B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Positive and Negative Signs: Fractions, Decimals, Ratios and Rates

B3.3 A1About 65 minutes

Learning goals

I can explain why −a/b, (−a)/b and a/(−b) are equal, and why (−a)/(−b) is positive.
I can compare and order negative fractions and decimals.
I can find a rate of change and explain what a positive or negative rate means in context.
I can predict the sign of a product or quotient of rational numbers and explain my reasoning.

Words to know

rational number
A number that can be written as a/b, where a and b are integers and b ≠ 0. It can be positive, negative or zero.

rate
A comparison of two quantities with different units, like km/h or °C per hour.

rate of change
Change in one quantity ÷ change in another, such as (change in temperature) ÷ (change in time).

ratio
A comparison of two quantities, like 3 : 4 or 3/4.

opposite
The number with the same size and the other sign: the opposite of −2/3 is 2/3.

Remember from Grade 8

In Grade 8 you learned the sign rules for multiplying and dividing integers: same signs give a positive answer, different signs give a negative answer. In Grade 9 you will apply them to fractions, decimals, ratios and rates, and explain what the sign means.

Where does the negative sign go?

A fraction is a division: −3/4 means −(3 ÷ 4). Look at three ways to write it:

(−3) ÷ 4 = −0.75
3 ÷ (−4) = −0.75
−(3 ÷ 4) = −0.75

So −3/4 = (−3)/4 = 3/(−4). One negative sign makes the whole fraction negative, wherever it sits. But (−3)/(−4) = 3/4, because a negative divided by a negative is positive. We usually write the sign in front: −3/4.

Sign rules for rational numbers

Two negatives (in a product or quotient) make a positive: (−0.6)(−0.5) = 0.3 and (−2/3) ÷ (−4) = 1/6.
One negative makes a negative: (−1.2)(3) = −3.6.
Count the negatives in a long product: an even number gives a positive, an odd number gives a negative.

Comparing negative numbers

On the number line, numbers get smaller as you move left. For negative numbers, the one that is farther from zero is smaller. Since 3/4 > 2/3, it follows that −3/4 < −2/3. Think of temperature: −10 °C is colder than −5 °C.

Negative fractions and decimals between −1 and 0. In order from least to greatest: −3/4 (−0.75), −0.7, −2/3 (about −0.667), −5/8 (−0.625). The one closest to zero is the greatest.

Worked example

Example 1: Equal or not?

Which are equal to −2/5? a) 2/(−5) b) (−2)/(−5) c) −0.4 d) −(−2/5)

a) One negative sign: 2 ÷ (−5) = −0.4 = −2/5. Equal.
b) Two negatives: (−2) ÷ (−5) = +2/5. Not equal.
c) −2/5 = −(2 ÷ 5) = −0.4. Equal.
d) The opposite of −2/5 is +2/5. Not equal.

a and c

Worked example

Example 2: Order negative rational numbers

Order from least to greatest: −3/4, −2/3, −0.7, −5/8.

Write each as a decimal: −0.75, −0.666…, −0.7, −0.625.
For negatives, the larger size is the smaller number: 0.75 > 0.7 > 0.666… > 0.625.
So −0.75 < −0.7 < −0.666… < −0.625.

−3/4, −0.7, −2/3, −5/8

Signs in rates of change

rate of change = (change in quantity) ÷ (change in time)

A positive rate means the quantity is increasing.
A negative rate means it is decreasing.
The size tells you how fast.

The same idea gives the slope of a line in the Linear and Non-Linear Relations unit: slope = (change in y) ÷ (change in x).

The temperature falls from 3 °C at noon to −7 °C at 4 p.m. The rate of change is −10 °C ÷ 4 h = −2.5 °C per hour. The negative sign means the line goes down from left to right. The dashed part assumes the same rate continues: −12 °C at 6 p.m.

Worked example

Example 3: Temperature rate

At noon it is 3 °C. At 4 p.m. it is −7 °C. Find the rate of change. If the rate stays the same, predict the temperature at 6 p.m.

Change in temperature: −7 − 3 = −10 °C. Change in time: 4 h.
Rate = −10 ÷ 4 = −2.5 °C/h. Negative: it is getting colder by 2.5 °C each hour.
From 4 p.m. to 6 p.m. is 2 h: 2 × (−2.5) = −5. So −7 + (−5) = −12 °C.

−2.5 °C/h; about −12 °C at 6 p.m.

Worked example

Example 4: A descending drone

A drone's height goes from 120 m to 45 m in 15 s. Find its rate of change in height. Then find its height after another 6 s at the same rate.

Change in height: 45 − 120 = −75 m.
Rate: −75 ÷ 15 = −5 m/s. It is descending 5 m every second.
After 6 more seconds: 45 + 6 × (−5) = 45 − 30 = 15 m.

−5 m/s; 15 m

Worked example

Example 5: Ratio of changes (slope)

A line passes through (−2, 5) and (2, −1). Find the ratio (change in y) : (change in x) as a fraction in lowest terms.

Change in y: −1 − 5 = −6. Change in x: 2 − (−2) = 4.
Ratio = −6/4 = −3/2.
Check by going the other way: (5 − (−1)) / (−2 − 2) = 6/(−4) = −3/2. Same answer.

−3/2: for every 2 units right, the line goes 3 units down.

Hands-on activity

Freezer cool-down

You need: a plastic cup of water, a kitchen or outdoor thermometer, a freezer (or a cold winter balcony), a timer.

Measure the starting temperature of the water.
Put it in the freezer. Measure every 10 minutes for 40 to 60 minutes. (Open the door quickly so the freezer stays cold. Use a plastic cup, not glass: glass can crack if the water starts to freeze.)
Find the rate of change for each 10-minute interval, in °C per minute. All should be negative. Are they the same?
Find the overall rate from start to finish.

Model answer: 20 °C to 8 °C in 40 min gives (8 − 20) ÷ 40 = −12 ÷ 40 = −0.3 °C/min. You will probably notice the water cools faster at the start than near 0 °C, so the rate is not constant.

Just for fun

Why did −3 and −4 stay friends after they divided?

Because together, they always came out positive.

Your turn

Write −7/8 in two other equivalent ways.

Is (−4)/(−9) positive or negative? Explain.

Which is greater: −5/6 or −4/5?

Calculate (−0.8)(−1.5) and (−2.4) ÷ 0.6.

A tank's volume changes from 80 L to 56 L in 8 min. Find the rate of change.

Without calculating, is (−1/2)(−2/3)(−3/4) positive or negative?

Find the ratio of change in y to change in x from (1, 4) to (7, 1).

More practice: warm-up

Write −0.25 as a fraction.

Is 3/(−5) positive or negative?

Calculate (−3/4)(4).

Which is less: −1/2 or −1/3?

A rate of −4 °C/h means the temperature is…

More practice: on your own

Order from least to greatest: −1.2, −6/5, −5/4, −1 1/8.

Calculate (−2/3) ÷ (−4/9).

A stock price changed by −$0.45 per day for 6 days. What was the total change?

A hot-air balloon descends from 850 m to 610 m in 3 min. Find the rate of change of height in m/min.

Explain why −(−3/7) is positive.

Challenge

For what values of x is x/(−3) positive?

A snowbank is 84 cm deep on Monday. It melts at a rate of −3.5 cm/day, but on Thursday night 12 cm of new snow falls. How deep is it at the end of Saturday (6 days of melting, Mon–Sat)?

The ratio a/b is negative and the ratio b/c is negative. Is a/c positive or negative? Explain.

Check your understanding

Quiz

Which is not equal to −5/9?

Which is the greatest?

What is (−1.5)(−0.4)?

The temperature goes from −2 °C to −14 °C in 4 h. What is the rate of change?

A product has four negative factors. Its sign is:

A negative rate of change means:

For parents and teachers

Big idea: Expectation B3.3 asks students to apply their understanding of integers to explain how positive and negative signs affect the values of ratios, rates, fractions and decimals. Rates of change with signs lead directly to slope in the Linear and Non-Linear Relations unit.

Common difficulties:

Ordering negatives like positives (thinking −3/4 > −2/3 because 3/4 > 2/3).
Thinking (−3)/(−4) is negative because it "has negatives".
Mixing up the sign of a quantity and the sign of its rate. A drone at 45 m (positive height) can have a negative rate.
Subtracting in different orders for the top and bottom of a rate. Always do final − initial for both.

How to help: Ask your teen to describe a rate they see in daily life (gas used per km, temperature dropping, a phone battery losing charge) and to say whether it is positive or negative and why.