Chapter 04 of 10

Integers & the Coordinate Plane

How do negative numbers describe direction and position?

Positive and negative numbers describe quantities on opposite sides of zero. The number line organizes rational numbers from least to greatest, absolute value measures distance from zero, and coordinate pairs locate points horizontally and vertically.

01

The big ideas

Understand these before you worry about speed.

Farther right means greater

On a number line, every number is greater than every number to its left. Among negative numbers, the value closer to zero is greater.

Example−3>−8\displaystyle -3>-8 because −3\displaystyle -3 lies to the right of −8\displaystyle -8.

Absolute value is distance

Absolute value measures a number's distance from zero, so it is never negative. Opposites have equal absolute values.

Example∣−7∣=7\displaystyle |-7|=7 and ∣7∣=7\displaystyle |7|=7.

Coordinates are ordered

In (x, y), move horizontally according to x and vertically according to y. Signs determine left or right and down or up.

Example(−4,3)\displaystyle (-4,3) is four units left and three units up from the origin.

Same-coordinate distances use absolute difference

When two points share an x- or y-coordinate, subtract the coordinates that differ and take the absolute value.

ExampleThe vertical distance from (2,−5)\displaystyle (2,-5) to (2,4)\displaystyle (2,4) is ∣4−(−5)∣=9\displaystyle |4-(-5)|=9.
Plotting points

Start at the origin. Read x first, then y.

The first coordinate controls horizontal movement. The second controls vertical movement.

  1. A
    (3, 2)

    Move right 3, then up 2.

    Quadrant I
  2. B
    (-4, 3)

    Move left 4, then up 3.

    Quadrant II
  3. C
    (-3, -2)

    Move left 3, then down 2.

    Quadrant III
  4. D
    (2, -4)

    Move right 2, then down 4.

    Quadrant IV
  5. E
    (0, 4)

    Do not move sideways; move up 4.

    On the y-axis

Notice: Point E lies on an axis, so it is not inside any quadrant.

02

Worked example

Follow the reason for each step—not just the symbols.

Problem

Order −2.5\displaystyle -2.5, 34\displaystyle \frac{3}{4}, −1\displaystyle -1, and 0.2\displaystyle 0.2 from least to greatest.

  1. Convert 34\displaystyle \frac{3}{4} to 0.75\displaystyle 0.75 so every value is easy to compare.
  2. Place the negative values first. The value farther left is smaller, so −2.5 comes before −1.
  3. Compare the positive values: 0.2<0.75\displaystyle 0.2<0.75.
Answer−2.5<−1<0.2<34\displaystyle -2.5<-1<0.2<\frac{3}{4}
03

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

1

Which is greater: −4\displaystyle -4 or −9\displaystyle -9?

Check answer−4\displaystyle -4
2

Evaluate ∣−12.6∣\displaystyle |-12.6|.

Check answer12.6\displaystyle 12.6
3

In which quadrant is (−3,−7)\displaystyle (-3,-7)?

Check answerQuadrant III
4

Find the distance between (−5,2)\displaystyle (-5,2) and (4,2)\displaystyle (4,2).

Check answer9\displaystyle 9 units
04

Practice and tools

Use the resource that matches what you need next.

Worksheet

Integers Practice

Work with signed numbers, operations, and powers.

Open resource Worksheet

Absolute Value

Interpret distance from zero and basic absolute-value statements.

Open resource Worksheet

Coordinate Plane Introduction

Plot and read points in all four quadrants.

Open resource Worksheet

Graphing Points on the Coordinate Plane

Plot and label 26 ordered pairs on large numbered grids.

Open resource