Algebraic Reasoning as Equality Representations and Proof Curriculum

Algebraic Reasoning as Equality, Representations and Proof Curriculum and Assessment Policy Branch, 2016

Learning Focus Participants will: • deepen mathematical content knowledge of algebraic reasoning • develop awareness of key concepts associated with algebraic reasoning, specifically: § Equality as a relationship between quantities § Representations § Proof • develop pedagogical knowledge for teaching algebraic reasoning 2 2 Curriculum and Assessment Policy Branch, 2016

Session Norms • Be engaged in the tasks and discussions as this enriches everyone’s experience. • Embrace the learning! • Actively seek connections between the research and your classroom experience. By better understanding student thinking we better understand the impact we can have on their learning. 3 Curriculum and Assessment Policy Branch, 2016

Agenda What is Algebraic Reasoning? Exploring Equality – a relationship between quantities Exploring Representation – double number lines and symbols Exploring Proof – generalizing mathematical properties and relationships Consolidate Use of Symbols Resources 4 Curriculum and Assessment Policy Branch, 2016

What is Algebraic Reasoning? Paying Attention to Algebraic Reasoning Algebraic reasoning permeates all of mathematics and is about describing patterns of relationships among quantities – as opposed to arithmetic, which is carrying out calculations with known quantities. In its broadest sense, algebraic reasoning is about generalizing mathematical ideas and identifying mathematical structures. 5 5 Curriculum and Assessment Policy Branch, 2016

Mathematical Generalization 6 THINK: to yourself PAIR: in your group SHARE: with the large group. Curriculum and Assessment Policy Branch, 2016

Generalization “Generalization is the heartbeat of mathematics and appears in many forms. If teachers are unaware of its presence, and are not in the habit of getting students to work at expressing their own generalizations, then mathematical thinking is not taking place. ” (John Mason, 1996) 7 Curriculum and Assessment Policy Branch, 2016

Equality 8 Curriculum and Assessment Policy Branch, 2016 8

The Equal Sign Consider the following: 9 9 Curriculum and Assessment Policy Branch, 2016

What does = mean? How do many students respond? The answer is 10 Curriculum and Assessment Policy Branch, 2016

Exploring Equality as a Relationship between Quantities - 1 Students were asked to determine if the following number sentences are true or false. Predict their responses: 3+4=7 Most students stated true. 11 Curriculum and Assessment Policy Branch, 2016

Exploring Equality as a Relationship between Quantities - 2 7=3+4 Students said false because it is backwards. 3+4=5+2 Students initially considered this expression to be nonsense. 12 Curriculum and Assessment Policy Branch, 2016

Exploring Equality as a Relationship between Quantities -3 5+1=7 Students say true if considering the format or false if paying attention to the calculation. 7=7 Many young students claim this is false and correct it by substituting 7 + 0 = 7. 13 Curriculum and Assessment Policy Branch, 2016

Exploring Equality as a Relationship between Quantities - 4 It has been well documented that students do not recognize that the equal sign denotes equality… Most students see the equal sign as a signal to do something – to carry out a calculation and put the answer after the equal sign. Paying Attention to Algebraic Reasoning p. 6 14 Curriculum and Assessment Policy Branch, 2016

The Equal Sign - Revisited 15 15 Curriculum and Assessment Policy Branch, 2016

Reasoning Algebraically about Arithmetic Instruction in primary classrooms should encourage students to explore equality as a relationship between two quantities. 16 Curriculum and Assessment Policy Branch, 2016

Exploring Equality as a Relationship between Quantities - 5 Questions posed by Grade 2 students 58 + ___ = 34 + 60 17 + 13 = 18 + ____ + 1580 = 1582 + 400 17 Curriculum and Assessment Policy Branch, 2016

Paying Attention To Algebraic Reasoning When students work with equations, it is imperative that they understand that the equal sign represents a relation between quantities … Students who develop this understanding can compare without having to carry out the calculations. They can focus on the equivalence … (algebraic reasoning) rather than comparing … answers (arithmetic reasoning). p. 7 18 18 Curriculum and Assessment Policy Branch, 2016

Something to think about: How can • varying the position of the equal sign, and • providing students with an opportunity to define the equal sign, support them with algebraic reasoning? 19 Curriculum and Assessment Policy Branch, 2016

Representation 20 20 Curriculum and Assessment Policy Branch, 2016

Sunny’s Jumps When Sunny jumps 4 times and takes 11 steps forward, he lands in the same place as when he jumps 5 times and takes 4 steps forward. How many steps long is Sunny’s jump? • All jumps are assumed to be equal in length. All steps are also assumed to be equal in length. 2007 Catherine Twomey Fosnot from Contexts for Learning Mathematics (Portsmouth, NH: Heinemann) 21 Curriculum and Assessment Policy Branch, 2016

Sunny’s Jumps: Double Number Line Representation When Sunny jumps 4 times and takes 11 steps forward, he lands in the same place as when he jumps 5 times and takes 4 steps forward. 1 jump = 7 steps 22 Curriculum and Assessment Policy Branch, 2016

Paying Attention To Algebraic Reasoning p. 7 At the heart of algebraic reasoning are generalizations as expressed by symbols. 23 23 Curriculum and Assessment Policy Branch, 2016

Sunny’s Jumps: Algebraic Representation 4 j + 11 s = 5 j + 4 s * 4 j – 4 j + 11 s = 5 j – 4 j + 4 s 11 s = j + 4 s 11 s – 4 s = j – 4 s 7 s = j When Sunny jumps 4 times and takes 11 steps forward, he lands in the same place as when he jumps 5 times and takes 4 steps forward. 1 jump = 7 steps 24 Curriculum and Assessment Policy Branch, 2016

The Pairs Competition -1 In the competition, two frogs jump. Each team gets two jumping sequences. The length of the jump for each frog is then determined and the lengths are added together for an overall result. The winners are the pair with the longest combined jumping distance (one jump each). Referee’s Rule: Each frog’s jumps are assumed to be equal in length. All steps are assumed to be equal in length. 2007 Catherine Twomey Fosnot from Contexts for Learning Mathematics (Portsmouth, NH: Heinemann) 25 25 Curriculum and Assessment Policy Branch, 2016

The Pairs Competition -2 Team # 1: When Huck jumps three times and Tom jumps once, their total is 40 steps, but when Huck jumps four times and Tom jumps twice, their total is 58 steps. 26 26 Curriculum and Assessment Policy Branch, 2016

Huck and Tom’s Results – Double Number Line Representation When Huck jumps three times and Tom jumps once, their total is 40 steps, but when Huck jumps 4 times and Tom jumps twice, their total is 58 steps. 40 steps Huck + Tom = 18 steps 27 Curriculum and Assessment Policy Branch, 2016 58 steps

Huck and Tom’s Results – Algebraic Representation 4 H + 2 T = 58 3 H + 1 T = 40 1 H + 1 T = 18 When Huck jumps three times and Tom jumps once, their total is 40 steps, but when Huck jumps 4 times and Tom jumps twice, their total is 58 steps. Huck’s jump + Tom’s jump = 18 steps 28 Curriculum and Assessment Policy Branch, 2016

The Pairs Competition -3 Team # 2: Hopper and Skipper have a different technique than the other pairs. First Hopper takes three jumps and lands in the same place as Skipper does when he takes four jumps. Then Hopper takes six jumps and nine steps to land in the same place as Skipper does when he takes nine jumps. 29 29 Curriculum and Assessment Policy Branch, 2016

Hopper and Skipper: Double Number Line Representation 30 Curriculum and Assessment Policy Branch, 2016

Hopper and Skipper: Algebraic Representation 31 31 Curriculum and Assessment Policy Branch, 2016

What the students did: 3 h = 4 s subtract again 32 32 6 h + 9 = 9 s 3 h = 4 s 3 h + 9 = 5 s 3 h = 4 s 9 = 1 s Curriculum and Assessment Policy Branch, 2016 3 h 3 h 3 h 1 h = = 4 s 4 x 9 36 12 1 h + 1 s = 9 + 12 1 h + 1 s = 21 steps

Something to think about Why is it important for students to ‘see’ solutions in a variety of representations? 33 Curriculum and Assessment Policy Branch, 2016

Proof 34 Curriculum and Assessment Policy Branch, 2016

Something to Prove! Choose three consecutive numbers and add them together. Try another set. What do you notice? Will this always be true? How do you know? 35 Curriculum and Assessment Policy Branch, 2016

One way of thinking about it Video: The Sum of Three Consecutive Numbers Listen to what this student is saying. What generalizations is she making? Are there other conjectures you can make about this sum? Is this a proof? 36 Curriculum and Assessment Policy Branch, 2016

Another way of seeing How do these symbols represent the problem? What does it prove? 37 Curriculum and Assessment Policy Branch, 2016

Notion of Proof Instructional programs from prekindergarten through grade 12 should enable all students to • Recognize reasoning and proof as fundamental aspects of mathematics; • Make and investigate mathematical conjectures; • Develop and evaluate mathematical arguments and proofs • Select and use various types of reasoning and methods of proof -NCTM 38 Curriculum and Assessment Policy Branch, 2016

Quote: “Convince yourself, convince a friend, convince a skeptic. ” Cathy Humphreys 39 Curriculum and Assessment Policy Branch, 2016

Process of Proving Specializing - trying specific cases Generalizing - detecting a pattern Conjecturing - articulating a pattern (the WHAT) Convincing - justifying and proving (the WHY) Mason, Burton, Scacey, Thinking Mathematically 40 Curriculum and Assessment Policy Branch, 2016

Exploring Properties and Relationships 2 + 4 = 6, 10 + 20 = 30 Conjecturing Proving a Properties of numbers Conjecture Even + Even = Even 5 + 3 = 8, 13 + 1 = 14 Odd + Odd = Even 5 + 8 = 13, 2 + 9 = 11 Odd + Even = Odd Specific Calculations 41 Curriculum and Assessment Policy Branch, 2016

Actions to Develop Algebraic Reasoning Offering and Testing Conjectures • A guess or prediction based on limited evidence Justifying and Proving • Is this always true? How do you know? • Example vs. Counter-Example 42 Curriculum and Assessment Policy Branch, 2016

Using Symbols, Including Letters, as Variables Algebraic reasoning is based on our ability to notice patterns and generalize from them. Algebra is the language that allows us to express these generalizations in a mathematical way. Paying Attention to Algebraic Reasoning p. 3 43 43 Curriculum and Assessment Policy Branch, 2016

Variables as Changing Quantities The Property of Adding Zero 44 44 Curriculum and Assessment Policy Branch, 2016

Variables as Unknowns 45 45 Curriculum and Assessment Policy Branch, 2016

Solve for ? 46 46 Curriculum and Assessment Policy Branch, 2016

Something More to Think About An algebraic expression can be treated as an object: • Even if students are comfortable with x + 5 = 9, it is more difficult for them to think of the expression x + 5 as an object by itself In our example (horse + butterfly) can be treated as an object to solve the problem horse + butterfly = 5 47 Curriculum and Assessment Policy Branch, 2016 Jacob, Fosnot; 2007

Student Solutions Across Grades p. 23 -26 http: //www. edugains. ca/resources/Learning. Materials/Con tinuum. Connection/Solving. Equations. pdf 48 Curriculum and Assessment Policy Branch, 2016

Resources – Adobe Presenter 49 49 Curriculum and Assessment Policy Branch, 2016

Resources mathies. ca 50 50 Edu. GAINS Mathematics Curriculum and Assessment Policy Branch, 2016