1 welcome to module 9 teaching basic facts and multidigit computations

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Welcome to Module 9

Teaching Basic Facts and Multidigit Computations

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Getting Started

Calculate 46 + 27 mentally.

Be prepared to explain how you got your answer.

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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?

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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.

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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.

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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.

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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.

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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.

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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.

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Key Messages

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

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Definitions and Approaches

Basic facts include the addition, subtraction, multiplication, and division of numbers from 0 to 9.

54÷6=___

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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.

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

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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.

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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.

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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.

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

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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.

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Create home groups of six. Number yourselves from 1 to 3. 1

2

3

Join your number group and read the section of the guide that covers your assigned topic.

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Expert Group 1Topic: Principles for Teaching the Facts

(p. 10.13) Expert Group 2Topic: Prior Learning / Developing

Computational Sense (pp. 10.14–10.16)

Expert Group 3Topic: Worksheets / Timed Tests (pp. 10.16–10.18)

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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.

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your main points with your table group.

Continue to use BLM 9.1 as a recording sheet.

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Working on It

Addition andSubtraction Facts

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Addition and Subtraction Facts

Work with a partner. Decide who will study the following topics:

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Using Models (p. 10.19)

Strategies (pp. 10.20–10.25)

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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.

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with your partner.


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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.

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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.

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* Ensure that any drill practice is focused on using strategies and not just on rote recall.

* Cluster facts and practice around strategies.

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* 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).

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Working on It

Multiplication and

Division Facts

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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.

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Multiplication can be thought of as

repeated addition,

+ +

as an array,

and as a collection of equal groups.

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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

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the zero property (0 X a is always 0);

3 groups of nothing is 0

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the commutative property (2 X 3 = 3 X 2);

6 cats = 6 cats

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the distributive property (4 X 3 = 2 X 3 + 2 X 3);

+

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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.

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the inverse relationship with division.

I can share 10 star cookies equally among four friends and me!

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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!

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Properties and strategies that help with a conceptual understanding of division include:

the use of 1 as a divisor (6 ÷ 1 = 6);

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The relationship of division to fractional sense(4 candies divided into 2 groups represents both 4 ÷ 2 and the whole divided into 2 halves);

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the inverse relationship with multiplication.

Dividing this set in half shows 2 groups

of 6

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Choose a different partner. Decide who will study the following topics:

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Using Models (pp. 10.26 – 10.27)

Strategies (pp. 10.28 – 10.32)

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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.

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with your partner.


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Working On It

MultidigitWhole NumberCalculations

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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 )

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Many teachers learned only one way to solve multidigit computations — using the standard North American algorithm taught in schools.

500

-129

But solving a multidigit computation can be done in many ways, depending on the context of the problem.

4 9 1

371

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Consider making change from a $10 bill for a $7.69 purchase.

Is the standard algorithm efficient in this situation?

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

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

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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.10.34.)

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Students who are encouraged to use their own flexible strategies for computing multidigit numbers develop:

A better sense ofnumber;

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More flexibility in solving problems;

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A stronger understanding of place value;

The 2 in ’26’ represents

twenty.

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Greater facility with mental calculations;

20 + 10 makes 30, plus 3 is 33!

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More ease in linking meaning to symbols in the traditional algorithms.

500

-129

4 9 1

371

When I check, I know that 3 + 1 = 4, but I

traded 100 to the tens, so that is 5 all

together.

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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:

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1. Algorithms (pp. 10.37 – 10.40)2. Student-Generated Algorithms

(pp. 10.40 – 10.42)3. An Investigative Approach

(pp. 10.42 – 10.43)4. Standard Algorithms

(pp. 10.49 – 10.51)5. Estimation (pp. 10.51 – 10.53)

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important ideas about your topic with your group.

Record what you learn from other group members on BLM 9.4a and BLM 9.4b.

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There are a number of activity ideas for multidigit computations on pp. 10.42 – 10.48.

The appendices contain instructions and blackline masters for the activities.

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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.

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Celebration!!!

for participating in this professional development opportunity!

1 welcome to module 9 teaching basic facts and multidigit computations

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