Deriving facts by doubling

Purpose

The purpose of this activity is to support students recognize that scaling applies to comparisons where one amount is less than another. For example, a 3-stack is “four times less” than a 12-stack. The use of “times” in such situations seems counter-intuitive for students who believe the multiplication makes bigger.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
Required Resource Materials
  • Materials to form equal sets, such as cubes, tiles, novelty counters, and play dollar coins.
  • Calculators
Activity
  1. Create two stacks of five cubes using a single colour.
    Mask the stacks with a sheet of paper.
    I have two stacks of five cubes. How many cubes are there altogether?
    What multiplication did you use to work the product out?
    Some students may have used doubles, 5 + 5 = 10. Record 2 x 5 = 10.
  2. cubesI am going to double the number of cubes under the paper. Watch.
    Make another two stacks of five cubes and slide the stacks under the paper.
    How many stacks do I have now?
    How many cubes are in each stack?
    How many cubes are under the paper, in total?
    What multiplication did you use to work the product out? (calculator
    Look for students to apply multiplication to find the product, that is 4 x 5 = 20. Record the two equations together:
    2 x 5 = 10
    4 x 5 = 20
    What patterns do you notice in the equation?
    Do students notice that the numbers of stacks doubles and so does the product?
  3. Extend the pattern by doubling four stacks of five to make eight stacks of five cubes, under the piece of paper.
    What multiplication will give the total number of cubes?
    What will the product be?
    Record the equations:
    2 x 5 = 10
    4 x 5 = 20
    8 x 5 = [ ]
    Can students use doubling to anticipate the product?
    Confirm their answer by checking the number of cubes visually and with a calculator.
  4. Pose similar problems, masking the stacks under paper, and using repeated doubling. Make the doubling as accessible as possible to encourage students to apply the strategy. Good examples might be:
    • 1 x 4 = 4      
      2 x 4 = 8
      4 x 4 = 16
      8 x 4 = [ ]
    • 2 x 3 = 6
      4 x 3 = 12
      8 x 3 = [ ]
      16 x 3 = [    ] (optional)
    • 1 x 6 = 6
      2 x 6 = 12
      4 x 6 = 24
      8 x 6 = 48
  5. Progress to just presenting pairs of equations with one product missing. Discuss the meaning of each equation and why the second product is twice (double) the first. Calculators might be used to confirm products once students have predicted first. Good examples might be:
    3 x 7 = 21, so 6 x 7 = [ ]
    5 x 7 = 35, so 10 x 7 = [ ]
    4 x 4 = 16, so 8 x 4 = [ ]

Next steps

  • Explore the meaning of equations when the second factor is doubled. For example, 5 x 3 = 15 means “five sets of three.” 5 x 6 means “five sets of six.” How are the products connected? Since the size of sets is doubled, the product is also doubled. Other examples might be 4 x 4 = 16 so 4 x 8 =  [ ] ,  8 x 3 = 24 so 8 x 6 = [ ], 7 x 2 = 14 so 7 x 4 = [ ].
  • Encourage students to learn their basic multiplication facts using the Number facts pathway in e-ako maths.
  • For extension, go beyond the range of basic facts to include examples with one factor more than ten, such as 7 x 3 = 21 so 14 x 3 = [ ], 4 x 8 = 32 so 4 x 16 = [ ], and  8 x 9 = 72 so 16 x 9 = [ ].
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Level Two