Arithmetic properties: Commutative property

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  1. Introduction to the commutative property of addition and multiplication:
  2. Showing that a + b = b + a
  3. Why is it called the "commutative" property?
  4. Showing that a × b = b × a
  5. The general formulas for the commutative property
  1. Commutative property equations
    Use the commutative property (for addition and multiplication) to fill in the blanks.
    1. 3 + __ = 2 + 3
    2. 810\frac{8}{10} + 510\frac{5}{10} + __ = 510\frac{5}{10} + 610\frac{6}{10} + 810\frac{8}{10}
    3. 0.8 × 0.5 = __ × 0.8
    4. 4 × 9 × aa = 9 × __ × 4
  2. Changing the order to add/multiply lists of numbers
    Do the operations in order, and then backwards order. Are the answers different?
    1. 0.8 + 0.2 + 0.6 + 0.9 =
    2. 15 + 23 + 37 + 44 =
    3. 25\frac{2}{5} × 610\frac{6}{10} × 89\frac{8}{9} =
    4. 5 × 3 × 4 × 2 × gg =
  3. Commutative property of addition word problem
    James and Scarlet both need to buy a pair of skates and a helmet. Store A sells the skates for $83.67. Store B sells the helmet for $54.29. James goes to store A first, then store B; Scarlet goes to store B first, then store A. Did they pay different amounts? Explain.
    1. Commutative property of multiplication word problem
      Felix has 24 pages of homework he must finish in 6 days. If he is going to do the same number of pages each day:
      1. How many pages must he complete each day?
      2. What other schedule(s) can he follow to finish his homework?
    2. Which wall will need more paint to cover its area?
      • A wall that is 5 12\frac{1}{2}m tall and 8m wide
      • A wall that is 8m tall and 5 12\frac{1}{2} m wide
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      Topic Notes

      In this lesson, we will learn:

      • What is the commutative property of addition?
      • What is the commutative property of multiplication?
      • How to write the general formulas/equations for the commutative properties
      • Changing the order of a list of addends/factors does not change the answer
      • How to solve word problems for the commutative property


      • The commutative property means that changing the order of numbers in an equation does NOT change the answer ONLY when you are performing addition or multiplication
        • The numbers can be any real number (whole numbers, fractions, decimals, integers, etc.)
        • To β€œcommute” can mean to travel between home and schoolβ€”when you switch places, the travel time does not change (going to school and going back home).
          • If you switch the places of the numbers in an addition or multiplication equation, it will not change the answer in the end.

      • For addition: the order of addends does not change the answer
        • Ex. 3 + 5 = 5 + 3 (equals 8 either way)
        • Ex. 0.3 + 0.5 = 0.5 + 0.3 (equals 0.8 either way)
        • Ex. 310\frac{3}{10} + 510\frac{5}{10} = 510\frac{5}{10} + 310\frac{3}{10} (equals 810\frac{8}{10} either way)

      • For multiplication: the order of factors does not change the answer
        • Ex. 3 Γ— 4 = 4 Γ— 3 (equals 12 either way)
        • Ex. 0.3 Γ— 0.4 = 0.4 Γ— 0.3 (equals 0.12 either way)
        • Ex. 310\frac{3}{10} + 410\frac{4}{10} = 410\frac{4}{10} + 310\frac{3}{10} (equals 12100\frac{12}{100} either way)

      • The general formulas (where a and b are variables that represent real numbers) for the commutative property are:

      Arithmetic Property

      Of Addition

      Of Multiplication


      a+b=b+aa + b = b + a

      a x b=b x aa \,x\,b = b\, x\, a

      • The commutative property does NOT work for subtraction nor division. In subtraction and division, the order of numbers DO matter and will change the answer
      • Ex. 5 – 2 = 3, but if you switch the order, the answer will change: 2 – 5 = -3
      • Ex. 10 Γ· 2 = 5, but if you switch the order, the answer will change: 2 Γ· 10 = 0.2