Challenge Time

All of the problems in this activity involve proportions and require students to have efficient multiplicative strategies at the advanced additive stage or beyond. Many of them can be efficiently solved using double number lines, as suggested for question 1. Using double number lines is one way of encouraging a systematic approach towards problem solving. However, students also need to realise that this is not the only way of solving proportion problems, and in some cases, other strategies are more effective. For example, question 5 is best approached using equivalent fractions,
and the odd numbers in question 6 call for a different sort of reasoning. Ratio tables are also an efficient scaffolding method.
In question 1, the task is to determine what the 100 has to be multiplied by to make it 450. The 10 must then be multiplied by the same number (4.5), so it becomes 45.
Encourage the students to draw up a double number line for question 2. If they are unsure where to start, ask them “How many pieces of toast are eaten each day?” “How many of these does Hamish eat?” “Where will you put the 2 and the 5 on your double number line?” Finding the required answer becomes simple if the students show on their double number line what they are multiplying by:

The students should be able to apply the same systematic approach to all the problems in this activity. To make sure the students are on the right track, have them compare their double number line with a classmate’s.
For question 3b, you need to check that the students have recognised the change from millilitres to litres.
In question 7, the students need to realise that the 10 cent coin is really irrelevant for solving the problem. A possible double number line for this problem is:

The 1 represents one $2 coin, and the 3 represents three 20 cent coins in change.
Questions 4, 5, 6, and 8 are a bit different from the others and require the students to exercise some logical thinking and reasoning.
For question 4, the students need to reason that the food should last Josh, Ani, and Philip 3 times as long as it would the 9 people, so it should last for 15 days.
In question 5, the students need to work out what fraction of the total bag the 4 remaining lollies represent (see the explanation in the Answers). To do this, they first need to convert the half and the third into sixths, the lowest common denominator.
In question 6, the 17 trucks and 31 wheels are not multiples of 2 and 4 respectively, so it is apparent that there are going to be some spare trucks and wheels when more skateboards are assembled. The students may find it helpful to make a simple sketch of a skateboard with an outline of the board itself, the 2 trucks, and the 4 wheels.
Question 8 involves ratio. This type of problem is more appropriate for students who have developed multiplicative reasoning skills and who are at the advanced additive stage or beyond. A ratio is a fraction that compares 2 quantities of the same item. Once the students realise that the 2:1 ratio means there are 3 pieces of chicken in a special combo and 1 of them is spicy baked, they should be able to think in terms of 3 x  = 27. There are 9 combos, which means 9 pieces of spicy baked chicken and 18 pieces of regular chicken.
Two units of measurement (millilitres and litres) are used in any given part of the problem in question 9. Some students may feel they need to convert one unit into the other, but this is not necessary when you are focusing on rates and the units used remain constant.
Many of the problems in this activity also lend themselves to exploring equivalent fractions. For earlier work on equivalent fractions, see Number, Figure It Out, Level 3, page 9, and Number: Book Three, Figure It Out, Level 3, pages 22–23.
 

Answers to Activity

1. 45 L
2. 20 pieces
3. a. 75 g
b. 150 g. To get this answer, you could extend the number line shown for 3a:

c. Answers and possible number lines are:

i. 1.4L

ii. 2.275 L

4. 15 days. (9 ÷ 3 = 3. 3 x 5 = 15)
5. 24 lollies. (1/2 + 1/3 is the same as 3/6 + 2/6, which is 5/6. So Mele’s 4 lollies is 1/6. Therefore, Ken gets 2/6 or 8, and Hineata gets 3/6 or 12.)
6. 7. (You would have 3 wheels and 3 trucks left over.)
7. $14
8. 9 pieces of spicy baked chicken; 18 pieces of regular chicken. (27 ÷ 3 = 9. 9 x 2 = 18)
9. Answers and possible number lines are: