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# Multiplication and Division Part A

Strand: Number
Outcomes: 4, 5

## Step 1: Identify Outcomes to Address

### Guiding Questions

• What do I want my students to learn?
• What can my students currently understand and do?
• What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies? See Sequence of Outcomes from the Program of Studies

Strand: Number  Specific Outcomes 10. Apply mental mathematics strategies and number properties, such as: using doubles making 10 using the commutative property using the property of zero thinking addition for subtraction for basic addition facts and related subtraction facts to 18. 11. Demonstrate an understanding of multiplication to 5 × 5 by: representing and explaining multiplication using equal grouping and arrays creating and solving problems in context that involve multiplication modelling multiplication using concrete and visual representations, and recording the process symbolically relating multiplication to repeated addition relating multiplication to division. 12. Demonstrate an understanding of division (limited to division related to multiplication facts up to 5 × 5) by: representing and explaining division using equal sharing and equal grouping creating and solving problems in context that involve equal sharing and equal grouping modelling equal sharing and equal grouping using concrete and visual representations, and recording the process symbolically relating division to repeated subtraction relating division to multiplication.

 Specific Outcomes 4. Apply the properties of 0 and 1 for multiplication and the property of 1 for division. 5. Describe and apply mental mathematics strategies, such as: skip counting from a known fact using doubling or halving using doubling or halving and adding or subtracting one more group using patterns in the 9s facts using repeated doubling to determine basic multiplication facts to 9 × 9 and related division facts.

 Specific Outcomes 3. Apply mental mathematics strategies and number properties, such as: skip counting from a known fact using doubling or halving using patterns in the 9s facts using repeated doubling or halving to determine, with fluency, answers for basic multiplication facts to 81 and related division facts. 4. Apply mental mathematics strategies for multiplication, such as: annexing then adding zero halving and doubling using the distributive property.

### Big Ideas

Developing basic multiplication facts to 9 × 9 and related division facts requires that the students have a strong foundation in patterns, number relationships, place value, and the meaning, relationships and properties of operations as described below.

• Many patterns are used in developing mental strategies, such as skip counting from a known fact and using the constant sum of the digits in products with the 9s facts.
• Number relationships are evident when using the properties of operations or other strategies, such as repeated doubling; e.g., 4 × 6 = (6 × 2) × 2 = 24.
• Place value is used extensively in various strategies, such as doubling and adding one more group; e.g., 3 × 7 = 2 × 7 + 7 = 14 + 7 = 21.
• The meaning of multiplication and division is crucial as the students develop understanding of multiplication and division facts. Word problems are the key in developing this understanding.
• The relation between multiplication and division is the foundation on which the students learn the division facts. Word problems are the key to develop this connection (Van de Walle 2001, p. 144).
• The properties of multiplication and division used in developing mental strategies include:
• commutative property of multiplication; e.g., 6 × 2 = 2 × 6
• associative property of multiplication; e.g., (2 × 2) × 6 = 2 × (2 × 6), used in repeated doubling when finding the product of 4 and 6
• distributive property; e.g., 7 × 7 = (7 × 5) + (7 × 2) = 35 + 14 = 49
• identity element for multiplication is 1; i.e., any number multiplied by 1 remains unchanged. Similarly, any number divided by 1 remains unchanged
• zero property; i.e., any number multiplied by zero is zero.

The students construct meaning of these properties by solving appropriate story problems and follow-up discussion.

Van de Walle lists the following as Big Ideas in helping children master number facts:

1. Number relationships can be used to help remember basic facts.
2. For subtraction facts, the concept "think addition" is the most important idea. (Similarly, for division facts, the concept "think multiplication" is the most important idea.)
3. There are patterns and relationships in basic facts. You can figure out new or unknown facts from the ones you already know.
4. All the facts can be learned with the help of efficient strategies.
Adapted from John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4/e. Published by Allyn and Bacon, Boston, MA. Copyright © 2001 by Pearson Education. Adapted by permission of the publisher.

To clarify the second point, he states:

Mastery of multiplication facts and connections between multiplication and division are the key elements of division fact mastery. Word problems continue to be a key vehicle to create this connection (Van de Walle 2001, p. 144).

He goes on to say that "an efficient strategy is one that can be done mentally and quickly" (Van de Walle 2001, p. 129). There is one rule for teaching number facts: "Do not subject any student to fact drills unless the student has developed efficient strategies for the facts being practiced" (Van de Walle 2001, p. 144).

Efficient strategies for number facts result when the students develop personal mental strategies that are built on their understanding of numbers, compare these strategies to those used by others and choose strategies that suit their strengths and the numbers being multiplied or divided (Willis et al. 2006, p. 152). Fact strategy instruction includes two important facets:

• learning how to use a strategy
• learning how to select an appropriate strategy when it is needed.
(Van de Walle 2001, p. 130) 