Token Gesture

This activity requires students to multiply a decimal by a whole number. It also involves a simple rate in dollars per token that makes the problem more demanding. In this activity, students use multiplicative strategies to calculate the amount earned by collecting tokens and bonuses and then identify patterns and relationships that could be used to predict earnings for any number of tokens collected.

To solve the calculations in this activity mentally, students need to be at least advanced multiplicative thinkers (stage 7) because they have to solve multiplication problems such as 28 x 40 and 55 x 40. Calculations such as those in question 1 require the stage 8 skills of multiplying decimal fractions (for example, 32 x $0.40), but this can also be done at stage 7 using whole numbers and then converting to dollars if necessary (for example, 32 x 40 cents = 1 280 cents or $12.80).

Set the scene with a guided teaching group by talking about promotions involving collecting tokens, receipts, or points that the students may have taken part in. Ask them what motivated them to collect the tokens, what tactics they used to get lots of them, and about any prizes they earned.

The calculations in question 1 give the students opportunities to apply and compare different multiplicative strategies. Get the students to discuss each question with a classmate or group of 3–4 first before sharing with the whole group. Possible strategies include:
• place value partitioning:
14 x $0.40 = (10 x 0.4) + (4 x 0.4)
= 4 + 1.6
= $5.60
14 x 40 cents = (10 x 40) + (4 x 40)
= 400 + 160
= 560 cents or $5.60
• using tidy numbers and compensating:
28 x $0.40 = (30 x 0.4) – (2 x 0.4)
= 12 – 0.8
= $11.20
28 x 40 cents = (30 x 40) – (2 x 40)
= 1 200 – 80
= 1 120 cents or $11.20
• doubling and halving repeatedly:
32 x 40 cents = 16 x 80
= 8 x 160
= 4 x 320
= 2 x 640
= 1 280 cents or $12.80
For question 2, get the students to discuss in their small groups how Mere could work out her bonus. Have the students share back with the whole group how they would apply their group’s strategies. They need to recognise that it is the multiples of 5 that will help Mere to work out her bonus because it’s not until she reaches each multiple of 5 that she gets another dollar. You could demonstrate this on a hundreds board by getting the students to flip or circle every fifth number as they count up, keeping a tally at the same time of how many bonus dollars Mere would get as she collects each extra group of 5. As the students count up towards the next
multiple of 5, you could ask: How much will Mere’s bonus be now that she’s got 18 tokens? 19 tokens? 20 tokens?
This relationship could be plotted on a graph:

 
For obvious reasons, this is known as a step function.
You could use questions 3 and 4 to make formative assessments of the students’ algebraic thinking. Listen as they work through the questions with their classmate or group to identify which students are merely using recursive (sequential) rules by adding another 40 cents or bonus repeatedly and which students have generalised by identifying the functional relationship, that is, the number of tokens x 0.4.
Some students may need to use a calculator when they are working out the money earned from 153 tokens so that they can concentrate on the algebraic aspects. Encourage them to evaluate for themselves when they need to use a calculator. Remind the students that number sense is very important when using a calculator because they need to check by “backwards estimation” that the answer they get is sensible and not the result of an incorrect entry.
The students who are able to solve the reversed problem in question 5 will show a deep understanding of the way the patterns work. A simple approach might be to look at the table to find the number of tokens that earns close to $69 (such as 100 tokens earns $60) and use a trial-and-improvement strategy to hit the correct number. An alternative approach is to recognise that every 5 tokens collected represent total earnings of $3 (from $0.40 x 5 tokens + $1 bonus).
This $3 is earned for every group of 5 tokens in a multiple of 5, for example, 20 tokens has 4 lots of 5 tokens, so Rewi earns 4 lots of $3. Using this approach, the students would see the $69 as being made up of 23 lots of $3 or 23 lots of 5 tokens (23 x 5 = 115).
As a challenge, you could extend the problem in question 5: Mere’s school wants to raise $3,000 for new sports equipment. How many tokens will the students have to collect?
• Every $3 earned needs 5 tokens, and there are 1 000 lots of $3 in $3,000, so they would need 1 000 lots of 5 tokens: 1 000 x 5 = 5 000 tokens.
• 10 tokens earns you $6, and 100 tokens earns you $60. 1 000 tokens would earn $600, and you’d need 5 times that to get $3,000. 5 x 1 000 tokens = 5 000 tokens. Working backwards to check:(5 000 tokens x $0.40) + (bonus of 5 000 tokens ÷ 5 x $1) = $2,000 + $1,000= $3,000.
For question 6, you might need to suggest that the students use a systematic method of comparison, such as making a table or drawing a graph, because the comparison is not consistently better for one company or the other until you collect over 50 tokens. The Breakfast Bonanza company pays bonuses on multiples of 5, and Serious Cereals pays out on multiples of 10, so another worthwhile strategy is to compare how much money would be earned from every 10 tokens collected ($6 for Breakfast Bonanza tokens compared to $6.50 for Serious Cereals tokens,
which means that, in the long term, Serious Cereals tokens would earn you more).
The students could use a computer graphing program to compare the two kinds of tokens. If they do, they will need to enter the data for each number of tokens (1, 2, 3, and so on up to at least 60) because the graphs are stepped. The steps can be copied in as “blocks” to make the data entry easier. The program will show the rise of each step to be steeply sloped, but it should be vertical. Most graph-drawing programs have this limitation. The graph shown below is not ideal, but it’s similar to the graphs you and your students will be able to produce. The inaccuracy
of the steps doesn’t detract from the main point, which is that, over time, the two lines are getting further apart. The graph should clearly demonstrate that the return on Serious Cereals tokens is always greater once the number of tokens is greater than 50.

Extension

The students could write their own token and bonus problem for another group member to solve.
 

Answers to Activity

1. a. $12.80. (32 x 40c = 1 280c, which is $12.80)
b. $11.20. (28 x 40c = 1 120c, which is $11.20)
c. $22.00. (55 x 40c = 2 200c, which is $22.00)
d. $5.60. (14 x 40c = 560c, which is $5.60)
2. a. Mere could be dividing the number of tokens she has collected by 5 and working out the answer with a remainder if it has one, for example, for 19 tokens: 19 ÷ 5 = 3 r4. To work out the bonus, Mere must ignore the remainder, because those 4 tokens are not enough to get her another bonus dollar. So for 19 tokens, she would get a $3 bonus. Mere could work out whether the number of tokens is a multiple of 5. If it is, the bonus can be worked out by dividing the number of tokens by 5, for example, 20 ÷ 5 = $4 bonus. If it’s not a multiple of 5, Mere could
find the multiple of 5 that is just before the number of tokens and divide that by 5
instead. For example, for 19 tokens: 19 isn’t a multiple of 5, but the multiple of 5 just
before 19 is 15. 15 ÷ 5 = $3 bonus.
b. i. $4. (23 ÷ 5 = 4 r3, so the bonus is still based on 20 tokens.)
ii. $7. (37 ÷ 5 = 7 r2, so the bonus is $7, based on 35 tokens.)
iii. $9. (48 ÷ 5 = 9 r3, so the bonus is $9, based on 45 tokens.)

3.

4. a. Answers may vary. Mere will lose out on possible bonuses each time she sends in tokens that aren’t multiples of 5. As long as she sends in multiples of 5, she won’t lose out on any bonuses, regardless of when in March she sends them in. If she waits until the end of the month, she will only miss out on 1 bonus at the most (if she has a set of less than 5 left over).
b. Answers may vary. One method is:
153 x 0.40 = $61.20 (or 153 x 40c = 6 120c, which is $61.20) plus the bonus.
153 ÷ 5 = 30 r3, so the bonus is $30.
61.20 + 30 = $91.20 raised
5. a. 115 tokens
b. Strategies will vary. One strategy is working from known quantities. You could perhaps start by estimating that it will be a bit more than 100 tokens because this would earn $60, according to the table in question 3.
The table also tells you that 15 tokens would earn $9. $60 + $9 = $69, and so the number of tokens must be 100 + 15 = 115. Another strategy is to start from the fact that Rewi needs 5 tokens for every $3 he earns. There are 23 lots of $3 in $69 (69 ÷ 3 = 23), so there would be 23 lots of 5 tokens; 23 x 5 = 115.
6. Explanations will vary. The Serious Cereals tokens are a better deal if you can complete lots of 10 because for every 10 tokens collected, they pay $6.50 (that is, 10 x 25c + $4 bonus), while Breakfast Bonanza pays only $6 (that is, 10 x 40c + $2 bonus). However, it is not until you collect over 50 tokens that the Serious Cereals
tokens will always give you a better deal. If you have collected 1–9, 14–19, 25–29, 35–39, or 47–49 tokens, Breakfast Bonanza would give you more money than Serious Cereals.