Fractions, Decimals and the Opposite of an Opposite | Grade 6 Math | LevelKite

You can already graph integers such as $-2$ , $0$ and $3$ . To graph a number means to mark its place on a number line. You can also find the opposite of a number: the number on the other side of $0$ that is the same number of steps away from $0$ . The opposite of $3$ is $-3$ , and zero is its own opposite.

Many numbers sit between the integers. A fraction such as $\frac{3}{4}$ and a decimal such as $-1.4$ each have exactly one place on the line. In this lesson you will find that place by counting equal steps from $0$ . Then you will use the same counting to find the opposite of the opposite of a number.

On the number lines in this lesson, not every tick has a printed number. You read the size of a step from the numbers the line does print. Pick two printed numbers that are next to each other and count the equal steps between them. Count the moves from tick to tick, not the ticks themselves. Because the steps are equal, you can divide the distance between the two numbers by that count to get the size of one step.

For example, suppose a line has 10 equal steps from $0$ to $1$ . The distance from $0$ to $1$ is 1 unit, so each step is $1 \div 10 = 0.1$ of a unit.

On the number line below, $0$ and $1$ are printed next to each other. Three ticks sit between them, so it takes 4 equal steps to go from $0$ to $1$ . Each step is $1 \div 4 = \frac{1}{4}$ of a unit.

Now a fraction has a clear meaning on this line. In $\frac{3}{4}$ , the $4$ says that each step is one of 4 equal parts of a unit. The $3$ says to count 3 of those steps. A decimal works the same way. On a line where each step is $0.1$ of a unit, $0.7$ is 7 steps from $0$ .

The sign gives the direction. Every number line in this lesson is horizontal and its numbers grow to the right, so positive numbers are to the right of $0$ and negative numbers are to the left of $0$ . Every number you graph in this lesson lands exactly on a tick, so you can always reach it by counting whole steps from $0$ .

There are 4 equal steps from

0

to

1

, so each step is

\frac{1}{4}

of a unit. Point A is 3 steps right of

0

. Point B is 5 steps left of

0

.

Worked example

Graph two fractions on a line marked in fourths

On the number line above, the printed numbers are $-2$ , $-1$ , $0$ , $1$ and $2$ , with 4 equal steps between printed numbers that are next to each other. Graph $\frac{3}{4}$ and $-1\frac{1}{4}$ , and say which marked point is the place for each one.

Find the size of one step. From 0 to 1 is 1 unit, and there are 4 equal steps in it. Divide 1 unit by 4. Each step is one fourth of a unit.
$1 \div 4 = \frac{1}{4}$
Graph the first number. Three fourths is 3 steps of one fourth. It is positive, and numbers grow to the right on this line, so count 3 steps to the right of 0. Point A marks that tick.
$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}$
Write the second number as a count of fourths. One whole unit is 4 fourths, so 1 unit and 1 more fourth is 5 fourths. This changes only how you write the number. Its size stays the same.
$1\frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}$
Graph the second number. It is negative, so count 5 steps to the left of 0. The first 4 steps reach -1. The 5th step lands on the next tick to the left of -1. Point B marks that tick.
$-\frac{1}{4},\; -\frac{2}{4},\; -\frac{3}{4},\; -1,\; -1\frac{1}{4}$
Check each place against the printed numbers. Point A is between 0 and 1, where three fourths belongs. Point B is between -2 and -1, 1 step to the left of -1, where 5 fourths to the left of 0 belongs.

Answer

Point A, 3 steps of $\frac{1}{4}$ unit to the right of $0$ , graphs $\frac{3}{4}$ . Point B, 5 steps of $\frac{1}{4}$ unit to the left of $0$ , graphs $-1\frac{1}{4}$ .

Worked example

Graph a decimal and its opposite

A number line has the printed numbers $-2$ , $-1$ , $0$ , $1$ and $2$ , with 5 equal steps between printed numbers that are next to each other. Graph $1.4$ and its opposite, $-1.4$ .

Find the size of one step. From 0 to 1 is 1 unit, and there are 5 equal steps in it, so each step is one fifth of a unit. One fifth equals two tenths, because 5 groups of two tenths make ten tenths, which is 1 unit. You write two tenths as 0.2.
$1 \div 5 = \frac{1}{5} = \frac{2}{10} = 0.2$
Count by 0.2 from 0 until you reach 1.4. It takes 7 counts, so 1.4 is 7 steps from 0.
$0.2,\; 0.4,\; 0.6,\; 0.8,\; 1.0,\; 1.2,\; 1.4$
Graph 1.4. It is positive, so count the 7 steps to the right of 0. The first 5 steps reach 1, and 2 more steps reach 1.4.
$7 \times 0.2 = 1.4$
Graph -1.4. It is the opposite of 1.4, so it is the same 7 steps from 0, on the left side. The first 5 steps reach -1, and 2 more steps to the left reach -1.4.
$-0.2,\; -0.4,\; -0.6,\; -0.8,\; -1.0,\; -1.2,\; -1.4$
Check both places against the printed numbers. The tick for 1.4 is 2 steps to the right of 1. The tick for -1.4 is 2 steps to the left of -1.

Answer

On the number line below, the marker at $1.4$ is 7 steps of $0.2$ unit to the right of $0$ . The marker at $-1.4$ is 7 steps of $0.2$ unit to the left of $0$ .

There are 5 equal steps from

0

to

1

, so each step is

0.2

of a unit. The markers at

1.4

and

-1.4

are each 7 steps from

0

.

Look again at the two markers on the number line above. The marker at $1.4$ is 7 steps to the right of $0$ . Its opposite keeps the 7 steps and switches sides, so it is the marker at $-1.4$ .

Now take the opposite a second time. Start at $-1.4$ , keep the 7 steps, and switch sides again. You land on $1.4$ , the number you started with.

Here is why. For any number that is not $0$ , taking the opposite keeps the count of steps from $0$ and only switches the side. Two switches put you back on the side where you began, with the same count of steps. That is the tick where you started, so the second opposite undoes the first.

You can write this with symbols. A $-$ sign in front of a number means the opposite of that number. So $-1.4$ is the opposite of $1.4$ , and $-(-1.4)$ is the opposite of $-1.4$ . That gives $-(-1.4) = 1.4$ .

Key concept

A side of 0 and a count of equal steps

On a number line with equal steps, two things fix the place of any number that is not $0$ : which side of $0$ it is on, and how many steps it is from $0$ . To graph a fraction or a decimal, you find both.

Taking the opposite keeps the count of steps and switches the side. So taking the opposite twice brings you back to the number you started with. Let $a$ stand for any number, positive or negative. Then the opposite of the opposite of $a$ is $a$ .

Zero fits the rule too. It is 0 steps from $0$ , so it is its own opposite, and the opposite of its opposite is still $0$ .

-(-a) = a

Worked example

The opposite of the opposite of a negative number

A number line has the printed numbers $-3$ , $-2$ , $-1$ , $0$ , $1$ , $2$ and $3$ , with 2 equal steps between printed numbers that are next to each other. Find the opposite of the opposite of $-2.5$ .

Graph the starting number. Each step is 1 unit divided by 2, which is 0.5 of a unit. Counting by 0.5 gives 0.5, 1.0, 1.5, 2.0, 2.5, so 2.5 is 5 steps from 0. The starting number is negative, so it is 5 steps to the left of 0.
$1 \div 2 = 0.5, \quad 5 \times 0.5 = 2.5$
Take the opposite once. Keep the 5 steps and switch to the right side of 0. You land on 2.5.
$-(-2.5) = 2.5$
Take the opposite again. Keep the 5 steps and switch back to the left side of 0. You land on -2.5, the tick where you started.
$-(2.5) = -2.5$
Write both moves in one line. Each minus sign outside a pair of parentheses means one use of the opposite. Work from the inside out.
$-(-(-2.5)) = -(2.5) = -2.5$
Check with the key idea. Two switches return you to the side where you started, and this number started on the left side of 0. So the result is negative. The opposite of the opposite is not always positive. It always equals the starting number.

Answer

The opposite of the opposite of $-2.5$ is $-2.5$ . The number line below shows both markers: the first opposite takes you from $-2.5$ to $2.5$ , and the second takes you back to $-2.5$ .

There are 2 equal steps from

0

to

1

, so each step is

0.5

of a unit. The markers at

-2.5

and

2.5

are each 5 steps from

0

, on opposite sides of

0

.

To graph a fraction or a decimal that lands on a tick, read the size of one step from two printed numbers that are next to each other, then count equal steps from $0$ . On a horizontal line whose numbers grow to the right, count right for a positive number and left for a negative number.

Taking the opposite keeps the count of steps and switches the side. So the opposite of the opposite of any number $a$ is $a$ itself: $-(-a) = a$ .

Next, you will work the other way around: you will name the number that a point shows, and you will use number lines to show real-world quantities.