Position and distance differ

What do −3 and 3 have in common?

−3 and 3 are opposite positions, each three units from zero. Distance is nonnegative.

Place a start marker at zero and flags three units left and right. Their positions are −3 and 3. The positions differ, but each is the same number of intervals from the start.

Numbers at equal distances on opposite sides of zero are opposites. The opposite of −3 is 3, and the opposite of 3 is −3. The opposite of zero is zero.

Counting the intervals from zero gives distance without direction. The distance from −3 to zero is 3, not −3. Distance is always at least zero.

The distance of a number from zero is called its absolute value: |−3|=3 and |3|=3. The two vertical bars mark absolute value. You may first read it as “three units from zero”.

−2 is greater than −5, but their distances from zero are 2 and 5. Thus −5 is farther away. Read whether the question asks about order or distance.

Naming position and distance together prepares you to read directions in coordinates. This does not yet cover all distances between two points or distances in a plane. Next, move along one line.

Listen one paragraph at a time

1 / 6

2 / 6

3 / 6

4 / 6

5 / 6

6 / 6

Hold on to this distinction

Position −4 includes a side; distance 4 counts only intervals. Do not copy a position’s sign into a distance answer.

Change it and check

Change the position and read its distance and opposite. Alternate between −3 and 3 to check equal distances.

Position A

Number line with one-unit intervals. A: −2, B: 2. Values increase to the right. On narrow screens, scroll the diagram horizontally.

Position A −2, opposite B 2. Distance from zero: |−2| = 2.

Distances here run from zero to one point on a line. No formula for distance in a plane is used.

Try with paper and objects

Fold a paper number line at zero so −2 and 2 overlap. Count the intervals from each to zero.

Check and explain another example

Check your idea 1

What is the opposite of −4?

Check your idea 2

How far is position −5 from zero?

Apply it in a new setting

Two flags are two units left and right of zero. They have…

Write an answer yourself

Return to the explanation

Compare with a solution

2 · number only

The position is −2, but it is two units from zero, so its distance is 2.

Does your explanation include these ideas?

Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.

Find a pair of opposites.Name position and distance separately.Distinguish greater from farther.

Continue, or pause here

Distinguish start, direction, number of steps and endpoint.