Twizzling Your Way Through the Common Core: A Focus on the Practice Standards

1. Twizzling Your Way Through the Common Core:A Focus on the Practice Standards KATM Fall Conference ~ 2011 Jerry Braun – USD489/Hays

2. Standards for Mathematical Practice

3. Grouping the Standards for Mathematical Practice 8. Look for and Express Regularity in Repeated Reasoning Creating patterns from repeated calculations and evaluating the reasonableness of generalizations and shortcuts. 4. Model with Mathematics Applying mathematics to solve problems arising in everyday life. Making sense of the results. 2. Reason Abstractly and Quantitatively Making sense of quantities and relationships in problem solving. Flexibility in using operations and properties. 7. Look for and Make Use of Structure Looking for and using patterns to solve problems. 3. Construct Viable Arguments and Critique the Reasoning of Others Analyzing problems, making conjectures and constructing arguments based on data. Listening to the arguments of others and asking useful questions to determine rightness. Justifying conclusions and communicating them to others. 1. Make Sense of Problems and Persevere in Solving Them Explaining the meaning of a problem and looking for entry points to its solution. Monitoring progress and changing course if necessary. Analyzing givens, constraints, relationships, and goals. 5. Use Appropriate Tools Strategically Using available tools to solve problems and recognizing the strengths and limitations of each. 6. Attend to Precision Communicating precisely using clear definitions in discussion with others and in own reasoning. Understanding the meaning of mathematical symbols and calculating efficiently and accurately.

4. Example Student Task Your math partner turns to you and says: I worked the problem 124 ÷ 8 on my paper and got 15 r 4. But when I did it on my calculator, it said 15.5. Which answer is right? How would you respond to your partner’s question?

5. Things to consider. . . • Using the term “groups”, restate what 124 ÷ 8 means. • In the context of the problem, are the quotients 15 r 4 and 15.5 the same or are they different? • What do the quotients mean? • What does it mean to have a remainder of 4 when you are dividing by 8? • What does 0.5 mean? • What could you do to convince your partner that the answers are the same or different?

6. Teacher Actions • Each person needs a Practice Standards Sorting Mat and a set of Teacher Action Sorting Cards. • Read each Action Card and match it with the correct Practice Standard.

7. Standards for Mathematical Practice

8. Make sense of problems and persevere in solving them. • Ask clarifying questions, such as, • “Which part do you agree or disagree with? • Suggest starting points, such as, “What answer do YOU get when you do the division?” • Provide appropriate tools for investigating the problem.

9. Reason abstractly and quantitatively. • Ask questions that focus the student’s thinking on the operation, such as, • “What does it mean to have a remainder of 4 when you are dividing by 8?” • Ask questions that focus the student’s thinking on the meaning of the numbers, such as, “What does 0.5 mean?”

10. Construct viable arguments and critique the reasoning of others. • Design tasks in a way that directly involves mathematical argumentation. • Ask questions that model the desired • thinking, such as, “Are you sure that the answers are different? Can you convince me that they are different? How can you explain to me what each answer means?”

11. Model with mathematics. • Show students how to represent their thinking with symbols, for example: • 0.5 = = and • 4 out of 8 is = 5 1 10 2 4 1 8 2

12. Use appropriate tools strategically. • Have a selection of tools (e.g. place value blocks, calculators, graph paper) readily available for students to use to represent their thinking. • Ask students to use a drawing or a model to show what they mean. • Have students critique the uses of tools by asking questions like, “What made the use of the calculator in this problem confusing for this student? When would using a calculator be helpful?”

13. Attend to precision. • Ask students to describe the quotient and remainder in words. • Ask students to describe the decimal quotient in words. • Ask students to compare and contrast situations in which you might decide to use a decimal quotient instead of a remainder.

14. Look for and make use of structure. • Ask questions that focus student thinking on the relationship of the remainder to the divisor, i.e. 4 is half of 8. • Ask questions that focus student thinking on the values of the digits in the quotient, i.e. in 15.5, the 5 to the right of the decimal point means 5 tenths.

15. Look for and express regularity in repeated reasoning. • Have students explore what happens with other numbers that when divided by 8 leave a remainder of 4, e.g. 12 ÷ 8 • 36 ÷ 8 • 804 ÷ 8 • Have students explore the remainders in other division problems that have a quotient that ends in .5, e.g. 9 divided by 6, 35 divided by 10, 126 divided by 12.

16. Teaching for Understanding What is one half of one half divided by one half? Use your Twizzler to model your understanding of this question.

17. What is of ? one whole

18. What does it mean? What is ÷ ? How many 2’s will “fit” into 6? What do we mean when we ask, “What is 6 ÷ 2?” 3

19. What is ÷ ? How many halves will “fit” into the fourth? 1/2 1/4 1/2 How much of the half will “fit” into the fourth?

20. Teacher Actions • Get out your Practice Standards Sorting Mat again. • Brainstorm actions from this brief “lesson” to match with an appropriate Practice Standard.

21. Integrating the Standards for Mathematical Practices • Not “Problem Solving Fridays” • Not “enrichment” for advanced students • Most are in the process of arriving at an answer, not necessarily in the answer itself • Every lesson should seek to build student expertise in both Content and Practice standards

22. The End! • Thank you for a great session  • Questions, Comments, Etc.?? • jbraun@usd489.com