Non-unit fraction of a non-unit fraction
The purpose of this activity is to extend student understanding of fraction as an operator to include finding a fraction of a fraction.
About this resource
New Zealand Curriculum: Level 3
Learning Progression Frameworks: Multiplicative thinking, Signpost 6 to Signpost 7
These activities are intended for students who have some previous experience with equal partitioning, such as finding lines of symmetry in shapes. It is preferable that they are also able to call on and apply their knowledge of addition and multiplication facts.
Non-unit fraction of a non-unit fraction
Achievement objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-5: Know fractions and percentages in everyday use.
Required materials
- sheets of paper (A4 photocopying paper is ideal)
- pens (highlighters or felt pens are preferable)
1.
Take a rectangular sheet of paper and follow the steps described below.
- Fold the paper in thirds lengthways, then shade in two of the thirds.
- Fold the paper in quarters widthwise, then shade in three quarters of the two thirds.
- What fraction of the whole rectangle is double-shaded? Explain how you know.
Emphasise that while the dimensions of the results are different, both area models show double-shaded areas that are one twelfth in size. This illustrates that 3/4 x 2/3 = 2/3 x 3/4.
Record, describe, and reflect on the relevant expressions. For example:
2/3 x 3/4 = 6/12, “two thirds of three quarters equals six twelfths.”
2.
Use paper folding and shading to explore the commutative property of multiplication with fractions.
Does shading two thirds lengthways and then three quarters widthways give the same result as shading three quarters lengthways and two thirds widthways?
Students may notice that, in the equation, the numerators 3 and 2 are multiplied to form the numerators of the answer. Similarly, the denominators are multiplied to form the denominator of the answer.
3.
Provide further opportunities for students to use paper folding and shading to find a non-unit fraction of a non-unit fraction. Allow students to work in groupings that will encourage peer scaffolding and extension. Some students might benefit from working independently, while others might need further support from the teacher. Good examples are shown below:
Three quarters of three fifths (3/4 x 3/5 = 9/20) Four fifths of two thirds (4/5 x 2/3 = 8/15) One fifth of three eighths (1/5 x 3/8 = 3/40).
Record 3/4 x 2/3 = 6/12.
Explain what this equation means. Look for students to recognise the equations as “three quarters of two thirds equals six twelve.” What patterns do you see in the denominators? Why do we end up with twelve equal parts? What patterns do you see in the numerators? Why do we end up with six parts double-shaded?
4.
Provide examples for students to solve without the support of paper folding and shading. This might begin with drawing diagrams to show the product of two non-unit fractions. For example, 3/4 x 2/5 = 6/20 might be drawn as:
5.
Progress to solving problems with equations only, such as those shown below. Provide paper folding for the examples if students have difficulty.
- 3/5 x 7/8 = [ ]
- 4/5 x 4/5 = [ ]
- 2/3 x 5/8 = [ ]
- 4/9 x 5/7 = [ ]
1.
Encourage students to apply a wider range of basic multiplication facts to these types of problems. For example,
- 7/8 x 5/9 = [ ]
2.
Use written expressions to formally generalise the multiplication of two unit fractions:
- What is the answer to □/◊ x ○/∆? (In general, the product is always □ x ○/◊ x ∆.)
3.
Compare the products of non-unit fractions. For example:
- Tim shades two fifths of three eights on his piece of paper. Jayden shades three quarters of two fifths of his paper. The pieces of paper are the same size. Who shades more, Tim or Jayden?
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