Fractions of Fractions
This unit uses one of the digital learning objects, the Fractions of Fractions Tool, to support students as they investigate finding fractions of fractions. It is suitable for students working at Stage 7-Advanced Multiplicative or Stage 8-Advanced Proportional of the Number Framework. It includes problems and questions that can be used by the teacher when working with a group of students on the learning object, and ideas for independent student work.
AO elaboration and other teaching resources
use an array model to visualise and solve problems which involve finding a fraction of a fraction
solve problems involving fractions, including fractions greater than 1
The strategy section of the New Zealand Number Framework consists of a sequence of global stages that students use to solve mental number problems. On this framework students working at different strategy stages use characteristic ways to solve problems. This unit of work and the associated learning object are useful for students in transition between Stage 7 and Stage 8 of the Number Framework, moving from Advanced Multiplicative to Advanced Proportional. At Stage 7 students use a range of strategies to solve problems that involve multiplying whole numbers, while at Stage 8 they can also apply a range of strategies to solve problems that involve multiplying fractions and decimals.
Working with the object with students (fractions of fractions < 1)
- Show students the learning object and explain that it provides a model for multiplying fractions.
- Discuss the first screen in the Fractions of Fractions Tool with students, ensuring that they understand what is meant by calling the shaded square a ‘unit square’. Students at this stage should be familiar with the concept of a unit, and it is important that they realise that for the purposes of this learning object, the 1 by 1 square represents the unit. It may be useful here to ensure that students are familiar with the array model for multiplication – ie. that a rectangle with a length of a and a width of b has an area of a x b.
- Explain to students that the learning object allows you to split up the whole area, including the unit square, both horizontally and vertically.
- Demonstrate by choosing ‘halves’ and splitting the vertical axis into halves.
- Click on 1/2 to shade everything below 1/2 in the learning object.
- Discuss the fact that half of the unit square is now shaded, that 1/2 of 1 is 1/2, and that you can also see that 1/2 of 2 is 1/2 + 1/2, which equals 1.
- Choose halves again and split the horizontal axis into halves as well.
- Click on 1/2 to shade everything to the left of the 1/2.
- Ask How many equal parts has the unit square been split into now? (4) What is the size of each part? (1/4)
- Ask students how much of the unit square is now shaded twice. They should recognise that 1/4 of it is shaded twice (green in colour).
- Fill in 1/4 in the solution box at the bottom of the screen.
- Ask the students to explain how this relates to the two ‘halves’ that are shaded. It is important that they realise that the 1/4 is 1/2 of 1/2. Reinforce the fact that in maths ‘of’ means the same thing as ‘times’. Use examples from whole numbers such as "4 lots of 5 is the same as 4 times 5". So 1/2 of 1/2 is the same as 1/2 x 1/2.
- Ask for a volunteer to show how the learning object could be used to show 2/3 of 1/2.
- Allow a volunteer to demonstrate.
- Ask whether anyone can see another relationship that is shown on the learning object at the same time. Hopefully someone will identify that 1/2 of 2/3 is the same as 2/3 of 1/2. If not, draw the students’ attention to the two ways of describing the twice (green) shaded area.
- Try a few more examples with fractions less than 1, emphasising the fact that there is always a pair of sums, and that the sums can be described using ‘of’ or ‘times’ interchangeably.
Working with the object with students (fractions of fractions > 1)
- Ask students "What is 4/3 of 6/4?" Hopefully they will see that the learning object can be used to work out fractions of fractions greater than 1 as well as less than 1. If not, work through a few examples greater than 1 as a class.
Notes regarding the standard algorithm for multiplication of fractions
The focus of mathematics, and in particular numeracy, education in New Zealand is increasingly on developing an understanding of number properties and how they can be used. The Fractions of Fractions Tool provides a model which students can use to visualize how to find fractions of fractions, however, in practice we want them to learn how to solve problems such as 4/3 of 6/4 without needing to rely on a tool or a diagram. Students should have enough experience with tools and models to discover the patterns that occur when they solve fractions of fractions problems. Once this level of undertsanding is achieved they can then be introduced to (if they haven't alreday derived) the algorithm for multiplying fractions.
- Either on the Fractions of Fractions Tool or on a whiteboard, create a unit square split into thirds in one direction and quarters in the other direction.
- Discuss how many parts the unit square is split into.
- Ask students to identify how this number relates to the numbers in the fractions (multiply the denominators).
- Work out and record several problems that involve multiplying thirds by quarters, keeping the answers in twelfths:
- 1/3 x 1/4 = 1/12
- 1/3 x 2/4 = 2/12
- 2/3 x 2/4 = 4/12
- 2/3 x 3/4 = 6/12
- 3/3 x 4/4 = 12/12
- Ask students to identify the pattern in the numerators (multiply across).
- The Fractions of Fractions Tool provides a useful array model to illustrate why you can multiply across both the numerators and the denominators to solve fraction multiplication problems.
Students working independently with the learning object
Use one of the contexts below to set problems for the students to solve independently, either on their own or in pairs, using the learning object.
- Money, for example:
- You withdraw 3/4 of the money from your bank account. You then spend half of the money you withdrew. What fraction of the money originally in your bank account did you spend?
- Jim has 4/3 the amount of money that Sally has. He spends half of his money. Sally spends the same amount of money. What fraction of her money does she spend?
- Food, for example:
- There were 6 quarter slices of orange left after the rugby game. 2/3 of them had mud on them from when they were dropped on the ground. What fraction of the oranges were muddy?
- Tracey counted that there were 12/8 of a pizza leftover the morning after the party. If she and John eat 2/3 of this how much do they eat?
You can also ask some problems without imbedding them in a context, for example:
What is 1/2 of 2/3?
What is 2/3 of 5/8?
What is 5/4 of 6/4?
Students working independently without the learning object
Independent activities that develop the same concepts as the learning object include:
- Students can work with A4 sheets of paper, experimenting with folding them into fractions in both directions. For example if they fold the paper in quarters one direction, and into thirds the other direction they can use this to find 2/3 of 3/4 of the sheet of paper. Discuss whether folding paper can be used to represent fractions greater than 1.
- Encourage students to experiment with recording strategies to show how they know that, for example, 5/4 of 1/3 equals 5/12.