Problem
How high would a pile of paper be if there were 1 million sheets of paper in the pile?

What is this problem about?
This problem provides a context for exploring large numbers, in this case a million. Having a sense of large numbers, as well as very small numbers, provides the foundation for making reasonable estimates. Our sense of numbers generally comes from experiences, but few of us have had a chance to experience very large numbers, so our sense of such numbers is limited. Working out how high a million sheets would be provides children with a sense of "how big a million is".

Achievement Objectives
Number (Level 4)
- make sensible estimates and check the reasonableness of answers
- write and solve problems involving decimal multiplication and division

Mathematical Processes
- devise and use problem solving strategies (guess and check, use equipment, be systematic)

Resources required
a ream of paper
rulers
a sheet a paper
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- explain a meaning of a million
- make sensible estimates of numbers using powers of 10 up to a million.
- multiply and divide with numbers up to a million (using a calculator)

1. Play a guessing game (with say a 100 sheets) – how many sheets do you think are in this pile? Get the children to make enter their guess in the "competition". The competition could be introduced to the class the day before the problem is posed and the "winner" revealed at the start of the lesson.
2. Pose the problem – how high would a million sheets be?
3. Get the children to work in groups of 2-3 to explore ideas for working out how high the stack would be.
4. Visit the groups as they work asking them to justify their approach and the answers they are finding.
   How do you know that the numbers you are using are correct?
   Do you think that your answer is reasonable? How do you know?
5. Require the children to record their solution.

6. Share solutions.

Extension to the problem
Get the children to invent a problem along similar lines.

Other contexts for the problem
Build a model of a million blocks (using the cm cube as a unit)
What size bag of rice contains one million grains?
How long does it take to count to a million?
How high would one million pens reach?

Solution to the problem
It will be necessary for the children to measure their own ream of paper. In this case you will need to have handy some reams of (a ream of paper is approximately 5.2 cm thick.)

There is more than one way to approach the calculation here. One way is to note that 2000 reams is 1 million sheets and so the height will be 2000 × 5.2. Another way is to divide 5.2 by 500 to get the thickness of one sheet. Then multiply the answer of that calculation by 1 million. The answer should be 10400 cm.

Once that is all done, it would be nice to convert the answer into metres. So the pile would be 104 metres high.

But is that reasonable? How could you possibly check the answer? Could you stack 2000 reams of photocopying paper on top of each other? If you can’t do something practical like that, then can sensible estimates be made? It ought to be easy enough to get 20 reams and stack them.