Multiplication as Repeated Addition on a Number Line - Third Grade Mathematics

Multiplication is often described as repeated addition, meaning you add the exact same number over and over again. When you visualize repeated addition on a number line, each addition step becomes a leap of equal size across the line. In this chapter, you will master connecting repeated addition sentences to multiplication equations and jumping along number lines to model multiplication.

Repeated Addition: Adding the Same Amount Again and Again

When you add unequal numbers, like 4 + 7 + 2, you are simply adding. But when every single addend is identical, you have repeated addition: 5 + 5 + 5 + 5 = 20

Look at what this means:
- The number 5 is being added.

- It is being added 4 times.
Instead of writing a long chain of plus signs, multiplication provides a shorthand code: 4 x 5 = 20 The 4 tells you how many times to add, and the 5 tells you what number you are adding each time!

Addition Sentence:        6 + 6 + 6 = 18
How many 6s?              3 of them!
Multiplication Sentence:  3 x 6 = 18

Modeling Multiplication on a Number Line

A number line is an open pathway of whole numbers starting at 0 and continuing to the right.

To model multiplication on a number line:
1. Always start at 0.

2. Determine your jump size (the second factor).

3. Determine your number of jumps (the first factor).

4. Hop along the line landing on each multiple until you reach the final product.


Example: Modeling 4 x 3 = 12

Here we take 4 jumps of size 3:

     Jump 1      Jump 2      Jump 3      Jump 4
   (+3 units)  (+3 units)  (+3 units)  (+3 units)
   +---------> +---------> +---------> +--------->
0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
  • Jump 1 lands on: 0 + 3 = 3
  • Jump 2 lands on: 3 + 3 = 6
  • Jump 3 lands on: 6 + 3 = 9
  • Jump 4 lands on: 9 + 3 = 12 The final landing spot is 12! Therefore, 4 x 3 = 12.

What Does 3 x 4 Look Like on the Number Line?

Let us reverse the factors: 3 jumps of size 4:

        Jump 1            Jump 2            Jump 3
      (+4 units)        (+4 units)        (+4 units)
   +------------->   +------------->   +------------->
0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
  • Jump 1 lands on 4.
  • Jump 2 lands on 8.
  • Jump 3 lands on 12. Both journeys land on the exact same spot: 12! One took 4 short hops of 3, while the other took 3 bigger hops of 4.

Connecting Skip Counting to the Number Line

Notice that the landing points on a number line during multiplication are the exact numbers you say when skip counting!

When modeling 5 x 2 on a number line, your feet land on: 2, 4, 6, 8, 10. Skip counting is literally the spoken soundtrack of walking a number line!

+-------------------+----------------------------+-----------------------+
| Multiplication    | Repeated Addition          | Number Line Landings  |
+-------------------+----------------------------+-----------------------+
| 3 x 2             | 2 + 2 + 2                  | 2, 4, 6               |
| 4 x 5             | 5 + 5 + 5 + 5              | 5, 10, 15, 20         |
| 5 x 3             | 3 + 3 + 3 + 3 + 3          | 3, 6, 9, 12, 15       |
| 2 x 8             | 8 + 8                      | 8, 16                 |
+-------------------+----------------------------+-----------------------+

Chapter Practice Exercises

  1. Write the repeated addition sentence as a multiplication sentence: a. 4 + 4 + 4 + 4 + 4 b. 7 + 7 + 7 c. 9 + 9 + 9 + 9 d. 2 + 2 + 2 + 2 + 2 + 2
  2. Write each multiplication sentence as a repeated addition sentence: a. 3 x 8 b. 5 x 6 c. 4 x 7
  3. Draw or describe the number line jumps for 3 x 5. Where does each jump land? What is the final product?
  4. A cricket hops 4 inches on every leap. If the cricket takes 6 consecutive leaps starting at 0, what number does it reach on the measuring tape? Write both an addition sentence and a multiplication sentence.
  5. A student sees 8 + 8 + 8 and writes 8 x 8 x 8. Explain the student's error and write the correct multiplication equation.

Solutions and Step-by-Step Answers

  1. Multiplication sentences: a. 4 is added 5 times -> 5 x 4 = 20. b. 7 is added 3 times -> 3 x 7 = 21. c. 9 is added 4 times -> 4 x 9 = 36. d. 2 is added 6 times -> 6 x 2 = 12.
  2. Repeated addition sentences: a. 3 x 8 = 8 + 8 + 8 = 24. b. 5 x 6 = 6 + 6 + 6 + 6 + 6 = 30. c. 4 x 7 = 7 + 7 + 7 + 7 = 28.
  3. For 3 x 5: Start at 0 and take 3 jumps of 5 units each.
  4. Jump 1 lands on 5.
  5. Jump 2 lands on 10.
  6. Jump 3 lands on 15. The final landing spot is 15.
  7. Cricket jumps:
  8. Repeated addition: 4 + 4 + 4 + 4 + 4 + 4 = 24 inches.
  9. Multiplication: 6 x 4 = 24 inches. The cricket lands at 24 inches.
  10. The student confused the multiplication symbol with repeated addition. Adding 8 three times means 3 groups of 8, which is written as 3 x 8 = 24. Writing 8 x 8 x 8 would mean multiplying 8 by itself three times (which equals 512), not repeated addition!