1. Chapter 08

2. 8 | 2 Copyright © Cengage Learning. All rights reserved. Fluency through Meaningful Practice Mathematical Routines & Algebraic Thinking

3. 8 | 3 Copyright © Cengage Learning. All rights reserved. Mathematical Routine: What is the rule and how would this look on a graph? X (in) Y (out) 2 3 4 7 6 11 3 5

4. 8 | 4 Copyright © Cengage Learning. All rights reserved. X (in) Y (out) 24 2 12 1 36 3 48 4

5. 8 | 5 Copyright © Cengage Learning. All rights reserved. Conversation in Mathematics Look at the different solution strategies students came up with for the problem 35-x=36-20. What understanding does each student have?

6. 8 | 6 Copyright © Cengage Learning. All rights reserved. Developing Fluency through Mathematical Routines Pedagogy

7. 8 | 7 Copyright © Cengage Learning. All rights reserved. Mathematically Powerful

8. 8 | 8 Copyright © Cengage Learning. All rights reserved. What is a mathematical routine? Purposefully structured activities that help children develop procedural fluency as well as reasoning and problem solving skills through meaningful practice. They are not just a part of the daily schedule. They are planned based on the needs of the children for that day—not commercially produced. They take place outside of the regular math time

9. 8 | 9 Copyright © Cengage Learning. All rights reserved. Types of Routines Routines can address any math concept, but tend to be used mostly for the development and support of early number concepts and the base ten system, as well as computational fluency with whole and rational numbers.

10. 8 | 10 Copyright © Cengage Learning. All rights reserved. Types of routines Early Number Algebraic Thinking Computational Fluency –Number Talks –Number Strings –Number Lines Data and Graphing

11. 8 | 11 Copyright © Cengage Learning. All rights reserved. Algebraic Thinking & Reasoning Content

12. 8 | 12 Copyright © Cengage Learning. All rights reserved. 1 - Equality 8 + 4 = __ + 5

13. 8 | 13 Copyright © Cengage Learning. All rights reserved.

14. 8 | 14 Copyright © Cengage Learning. All rights reserved. Have you ever seen or done this? 4 + 5 = 9 - 1 = 8 + 4 = 12 Instead, we should record our thinking like this: 4 + 5 = 9 9 - 1 = 8 8 + 4 = 12

15. 8 | 15 Copyright © Cengage Learning. All rights reserved. Introducing the Equal Sign

16. 8 | 16 Copyright © Cengage Learning. All rights reserved. 4 + 5 = 9 True/False Questions 9 = 4 + 5 9 = 9 4 + 5 = 4 + 5 4 + 5 = 5 + 4 4 + 5 = 6 +3

17. 8 | 17 Copyright © Cengage Learning. All rights reserved. Video Watch the kindergarten video around equality. How does the teacher make this abstract concept accessible to her students?

18. 8 | 18 Copyright © Cengage Learning. All rights reserved. Inequalities Kill the alligator story!

19. 8 | 19 Copyright © Cengage Learning. All rights reserved. 2- Relational Thinking 7 + 6 = ___ + 5

20. 8 | 20 Copyright © Cengage Learning. All rights reserved. 37 + 56 = 39 + 54

21. 8 | 21 Copyright © Cengage Learning. All rights reserved. 33 - 27 = 34 - 26

22. 8 | 22 Copyright © Cengage Learning. All rights reserved. 3-Growing Patterns & Functions Task: A contractor is designing square swimming pools with a square center. He uses blue tiles to represent the water. Around each pool there is a border of yellow tiles. He wants to figure out a way to know how many blue and yellow tiles there are in a pool of any size.

23. 8 | 23 Copyright © Cengage Learning. All rights reserved. Pool 1Pool 2Pool 3 What will the size (yellow, blue, and total area) of the 4th pool be? The tenth pool? Any pool?

24. 8 | 24 Copyright © Cengage Learning. All rights reserved. Creating conjecture charts Editing conjectures throughout the year 4-Making Conjectures

25. 8 | 25 Copyright © Cengage Learning. All rights reserved. General forms of Justification –Restating the conjecture –Concrete examples that are more than examples: Building on basic concepts –Use of counter-examples 5-Justification & Proof

26. 8 | 26 Copyright © Cengage Learning. All rights reserved. Questions to Elicit Justification How do you know? Does that always work? Does that work with all numbers? How can you be sure?

27. 8 | 27 Copyright © Cengage Learning. All rights reserved. Criteria for Representation Based Proof 1-The meaning of the operation can be shown in diagrams, manipulatives, or in a story context. 2-The representation will work with a class of numbers (whole numbers, etc.) 3-The conclusion matches the representation. From: Reasoning Algebraically about operations Casebook, Education Development Center, Inc. and TERC, 2005.

28. 8 | 28 Copyright © Cengage Learning. All rights reserved. Task With a small group, you will design a routine around one of the following algebraic concepts: –Equality –Relational thinking –Growing patterns