Boxed biscuits

In the warm-up phase of the lesson, before the students begin this activity, have them review factors. They could list the factors of 24, 27, 36, and 100 on a chart. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Introduce the activity and ask the students to discuss how the factors of the numbers on the boxes can help them to decide the size and number of the smaller packs of biscuits.

They could start by listing the combinations. For example:

Total number	Number of packs	Biscuits in each pack
24	4	6
24	6	4
24	8	3
24	3	8
24	12	2
24	2	12

The students may need help to develop a chart to solve the problems. When they have worked out the number of biscuits in each pack, they need to focus on the fractions of each kind of biscuit. For example, in the list above, there are only two kinds of biscuit, so the three biscuits in eight packs won’t work. Ensure that they have headings like: Packs, Mixture, and Fraction. The students may like to use a spreadsheet to make their chart.

This activity features a “set” model for the whole. The size of the whole changes with each box. This is an opportunity to show how a fraction like 1/2 can be different in size as the whole set changes. A fraction is always relative to the whole. So, despite the change in size of the packs in the 24 box, the proportion that is apricot is still 1/2.

Students who can explain equivalent fractions using materials and images should be challenged to see how to find equivalences using numbers. This will help them to use and understand the number property used to calculate equivalences. The key idea here is that the relationship between the fractions needs to be kept equivalent, so the operation used to change the numbers is multiplication or division by 1. The factor of 1 needs to be written in a suitable equivalent fraction form. For example, to change 1/4 to an equivalent such as /20, use 1/4 x 5/5 = 5/20. Point out that 5/5 is a form of 1, so 1/4 is being multiplied by 1 to keep the equivalence.

1.

a. - b

Packs	Mixture	Fraction
		Apricot	Berry
2	6 apricots, 6 berry	6/12 or 1/2	6/12 or 1/2
3	4 apricots, 4 berry	4/8 or 1/2	4/8 or 1/2
6	2 apricots, 2 berry	2/4 or 1/2	2/4 or 1/2
12	1 apricots, 1 berry	1/2	1/2

2.

27 box

Packs	Fractions
	Chocolate	Berry
3	6/9 or 2/3	3/9 or 1/3
9	2/3	1/3

36 box

Packs	Fractions
	Date	Apricot
3	9/12 or 3/4	3/12 or 1/4
9	3/4	1/4

100 box

Packs	Fractions
	Chocolate	Date	Berry	Apricot
2	20/50 or 2/5	15/50 or 3/10	10/50 or 1/5	5/50 or 1/10
5	8/20 or 2/5	6/20 or 3/10	4/20 or 1/5	2/20 or 1/10
10	4/10 or 2/5	3/10	2/10 or 1/5	1/10