Multiplication as Equal Groups - Third Grade Mathematics

Multiplication is a fast, efficient mathematical operation created to count items organized into groups of the exact same size. Whenever you see several bags with the same number of apples, several nests with the same number of eggs, or several shelves with the same number of books, you are looking at equal groups. In this chapter, you will discover how equal groups define multiplication and how to write mathematical equations that describe them.

What Are Equal Groups?

Equal groups are collections of objects where every single collection contains the exact same number of items. If one group has 4 marbles and another has 5 marbles, they are unequal, and you cannot use simple multiplication. But when all groups are identical in size, multiplication comes to life!

Equal Groups: 3 groups of 4 stars each
Group 1: [ * * * * ]
Group 2: [ * * * * ]
Group 3: [ * * * * ]
Total: 12 stars!

Unequal Groups:
Group A: [ * * * ]       (3 items)
Group B: [ * * * * * ]   (5 items)
Not equal! Multiplication does not directly apply.

The Parts of a Multiplication Equation

A multiplication equation has special vocabulary that describes the groups, the size of each group, and the final total.

   3       x       4       =      12
(Factor)        (Factor)       (Product)
 Number          Size of         Total
of Groups       Each Group       Count
  • Factors: The numbers you multiply together. In equal groups, the first factor represents the number of groups, and the second factor represents the number of items in each group.
  • Multiplication Sign (x): Read as "groups of" or "times."
  • Product: The answer or total amount resulting from the multiplication.

When you read "3 x 4 = 12," say out loud: "3 groups of 4 equals 12."

Translating Pictures into Multiplication Sentences

To write a multiplication sentence from a picture, follow these three steps:
1. Count how many groups (circles, bags, plates, nests) there are in total. This is your first factor.

2. Count how many items are inside just one group. This is your second factor.

3. Count the total items across all groups to find the product.


Example:
Plate 1: (o o o o o)
Plate 2: (o o o o o)
Plate 3: (o o o o o)
Plate 4: (o o o o o)

Step 1: Number of plates = 4
Step 2: Cookies on each plate = 5
Step 3: Multiplication equation: 4 x 5 = 20 cookies in all!

Creating Equal Groups from a Total

You can also work in reverse: start with a total count and arrange them into equal groups.

Suppose you have 18 counters and want to make groups of 6:
- Group 1: 6 counters (12 remain)

- Group 2: 6 counters (6 remain)

- Group 3: 6 counters (0 remain)
You created 3 groups of 6, which gives the equation: 3 x 6 = 18.

What if you instead made groups of 3? You would get 6 groups of 3: 6 x 3 = 18. Both ways represent equal groups of the exact same 18 counters!

Chapter Practice Exercises

  1. Write an equal groups addition sentence and a multiplication sentence for each situation: a. 4 boxes with 3 toy cars in each box b. 5 baskets with 6 apples in each basket c. 2 packs with 8 juice boxes in each pack d. 6 bowls with 4 goldfish in each bowl
  2. Look at this drawing and write the multiplication sentence: Group 1: [ @ @ @ @ @ @ ] Group 2: [ @ @ @ @ @ @ ] Group 3: [ @ @ @ @ @ @ ]
  3. Draw or describe an equal groups model for the equation 3 x 7 = 21. State what each number represents.
  4. An insect has 6 legs. How many legs do 4 insects have altogether? Show your equal groups thinking.
  5. Jackson has 15 toy soldiers. He wants to put them into equal groups with none left over. Name two different ways Jackson can group his soldiers.

Solutions and Step-by-Step Answers

  1. Sentences: a. Addition: 3 + 3 + 3 + 3 = 12. Multiplication: 4 x 3 = 12 toy cars. b. Addition: 6 + 6 + 6 + 6 + 6 = 30. Multiplication: 5 x 6 = 30 apples. c. Addition: 8 + 8 = 16. Multiplication: 2 x 8 = 16 juice boxes. d. Addition: 4 + 4 + 4 + 4 + 4 + 4 = 24. Multiplication: 6 x 4 = 24 goldfish.
  2. There are 3 groups, and each group contains 6 symbols. The multiplication sentence is 3 x 6 = 18.
  3. Equal groups model: Draw 3 large circles (groups). Inside each circle, draw 7 dots. The 3 represents the number of groups, the 7 represents the items per group, and 21 represents the total dots.
  4. There are 4 insects (groups) and each insect has 6 legs (items per group). Equation: 4 x 6 = 24 legs in all.
  5. Jackson can group his 15 soldiers as:
  6. 3 equal groups of 5 soldiers (3 x 5 = 15).
  7. 5 equal groups of 3 soldiers (5 x 3 = 15).
  8. 1 group of 15 soldiers (1 x 15 = 15).