Doubling and halving to find factors

Purpose

The purpose of this activity is to support students to find factors of a given number using doubling and halving.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
Required Resource Materials
  • Square tiles or squared paper
  • Calculators
Activity
  1. Ask students about quilt making or tapa cloth. Look up quilt patterns or tapa cloth online.
    Do you know someone who makes quilts/tapa cloth?
    Discuss how the quilts/tapa cloth are made.
    The main point is that squares are joined in a tessellation.
  2. Imagine that you have 24 squares to make a quilt or tapa cloth. The finished quilt/cloth must be a rectangle.
    How big might the quilt/cloth be?
    Let students experiment with 24 square tiles or grid paper to find a possible rectangle with area of 24 squares. After an appropriate time gather the group to discuss possible answers. Possibilities include:
    arrays
    How can we write the size of each quilt/tapa cloth?
    Can we use multiplication?
    Record the sizes as 1 x 24, 2 x 12, 3 x 8, 4 x 6.
    Do you think we have found all the possible sizes? How do you know?
    Students may not regard 1 x 24 and 2 x 12 as viable quilts/ tapa cloths.
  3. Now that we have found some answers I wonder if there was an easy way to find them.
    1 x 24 is easy to find. Do you notice that 2 x 12 is also a size?
    What pattern do you notice?
    Record:
    1 x 24
    2 x 12
    Students might notice that doubling 1 to get 2 and halving 24 to get 12 gives another answer.
    You might show the two arrays that match the facts so students can compare them visually.
    What would happen if we doubled and halved again? (4 x 6)
    What would happen if we doubled and halves with 4 x 6? (8 x 3)
  4. You may have noticed that a 24-square quilt/tapa cloth is a bit small. So let’s double the number of squares.
    What is double 24?
    Agree that double 24 equals 48 squares.
    Before you start finding the sizes of the quilt/tapa cloths think about what you learned from the 24 square quilt. You can use the calculator if that helps.
  5. Let students attempt to find all the possible 48-square quilts/tapa cloths using tiles, grid paper, and calculator. Roam as they work. Look for:
    • Do they represent possible sizes as multiplication?
    • Do they use the sizes for 24 squares to pre-empt what will work for 48 squares?
    • Do they start with 1 x 48 and use doubling and halving?
    • Do they scan multiplication facts for possible sizes, such as try x 4 facts?
  6. Gather the group after a suitable time. Discuss how to find all the possible sizes for the quilt/tapa cloth.​​​​​​​
    Remember that we found 1 x 24, 2 x 12, 4 x 6, and 8 x 3 for the 24 square quilt.
    Did anyone use those facts to look for sizes with 48 squares?
    Did anyone use doubling and halving?
    Record sizes as multiplication as students offer then, whether in visual form or not. Record the facts systematically to enhance the chance of pattern spotting:
    1 x 48
    2 x 24
    4 x 12
    8 x 6
    16 x 3
    Have we found them all? How do you know?
    Can we make a 5 x size quilt/tapa with 48 squares?

Next steps

  • Explore other quilt/tapa cloth sizes with numbers of squares that have many factors, such as 36 squares and 60 squares.
  • Progress from using materials to anticipating sizes using multiplication facts.
  • Explore the division operation on the calculator to check for possible factors, such as dividing 48 by different numbers to see what factor occurs, e.g. 48 ÷ 3 = 16.
  • Investigate the commutative property. Do 6 x 8 and 8 x 6 represent different quilts/tapa cloths or the same one?
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Level Two