# Negative Numbers on the Number Line

Visualization is an important focus in Singapore math and the number line is one tool we used frequently to visualize relationships between numbers. However in 6th or 7th grade, when students are introduced to negative numbers, things can get a bit tricky when dealing with negative numbers on the number line. For example,

-3 – (-4) = ?

Rather than introducing rules such as double negative makes positive, it is better to let students have a consistent way to visualize the operation on the number in the same way they understood basic addition and subtraction in 1st grade.

## A Simple Analogy

In this post, we use a simple analogy to help students deal with negative numbers, especially when the addend/subtrahend is negative, like the example above.

A simple math expression is divided into three components – two numbers and an operator. Besides the operator (+ or -), both numbers can also be positive (regular numbers) or negative.

## The Starting Point

Now imagine you are standing on the number line. The first number tells you where to start.

## Which Direction to Face

The second number tells you which direction to face. According to convention, the right side of the number line extends to +∞ or positive infinity, while the left side extends to -∞. So, when adding (+ operator), you would stand facing to the right (towards positive infinity +∞). Conversely, if dealing with a subtraction operator, you should face to the left (towards negative infinity -∞).

## How Far to Walk

The final number tells you how far to walk. If the number is positive, you’ll walk forward (towards the direction you are facing). If the number is negative, you’ll walk backward (away from the direction you are facing).

## Put It All Together

So, in the above example, you would start from negative two (-3), turn to face the left (subtraction operator -) and then walk three steps backward (-4), to end up on +1 or just 1. So,

-3 – (-4) = 1

## Never be Confused Again

With this simple picture in their mind, students gets less confused when dealing with double negatives as they are able to use the familiar number line to visualize the operation!

### 2 thoughts on “Negative Numbers on the Number Line”

1. How do you solve this question, using the most simple solution?
How many different pairs (a, b) can be formed using
numbers from the list of integers {1, 2, 3, …, 10} such
that a < b and a + b is even?

1. For each possible choice of a, b can be one of 10-a choices, out of which only half (rounded down) will give an even a+b.

For example,
If a is 1, b can be one of 9 choices (2,3,4,…10), out of which 4 pairs result in an even a+b (3,5,7,9).
If a is 2, b can be one of 8 choices (3,4,5,…10), out of which 4 pairs result in an even a+b (4,6,8,10).
If a is 3, b can be one of 9 choices (4,5,6,…10), out of which 3 pairs result in an even a+b (5,7,9).
If a is 4, b can be one of 9 choices (5,6,7,…10), out of which 3 pairs result in an even a+b (6,8,10).

If a is 9, b can only be 10, but a+b is odd (not a solution).
If a is 10, there are no solutions for b.

So, summing up all the possible choices, 4+4+3+3+2+2+1+1+0+0, there are 20 possible pairs.

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