# 3.4 - Multiplication and Division

The student will

a) represent multiplication and division through 10 x 10, using a variety of approaches and models;

b) create and solve single-step practical problems that involve multiplication and division through 10 x 10;

c) demonstrate fluency with multiplication facts of 0, 1, 2, 5 and 10; and

d) solve single-step practical problems involving multiplication of whole numbers, where one factor is 99 or less and the second factor is 5 or less.

### BIG IDEAS

• So that I can add and subtract numbers more quickly (multiplication is repeated addition and division is repeated subtraction)
• So that I can understand and explain how multiplication and division are related
• So that I can figure out how many cupcakes I need when planning a party for a certain amount of guests

### UNDERSTANDING THE STANDARD

• Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and practical problems involving equal-sized groups, arrays, and length models.
• To extend the understanding of multiplication, three models may be used:To extend the understanding of multiplication, three models may be used:
• The equal-sets or equal-groups model lends itself to sorting a variety of concrete objects into equal groups and reinforces the concept of multiplication as a way to find the total number of items in a collection of groups, with the same amount in each group, and the total number of items can be found by repeated addition or skip counting. • The array model, consisting of rows and columns (e.g., three rows of four columns for a 3-by-4 array), helps build an understanding of the commutative property. • The length model (e.g., a number line) also reinforces repeated addition or skip counting. • The terms associated with multiplication are listed below:
•  factor   →         54
• factor    →        x 3
• product   →      162
• There is an inverse relationship between multiplication and division.
• The number line model can be used to solve a multiplication problem such as 3 x 6. This is represented on the number line by three jumps of six or six jumps of three, depending on the context of the problem. • The number line model can also be used to solve a division problem such as 6 divided by 3 and is represented on the number line by noting how many jumps of three go from six to zero.
• • The number line model above shows two jumps of three between 6 and 0, answering the question of how many jumps of three go from 6 to 0; therefore, 6 divided by 3 = 2.
• Computational fluency is the ability to think flexibly in order to choose appropriate strategies to solve problems accurately and efficiently.
• The development of computational fluency relies on quick access to number facts. There are patterns and relationships that exist in the facts. These relationships can be used to learn and retain the facts. By studying patterns and relationships, students build a foundation for fluency with multiplication and division facts.
• Beginning with learning the foundational multiplication facts for 0, 1, 2, 5, and 10 allows students to utilize prior skip counting skills and the use of doubles to solve problems. Understanding and using the foundational facts can be helpful in deriving and learning all multiplication facts. For example, decomposing one of the factors in 7 x 6, allows for the use of the foundational facts of 5s and 2s. This knowledge can be combined to learn the facts for 7 (e.g., 7 x 6 can be thought of as (5 x 6) + (2 x 6)).
• As students work to solve multiplication and division problems, they naturally tend to utilize strategies that involve place value understanding and properties of the operations. Applying the commutative property of multiplication (e.g., 5 x 8 = 8 x 5) reduces in half the number of multiplication facts that students must learn.  The distributive property of multiplication allows students to find the answer to a problem such as 6 x 7 by decomposing 7 into 3 and 4 (e.g., 6 x 7= 6 x (3 + 4)) allowing them to think about (6 x 3) + (6 x 4) = 18 + 24 = 42.
• Strategies that allow students to derive unknown facts from facts they do know include:  doubles (2s facts), doubling twice (4s facts), five facts (half of ten), decomposing into known facts (e.g., 7 x 8 can be thought of as (5 x 8) + (2 x 8)).
• Strategies for solving problems that involve multiplication or division may include mental strategies, partial products, the standard algorithm, and the commutative, associative, and distributive properties.
• An algorithm is a step-by-step method for computing.
• The least number of steps necessary to solve a single-step problem is one.
• Extensive research has been undertaken over the last several decades regarding different problem types. Many of these studies have been published in professional mathematics education publications using different labels and terminology to describe the varied problem types.
• Students should experience a variety of problem types related to multiplication and division. Some examples are included in the following chart: • Investigating arithmetic operations with whole numbers helps students learn about several different properties of arithmetic relationships. These relationships remain true regardless of the numbers.

• Grade three students should explore and apply the properties of multiplication and addition as strategies for solving multiplication and division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols).

• The properties of the operations are “rules” about how numbers work and how they relate to one another.  Students at this level do not need to use the formal terms for these properties but should utilize these properties to further develop flexibility and fluency in solving problems.  The following properties are most appropriate for exploration at this level:

• The commutative property of multiplication states that changing the order of the factors does not affect the product (e.g., 2 x 3 = 3 x 2).
• The identity property of multiplication states that if a given number is multiplied by one, the product is the same as the given number.
• The associative property of addition states that the sum stays the same when the grouping of addends is changed (e.g., 15 + (35 + 16) = (15 + 35) + 16).
• The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products:
• 8 x 7 = 8 (5 +2)           (8 x 5) + (8 x 2)           40 + 16          56
• 5 x 23 = 5 (20 + 3)      (5 x 20) + (5 x 3)         100 + 15        115

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Represent multiplication using a variety of approaches and models (e.g., repeated addition, equal-sized groups, arrays, equal jumps on a number line, skip counting). (a)
• Represent division using a variety of approaches and models (e.g., repeated subtraction, equal sharing, equal groups). (a)
• Write three related equations (fact sentences) when given one equation (fact sentence) for multiplication or division (e.g., given 6 x 7 = 42, write 7 x 6 = 42, 42 ÷ 7 = 6, and 42 ÷ 6 = 7. (a)
• Create practical problems to represent a multiplication or division fact. (b)
• Use multiplication and division basic facts to represent a given situation, using a number sentence. (b)
• Recognize and use the inverse relationship between multiplication and division to solve practical problems. (b)
• Solve single-step practical problems that involve multiplication and division of whole numbers through 10 ´ 10. (b)
• Demonstrate fluency with multiplication facts of 0, 1, 2, 5, and 10. (c)
• Solve single-step practical problems involving multiplication of whole numbers, where one factor is 99 or less and the second factor is 5 or less. (d)
• Apply strategies, including place value and the properties of multiplication and/or addition when multiplying and dividing whole numbers. (a, b, c, d)

### KEY VOCABULARY

Updated: Aug 22, 2018