Non-unit fraction of a non-unit fraction

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 widthways, then shade in three quarters of the two thirds.
  
  
  
  What fraction of the whole rectangle is double shaded? Explain how you know.
  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?
  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.
   

2. 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, whilst 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).
      
3. Use paper folding and shading to explore the commutative property of multiplication with fractions.
   Does shading two thirds lengthways then three quarters width ways give the same result as shading three quarters lengthways and two thirds widthways?
   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.” 
   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.
   
   
    
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 = [     ]

Next steps for further learning

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?