Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Foundations of Algebra Module 2: Arithmetic to Algebra These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 Module 2: Arithmetic to Algebra Contents FOUNDATIONS OF ALGEBRA REVISION SUMMARY ......................................................... 4 MATERIALS LIST ........................................................................................................................ 5 OVERVIEW ................................................................................................................................... 8 STANDARDS FOR MATHEMATICAL CONTENT ................................................................... 9 STANDARDS FOR MATHEMATICAL PRACTICE ................................................................ 10 ENDURING UNDERSTANDINGS ............................................................................................ 13 ESSENTIAL QUESTIONS .......................................................................................................... 14 SELECTED TERMS AND SYMBOLS ....................................................................................... 14 INTERNET INVESTIGATIONS FOR TEACHERS .................................................................. 15 Internet Based Virtual Manipulatives .............................................................................. 15 INTERVENTION TABLE ........................................................................................................... 16 SCAFFOLDED INSTRUCTIONAL LESSONS ......................................................................... 18 Arithmetic to Algebra ......................................................................................................... 20 Olympic Cola Display ......................................................................................................... 28 Distributing Using Area ..................................................................................................... 47 Triangles and Quadrilaterals ............................................................................................. 57 Tiling Lesson........................................................................................................................ 75 Conjectures About Properties ........................................................................................... 92 Quick Check I .................................................................................................................... 104 Visual Patterns .................................................................................................................. 109 Translating Math .............................................................................................................. 121 Exploring Expressions ...................................................................................................... 134 A Few Folds ....................................................................................................................... 147 Bacterial Growth ............................................................................................................... 153 Excursions with Exponents .............................................................................................. 165 Squares, Area, Cubes, Volume, Roots….Connected? .................................................. 172 Quick Check II .................................................................................................................. 185 What’s the “Hype” About Pythagoras? .......................................................................... 190 Fabulous Formulas ........................................................................................................... 201 Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 2 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 The Algebra of Magic ....................................................................................................... 209 APPENDIX OF RELEASED SAMPLE ASSESSMENT ITEMS ............................................. 224 Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 3 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 FOUNDATIONS OF ALGEBRA REVISION SUMMARY The Foundations of Algebra course has been revised based on feedback from teachers across the state. The following are changes made during the current revision cycle:  Each module assessment has been revised to address alignment to module content, reading demand within the questions, and accessibility to the assessments by Foundations of Algebra teachers.  All module assessments, as well as the pre- and posttest for the course, will now be available in GOFAR at the teacher level along with a more robust teacher’s edition featuring commentary along with the assessment items.  All modules now contain “Quick Checks” that will provide information on mastery of the content at pivotal points in the module. Both teacher and student versions of the “Quick Checks” will be accessible within the module.  A “Materials List” can be found immediately after this page in each module. The list provides teachers with materials that are needed for each lesson in that module.  A complete professional learning series with episodes devoted to the “big ideas” of each module and strategies for effective use of manipulatives will be featured on the Math Resources and Professional Learning page at https://www.gadoe.org/Curriculum- Instruction-and-Assessment/Curriculum-and-Instruction/Pages/Mathematics.aspx. Additional support such as Module Analysis Tables may be found on the Foundations of Algebra page on the High School Math Wiki at http://ccgpsmathematics9- 10.wikispaces.com/Foundations+of+Algebra. This Module Analysis Table is NOT designed to be followed as a “to do list” but merely as ideas based on feedback from teachers of the course and professional learning that has been provided within school systems across Georgia. Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 4 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 MATERIALS LIST Lesson Materials 1. Arithmetic to Algebra NA 2. Olympic Cola Display  Act 1 picture -Olympic Cola Display  Pictorial representations of the display  Student recording sheet 3. Distributing Using Area  Student activity sheet  Optional: Colored Sheets of paper cut into rectangles. These can be used to introduce the concepts found in this lesson and to create models of the rectangles as needed. 4. Triangles and  Student handout for www.visualpatterns.org Quadrilaterals activator  Cut out triangles and quadrilaterals on template  Envelopes  Student activity sheet  Match up cards for closing activity  Template for Like Terms closing activity 5. Tiling Lesson  patty paper units for tiling  (Teacher) 3 unit × 2 unit rectangle  (Students) 5 large mystery rectangles lettered A– E (1 of each size per group)  Student activity sheet 6. Conjectures about  Student activity sheet Properties  Optional: manipulatives to show grouping (put 12 counters into groups of zero) 7. Quick Check I  Student sheet 8. Visual Patterns  Various manipulatives such as two color counters  Color tiles  Connecting cubes  Visual Patterns Handout 9. Translating Math  Sticky notes may be offered as a way to build tape diagrams (bar models) Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 5 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 10. Exploring Expressions  Optional: Personal white boards (or sheet protectors)  Student lesson pages  Marbles and a bag (if using the differentiation activity) 11. A Few Folds  Student activity sheet for each student/pair of students/or small group  Paper for folding activity in Part 1 12. Bacterial Growth  Student activity sheet 13. Excursions with  Student handout/note taking guide Exponents 14. Squares, Area, Cubes,  One box of Cheez-Its per team (algebra tiles or Volume, other squares may be substituted) Roots…Connected?  One box of sugar cubes per team (average 200 cubes per one pound box) (algebra cubes, linking cubes, or other cubes may be substituted)  Graphic Organizer for Squares  Graphic Organizer for Cubes  2 Large number lines (using bulletin board paper) to display in the class; one for square roots and one for cube roots 15. Quick Check II  Student sheet 16. What’s the “Hype” about  Student handout Pythagoras?  Calculators  Sticky notes  Link or download version of Robert Kaplinksy’s “How Can We Correct the Scarecrow?” video http://robertkaplinsky.com/work/wizard-of-oz 17. Fabulous Formulas  Formula sheet  Application problems  Calculators 18. The Algebra of Magic  Computer and projector or students with personal technology (optional)  Directions for mathematical magic tricks  Counters Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 6 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2  Sticky notes or blank pieces of paper-all the same size Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 7 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 OVERVIEW As the journey into Module 2 of Foundations of Algebra begins, the teacher is charged with a pretty daunting task: create a connection from arithmetic skills to operations in algebra. The teacher will build bridges between the “known” world of numbers and the “unknown” world of variables. Students, however, do not come to class as “blank slates” ready to absorb all that is attempted in order to make connections. Instead, many of them arrive as wounded survivors of a battle they have been fighting for over eight years. This battle has been against MATH. Many of them feel that they “can’t do math” and that belief has been validated by peers (“How can you not understand that problem?…..It’s easy”), parents (“I was never good at math either...We just don’t have the math GENE”), teachers (“I have explained that to you a hundred times...what don’t you get?”), and society (“Some people just aren’t MATH people”). Research by such leaders as Jo Boaler from Stanford University has dispelled all of the “excuses” above to prove that ALL students CAN learn math. In Module 2 of Foundations of Algebra, students will draw conclusions from arithmetic’s focus on computation with specific numbers to build generalizations about properties that can be generalized to all sets of numbers. Students will look beyond individual problems to see patterns in mathematical relationships. They will apply properties of operations with emphasis on the commutative and distributive properties as they build and explore equivalent expressions with various representations. A critical emphasis on the area model for multiplication and the distributive property will provide a concrete connection to the multiplication of monomial and polynomial expressions. Students will experiment with composition and decomposition of numbers as they perform operations with variable expression. Students will connect concepts from Module 1’s focus on operations with numbers to algebraic operations. In Module 2 of Foundations of Algebra, students will also interpret and apply the properties of exponents. Students will explore and evaluate formulas involving exponents with attention to units of measure. Using concrete models, students will build area and volume models to explore square roots and cube roots. The module will conclude with specific applications of the Pythagorean Theorem. This module supplies multiple opportunities for students to explore and experience behind the standards amalgamated for its development. Extra practice and review opportunities along with sample released assessment items are provided for course instruction. Teachers should select the materials most appropriate for his/her students as they journey toward the connection between what is mathematically true for specific cases (arithmetic) to what can be generalized for multiple situations (algebra). Module 2 also sets the essential groundwork for future modules on proportional reasoning, equations and inequalities, and functions. Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 8 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 In this module, students will formally examine the connections between operations in arithmetic from elementary school to algebraic operations introduced in middle school in preparation for high school standards. The standards for mathematical content listed below will serve as the connection and focus for Module 2 of Foundations of Algebra. STANDARDS FOR MATHEMATICAL CONTENT 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. (MGSE7.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). (MGSE3.MD.7) c. Model numerical expressions (arrays) leading to the modeling of algebraic expressions. (MGSE7.EE.1,2; MGSE9-12.A.SSE.1,3) d. Add, subtract, and multiply algebraic expressions. (MGSE6.EE.3, MGSE6.EE.4, MC7.EE.1, MGSE9-12.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”. (MGSE4.0A.2; MGSE6.EE.3, MGSE7.EE.1, 2, MGSE9-12.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. (MGSE6.EE.2) MFAAA2. Students will interpret and use the properties of exponents. a. Substitute numeric values into formulas containing exponents, interpreting units consistently. (MGSE6.EE.2, MGSE9-12.N.Q.1, MGSE9-12.A.SSE.1, MGSE9-12.N.RN.2) b. Use properties of integer exponents to find equivalent numerical expressions. For example, 32 𝑥 3−5 = 3−3 = 1 = 1. (MGSE8.EE.1) 33 27 c. Evaluate square roots of perfect squares and cube roots of perfect cubes (MGSE8.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. (MGSE8.EE.2) e. Use the Pythagorean Theorem to solve triangles based on real-world contexts (Limit to finding the hypotenuse given two legs). (MGSE8.G.7) Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 9 of 232 Georgia Department of Education Georgia Standards of Excellence Framework GSE Arithmetic to Algebra  Module 2 STANDARDS FOR MATHEMATICAL PRACTICE 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 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. Mathematics  GSE Foundations of Algebra Module 2: Arithmetic to Algebra July 2017  Page 10 of 232