Welcome to Module 9

1. Welcome to Module 9 Teaching Basic Facts and Multidigit Computations

2. Getting Started Calculate 46 + 27 mentally. Be prepared to explain how you got your answer.

3. Getting Started Discuss... Is there one right way to solve this question? Why do people use different methods, and not just the traditional algorithm, to solve questions like this? What are the implications for teaching basic facts and multidigit computations?

4. Key Messages Learning the basic facts conceptually involves developing an understanding of the relationships between numbers and how these relationships can be evolved into strategies for doing the computations in a meaningful and logical manner.

5. Key Messages Providing students with problem-solving contexts that relate to the basic facts will allow them to develop a meaningful understanding of the operations.

6. Key Messages Research evidence suggests that the use of conceptual approaches in computation instruction results in improved achievement, good retention, and a reduction in the time students need to master computational skills.

7. Key Messages Students’ development of computational sense goes through several stages and is improved by exposure to a range of computational strategies, through guided instruction by the teacher and shared learning opportunities with other students.

8. Key Messages Using word problems to introduce, practise, and consolidate the basic facts is one of the most effective strategies teachers can use to help students link the mathematical concepts to the abstract procedures.

9. Key Messages Students who can work flexibly with numbers are more likely to develop efficient strategies, accuracy, and a strong foundation for understanding other standard algorithms.

10. Key Messages When standard algorithms are being introduced, it is important that students develop an understanding of the operations rather than just memorize rules.

11. Definitions and Approaches Basic facts include the addition, subtraction, multiplication, and division of numbers from 0 to 9. __-9=9 3+8=___ 6x__=36 54÷6=___

12. Definitions and Approaches By Grade 3, students are expected to develop proficiency in single-digit addition and subtraction, and in multiplication and division up to the 7 times table. 7 rows of seven pennies are 49 pennies all together.

13. Definitions and Approaches Multidigit computations include all combinations of two or more digits in addition, subtraction, multiplication, and division.

14. Definitions and Approaches By Grade 3, students are expected to develop efficiency with the addition and subtraction of multidigit numbers (up to three digits) and to use these computations in problem-solving situations.

15. Definitions and Approaches In the past, the emphasis in teaching was on memorization of the basic number facts, sometimes to the exclusion of establishing a firm conceptual understanding of the underlying number structures.

16. Definitions and Approaches Learning the basic facts conceptually involves developing an understanding of the relationships between numbers (e.g., 7 is 3 less than 10 and 2 more than 5) and how these relationships can be developed into strategies for doing the computations in a meaningful and logical manner.

17. Definitions and Approaches Providing students with problem-solving contexts that relate to the basic facts will allow them to develop a meaningful understanding of the operations.

18. Definitions and Approaches Games, active learning experiences, and investigations provide students with opportunities to use manipulatives and to interact with their peers as they rehearse basic fact strategies and practise multidigit computations.

19. Definitions and Approaches Create home groups of six. Number yourselves from 1 to 3. 1 3 2 Join your number group and read the section of the guide that covers your assigned topic.

20. Definitions and Approaches Expert Group 1 Topic: Principles for Teaching the Facts (p. 10.13) Expert Group 2 Topic: Prior Learning / Developing Computational Sense (pp. 10.14–10.16) Expert Group 3 Topic: Worksheets / Timed Tests (pp. 10.16–10.18)

21. Definitions and Approaches Groups summarize the main points for their section and record them on BLM 9.1. Be prepared to share your points with your table groups.

22. Definitions and Approaches Share... your main points with your table group. Continue to use BLM 9.1 as a recording sheet.

23. Working on It Addition and Subtraction Facts

24. Addition and Subtraction Facts Work with a partner. Decide who will study the following topics:

25. Addition and Subtraction Facts Using Models (p. 10.19) Strategies (pp. 10.20–10.25)

26. Addition and Subtraction Facts Each person should record a summary of main ideas on BLM 9.2. Each participant should also conduct a brief activity with his or her partner to help to explain the topic. The summary may be displayed before or after the activity. Suggestion: The “strategies” person could list some of the strategies along with a brief explanation or example of each, and then model one of the examples.

27. Addition and Subtraction Facts Share... with your partner. Continue to use BLM 9.2 as a recording sheet.

28. Addition and Subtraction Facts To help students practise selecting and using strategies for addition and subtraction facts, try the following approaches: * Use problem solving as the route to practising the facts. * Model problems (e.g., using counters) when needed.

29. Addition and Subtraction Facts To help students practise using and selecting strategies for addition and subtraction facts, try the following approaches: * Recognize that the level of strategy development for recalling the facts is rarely the same for all students. * Use games, repetition of worthwhile activities or songs, and mnemonic devices to individualize strategy development.

30. Addition and Subtraction Facts To help students practise using and selecting strategies for addition and subtraction facts, try the following approaches: * Ensure that any drill practice is focused on using strategies and not just on rote recall. * Cluster facts and practice around strategies.

31. Addition and Subtraction Facts To help students practise using and selecting strategies for addition and subtraction facts, try the following approaches: * Have students make their own strategy list for the facts they find hardest. * Help students make connections between the facts (e.g., by using triangular flashcards).

32. Working on It Multiplication and Division Facts

33. Multiplication and Division Facts Thinking About Multiplication and Division It is important to realize that there are several different ways to think about these operations.

34. Multiplication and Division Facts Multiplication can be thought of as repeated addition, + + as an array, and as a collection of equal groups.

35. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of multiplication include: 1 group of 9 happy faces is 9 1 x 9 = 9 the identity property (1 X a is always a);

36. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of multiplication include: 3 groups of nothing is 0 the zero property (0 X a is always 0);

37. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of multiplication include: 6 cats = 6 cats the commutative property (2 X 3 = 3 X 2);

38. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of multiplication include: + the distributive property (4 X 3 = 2 X 3 + 2 X 3);

39. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of multiplication include: 60 minutes is the same amount of time as 2 periods of 30 minutes. the associative property (5 X 12 is the same as 5 X 6 X 2);

40. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of multiplication include: I can share 10 star cookies equally among four friends and me! the inverse relationship with division.

41. Multiplication and Division Facts Division can be thought of as repeated subtraction, I’ll give two to you, two to you, and two to me! as equal partitioning, or as sharing. 6 brownies – we each get 3!

42. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of division include: the use of 1 as a divisor (6 ÷ 1 = 6);

43. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of division include: The relationship of division to fractional sense (4 candies divided into 2 groups represents both 4 ÷ 2 and the whole divided into 2 halves);

44. Multiplication and Division Facts Properties and strategies that help with a conceptual understanding of division include: Dividing this set in half shows 2 groups of 6 the inverse relationship with multiplication.

45. Multiplication and Division Facts Choose a different partner. Decide who will study the following topics:

46. Multiplication and Division Facts Using Models (pp. 10.26 – 10.27) Strategies (pp. 10.28 – 10.32)

47. Multiplication and Division Facts Each person should record a summary of main points of the topic on BLM 9.3. Each participant should also conduct a brief activity with his or her partner to help to explain the topic. The summary may be displayed before or after the activity. Suggestion: The “strategies” person could list some of the strategies along with a brief explanation or example of each, and then model one of the examples.

48. Multiplication and Division Facts Share... with your partner. Continue to use BLM 9.3 as a recording sheet.

49. Working On It Multidigit Whole Number Calculations

50. Multidigit Whole Number Calculations “There is mounting evidence that students both in and out of school can construct methods for adding and subtracting multidigit numbers without explicit instruction.” - Carpenter, Franke, Jacobs, Fennema, & Empson, A Longitudinal Study of Invention and Understanding of Children’s Multidigit Addition and Subtraction, Journal for Research in Mathematics Education, 1998, p. 4)