This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.
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solve ratio problems
compare ratios
Number Framework Links
Use this activity to help students learn to express fractions as ratios and to express ratios in their simplest form by dividing by common factors (stages 6 and 7).
Counters or cubes
Question 1 involves straightforward proportional thinking and can be solved either by counting chocolates and comparing numbers or by making connections between the sizes of the boxes. In question 1a, the students might notice that if they turned box A 90 degrees, it would fit twice into box B, so 2A = B. Box B would fit neatly over the top of box C, leaving 12 chocolates uncovered at the bottom, which is the number of chocolates in box A, so C = B + A or C = 3A. Once they know the relationship between the sizes of the boxes, it’s easy for them to determine the weight
and cost of each.
Question 2 requires the students to work out Tania’s best option. This can be done by counting the strawberry hearts for each combination of boxes that will make up 1 kg. Box A gives Tania 3:9 (or 3/12), box B gives her 8:16 (or 8/24), and box C gives her 13:23 (or 13/36).
If your students have had little experience working with ratios, you will need to show them carefully how the ratio 3:9 (favourites to non-favourites) can equally well be expressed as the fraction 3/12 (favourites out of total number of chocolates in the box).
There are four combinations of boxes that will make up 1 kg: 4 of box A (4A), 2 of box A and 1 of box B (2A + B), 2 of box B (2B), or 1 of box A plus 1 of box C (A + C). The last two of these options will give Tania the same number of strawberry hearts: 16. Expressed as a ratio of favourites to non-favourites, this is 16:32 or 1:2.
Give your students time to explain their strategies. Make sure they understand that the ratios 16:32 and 1:2 are identical. They may think that the ratio 16:32 contains within itself the number of chocolates and 1:2 doesn’t. This is a misconception. A ratio never means anything without its context. To make sense of the 16:32 (or 1:2), we need to know the “story” that goes with the numbers: in this case, the strawberry hearts and the number of chocolates in the boxes.
Question 3 doesn’t introduce any new information or processes but gives students a chance to apply and consolidate the ideas they have met in question 2. They may find it useful to collate the information and answers for both questions in a table.
In question 4, the students need to work out which combination of boxes would give each of the three friends the same number of favourite chocolates. Encourage them to make use of the work they have already done. There are only four combinations to be considered, and they know from question 2 that both 2B and A + C give Tania 16 of her favourites. The only remaining task is to find which of these two combinations also gives Atama and Chloe 16 of their favourites.
Answers to Activity
1. a. 500 g and 750 g
b. Box A (250 g) should cost $4.30.
Box B (500 g) should cost $8.60.
2. a. There are two possibilities: 2B or A + C. Both would give Tania 16 strawberry hearts.
b. The ratio of favourites to non-favourites would be 16:32 = 1:2.
3. Atama would get most caramel circles (18) by choosing 2 x B. The ratio of favourites to non favourites would be 18:30 = 3:5.
Chloe would get most peppermint squares (20) by choosing 4 x A. The ratio of favourites to non favourites would be 20:28 = 5:7.
4. If they buy boxes A and C, each person will get 16 of their favourite kind.