1. Visual Models in Math
Connecting Concepts with Procedures for Whole Number & Decimal Multiplication and Division
Tuesday, April 7, 2015
Presented by Sara Delano Moore, Ph.D.
Director of Mathematics and Science at ETA hand2mind
Join our community on edWeb.net
Building Understanding in Mathematics

2. Building Understanding in Mathematics
Join our community on edWeb.net
Building Understanding in Mathematics
Invitations to upcoming webinars
Webinar recordings and resources
CE quizzes
Online discussions
Join the community

3. Recognition for your participation today!
Attending Live?
Your CE Certificate will be
ed to you within 24 hours.
Viewing the Recording?
Join the community at
Go to the Webinar Archives folder
Take the CE Quiz to get a personalized CE Certificate
CE Certificate provided by

4. Webinar Tips
For better audio/video, close other applications (like Skype) that use bandwidth.
If you are having any audio or video issues, try refreshing your browser.
Maximize your screen for a larger view by using the link in the upper right corner.
Tweet with #edwebchat

5. Visual Models in Math: Connecting Concepts with Procedures
April 7, 2015: Base Ten
Multiplication & Division
Sara Delano Moore, Ph.D.
Director of Mathematics & Science
ETA hand2mind

6. Visual Models in Math: Series Overview
January 6: Connecting Concepts with Procedures Overview
February 3: Connecting Concepts with Procedures for Whole Number & Decimal Addition & Subtraction
March 3: Connecting Concepts with Procedures for Fraction Addition & Subtraction
April 7: Connecting Concepts with Procedures for Whole Number & Decimal Multiplication & Division
May 5: Connecting Concepts with Procedures for Fraction Multiplication & Division

7. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.
PtA, page 42

8. Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently.
PtA, page 42

9. Hands-On Learning Instructional Cycle
Concrete
Representational
This instructional cycle is how we make the connections between work with manipulatives (the Concrete gear in the graphic) and abstract mathematics (the A gear in the graphic). If you remove the representations gear, the concrete and abstract ones don’t connect. When we use this instructional method with fidelity, what do we see in terms of student learning?
[Research supporting this CRA cycle appears in Appendix D of the RTI report.]
Abstract

10. Key Ideas for Base Ten Multiplication & Division
Reminder: procedural focus in this series
Models for Multiplication
Repeated addition
Area model
Models for Division
How many groups?
How many in each group?

11. Strategies and Methods: Multiplication
Repeated Addition
Opportunity to show young learners about efficiency – “there has to be a better way”
Good transition from addition to multiplication
See Spring 2014 series for more information
This method becomes inefficient with larger numbers and numbers which are not whole numbers.
Arrays serve as a transition to area model.

12. Repeated Addition Example
Susan gets 10₵ per day allowance. How much allowance does she receive in one week?
10₵ + 10₵ + 10₵ + 10₵ + 10₵ + 10₵ + 10₵ = 70₵
For 7 days, she receives 10₵ each day.
7 × 10₵ = 70₵
What happens over two weeks?

13. Strategies and Methods: Multiplication
Area Model
Connects the operation to one of its applications
Translates more easily to a wider range of factor types
e.g., fractions or algebraic expressions
Most effective for one or two terms/digits in each factor

14. Area Model Examples
7 × 10 = 70

15. Area Model Examples 7 × 10 = 70 12 × 14 = 168 12 × (10 + 4) =
(12 × 10) + (12 × 4) =
= 168

16. Moving from Concrete to Abstract: Partial Products Multiplication12
× 14 20 8 2 + 10 8 40 20 +
100
100 40
168

18. Strategies and Methods: Division
Repeated subtraction
Inverse of repeated addition
Find the missing factor
Inverse of area model
Both models are appropriate for both forms of division
How many groups?
How many in each group?
As with multiplication, each model has situations where it is a better fit.

19. Repeated Subtraction Example
30 ÷ 6 = 5
How many times can I subtract 6 from 30? 1 2 3 4 5

20. Missing Factor Example
30 ÷ 6 = 5
6 × = 30

21. Moving from Concrete to Abstract: Partial Quotients Division21
12)252 10
-120
132 10
-120 1 12
- 12

22. Working with Decimals Multiplication
Extend the models we have to more decimal places.

23. Multiplying Decimals
1.2 × 32.4 = 1 + 30 2 6
= 38.88

24. Working with Decimals Multiplication Division
Extend the models we have to more decimal places.
Division
Understand the fraction bar as a representation of division.
Use knowledge of equivalent fractions & multiplicative identity.

25. Dividing Decimals
25.2 ÷ 1.2 =
× = 12

26. Visual Models in Math: Series Overview
January 6: Connecting Concepts with Procedures Overview
February 3: Connecting Concepts with Procedures for Whole Number & Decimal Addition & Subtraction
March 3: Connecting Concepts with Procedures for Fraction Addition & Subtraction
April 7: Connecting Concepts with Procedures for Whole Number & Decimal Multiplication & Division
May 5: Connecting Concepts with Procedures for Fraction Multiplication & Division

28. Building Understanding in Mathematics
Join our community on edWeb.net
Building Understanding in Mathematics
Invitations to upcoming webinars
Webinar recordings and resources
CE quizzes
Online discussions
Join the community

29. Recognition for your participation today!
Attending Live?
Your CE Certificate will be
ed to you within 24 hours.
Viewing the Recording?
Join the community at
Go to the Webinar Archives folder
Take the CE Quiz to get a personalized CE Certificate
CE Certificate provided by

30. Visual Models in Math: Join us for the next webinar
Tuesday, May 5th – 4 PM Eastern Time
Visual Models in Math:
Connecting Concepts with Procedures for Fraction Multiplication and Division
For an invitation to the next webinar
Join Building Understanding in Mathematics

31. Thank you!