Problem
Mr Morrison and his daughter Rose both have very bad coughs. They go to the pharmacy and they each get a bottle of QwikQure Cough Medicine. Mr Morrison’s bottle has 300ml and he has to take 10ml every 3 hours. On the other hand, Rose is given a smaller bottle with only 90ml and she has to take 5ml every 6 hours. They have to take the medicine between the hours of 7am and 7pm.

Whose bottle lasts longest?

What is this problem about?
This problem can be done in a number of ways. We suggest that a graph or a table is the best way to solve it. Both of these approaches produce the conceptual knowledge that is required to solve more complicated algebraic problems in secondary school and beyond.

The aim of the problem is to see how different rates of usage can be compared. This is typical of many real world problems such as investment problems, where there are different rates of return, and travel problems where different methods of travel cost different amounts.

Beware! There is a slight trick in this problem. Both of the sickies will take their first dose at 7am! So, for example, Mr Morrison takes his medicine 4 times over 12 hours (once every three hours) plus the dose at 7am.

Achievement Objectives
Algebra (Level 4)
- sketch and interpret graphs on whole number grids which represent everyday situations

Mathematical Process
- devise and use problem solving strategies (draw a picture, make a table, think)

Resources
Graph paper
Blackline master of the problem (English)

Specific Learning Outcomes
The children will be able to:
- draw a graph accurately
- be able to use the graph to obtain information

1. Introduce the problem using a bottle of medicine. Ask the children to estimate the volume of the medicine and to predict what the instructions on the bottle will read.
2. Read the problem to the class or you could present the problem using 2 bottles labelled according to the problem.
3. Get the children to think about how they might solve the problem before asking them to work on the problems in pairs.
4. As the children solve the problem, ask questions that focus their thinking on the strategy that they have used to solve the problem:
   How are you solving the problem?
   Why did you decide to solve it that way?
   What can you tell me about what you have found out?
   How are you going to record your solution?
   Do you think that you can convince others that you have found the solution? What will you say?
5. Share solutions.

Extension Problem
Mrs Morrison also develops a bad cough. The pharmacist says that she should take 10ml every 4 hours. If her bottle lasts as long as her husband’s, how much medicine does her bottle hold?
Why does Rose have the smallest dosage?
How large is 10ml? How many doses would go into a cup/glass/soft drink can?
How realistic is this problem? What are typical doses of cough medicine?

Other Contexts for the Problem
Walking different distances at different speeds
Use of other resources such as money

Solution
Use a graph with the horizontal axis marked from 7am to 7pm in hours for about 7 days. Mr Morrison will then use all his medicine up on the sixth day at 7pm. Rose will do the same.