Turn-Around Facts (Commutative Property) - First Grade Mathematics

Have you ever put your left shoe on before your right shoe, or your right shoe on before your left shoe? Either way, both shoes end up on your feet! Addition works the exact same way. When you add two numbers together, you can switch their order, and the total will always be the same. In first grade, we call these related pairs "turn-around facts." In higher mathematics, this rule is known by its formal name: the Commutative Property of Addition. In this chapter, you will explore how turn-around facts work and how they cut the number of addition facts you need to memorize in half!

The Magic of Turning Around Facts

A turn-around fact means that changing the order of the addends does not change the sum.

The Turn-Around Demonstration with Connecting Cubes:

Train 1:  3 Red Cubes + 4 Blue Cubes
          [R][R][R]   + [B][B][B][B]   = 7 Cubes in all!  (3 + 4 = 7)

Now pick up the train and TURN IT AROUND!

Train 2:  4 Blue Cubes + 3 Red Cubes
          [B][B][B][B] + [R][R][R]     = 7 Cubes in all!  (4 + 3 = 7)

Both trains have the exact same length: 7!

The Word "Commutative"

The formal name "commutative" comes from the word "commute," which means to move around or travel. The numbers are allowed to move around and switch seats, but the sum stays unchanged.

The Commutative Rule:
Part A + Part B = Part B + Part A

How Turn-Around Facts Save Brain Power

Imagine you are learning all your addition facts up to 10. There are dozens of combinations! But with turn-around facts, every time you learn one fact, you get a second fact for free!

Buy One, Get One Free!
If you know:       Then you automatically know:
  2 + 5 = 7   -->    5 + 2 = 7
  1 + 8 = 9   -->    8 + 1 = 9
  3 + 6 = 9   -->    6 + 3 = 9
  4 + 6 = 10  -->    6 + 4 = 10

The Only Exceptions: Doubles Facts

Doubles facts like 3 + 3 = 6 do not have a different turn-around fact because both addends are already identical. When you turn 3 + 3 around, it is still 3 + 3!

Using Turn-Around Facts to Add Faster

Remember the count-on strategy from earlier in this unit? Turn-around facts make counting on much easier when the small number is written first.

Solving 2 + 8:
Don't start at 2 and count on 8 times!
Turn it around: 8 + 2!
Lock 8 in your head: "8... 9, 10!"
Done in two hops!

Turn Around in Your Head

Whenever you see a small number plus a big number, immediately turn the problem around in your mind. Start with the bigger number and count on the smaller amount.

Practice Exercises

Exercise 1: Write the turn-around fact for each of these addition equations: 3 + 5 = 8 2 + 7 = 9 4 + 1 = 5 8 + 2 = 10

Exercise 2: You have a cube train made of 6 green cubes connected to 2 yellow cubes. If you flip the train around so the yellow cubes are first, how many total cubes are in the train? Write both addition equations.

Exercise 3: Explain how the turn-around rule helps you solve 1 + 9 using the count-on strategy.

Exercise 4: Does the doubles fact 4 + 4 = 8 have a different turn-around fact? Explain why or why not.

Solutions and Explanations

Solution to Exercise 1: The turn-around fact for 3 + 5 = 8 is 5 + 3 = 8. The turn-around fact for 2 + 7 = 9 is 7 + 2 = 9. The turn-around fact for 4 + 1 = 5 is 1 + 4 = 5. The turn-around fact for 8 + 2 = 10 is 2 + 8 = 10.

Solution to Exercise 2: There are still 8 cubes in the train. The two addition equations are 6 + 2 = 8 and 2 + 6 = 8.

Solution to Exercise 3: Instead of starting at 1 and counting forward 9 times, you can turn the fact around to 9 + 1. Then you simply start at 9 and count on 1 hop ("9... 10"), which gives the answer in a single step.

Solution to Exercise 4: No, 4 + 4 = 8 does not have a different turn-around fact because both addends are the same number (4). When you switch the two 4s, the equation remains 4 + 4 = 8.