Repeated Addition as Multiplication - Second Grade Mathematics

Addition is one of your oldest mathematical friends, but when you need to add the exact same number over and over again, writing repeated addition equations can become lengthy and tiring. Repeated addition is the repeated joining of equal groups, and it serves as the direct mathematical gateway to multiplication. When you write 4 + 4 + 4 = 12, you are adding three equal groups of 4. In this chapter, you will master writing repeated addition sentences from visual models, discover how repeated addition translates into multiplication, and use number lines to see equal jumps in action.

From Equal Groups to Repeated Addition

When equal groups are joined together, the total can be recorded as an addition equation where every addend is identical.

Visualizing Repeated Addition:
Three flowerpots with 5 flowers each:
| Y Y Y Y Y |   | Y Y Y Y Y |   | Y Y Y Y Y |
  Pot 1: 5        Pot 2: 5        Pot 3: 5

Repeated Addition Sentence:
5 + 5 + 5 = 15

Word Description:
3 groups of 5 equals 15.

The Structure of the Equation

Notice that in 5 + 5 + 5 = 15: The number being added (5) is the size of each group. The number of times 5 is written (3 times) is the number of groups! If you have 6 groups of 2, you write the number 2 six times: 2 + 2 + 2 + 2 + 2 + 2 = 12.

The Bridge to the Multiplication Symbol (x)

The multiplication symbol (x) was invented by mathematicians as a brilliant shorthand for repeated addition.

The Translation Bridge:
Repeated Addition:     4 + 4 + 4 + 4 + 4 = 20
Number of Groups:      5 groups
Size of Each Group:    4 items in each group
Multiplication:        5 x 4 = 20
Read Aloud:            "5 times 4 equals 20" (or "5 groups of 4 equals 20")

Reading the Multiplication Sentence

When you see the expression 5 x 4, read the "x" symbol as "groups of" or "times": 5 x 4 literally means "5 groups of 4." It tells you to take the number 4 and add it together 5 times. Multiplication does not replace addition; it is simply a fast, elegant way to write repeated addition!

Visualizing Repeated Addition on a Number Line

On a number line, repeated addition appears as a series of equal-sized hops traveling to the right.

Modeling 4 Hops of 3 (4 x 3 = 12):
        +3              +3              +3              +3
   +---------->    +---------->    +---------->    +---------->
---+-----------+---+-----------+---+-----------+---+-----------+--->
   0           3               6               9              12

The Rhythm of the Hops

Notice that every hop is identical in width: Hop 1: Lands on 3 Hop 2: Lands on 6 Hop 3: Lands on 9 Hop 4: Lands on 12 The number of hops represents the first factor (4 hops), the size of each hop represents the second factor (3 units), and the final landing point is the total product (12).

Practice Exercises

Exercise 1: Write a repeated addition equation and find the total for: 4 boxes of crayons with 6 crayons in each box.

Exercise 2: Look at the repeated addition sentence: 7 + 7 + 7 + 7 = 28. State how many groups there are, the size of each group, and rewrite it as a multiplication sentence using the "x" symbol.

Exercise 3: Draw or describe how you would model 3 x 4 on an open number line starting at 0. State the landing point of each hop.

Exercise 4: A student wrote 3 + 3 + 3 + 3 + 3 = 15. Explain how this equation can be read using both the phrase "groups of" and the word "times."

Solutions and Explanations

Solution to Exercise 1: For 4 boxes with 6 crayons each, write the number 6 four times: 6 + 6 + 6 + 6 = 24. There are 24 crayons in all.

Solution to Exercise 2: In 7 + 7 + 7 + 7 = 28, the number 7 appears 4 times. Therefore, there are 4 groups, and the size of each group is 7. Rewritten as a multiplication sentence: 4 x 7 = 28.

Solution to Exercise 3: To model 3 x 4 on an open number line, start at 0 and make 3 equal hops of 4 units each. Hop 1 lands on 4. Hop 2 lands on 8. Hop 3 lands on 12. The final landing point is 12, showing that 3 x 4 = 12.

Solution to Exercise 4: In 3 + 3 + 3 + 3 + 3 = 15, the number 3 is added 5 times. Using "groups of", it is read as "5 groups of 3 equals 15." Using "times", it is read as "5 times 3 equals 15." Both phrases express that 5 collections of 3 items combine to make 15.