Faster Isn’t Smarter Message 17: Constructive Struggling Amanda Graybill, Fulton Elementary School

Common Core 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 Common Core State Standards for Mathematics, 2010, pp.6-7

Importance of Mathematical Practices “Developing students’ capacity to engage in the mathematical practices specified in the Common Core State Standards will ONLY be accomplished by engaging students in solving challenging mathematical tasks, providing students with tools to support their thinking and reasoning, and orchestrating opportunities for students to talk about mathematics and make their thinking public. It is the combination of these three dimensions of classrooms, working in unison, that promote understanding.” Adapted from Peg Smith’s Workshop, “Tasks, Tools and Talk”, Teachers Development Group Leadership Seminar, February 2013

Adding Odd Numbers – Version 1 MAKING CONJECTURES: Complete the conjecture based on the pattern you observe in the specific cases. #29. Conjecture: The sum of any two odd numbers is ____?____. 1+1 = 2 7+11 = 18 1+3 = 4 13+19 = 22 3+5 = 8 201+305=506 #30. Conjecture: The product of any two odd numbers is ____?____. 1x1 = 1 7x11 = 77 1x3 = 3 13x19 = 247 3x5 = 15 201x305=61,305

Adding Odd Numbers – Version 2 For problems 29 and 30, complete the conjecture based on the pattern you observe in the examples. Then explain why the conjecture is always true or show a case in which it is not true. MAKING CONJECTURES: Complete the conjecture based on the pattern you observe in the specific cases. #29. Conjecture: The sum of any two odd numbers is ____?____. 1+1 = 2 7+11 = 18 1+3 = 4 13+19 = 22 3+5 = 8 201+305=506 #30. Conjecture: The product of any two odd numbers is ____?____. 1x1 = 1 7x11 = 77 1x3 = 3 13x19 = 247 3x5 = 15 201x305=61,305

Require justification or explanation Characteristics of Tasks That Align with CCSS Standards for Mathematical Practice and Promote “Constructive Struggling” High cognitive demand Significant content Require justification or explanation Make connections between two or more representations Open-ended Multiple ways to enter the task and show competence Problems adapted from Peg Smith’s Workshop, “Tasks, Tools and Talk”, Teachers Development Group Leadership Seminar, February 2013

Tiling a Patio – Version 1 Alfredo Gomez is designing patios. Each patio has a rectangular garden area in the center. Alfredo uses black tiles to represent the soil of the garden. Around each garden, he designs a border of white tiles. The pictures shown below show the three smallest patios he can design with black tiles for the garden and white tiles for the border. Patio 1 Patio 2 Patio 3 Write an equation that could be used to determine the number of white tiles needed for any patio.

Tiling a Patio – Version 2 Alfredo Gomez is designing patios. Each patio has a rectangular garden area in the center. Alfredo uses black tiles to represent the soil of the garden. Around each garden, he designs a border of white tiles. The pictures shown below show the three smallest patios he can design with black tiles for the garden and white tiles for the border. Patio 1 Patio 2 Patio 3 Draw patio 4 and patio 5. How many white tiles are in Patio 4? Patio 5? Make some observations about the patios that could help you describe larger patios. Describe a method for finding the total number of white tiles needed for patio 50, without constructing it. Write a rule that could be used to determine the number of white tiles needed for any patio. Explain how your rule relates to the visual representation of the patio. Write a different rule that could be used to determine the number of white tiles needed for any patio. Explain how your rule relates to the visual representation of the patio.

Require justification or explanation Characteristics of Tasks That Align with CCSS Standards for Mathematical Practice and Promote “Constructive Struggling” High cognitive demand Significant content Require justification or explanation Make connections between two or more representations Open-ended Multiple ways to enter the task and show competence Problems adapted from Peg Smith’s Workshop, “Tasks, Tools and Talk”, Teachers Development Group Leadership Seminar, February 2013

Tiling a Patio – Standards for Math Practice High cognitive demand - no specified pathway to follow, requires students to explore relationships Significant content - equivalence, rate of change Require justification or explanation - explain in d and e Make connections between two or more representations -connect rule to visual; could also connect with tables and graphs Open-ended - different descriptions and rules can be written and in different forms Multiple ways to enter the task and to show competence - build patios, draw pictures, make tables, write equations, draw graphs