The number line goes both ways

Integers are the whole numbers extended in both directions, so the line runs on for ever to the left as well as to the right.

Where negative numbers are ordinary

Minus 5 degrees, level minus 2 in a car park, a bank balance of minus 3000 rupees. The minus sign is not saying something is wrong; it says which side of zero you are on.

Comparing integers

Further right always means larger. So minus 2 is larger than minus 7. This feels wrong at first, but a debt of 7 rupees is worse than a debt of 2.

Bigger digits do not mean a bigger number

Among negative numbers, the larger the digits, the further left it sits and the smaller it is. Picture the line, or ask which is colder.

Adding and subtracting

What you do	Which way you move	Example
Add a positive	Move RIGHT	−3 + 5 = 2
Subtract a positive	Move LEFT	2 − 5 = −3
Add a negative	Move LEFT	2 + (−5) = −3
Subtract a negative	Move RIGHT	2 − (−5) = 7

Two minus signs together turn into a plus

Taking away a debt makes you richer. If someone owes 500 rupees and that debt is cancelled, they are 500 rupees better off. So removing a negative moves you upwards.

Multiplying and dividing

Two signs the same give a positive answer; two signs different give a negative one. So minus 3 times minus 4 is 12, while minus 3 times 4 is minus 12.

Why two negatives multiply to a positive

Minus 3 times 3 is minus 9, then minus 6, then minus 3, then 0. Each answer goes up by 3. Carrying that on, minus 3 times minus 1 must be 3. The rule is the only continuation that keeps the pattern working.

The rules are not interchangeable

Minus 3 plus minus 4 is minus 7, not 7, because adding two debts makes a bigger debt. But minus 3 times minus 4 is 12. Before touching the signs, look at which operation you are doing.

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Worked Example 1

The temperature is minus 6 degrees at night and rises by 11 degrees by midday.

Set it upRising means moving right: −6 + 11

Count along the lineSix steps right reaches zero, and 5 of the 11 are left.

Answer5 degrees. The rise was bigger than the distance below zero, so the answer is positive.

Worked Example 2

An account is 2400 rupees overdrawn, and a debt of 900 rupees is cancelled.

Write it in symbols−2400 − (−900)

Two minus signs make a plus−2400 + 900 = −1500

Does it make sense?Still overdrawn, but by less than before, which is what cancelling part of a debt should do.

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Activity 13.1

Build a human number line and walk the calculations

What you needChalk or cards numbered from minus 10 to 10, laid in a straight line with zero in the middle.

The two rulesAlways FACE the positive direction. Adding means walking forwards, subtracting means walking backwards without turning.

MethodOne student walks the calculation while the class calls the moves. Do all four cases.

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Integers are the whole numbers extended below zero as well as above it.

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1Why is minus 2 larger than minus 7, when 7 is larger than 2?

2Explain why subtracting a negative is the same as adding, using a real situation.

3Why do two negative numbers multiply to give a positive answer?

4Why is (−3) + (−4) equal to minus 7 rather than 7?

5Give three real situations where negative numbers are used, and say what zero means in each.

Question 1 of 10

1Which is larger?

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Unit 13 · Integers

Integers

Mathematics · Grade 7

What You Will Be Able To Do

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Compare integers correctly, including two negatives

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Add and subtract integers by moving along the number line

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Multiply and divide them, and say why two negatives give a positive

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Use them in real situations: temperature, debt, depth below sea level

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The minus sign is not saying something is wrong. It is saying which side of zero you are on, and zero is the agreed starting point, not the smallest possible value.

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Presenter notes: Open by asking whether zero is the smallest number there is. Most of the class will say yes, because every number they have ever counted has been zero or above. Then ask for the temperature in Jumla on a January night, or what floor a basement car park is on, or what happens to a bank balance when somebody spends more than they have. Each answer needs a number below zero. The line does not stop at zero, and today it gets extended.