The present publication was developed under grant 84.373X100001 from the

2. An Overview of the Common Core State Standards for Mathematical Practicefor use with the Common Core Essential Elements The present publication was developed under grant 84.373X100001 from the U.S. Department of Education, Office of Special Education Programs. The views expressed herein are solely those of the author(s), and no official endorsement by the U.S. Department should be inferred.

3. 8 Standards of Mathematical Practice Standard 1: Make sense of problems and persevere in solving them Standard 2: Reason abstractly and quantitatively Standard 3: Construct viable arguments and critique the reasoning of others Standard 4: Model with mathematics Standard 5: Use appropriate tools strategically Standard 6: Attend to precision Standard 7: Look for and make use of structure Standard 8: Look for and express regularity in repeated reasoning

4. Grouping the Standards of Mathematical Practice (McCallum, 2011)

5. Standard 1: Make sense of problems and persevere in solving them

6. Mathematically proficient students: • explain the meaning of a problem and look for solutions • analyze • make conjectures about the form and meaning of the solution to plan a solution pathway • monitor and evaluate their progress • explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams to show relationships or trends. • check their answers to problems using a different methods

7. Standard 1: What might it look like for a student with significant cognitive disabilities?

8. “Key Words” as an example of the “suspension of sense making” Key words don’t work!

9. We tell them—more means add Erin has 46 comic books. She has 18 more comic books than Jason has. How many comic books does Jason have. But is our answer really 64 which is 46 + 18?

10. Standard 2: Reason abstractly and quantitatively

11. Mathematically proficient students: • make sense of quantities and their relationships • bring two complementary abilities to bear on problems involving quantitative relationships: • the ability to decontextualize • the ability to contextualize • Quantitative reasoning entails habits of creating a coherent representation of the problem at hand

12. Standard 3: Construct viable arguments and critique the reasoning of others

13. Mathematically proficient students: • understand and use stated assumptions, definitions, and previously established results • make conjectures • analyze situations by breaking them into cases and, can recognize and use counterexamples. • reason inductively about data • compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed • listen or read the arguments of others, decide whether they make sense

14. Grouping the Standards of Mathematical Practice (McCallum, 2011)

15. Standards 2 & 3: What might they look like for a student with significant cognitive disabilities?

16. Why does this work? If Sam can mow one lawn in 2 hours, how many lawns can he mow in 8 hours? Write a proportional relationship that represents the situation.

17. Representational proportions =

18. Concrete Proportions = X = X = X =

19. Standard 4: Model with mathematics

20. Mathematically proficient students: • apply the mathematics • identify important quantities in a practical situation and map their relationships • analyze relationships mathematically to draw conclusions • Routinely interpret their mathematical results in the context of the situation

21. Standard 5: Use appropriate tools strategically

22. Mathematically proficient students: • consider the available tools when solving a mathematical problem • are sufficiently familiar with tools appropriate for their grade or course • know that technology can enable them to visualize the results • identify relevant external mathematical resources, and use them to pose or solve problems • are able to use technological tools to explore and deepen their understanding of concepts

23. Grouping the Standards of Mathematical Practice (McCallum, 2011)

24. Standards 4 & 5: What might they look like for a student with significant cognitive disabilities?

25. Your New Car! You are buying a new car that is on sale for $27,000. This is 80% of the Original cost of the car. What was the Original cost of the car?

26. Using Hundreds Board to Solve Relatively Difficult Problems

27. Using Hundreds Board to Solve Relatively Difficult Problems Original Cost: 100%

28. Using Hundreds Board to Solve Relatively Difficult Problems

29. Using Hundreds Board to Solve Relatively Difficult Problems 27,000 Sale Cost Original Cost: 100%

30. Using Hundreds Board to Solve Relatively Difficult Problems 27,000 How much is each 10th of the whole?

31. Using Hundreds Board to Solve Relatively Difficult Problems 3,375 3,375 How much is each 10th of the whole? 3,375 3,375 3,375 3,375 3,375 3,375

32. Using Hundreds Board to Solve Relatively Difficult Problems 3,375 x 10 Original Cost: 100%

33. Standard 6: Attend to precision

34. Mathematically proficient students: • try to communicate precisely to others • use clear definitions in discussion with others and in their own reasoning • state the meaning of the symbols they choose • are careful about specifying units of measure • calculate accurately and efficiently

35. Standard 6: What might they look like for a student with significant cognitive disabilities?

36. We also use the word “same” when it doesn’t really apply.

37. Are these the same? 4+4 = 7+1

39. Are these the same?

40. Standard 7: Look for and make use of structure

41. Mathematically proficient students: • look closely to discern a pattern or structure • recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. • can step back for an overview and shift perspective • can see complicated things as single objects or as being composed of several objects

42. Standard 8: Look for and express regularity in repeated reasoning

43. Mathematically proficient students: • notice if calculations are repeated, and look both for general methods and for shortcuts • maintain oversight of the process, while attending to the details • continually evaluate the reasonableness of their intermediate results

44. Grouping the Standards of Mathematical Practice (McCallum, 2011)

45. Standards 7 & 8: What might they look like for a student with significant cognitive disabilities?

46. Multiplication 6 * 7= ?

47. Multiplication 6 * 7 = 6 (5) + 6 (2) = 42 6 * 7 = 42 or 30 + 12 = 42 ----------OR----------- 6 * 7 = 6 (6) + 6 (1) = 42 6*7 = 42 or 36 + 6 = 42

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