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Math /Algebra Talks: Mental Math Strategies Session Goal To consider discussion-based activities that: Develop targeted CCSS Standards for Mathematical Practice (SMPs). Expand number sense to develop algebraic thinking. Build in formative assessment. Session Outline Watch and discuss an elementary number talk. Try some number talks and algebra talks together. Understand the purpose and structure of number talks. Key Components of Number Talks Classroom Environment and Community Classroom Discussions The Teacher’s Role The Role of Mental Math Purposeful Computation Problems Overarching Goals of Number Talks Number Sense Place Value Fluency Properties Connecting Mathematical Ideas Number Talks Whole-class activities centered around mental math tasks. Students explain and justify multiple solution strategies. Teacher acts as a facilitator. Time required: 5-10 minutes. Let’s try this problem: 26 + 27 Remember this is a mental math problem… No paper or pencil… An Elementary Number Talk How would you mentally calculate. 32 x 15? Try to find the product in two or more ways. Video Clip Classroom Discussions: Using Math Talk to Help Students Learn Students find 32 x 15 How does this teacher… Use wait time? Assess understanding? Record student thinking? How do these students… Meet the Standards for Mathematical Practice? Show algebraic thinking? Extend student thinking? What aspects of this activity would you use in your classroom? Which would you change? Compare and explain your reasoning. 7 8 12 13 6 7 10 10 1 3 1 8 2 5 1 4 5 18 6 16 2 5 2 8 6 8 7 9 3 5 1 2 7 8 12 13 What Makes it a Math Talk as opposed to a Lesson? • Understanding how numbers work, rather than learning various skills. • Empowers students to examine problems in their own way. • Short term practice toward long term goals. • Increased difficulty levels - encourages students to find more efficient ways to solve problems. • Never expect students to see the problem the “teacher’s” way. • Not predictable. • Don’t replace current curriculum or lesson; only 10-15 minutes of each day. Algebra Talks At the secondary level, build on number sense to make connections to algebra. Choose a topic and build a scaffolded “string” of mental math tasks. Examples: percents, function concepts, solving equations, and factoring patterns. Percent String Find and compare each pair of numbers. Be ready to explain how you arrived at your answers. 60% of 40 and 40% of 60 25% of 80 and 80% of 25 5% of 110 and 110% of 5 n% of 100 and 100% of n Describe the pattern. Will this hold every time? Why? Guess My Rule Input Output Guess My Rule Input Output 1 0 2 3 -3 8 5 24 x x2 – 1 Equation String Use only mental math to find a value for the variable that makes the equation true. Be prepared to explain your solution. Equation String x+1=5 x + ½ = 4½ 2x + ½ = 4½ 2(x + ½ ) = 9 5.5 = 3x + 2.5 Product String Estimate each product first. Do not calculate until told to do so! Product String Estimate First 19 x 21 99 x 101 199 x 201 39 x 41 299 x 301 (n – 1)(n + 1) Standards for Mathematical Practice 1. Make sense of problems, and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments, and critique the reasoning of others. 4. Model with mathematics. 5. 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. Standards for Mathematical Practice 1. Make sense of problems, and persevere in solving them. 2. Reason abstractly, and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. 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. How to Get Started Choose a topic, skill or problem that will be taught in the next two weeks. This will be your target. Choose a starter question. It should involve a prerequisite skill or topic. Make it accessible! How will you scaffold questions to build complexity towards the target? What tools will be available to students? What will you listen for in student responses? Some Examples Talk Starters, and Targets Which is larger, 4/7 or 3/8? Estimate the number of hairs on your head. Find 13% of 30 mentally. (x – 7)(x + 2) = 0 x2 + x Simplify 2 x + 2x +1 Use estimation. Suggest the equation of a parabola passing through the points below: I paid $54 for an item that was discounted 40%. What was my savings? Write a numerical expression equal to 46. Find as many as you can. Write an equation equivalent to x = 6. Find as many as you can. Benefits of Math Talks Clarify thinking (MP 1,2). Consider and test other strategies to see if they make sense (MP 1). Investigate and apply mathematical relationships (MP 2,3,7,8). Build a repertoire of efficient strategies (MP 1,3,5,8). Make decisions about choosing efficient strategies for specific problems (MP 5,7,8). Teachers don’t always understand the student’s thinking. It’s OK to say, “I’d like to study this further and get back with you.” “When kids listen to each other, they understand it better than when they hear it directly from me. It makes more sense to them…I really see a lot of learning going on by children listening to the other children, I really do. I mean I see some of the slower kids really picking up on concepts…really learning a lot from listening to other kids.” Susan Gehn, first and third-grade teacher Children’s Mathematics, CGI (98) Thank you! Teachers are the key to changing the way students learn mathematics -Dana and Yendol-Silva Questions? Contact us: Madeleine Jetter Department of Mathematics Cal State University, San Bernardino [email protected] Vicky Kukurda Instructional Services Riverside County Office of Education [email protected] Recommended Resources Chapin, O’Connor and Anderson, Classroom Discussions: Using Math Talk to Help Students Learn. Math Solutions Anderson and Schuster, Good Questions for Math Teaching: Why Ask Them and What to Ask, Grades 5-8. Math Solutions Classroom Video Visits at www.insidemathematics.org Parrish, Sherry, Number Talks: Helping Children Build Mental Math and Computation Strategies, Grade K-5. Math Solutions.