Arithmetic properties: Commutative property

Arithmetic properties: Commutative property

Lessons

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

Notes:

  • 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 “commute” can mean to travel between home and school—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 × 4 = 4 × 3 (equals 12 either way)
    • Ex. 0.3 × 0.4 = 0.4 × 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

Commutative
Property

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

axb=bxaa \,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 ÷ 2 = 5, but if you switch the order, the answer will change: 2 ÷ 10 = 0.2
  • Introduction
    Introduction to the commutative property of addition and multiplication:
    a)
    Showing that a + b = b + a

    b)
    Why is it called the "commutative" property?

    c)
    Showing that a × b = b × a

    d)
    The general formulas for the commutative property


  • 1.
    Commutative property equations
    Use the commutative property (for addition and multiplication) to fill in the blanks.
    a)
    3 + __ = 2 + 3

    b)
    810\frac{8}{10} + 510\frac{5}{10} + __ = 510\frac{5}{10} + 610\frac{6}{10} + 810\frac{8}{10}

    c)
    0.8 × 0.5 = __ × 0.8

    d)
    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?
    a)
    0.8 + 0.2 + 0.6 + 0.9 =

    b)
    15 + 23 + 37 + 44 =

    c)
    25\frac{2}{5} × 610\frac{6}{10} × 89\frac{8}{9} =

    d)
    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.

  • 4.
    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:
    a)
    How many pages must he complete each day?

    b)
    What other schedule(s) can he follow to finish his homework?


  • 5.
    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