Multiplication and Division Information Guide
MA.3.2.1 The student will recognize and use inverse relationships between
multiplication/division to complete basic fact sentences and to solve problems
Inverse Relationships:
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Addition and subtraction are inverse operations, as are multiplication and division.
To make understanding of division simpler, the division fact should be tied to the related
multiplication fact.
Multiplication and division should be taught concurrently in order to develop
understanding of the inverse relationship.
Fact Families and inverse relationship:
5, 4, 20 (3 numbers make a fact family, 4 number sentences, 2 multiplication, 2 division)
1.
2.
3.
4.
5 x 4=20
4 x 5=20
20÷ 5 = 4
20÷ 4 = 5
Using Fact families to answer questions such as:
1. Which of the following could you solve knowing 6 x 8 = 48?
a. 6 + 8
b. 48 x 6
c. 48 ÷ 6
d. 48 x 8
2. Knowing 72 ÷ 9 = 8 would help you solve which of the problems below?
a. 9 x 8
b. 8 + 9
c. 8 + 72
d. 72 x 9
3. Given 3 x 9 = 27, circle all the related math facts below.
9 x 3 = 27
27 ÷ 1 = 27
27 ÷ 3 = 9
27 ÷ 9 = 3
3 + 9 = 12
MA.3.8.1 The student will investigate and identify examples of the identity and
commutative properties for multiplication.
Algebraic Properties:
Investigating arithmetic operations with whole numbers helps students learn about several
different properties of arithmetic relationships. These relationships remain true regardless of the
numbers.

The commutative property for multiplication states that changing the order of the factors
does not affect the product
(e.g., 2  3 = 3  2).

The array, by contrast, is quite powerful in illustrating the commutative property, as shown
below. Children should build or draw arrays and use them to demonstrate why each array
represents two equivalent products.
3x 6= 6x3

The identity property of multiplication states that if a given number is multiplied by one,
the product is the same as the given number. (5 x 1=5)
MA.3.2.5 The student will represent multiplication and division of two whole numbers, one
factor 99 or less and the second factor 5 or less, using area, set, and number line models.
MA.3.2.4 The student will recall multiplication facts through the twelves table and the
corresponding division facts.
Multiplication and division models:

To extend the understanding of multiplication, three models may be used:
SET MODELo
o
The equal-sets or equal-groups model lends itself to sorting a variety of concrete
objects into equal groups and reinforces repeated addition or skip-counting.
Examples:
ARRAY OR AREA MODELo
The array model, consisting of rows and columns (e.g., 3 rows of 4 columns for a
3-by-4 array) helps build the commutative property.
Multiplication Fact: 7 x 8 = 56
Division fact: 56 ÷ 7 = 8 or 56÷8= 7
NUMBER LINEo
The length model (e.g., a number line) also reinforces repeated addition or skipcounting.
**5 hops of 4, multiplication arrows face FORWARD**
Division fact : 6÷ 3 = 2
**division arrows face BACKWARDS**
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The multiplication and division facts through the twelves tables should be modeled.
Multiplication is a shortcut for repeated addition. The terms associated with
multiplication are listed below:
factor  54
factor   3
product 162
Students also need to be able to multiply a 2 digit number by a 1 digit number. (56 x 5)
MA.3.2.6 The student will create and solve real-world problems that involve multiplication
of two whole numbers, one factor 99 or less and the second factor 5 or less.
Examples:
Lawrence was planting a garden. He had enough seeds to plant 3 rows of carrots if he used 12
seeds in each row. How many carrot seeds did Lawrence plant?
Pedro is baking cookies. He is making 48 cookies. He plans to put 5 chocolate chips in each
cookie. How many chocolate chips will he need?
The Big Ideas of Multiplication
It is important that students are able to decompose (break apart) and compose (put back
together) numbers in a variety of ways. This helps students ascertain flexible methods for
computing. Students do not need to know there are two different types of multiplication
problems.
EQUAL GROUP PROBLEMS:
COMPARISON PROBLEMS:
When working in multiplication, equal group
problems are when either the number of sets or
the size of the sets is unknown.
For example:
Matthew bought four packs of tennis balls.
There are 6 tennis balls in each bag. How
many tennis balls does Matthew have?
When working in multiplication, comparison
problems involve two different sets. The
comparison is based off one set being a
multiple of another.
For example:
Matthew bought four tennis balls. Jacob
bought six times as many tennis balls as
Matthew. How many tennis balls did Jacob
buy?
COMMON MULTIPLICATION STRATEGIES
REPEATED ADDITION:
SET MODEL:
6 + 6 + 6 + 6 = 24 tennis
balls
NUMBER LINES:
ARRAYS:
XXXXXX
XXXXXX
XXXXXX
XXXXXX
ALGORITHM
PARTIAL PRODUCTS
This is when being able to compose and decompose numbers becomes useful. A
student that can decompose 24 into 20 + 4 is going to be able to use and
understand why the partial products method works.