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# How High?

Achievement Objectives:

Achievement Objective: NA4-6: Know the relative size and place value structure of positive and negative integers and decimals to three places.
AO elaboration and other teaching resources

Specific Learning Outcomes:

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)

Description of mathematics:

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 of paper would be provides students with a sense of "how big a million is".

Required Resource Materials:
A ream of paper
Rulers
A sheet a paper
Copymaster of the problem (English)
Copymaster of the problem (Māori)
Activity:

### The Problem

How high would a million sheets of paper be?

#### Teaching Sequence

1. Play a guessing game (with say a 100 sheets) – how many sheets do you think are in this pile? Get the students 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 students 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 students to record their solution.
6. Share solutions.

#### Extension to the problem

Get the students 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

It will be necessary for the students 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.

(There is a minor difficulty here. Surely the paper would squash. That means that any calculation will give an overestimate. It also means that lying 2000 reams side by side will also give an overestimate. Although it would be fun.)

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HowHigh.pdf55.74 KB
HowHighMaori.pdf53.54 KB