Why Subtraction Isn't Commutative or Associative — Math Worksheet for 8-Year-Olds (Grade 3)

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Instructions

Hello, Math Detective! Today you are on a special mission to investigate two important math rules. These rules work for addition, but we need to find out if they also work for subtraction. Grab your pencil and let's solve the case!

Part 1: The Order Rule

In addition, the Order Rule says you can swap numbers around and still get the same answer. For example, 4 + 5 = 9 is the same as 5 + 4 = 9. Does this work for subtraction? Let's investigate!

Solve both problems in each pair. Then, circle if the answers are the same.

1. Case #1

A)   8   -   3   =   ______

B)   3   -   8   =   ______

(Hint: If you can't subtract a bigger number from a smaller one, you can write "Can't do!")

Are the answers in A and B the same?     Yes   /   No


2. Case #2

A)   12   -   7   =   ______

B)   7   -   12   =   ______

Are the answers in A and B the same?     Yes   /   No


Detective's Conclusion: Based on your investigation, does the Order Rule work for subtraction?

Yes   /   No

Part 2: The Grouping Rule

In addition, the Grouping Rule says you can change how you group the numbers and the answer stays the same. The numbers in the brackets ( ) are always done first! For example, (2 + 3) + 5 = 10 is the same as 2 + (3 + 5) = 10. Let's test this rule on subtraction!

1. Case #3

A)   (10 - 5) - 2   =   ______ - 2   =   ______

B)   10 - (5 - 2)   =   10 - ______   =   ______

Are the final answers in A and B the same?     Yes   /   No


2. Case #4

A)   (15 - 8) - 4   =   ______ - 4   =   ______

B)   15 - (8 - 4)   =   15 - ______   =   ______

Are the final answers in A and B the same?     Yes   /   No


Detective's Conclusion: Based on your investigation, does the Grouping Rule work for subtraction?

Yes   /   No

Case Closed!

Great work, Detective! You have proven that for subtraction, the order of the numbers and how they are grouped is very important. The Order Rule and the Grouping Rule DO NOT work for subtraction!





Answer Key

Part 1: The Order Rule

1. Case #1
A) 8 - 3 = 5
B) 3 - 8 = Can't do!
Are the answers the same? No

2. Case #2
A) 12 - 7 = 5
B) 7 - 12 = Can't do!
Are the answers the same? No

Detective's Conclusion: Does the Order Rule work for subtraction? No

Part 2: The Grouping Rule

1. Case #3
A) (10 - 5) - 2 = 5 - 2 = 3
B) 10 - (5 - 2) = 10 - 3 = 7
Are the final answers the same? No

2. Case #4
A) (15 - 8) - 4 = 7 - 4 = 3
B) 15 - (8 - 4) = 15 - 4 = 11
Are the final answers the same? No

Detective's Conclusion: Does the Grouping Rule work for subtraction? No

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