Depth · one concept from first principles · step 2 of 4

Turn directions and distances into signed coordinates

A playground is $100$ m from our house. A flower shop is $150$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $-100$ , what is the coordinate of the bus stop?

Number system

Your answer

How to solve

Strategy Draw a Diagram — Draw the number line and fix which way is negative. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

1STEP 1

Fix the directions

The playground at -100 is on the negative side, so toward the playground is negative and toward the house is positive.

2STEP 2

Place the flower shop

The flower shop is 150 m farther in the negative direction from the playground.

-100 - 150 = -250

3STEP 3

Measure house to flower shop

The distance between 0 and -250 is the absolute value of their difference.

|-250 - 0| = 250

4STEP 4

Place the bus stop

From the playground, move 250 m in the positive direction, toward the house. The trip passes the house and ends 150 m on the other side.

-100 + 250 = 150

Answer

150

The bus stop at 150 is 250 m from the playground at -100, since 150-(-100)=250, which matches the house-to-flower-shop distance.

Takeaway

Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Fix the directions
Place the flower shop
Measure house to flower shop
Place the bus stop

Where next?

Level upsuggested

Opposite signs turn |b|=2|a| into b=-2a Combine sign cases to get extreme distances

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Split by common divisors, count |b| values Let the sign decide the exponent's parity

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▶ Practice — 12 problems