Practice Commutativity in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A property where swapping the order of two operands does not change the result: a ⋆ b=b ⋆ aa \ \star\ b = b\ \star\ a.

3+5=5+33 + 5 = 5 + 3 and 3×5=5×33 \times 5 = 5 \times 3. Swapping the order doesn't change the answer.

Showing a random 20 of 50 problems.

Example 1

medium
Lucy says 7÷1=1÷77 \div 1 = 1 \div 7 because division is commutative. Is she correct?

Example 2

medium
Does the order in 3×4×53 \times 4 \times 5 matter for the product?

Example 3

challenge
Define a⋆b=a+b+aba\star b = a+b+ab. Is ⋆\star commutative? Justify.

Example 4

easy
Is 10÷2=2÷1010\div 2 = 2\div 10?

Example 5

medium
Does commutativity work for subtraction? Check with 10−310 - 3 and 3−103 - 10.

Example 6

medium
A student writes 12−5=5−1212-5 = 5-12. Explain the error in one property term.

Example 7

easy
Use commutativity to rewrite 15+3615 + 36 so the smaller addend comes first.

Example 8

easy
Rewrite 6+116+11 using commutativity.

Example 9

easy
Which property lets you rewrite 3+53+5 as 5+35+3?

Example 10

medium
Does forming a fraction commute: is 23=32\frac{2}{3}=\frac{3}{2}?

Example 11

easy
A 3×53 \times 5 array of dots has the same number of dots as a 5×35 \times 3 array. How many dots?

Example 12

medium
Is the operation a⋆b=aba\star b = a^b commutative? Test with 22 and 33.

Example 13

medium
Use commutativity to compute 7+23+37 + 23 + 3 quickly.

Example 14

medium
Use commutativity to compute 25×17×425\times 17\times 4 quickly.

Example 15

hard
Is the operation 'subtract then negate', defined as a⋆b=−(a−b)a \star b = -(a - b), commutative?

Example 16

medium
Rewrite 3×4+4×33\times 4 + 4\times 3 as a single product using commutativity.

Example 17

easy
Does swapping the order in 5+05+0 to 0+50+5 change the value?

Example 18

hard
Use commutativity to make 5×13×25 \times 13 \times 2 easier, and compute the product.

Example 19

medium
Is matrix-style 'subtract then negate' a⋆b=−(a−b)a\star b=-(a-b) commutative? Test 5,25,2.

Example 20

medium
True or false: for all numbers a,ba, b, a+b=b+aa + b = b + a.
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