1. Rectangles as Problem- Solving Tools Use Area Models to Teach Math Concepts at All Levels www.elevatingadulteducation.com1

2. Introductions Amy Vickers www.elevatingadulteducation.com amyjvickers@gmail.com Are you familiar with area models? www.elevatingadulteducation.com2

3. Objectives Participants will be able to…  Gain an understanding of the value of area models  Build area models and discuss their components  Use area models to teach relevant mathematical concepts Today we will….  Become familiar with area models and how they work.  Consider area models as tools for teaching conceptual understanding.  Use area models as tools for exploring various topics. www.elevatingadulteducation.com3

4. Why this topic?  College and Career Readiness Standards for Adult Education  Math 140  Rigor, Coherence, Focus www.elevatingadulteducation.com4

5. CCRS Connections  Level B: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.  Level B: Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.  Level C: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.  Level C: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. www.elevatingadulteducation.com5

6. Levels of Knowing Math Mahesh Sharma www.elevatingadulteducation.com6 Communication Application Abstract Pictorial Concrete Intuitive

7. Numeracy vocabulary gives learners the tools to make math their own www.elevatingadulteducation.com7

8. How many eggs? www.elevatingadulteducation.com8

9. How many eggs? How did you know? www.elevatingadulteducation.com9

10. How many muffins are in this pan? www.elevatingadulteducation.com10

11. How many muffins? How did you know? www.elevatingadulteducation.com11

12. Rectangular arrays www.elevatingadulteducation.com12

13. How many rectangular arrays do you see in this photo? Envision Plans to Install 2,300 Rotating Solar Tree Car Shelters in South Carolina www.elevatingadulteducation.com13

14. What is a rectangular array? How can we use them for teaching math? www.elevatingadulteducation.com14

15. Directions  Clear a work space at your table. You will need one bag of tiles, a sheet of paper, and a writing utensil.  Count your tiles. Let me know if you do not have 25 tiles.  Arrange the tiles in a rectangular array to demonstrate 8 x 3 = 24.  Look around the table and help others as needed. www.elevatingadulteducation.com15

16. Discuss at your table:  Describe your array.  Do you see an array that is different than yours? How is it different?  What equations can you write to describe the array that use multiplication?  What equations can you write to describe the array that use division?  What is the relationship between multiplication and division?  What shape is your array?  What are the properties of that shape? www.elevatingadulteducation.com16

17. What are the factors of 12? Show us that they are factors by using rectangular arrays. www.elevatingadulteducation.com17

18. Discuss at your table:  What is a factor?  How would you explain a factor in the context of rectangular arrays?  What are the factors of 12?  What are a few examples of numbers that are not factors of 12? How would this look in a rectangular array?  What equations can you write to describe the arrays that use multiplication?  What equations can you write to describe the arrays that use division?  What is the relationship between multiplication and division?  For what other math topics is an understanding of factors essential? www.elevatingadulteducation.com18

19. A measured area model is a type of rectangular array www.elevatingadulteducation.com19

20. Practice with area models  Put the tiles back in the bag and take out a piece of graph paper.  On a piece of graph paper, use an area model to multiply 9 x 4.  Write 2 multiplication and 2 division equations to describe the model.  On a piece of graph paper, use an area model to multiply 14 x 25.  Now multiply 135 x 29.  Try 14 x 25 using an abstract area model. www.elevatingadulteducation.com20

21. What is an abstract area model? When is it useful? www.elevatingadulteducation.com21

22. As you share this with learners, do not skip steps in conceptual understanding! Rectangular array Measured area model Abstract area model www.elevatingadulteducation.com22

23. Levels of Knowing Math Mahesh Sharma www.elevatingadulteducation.com23 Communication Application Abstract Pictorial Concrete Intuitive

24. Activities with area models will give opportunities for …  Thinking about math  Reasoning through problems using mathematical thinking  Hearing math vocabulary  Using math vocabulary  Explaining the process  Examining various solutions  Asking and answering clarifying questions  Asking and answering follow-up questions www.elevatingadulteducation.com24

25. The purpose of using area models…  Is to add a layer of conceptual understanding to the learner’s experience…  Not to replace all computation methods. www.elevatingadulteducation.com25

26. What does this have to do with math anxiety?  Math has structure.  Math fits with what I understand.  Math is predictable.  I can trust it.  I don’t always need to memorize steps.  Important ideas work at all levels. www.elevatingadulteducation.com26

27. Using area models in various settings How can this work for me? www.elevatingadulteducation.com27

28. Multiplying Fractions Multiply 2/3 by 3/5. www.elevatingadulteducation.com28

29. Multiplying Fractions Multiply 1/4 by 4/5. www.elevatingadulteducation.com29

30. Division Divide 12 by 3. Write an equation that shows how to check your answer. www.elevatingadulteducation.com30

31. Division Divide 17 by 3. Write an equation that shows how to check your answer. www.elevatingadulteducation.com31

32. Division Divide 67 by 5. Write an equation that shows how to check your answer. www.elevatingadulteducation.com32

33. Division Divide 816 by 5. Write an equation that shows how to check your answer. www.elevatingadulteducation.com33

34. Division Divide 529 by 7. Write an equation that shows how to check your answer. www.elevatingadulteducation.com34

35. Examining properties Simplify 4(3 + 2) by using an area model. What property does this illustrate? www.elevatingadulteducation.com35

36. Examining properties Evaluate x(y + 4) by using an area model. What property does this illustrate? www.elevatingadulteducation.com36

37. Examining properties Use an area model to illustrate the commutative property of multiplication. www.elevatingadulteducation.com37

38. Multiplying polynomials Find the product of (x+3) and (x-5). www.elevatingadulteducation.com38

39. www.elevatingadulteducation.com39

40. www.elevatingadulteducation.com40

41. Objectives Participants will be able to…  Gain an understanding of the value of area models  Build area models and discuss their components  Use area models to teach relevant mathematical concepts To take with you…  Set of 25 tiles  Sample facilitation questions to use with tiles www.elevatingadulteducation.com41

42. Thank you! www.elevatingadulteducation.com42