graphing technology. (MCC7.NS.3, MCC7.E.3, MCC9-12.N.Q.3)
Arithmetic to Algebra
Students will extend arithmetic operations to algebraic modeling.
MFAAA1. Students will generate and interpret equivalent numeric and algebraic expressions. a. Apply properties of operations emphasizing when the commutative property applies. (MCC7.EE.1) b. Use area models to represent the distributive property and develop understandings of addition and multiplication (all positive rational numbers should be included in the models). (MCC3.MD.7) c. Model numerical expressions (arrays) leading to the modeling of algebraic expressions. (MCC7.EE.1,2; MCC9-12.A.SSE.1,3) d. Add, subtract, and multiply algebraic expressions. (MCC6.EE.3, MCC6.EE.4, MC7.EE.1, MCC9-12.A.SSE.3) e. Generate equivalent expressions using properties of operations and understand various representations within context. For example, distinguish multiplicative comparison from additive comparison. Students should be able to explain the difference between "3 more" and "3 times". (MCC4.0A.2; MCC6.EE.3, MCC7.EE.1,2;MCC9-12.A.SSE.3) f. Evaluate formulas at specific values for variables. For example, use formulas such as A = l x w and find the area given the values for the length and width. (MCC6.EE.2)
MFAAA2. Students will interpret and use the properties of exponents. a. Substitute numeric values into formulas containing exponents, interpreting units consistently. (MCC6.EE.2, MCC9-12.N.Q.1, MCC9-12.A.SSE.1, MCC9-12.N.RN.2) b. Use properties of integer exponents to find equivalent numerical expressions. For example, 32 x 3 -5 = 3 -3 = = . (MCC8.EE.1) c. Evaluate square roots of perfect squares and cube roots of perfect cubes (MCC8.EE.2) d. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. (MCC8.EE.2) e. Use the Pythagorean Theorem to solve triangles based on real-world contexts (Limit to finding the hypotenuse given two legs). (MCC8.G.7)
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Georgia Department of Education
Georgia Performance Standards in Mathematics Foundations of Algebra
Proportional Reasoning
Students will use ratios to solve real-world and mathematical problems.
MFAPR1. Students will explain equivalent ratios by using a variety of models. For example, tables of values, tape diagrams, bar models, double number line diagrams, and equations. (MCC6.RP.3)
MFAPR2. Students will recognize and represent proportional relationships between quantities. a. Relate proportionality to fraction equivalence and division. For example, is equal to because both yield a quotient of and, in both cases, the denominator is double the value of the numerator. (MCC4.NF.1) b. Understand real-world rate/ratio/percent problems by finding the whole given a part and find a part given the whole. (MCC6.RP.1,2,3;MCC7.RP.1,2) c. Use proportional relationships to solve multistep ratio and percent problems. (MCC7.RP.2,3)
MFAPR3. Students will graph proportional relationships. a. Interpret unit rates as slopes of graphs. (MCC8.EE.5) b. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. (MCC8.EE.6) c. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. (MCC8.EE.5)
Equations and Inequalities
Students will solve, interpret, and create linear models using equations and inequalities.
MFAEI1. Students will create and solve equations and inequalities in one variable. a. Use variables to represent an unknown number in a specified set. (MCC.6.EE2,5,6) b. Explain each step in solving simple equations and inequalities using the equality properties of numbers. (MCC9-12.A.REI.1) c. Construct viable arguments to justify the solutions and methods of solving equations and inequalities. (MCC9-12.A.REI.1) d. Represent and find solutions graphically. e. Use variables to solve real-world and mathematical problems. (MCC6.EE.7,MCC7.EE.4)
MFAEI1I2. Students will use units as a way to understand problems and guide the solutions of multi-step problems.
a. Choose and interpret units in formulas. (MCC9-12.N.Q.1) b. Choose and interpret graphs and data displays, including the scale and comparisons of data.
(MCC3.MD.3, MCC9-12.N.Q.1) c. Graph points in all four quadrants of the coordinate plane. (MCC6.NS.8)
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Georgia Performance Standards in Mathematics Foundations of Algebra
MFAEI2. Students will create and solve equations and inequalities in one variable. a. Use variables to represent an unknown number in a specified set. (MCC.6.EE2,5,6) b. Use variables to solve real-world and mathematical problems. (MCC6.EE.7,MCC7.EE.4) c. Use variables to represent two quantities in a real-world problem that change in relationship to one another (conceptual understanding of a variable). (MCC6.EE.9) d. Represent and find solutions graphically. (MCC9-12.A.REI.6) e. Find approximate solutions using technology to graph, construct tables of values, and find successive approximations. (MCC9-12.A.REI.10,11) f. Explain each step in solving simple equations and inequalities using the equality properties of numbers. (MCC9-12.A.REI.1) g. Construct viable arguments to justify the solutions and methods of solving equations and inequalities. (MCC9-12.A.REI.1)
MFAEI3. Students will create algebraic models in two variables. a. Create an algebraic model from a context using two-variable equations and inequalities. (MCC6.EE.6,8; MCC8.EE.8; MCC9-12.A.CED.2) b. Find approximate solutions using technology to graph, construct tables of values, and find successive approximations. (MCC9-12.A.REI.10,11) c. Represent solutions to systems of equations and inequalities graphically or by using a table of values. (MCC6.EE.5; MCC7.EE3; MCC8.EE.8; CC9-12.A.CED.2) d. Analyze the reasonableness of the solutions of systems of equations and inequalities within a given context. (MCC6.EE.5,6,MCC7.EE34)
MFAEI4. Students will solve literal equations. a. Solve for any variable in a multi-variable equation. (MCC6.EE.9,MCC9-12.A.REI.3) b. Rearrange formulas to highlight a particular variable using the same reasoning as in solving equations. For example, solve for the base in A = bh. (MCC9-12.A.CED.4)
Quantitative Reasoning with Functions
Students will create function statements and analyze relationships among pairs of variables using graphs, table, and equations.
MFAQR1. Students will understand characteristics of functions. a. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range.(MCC9-12.F.IF.1) b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (MCC9-12.F.IF.5) c. Graph functions using sets of ordered pairs consisting of an input and the corresponding output. (MCC8.F.1, 2)
MFAQR2. Students will compare and graph functions. a. Calculate rates of change of functions, comparing when rates increase, decrease, or stay constant. For example, given a linear function represented by a table of values and a linear
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Georgia Department of Education
Georgia Performance Standards in Mathematics Foundations of Algebra
function represented by an algebraic expression, determine which function has the greater rate of change. (MCC6.RP.2;MCC7.RP.1,2,3;MCC8.F.2,5; MCC9-12.F.IF.6) b. Graph by hand simple functions expressed symbolically (use all four quadrants). (MCC912.F.IF.7) c. Interpret the equation y = mx + b as defining a linear function whose graph is a straight line. (MCC8.F.3) d. Use technology to graph non-linear functions. (MCC8.F.3, MCC9-12.F.IF.7) e. Analyze graphs of functions for key features (intercepts, intervals of increase/decrease, maximums/minimums, symmetries, and end behavior) based on context. (MCC9-12.F.IF.4,7) f. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the great rate of change. (MCC8.F.2)
MFAQR3. Students will construct and interpret functions. a. Write a function that describes a relationship between two quantities. (MCC8.F.4, MCC912.F.BF.1) b. Use variables to represent two quantities in a real-world problem that change in relationship to one another (conceptual understanding of a variable). (MCC6.EE.9) c. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of context. (MCC9-12.F.IF.2)
Mathematics | Standards for Mathematical Practice
Mathematical Practices are listed with each grade's mathematical content standards to reflect the need to connect the mathematical practices to mathematical content in instruction.
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council's report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy).
1. Make sense of problems and persevere in solving them. High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform
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Georgia Department of Education
Georgia Performance Standards in Mathematics Foundations of Algebra
algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
2. Reason abstractly and quantitatively. High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others. High school students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. High school students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and--if there is a flaw in an argument--explain what it is. High school students learn to determine domains to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics. High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. High school students making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
5. Use appropriate tools strategically. High school students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing
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Georgia Performance Standards in Mathematics Foundations of Algebra
both the insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
6. Attend to precision. High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure. By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose functions, and transform figures.
8. Look for and express regularity in repeated reasoning. High school students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x2 + x + 1), and (x 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics should engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.
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Georgia Performance Standards in Mathematics Foundations of Algebra
The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who do not have an understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
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Georgia Department of Education Item for State Board of Education Approval
Curriculum Adoption & Revision
Item Name CI Curriculum - High School Mathematics Course Options
Recommendation Action Curriculum Adoption It is recommended that the State Board of Education adopt the Georgia Performance Standards (GPS) for Foundations of Algebra, Algebra I, and Geometry courses.
Rationale The mathematics standards for these three courses support College and Career Readiness and student academic achievement
Details High school mathematics standards for the current Coordinate Algebra and Analytic Geometry courses have been resorted to create a more discrete mathematics course sequence. These discrete course titles will be Algebra I and Geometry. In addition, a Foundations of Algebra course has been created to meet the academic needs of ninth grade students who struggle significantly in middle school mathematics as determined by the analysis of assessment and other student achievement data. Students eligible to enroll in the Foundations of Algebra course will be identified based on K-8 data analysis. Foundations of Algebra course completers will be able to earn one core unit of credit. District enrollment in the Foundations of Algebra course will be closely monitored to ensure that eligibility criteria are applied. Performance in subsequent courses following the Foundations of Algebra course will also be closely monitored. Funding will be utilized to develop resources and professional learning to support teachers in the implementation of all three of these courses and for differentiating instruction based on student need.
Contacts Sandi Woodall, Mathematics Program Manager Pam Smith, Director of Curriculum and Instruction Dr. Martha Reichrath, Deputy State School Superintendent for Curriculum and Instruction
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