This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.
use simple proportions to solve problems
Number Framework Links
Use this activity to:
• help the students to develop advanced additive strategies (stage 6) in the domain of multiplication and division
• help the students using advanced additive strategies (stage 6) to develop factor thinking to assist them in their transition to the advanced multiplicative stage
• develop the students’ knowledge of basic facts.
The trays of sweets model the array concept for multiplication, in which two factors define the size of the array. The width of each tray is the same. This is an important element in the solution to the problems.
If necessary, use materials to introduce the activity by having the students cut out a model of each tray using grid paper or the Fraction Grid, Numeracy Project material master 4-27 (see Related Resources). The students can use these models to explore question 1 and see the common factor of 7. Try these questions to focus the exploration:
Compare the size of the trays. Do you notice anything similar?
How would you stack the trays so that they look tidy and fit together?
If you know there are 28 squares of coconut ice, can you use that to work out the number of squares of fudge and toffee?
Check that the students recognise that the tray of fudge holds twice as much as the coconut ice tray and that the toffee tray is 1 times the size of the coconut ice tray.
Before the students attempt question 2, you may need to explain the meaning of the phrase “same mix”. Each bag needs to have the same ratio of fudge to toffee to coconut ice. A ratio is a multiplicative recipe for combining different things, for example, 2 coconut ice : 3 toffee : 4 fudge.
Send the students into small groups to work out the mix for each bag of sweets. When they succeed, ask them to compare the amount of each type of sweet per bag with the length of each side of the tray for each type. By recording each tray as factors, they should see that the mix is the same as the different factor for each tray:
Have the students use a variety of ways to estimate the solution for question 3. A double number line may be useful, for example:
Where would $175 be on the line?
How far away from $140 and $210 is $175?
So what would be the price per bag at that point?
Explore other tray sizes that have proportional relationships, for example, 6 x 6, 9 x 3, 9 x 6, and 2 x 9. Encourage the students to use both spatial and number sense to find the relationships. For example:
Answers to Activity
1. All the trays are arranged in multiples of 7.
Coconut ice = 7 x 4
Fudge = 7 x 8
Toffee = 7 x 6
2. a. They used the other factor for each tray: 4 coconut ice + 8 fudge + 6 toffee = 18 pieces.
b. There are 7 lots of 4, 6, and 8 altogether, so they could make 7 bags of 18 pieces out of each set of trays.
c. 70. (10 x 7)
3. a. $2.50. (175 ÷ 70)
b. Strategies will vary. For example, 70 + 70 = 140. 175 – 140 = 35, which is 1/2
of 70. So 175 ÷ 70 is 2 1/2 or $2.50.