1 Welcome to Module 9 Teaching Basic Facts and Multidigit Computations

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

3 Getting Started 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. 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 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 ) Expert Group 2 Topic: Prior Learning / Developing Computational Sense (pp – 10.16) Expert Group 3 Topic: Worksheets / Timed Tests (pp –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 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 ) Strategies (pp –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 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: the identity property (1 X a is always a); 1 group of 9 happy faces is 9 1 x 9 = 9

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

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

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 X 3); +

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

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

41 Multiplication and Division Facts Division can be thought of as I’ll give two to you, two to you, and two to me! repeated subtraction, 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: the inverse relationship with multiplication. Dividing this set in half shows 2 groups of 6

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

46 Multiplication and Division Facts Using Models (pp – 10.27) Strategies (pp – 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 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 )

51 Multidigit Whole Number Calculations Many teachers learned only one way to solve multidigit computations — using the standard North American algorithm taught in schools But solving a multidigit computation can be done in many ways, depending on the context of the problem

52 Multidigit Whole Number Calculations Consider making change from a $10 bill for a $7.69 purchase. Is the standard algorithm efficient in this situation?

53 Multidigit Whole Number Calculations Students who can work flexibly with numbers are more likely to develop efficient strategies, accuracy, and a strong foundation for understanding other standard algorithms.

54 Multidigit Whole Number Calculations The standard North American algorithms were established to help make calculations fast. Such shortcuts are practical and useful for those who understand the algorithm.

55 Multidigit Whole Number Calculations For students who have not been taught the underlying concepts, memorizing the abstract algorithm is often the beginning of their belief that mathematics “doesn’t make sense” and is dependent on memorizing rules or routines. (See example on p )

56 Multidigit Whole Number Calculations Students who are encouraged to use their own flexible strategies for computing multidigit numbers develop: A better sense of number;

57 Multidigit Whole Number Calculations Students who are encouraged to use their own flexible strategies for computing multidigit numbers develop: More flexibility in solving problems;

58 Multidigit Whole Number Calculations Students who are encouraged to use their own flexible strategies for computing multidigit numbers develop: A stronger understanding of place value; The 2 in ’26’ represents twenty.

59 Multidigit Whole Number Calculations Students who are encouraged to use their own flexible strategies for computing multidigit numbers develop: Greater facility with mental calculations; makes 30, plus 3 is 33!

60 Multidigit Whole Number Calculations Students who are encouraged to use their own flexible strategies for computing multidigit numbers develop: More ease in linking meaning to symbols in the traditional algorithms When I check, I know that = 4, but I traded 100 to the tens, so that is 5 all together.

61 Multidigit Whole Number Calculations Form groups of five. Each person reads about one of the following topics and records important ideas on BLM 9.4a or BLM 9.4b:

62 Multidigit Whole Number Calculations 1. Algorithms (pp – 10.40) 2. Student-Generated Algorithms (pp – 10.42) 3. An Investigative Approach (pp – 10.43) 4.Standard Algorithms (pp – 10.51) 5. Estimation (pp – 10.53)

63 Multidigit Whole Number Calculations important ideas about your topic with your group. Record what you learn from other group members on BLM 9.4a and BLM 9.4b.

64 Multidigit Whole Number Calculations There are a number of activity ideas for multidigit computations on pp – The appendices contain instructions and blackline masters for the activities.

65 Reflecting and Connecting Consider the variety of strategies and activities that were examined today. Choose a new strategy or activity to try in your classroom.

66 Celebration!!! for participating in this professional development opportunity!