# Representing the Laws of Arithmetic

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1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing the Laws of Arithmetic Mathematics Assessment Resource Service University of Nottingham & UC Berkeley For more details, visit: 0 MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Representing the Laws of Arithmetic MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to: Perform arithmetic operations, including those involving whole-number exponents, recognizing and applying the conventional order of operations. Write and evaluate numerical expressions from diagrammatic representations and identify equivalent expressions. Apply the distributive and commutative properties appropriately. Use the method for finding areas of compound rectangles. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: 6.EE: 6.G: Apply and extend previous understandings of arithmetic to algebraic expressions. Solve real-world and mathematical problems involving area, surface area, and volume. This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practices, 3, and 7:. Make sense of problems and persevere in solving them.. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others.. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. INTRODUCTION The lesson unit is structured in the following way: Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You then review their responses and create questions for students to consider when improving their work. After a whole-class introduction, students work collaboratively on a card matching activity. Towards the end of the lesson there is a whole-class discussion. In a follow-up lesson, students work alone on a similar task to the introductory assessment task. MATERIALS REQUIRED Each student will need a copy of the two assessment tasks: Expressions and Areas and Expressions and Areas (revisited), a mini-whiteboard, a pen, and an eraser. Each small group of students will need cut-up copies of Card Set: Expressions and Card Set: Area Diagrams, a large sheet of paper for making a poster, and a glue stick. There is a projector resource to support whole-class discussions. You may also want to copy the card sets onto transparencies to be used on an overhead projector to support discussions. TIME NEEDED minutes before the lesson, an 8-minute lesson (or two 0-minute lessons), and minutes in a follow-up lesson. Timings are approximate. Exact timings will depend on the needs of your class. Teacher guide Representing the Laws of Arithmetic T-

4 Common issues Does not find the areas of the rectangles Does not recognize the function of parentheses For example: The student selects 3 + and/or 3 + as an appropriate expression (Qa). Or: The student selects as an appropriate expression (Qb). Does not understand the distributive law of multiplication (or division) For example: The student believes that 0 (4 + ) and are equivalent (Qa). Fails to recognize the commutative property For example: Student does not select as a correct expression (Qb) and does not recognize 3 and 3 as equivalent. Does not see the link between multiplication and addition For example: The student does not select as an appropriate expression (Qa). Assumes that squaring a number is the same as multiplying by two For example: The student states that 8 + is the same as (8 + ) (Qb) and that both are equal to 0. Does not understand the significance of the fraction bar For example: The student is unable to explain the difference between 8+ 6 and Completes the task The student needs an extension task. Suggested questions and prompts How do you calculate the area of a rectangle? Does this expression describe the area of both rectangles in the compound area diagram? Which operation would you do first in the expression 3 +? How do you know? Work it out using a calculator. Is the answer what you expected? What does ( ) mean? Can you write this expression without parentheses: (3 + )? Try typing these into a calculator. What happens in each case? Can you explain this? Try drawing an area diagram for each expression. What is the same and what is different? Can you write an expression for each of the four sections in the compound area diagram? Do any of the sections have the same area? How do you know? How could you check? Can you draw an area diagram for (3 + )? What expression would describe this area diagram in the simplest way possible? If you put two of these diagrams together, what would the expression for the area be? What is the difference between 8 and 8? Can you write these more succinctly: and 8 8 8? How would you say 8+ 6 in words? What is the difference between 8+ 6 and and 8 + 6? What is divided by two in each expression? Can you draw area diagrams to represent the expressions in question? Can you draw two different area diagrams that represent 4? Teacher guide Representing the Laws of Arithmetic T-3

6 Display Slide P- of the projector resource: Compound Area Diagrams Area A Area B Area C Which compound area diagram represents the expression: + 4 x? Projector Resources On your whiteboard write the letter A, B or C to indicate which compound area diagram you think represents the expression + 4. Can you give me a different expression for this same area? [4 + ; 4 + ; + 4 ] Spend some time discussing each area diagram with the class and help them to recognize why Area A is the correct area diagram. Can you show me an expression that represents Area B or Area C? [ + 4; 4 + ] Point out the equivalence of B and C and the commutative nature of addition in the two expressions. Collaborative activity: Matching Cards (3 minutes) Organize the class into groups of two or three students and give each group Card Set: Expressions, Card Set: Area Diagrams (all already cut up), a piece of poster paper, and a glue stick. Explain to students how they are to work collaboratively: You are now going to work together, taking turns to find cards that match. Each time you do this, explain your thinking clearly and carefully to your partner. You both need to be able to agree on and explain the placement of every card. For each Area card find at least two Expressions cards. If you think there is no suitable card that matches, use a blank card to write one of your own. Once agreed, stick the matched cards onto the poster paper writing any relevant calculations and explanations next to the cards. These instructions are summarized on Slide P-3 of the projector resource, Matching Cards. The purpose of this structured work is to encourage students to engage with each others explanations and take responsibility for each others understanding. While students work in small groups you have two tasks: to note different student approaches to the task and to support student reasoning. Note different student approaches to the task Laws of Arithmetic Listen and watch students carefully. In particular, notice how students make a start on the task, where they get stuck, and how they overcome any difficulties. Do students first match equivalent expressions, or do they match an expression with an area diagram? Do they use the area diagram to find a value for the area or do they calculate the values for the matched expressions to check they are the same? Do they make any attempts to generalize? Notice any interesting ways of explaining a match and whether or not students check their matches. Teacher guide Representing the Laws of Arithmetic T- P-

10 SOLUTIONS Assessment task: Expressions and Areas a. 3 Expressions (ii) ( 3 + ), (iv) ( ) and (v) ( (3 + )) represent the value of the area of the diagram. The expression (iv) ( ) is not directly evident, but may be seen as the area made from two halves (3 + ) + (3 + ). Expression (i) 3 + can be represented by: 3 Expression (iii) 3 + can be represented by: Students indicating that expressions (i) or (iii) represent the area may be having difficulty with the order of operations. b Expressions (i) ; (iv) ( + 3) and (v) ( + 3) ( + 3) represent the area of the above diagram. Expression (ii) + 3 represents only two out of the four sections of the compound rectangle diagram above and can be represented by: 3 3 Expression (iii) can be represented by: 3 Teacher guide Representing the Laws of Arithmetic T-9

11 a. The student may be able to express in words that 0 (4 + ) involves multiplying the whole sum 4 + by 0, whereas in 0 4 +, only the 4 is multiplied by the 0. A suitable diagram to explain the difference between and 0 (4 + ) is as follows: b. The student may explain that the first expression 8 + involves squaring each part then adding the result to obtain 68, whereas (8+) involves adding first then squaring to obtain 00. A suitable diagram explaining the difference is: c. The student may explain that the first expression, 8+ 6 involves halving the result of 8+6, giving the answer of 7, whereas, involves first halving 6 then adding the result to 8, giving the answer. A suitable area diagram explaining the difference is: In question, we would not expect many students to be able to draw such diagrams on a first attempt, but when revisiting the task they may be able to do so. Collaborative Activity: Card Matching Examples for completion of blank cards are shown in bold. A E7 E A E Blank (3 + 4) Teacher guide Representing the Laws of Arithmetic T-0

12 A3 E A4 E4 E E A E (3 4) Blank A6 E9 E (3 + 4) A7 E E Blank (4 + 3) 3 4 Teacher guide Representing the Laws of Arithmetic T-

13 Blank A8 E6 E E (3+ 4) 3+ 4 Assessment task: Expressions and Areas (revisited) a. 6 Expressions (i) (+ 6), (ii) + 6, (iv) 6 + and (v) 6 + represent the area of the above diagram. Students will need to recognize the commutative property of addition when identifying expressions (i) and (ii) as correct. Expression (iii) 6 + can be represented by: 6 Students indicating that expression (iii) represents the diagram given are failing to recognize the need for parentheses to override the conventional order of multiplication over addition. b. Expressions (i) 3, (iii) 3 and (v) 6 represent the area of the above diagram. Expression (ii) 3 can be represented by: Teacher guide Representing the Laws of Arithmetic T-

14 Expression (iv) (3 ) can be represented by:. Expressions (ii) 6 + and (iii) (6 + ) 6 are equivalent. This equivalence is shown in the diagrams below: (ii) 6 + (iii) (6 + ) with two 6 rectangles removed Possible diagrams for the remaining expressions are: (i) ( ) (iv) ( 6 ) (v) ( + 6) ( + 6) Teacher guide Representing the Laws of Arithmetic T-3

15 Expressions and Areas. (a) Check ( ) everyexpressionthatrepresentsthearea shaded in the following diagram: i 3+ ii 3+ iii 3+ iv v (3+ ) Explain your choices: (b) Check ( ) everyexpressionthatrepresentsthearea shaded in the following diagram: i ii + 3 iii iv (+ 3) v (+ 3) (+ 3) Explain your choices: Student materials Representing the Laws of Arithmetic S- 0MARS,ShellCenter,UniversityofNottingham

16 . (a) Explain the difference in meaning between 0 4+and0 (4 + ). Use words or diagrams to help your explanation. (b) Explain the difference in meaning between 8 + and (8 + ). Use words or diagrams to help your explanation. (c) Explain the difference in meaning between Use words or diagrams to help your explanation. and #. Student materials Representing the Laws of Arithmetic S- 0MARS,ShellCenter,UniversityofNottingham

17 Card Set: Expressions E E E3 E E (3 4) E E7 E Student materials Representing the Laws of Arithmetic S-3 0MARS,ShellCenter,UniversityofNottingham

18 Card Set: Expressions (continued) E9 E0 (3+ 4) E (3+ 4) E E E E E E7 E8 Student materials Representing the Laws of Arithmetic S-4 0MARS,ShellCenter,UniversityofNottingham

19 Card Set: Area Diagrams Student materials Representing the Laws of Arithmetic S- 0MARS,ShellCenter,UniversityofNottingham

20 Expressions and Areas (revisited). (a) Check ( ) everyexpressionthatrepresentsthearea shaded in the following diagram: i ii (+ 6) + 6 iii iv v 6 + Explain your choices: (b) Check ( ) everyexpressionthatrepresentsthearea shaded in the following diagram: i 3 ii 3 iii 3 iv (3 ) v 6 Explain your choices: Student materials Representing the Laws of Arithmetic S-6 0MARS,ShellCenter,UniversityofNottingham

21 . Two of the following expressions are the same. Identify the two expressions and draw area diagrams to show that they are the same: i ii 6 + iii (6 + ) - 6 iv 6 v (+ 6) (+ 6) Area diagrams: Student materials Representing the Laws of Arithmetic S-7 0MARS,ShellCenter,UniversityofNottingham

22 Writing Expressions 4 3 Write an expression to represent the total area of this diagram Projector Resources Representing the Laws of Arithmetic P-

23 Compound Area Diagrams Area A Area B 4 4 Which compound area diagram represents the expression: + 4 x? Area C 4 Projector Resources Representing the Laws of Arithmetic P-

24 Matching Cards. Take turns at matching pairs of cards that you think belong together. For each Area card there are at least two Expressions cards.. Each time you do this, explain your thinking clearly and carefully. Your partner should either explain that reasoning again in his/her own words or challenge the reasons you gave. 3. If you think there is no suitable card that matches, write one of your own on a blank card. 4. Once agreed, stick the matched cards onto the poster paper writing any relevant calculations and explanations next to the cards. You both need to be able to agree on and to be able to explain the placement of every card. Projector Resources Representing the Laws of Arithmetic P-3

26 Mathematics Assessment Project Classroom Challenges These materials were designed and developed by the Shell Center Team at the Center for Research in Mathematical Education University of Nottingham, England: Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead We are grateful to the many teachers and students, in the UK and the US, who took part in the classroom trials that played a critical role in developing these materials The classroom observation teams in the US were led by David Foster, Mary Bouck, and Diane Schaefer This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) by Alan Schoenfeld at the University of California, Berkeley, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the University of Nottingham Thanks also to Mat Crosier, Anne Floyde, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularly grateful to Carina Wong, Melissa Chabran, and Jamie McKee The full collection of Mathematics Assessment Project materials is available from 0 MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact if this license does not meet your needs.

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