Arithmetic properties: Associative property

Arithmetic properties: Associative 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 associative property means that changing the grouping of numbers in an equation does NOT change the answer when you are performing ONLY addition or ONLY multiplication
    • The numbers can be any real number (whole numbers, fractions, decimals, integers, etc.)
    • To “associate” can mean to interact with a group of people/friends or to group together.
    • No matter how you want to group (using brackets) the numbers in an addition or multiplication equation, it will not change the answer in the end.

  • For addition: the grouping of addends does not change the answer
    • Ex. (1 + 2) + 3 = 1 + (2 + 3) will equal 6 either way
      • Because (1 + 2) + 3 = (3) + 3 = 6
      • As well, 1 + (2 + 3) = 1 + (5) = 6

  • The associative property for addition can make shortcuts for adding whole numbers and decimals by making sums of 10 (i.e. 1 + 9, 2 + 8, 3 + 7, 4 + 6, and 5 + 5)
    • Ex. 8 + 6 + 2 + 4 + 5 + xx
      • Group as: (8 + 2) + (6 + 4) + 5 + x = (10) + (10) + 5 + xx = 25xx
    • Ex. 0.9 + 0.7 + 0.3 + 0.1
      • Group as: (0.9 + 0.1) + (0.7 + 0.3) = (1.0) + (1.0) = 2.0

  • Shortcuts for adding fractions is also possible with the associative property by making wholes (i.e. same numerator and denominator; 44,22,1010\large \frac{4}{4}, \frac{2}{2},\frac{10}{10})
    • Ex. 34+24+14\large \frac{3}{4} + \frac{2}{4} + \frac{1}{4}
      • Group as: (34+14)+24=44+24=1+24=124\large (\frac{3}{4} + \frac{1}{4}) + \frac{2} {4} = \frac{4} {4} + \frac{2} {4} = 1 + \frac{2} {4} = 1 \frac{2}{4}
    • Ex. 29+25+79+35+14\large \frac{2}{9} + \frac{2}{5} + \frac{7}{9} + \frac{3}{5} + \frac{1}{4}
      • Group as: (29+79)+(25+35)+14=(99)+(55)+14=1+1+14=214\large (\frac{2}{9} + \frac{7}{9}) + (\frac{2} {5} + \frac{3} {5}) + \frac{1} {4} = (\frac{9} {9}) + (\frac{5}{5}) + \frac{1}{4} = 1 + 1 + \frac{1}{4} = 2 \frac{1}{4}

  • For multiplication: the grouping of factors does not change the answer
    • Ex. (2 × 3) × 4 = 2 × (3 × 4) will equal 24 either way
      • Because (2 × 3) × 4 = (6) × 4 = 24
      • As well, 2 × (3 × 4) = 2 × (12) = 24

  • The associative property for multiplication can make shortcuts for multiplying any real numbers by making multiples of 10 (i.e. 10, 20, 30, 40…)
    • Ex. 2 × 8 × 5 × ee
      • Group as: (2 × 5) × 8 × ee = (10) × 8 × ee = 80 × ee = 80ee
    • Ex. 0.9 × 0.5 × 0.6
      • Group as: (0.5 × 0.6) × 0.9 = (0.30) × 0.9 = 0.270
    • Ex. 52\large \frac{5}{2} × 913\frac{9}{13} × 450\frac{4}{50}
      • Group as: 5x9x42x13x50\large \frac{5 \, x \, 9 \, x \, 4}{2 \, x \, 13 \, x \, 50} = (5x4)x9(2x50)x13\large \frac{(5 \, x \, 4) \, x \, 9}{(2 \, x \, 50) \, x \, 13} = (20)x9(100)x13=1801300\large \frac{(20) \, x \, 9 }{(100) \, x \, 13 } = \frac{180}{1300}

  • The general formulas (where aa, bb and cc are variables that represent real numbers) for the associative property are:

Arithmetic Property

Of Addition

Of Multiplication

Associative Property

(a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

(a×b)×c=a×(b×c) (a × b) × c = a × (b × c)

  • Introduction
    Introduction to the associative property of addition and multiplication:
    a)
    Showing that (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

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

    c)
    Addition shortcuts using the associative property

    d)
    Showing that (a × b) × c = a × (b × c)

    e)
    Multiplication shortcuts using the associative property

    f)
    The general formulas for the associative property


  • 1.
    Associative Property Equations
    Use the associative property (for addition and multiplication) to fill in the blanks.
    a)
    (7 + 2) + 8 = 7 + (2 + __)

    b)
    14×(16×15)=(14×\large \frac{1}{4} \, \times \, (\frac{1}{6} \, \times \, \frac{1}{5}) = (\frac{1}{4} \, \times \, ______ )×15\large ) \, \times \, \frac{1}{5}

    c)
    (xx + 3 + 8) + __ = (3 + 9 + x) + 8


  • 2.
    Changing the grouping to add lists of numbers
    Decide how to group the addends as a shortcut for addition. Double check your answer by adding without groups.
    a)
    0.1 + 0.6 + 0.9 + 0.4 =

    b)
    210+310+110+410+910+710+610+810=\large \frac{2}{10} + \frac{3}{10} + \frac{1}{10} + \frac{4}{10} + \frac{9}{10} + \frac{7}{10} + \frac{6}{10} + \frac{8}{10} =

    c)
    23+59+13+49+67=\large \frac{2}{3} + \frac{5}{9} + \frac{1}{3} + \frac{4}{9} + \frac{6}{7} =


  • 3.
    Changing the grouping to multiply lists of numbers
    Decide how to group the factors as a shortcut for multiplication. Double check your answer by multiplying without groups.
    a)
    20 × 15 × 5

    b)
    0.8 × 0.9 × 0.5

    c)
    6 × 20 × 5 × 2


  • 4.
    Associative property of addition word problem
    Ryan added these numbers together and his answer is correct. Show another way of adding numbers (with grouping) using the associative property!

    Associative Property

  • 5.
    Associative property of multiplication word problem
    Explain which choice is NOT an equal statement to: (6 × 8) × 5
    1. 6 × 40
    2. 48 × 5
    3. 6 × (8 × 5)
    4. 14 × 5

  • 6.
    Associative property and volume
    The formula for the volume of a rectangular prism is given by:

    Volume = length × width × height


    Use the associative property of multiplication to show 2 ways to solve for this prism.
    Associative Property