A Dollar To Spend

Task 80 ... Years 4 - 8

Summary

Students are offered several challenges based on money values - one they make for themselves, one is based on the materials list for the task and the harder one involves keeping track of two conditions. The objects are important for some students because they can place them and count in money units as they do so, for example, placing pencils one at a time and counting 10¢, 20¢, 30¢... Without these aids, some students would be less likely to enter the problem or perhaps less successful if they did.

Materials

10 pencils
20 hairpins
100 paper clips

Content

money calculations
problem solving strategies such as guess, check & improve, trying every possible case, breaking the problem into smaller parts, make a list or a table

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Answers to Question 1 will vary. However, you can always ask questions like:

For your collection, how much change would you get if you only had $1 coin and a 50¢ coin (or other appropriate amount)?
If mum sent you to the shop to buy one of each of your collection for yourself and each of your brothers and sisters, what is the smallest collection of paper money she could give you to pay for it? How much change would you get?
Select a new collection. How close is its cost to $2?

The second question is the total of 10 x 10¢ plus 20 x 5¢ plus 50 x 1¢, which is $2·50.

Can you check your answer another way?
How do you do this calculation on a calculator?

Josephine's problem is more challenging. Perhaps students will see limits:

It can't be more than 10 pencils, because that's worth $1. But it's only 10 items.
It can't be just the paper clips because that would be 100 things but only 50¢.
We have to trade stuff.
You can't buy odd numbers of paper clips.
One pencil is worth 2 hairpins or 20 paper clips
One hairpin is worth 10 paper clips

One approach from here is to guess and check. Another is to break the problem into parts, make a table and try every possible case.

Pencils H/Pins P/Clips Items Cost

0 20 0 20 $1·00
That's it for 20 pins.

0 19 2 21 $0·96 - No

0 19 4 23 $0·97 - No

0 19 6 25 $0·98 - No

0 19 8 27 $0·99 - No

0 19 10 29 $1·00
That's it for 19 pins.

0 18 2 20 $0·91 - No

... ... ... ... ...

1 18 0 19 $1·00 - No

... ... ... ... ...

1 9 90 100 $1·00 - Yes

... ... ... ... ...

Using this approach not only answers the question, but provides a way to answer the question Is this the only solution?.

Could Josephine spend $0.50 and get 50 items? $0.90 and get 90 items?
Could a similar problem be invented using different prices for the same objects?

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

It is possible to run the whole class lesson for this task with just the equipment in the bag. Use it to set up a 'shop' with cards showing the prices of the items. Also draw up a poster showing Josephine's problem. After the students have gathered around the shop to get a feel for it, invite each group in turn to send a representative to make a selection of items and write it on the board. All groups then work out the cost of the collection and the teacher orchestrates discussion of ways of calculating, possible change questions and so on.

Next step is to display the poster of Josephine's Problem and run the session as a Poster Problem Clinic. The focus is as much on how to go about finding an answer as it is on finding the answer(s) itself. The materials are still there in the shop if any students need them, and every student is now supported by working in a small group.

If you explore the table approach above, the work can be shared between groups. Once it's all done, remember to review with the question How have we worked like a mathematician?.

At this stage, A Dollar To Spend does not have a matching lesson on Maths300, however, Lesson 14, The Farmer's Puzzle, is based around an equivalent problem used as a Poster Problem Clinic. For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The A Dollar To Spend task is an integral part of:

MWA Number & Computation Years 5 & 6

Follow this link to Task Centre Home page.