Problem
Sally and David have agreed to work for their mother over the holidays. The pay they get will vary though. Sally will get $10 for the first day she works and two more dollars for every day she works after that. David, on the other hand, will get $1 for the first day he works but for each day he works from then on his pay will be doubled. Who would you rather be and why?

What is this problem about?
The mathematics behind this question is a comparison of the rate at which adding a fixed number increases the total compared to continual doubling of a number. In fact no matter what whole numbers you start with, doubling will eventually win.

Achievement Objectives
Number (Level 3)
- explain the meaning of the digits in any whole number
- write and solve problems which involve whole numbers and decimals and require a choice of one or more of the four arithmetic operations.

Mathematical Processes
- devise and use problem solving strategies (look for patterns, use equipment)

Resources
calendar
play money
Blackline master of problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- add a series of two digit numbers
- double numbers
- systematically keep track of their calculations to compare numbers.

Teaching Sequence

1. Read the problem to the class. Give them time to think about the problem by themselves.
2. Ask them to highlight the important pieces of information. Encourage them to explain why they consider them important.
3. Brainstorm for ways to solve the problem. Remind them that it is important to plan ways to keep track of their working.
4. If the children are beginning problem solvers you may work out the first couple of days together. Once more encourage the children to plan ways to keep track of the totals.
5. Let the children work on the problem in small groups. As they work ask questions that require them to compare the daily totals. Ask them to justify the number operations that they are using and to explain the steps they are taking to find the answers.
6. Sharing of solutions.

Extension problem
After how many days will David’s mother run out of money?

Solution
This problem actually doesn’t have a definite answer. Nowhere in the problem does it say how many days the two children work. If they work for less than 6 days then Sally will earn most money. If they work for more than 6 days, then David will get the most.