What Integer Products and Quotients Mean in Real Life | Grade 7 Math | LevelKite

A positive or negative number in a real situation tells you two things: how much and which way. A temperature change of $-3$ degrees is a drop of 3 degrees. A temperature change of $3$ degrees is a rise of 3 degrees.

You can already multiply and divide integers (the counting numbers, their opposites and zero, such as $7$ , $-7$ and $0$ ). In this lesson you will say what the answer means. You will take a product (the result of multiplying) or a quotient (the result of dividing) and turn it into a clear sentence about the situation. For example, you will use a product to find where a sinking balloon was 6 minutes ago.

A sign has a meaning only after you make two choices. First choose a reference: the starting point that counts as zero, such as right now or no change. Then choose which direction is positive, so that the opposite direction is negative. Name the unit too, such as meters or dollars.

The table shows the choices this lesson uses.

Choosing what positive and negative mean

Quantity and unit	Zero means	Positive means	Negative means
Change in height, in meters	no change in height	up	down
Change in temperature, in degrees Celsius	no change in temperature	warmer	colder
Change in an account balance, in dollars	no change in the balance	balance goes up	balance goes down
Time, in minutes, hours or weeks	right now, or the starting moment	later	earlier

Many real situations have a rate: how much one quantity changes for each 1 unit of another quantity, such as meters per minute or dollars per week.

Let $t$ be the time and $c$ the total change, both measured from the reference, and let $r$ be the rate. When the rate stays the same the whole time, every unit of time brings the same change. So the product of the time and the rate is the total change: $c = t \times r$ .

Division undoes that multiplication when the divisor (the number you divide by) is not $0$ :

$r = c \div t$ when $t \neq 0$ : the total change divided by the time gives the rate.
$t = c \div r$ when $r \neq 0$ : the total change divided by the rate gives the time.

Division by $0$ is undefined, which means it has no answer. Check that each divisor is not $0$ before you divide.

Key concept

The sign gives the direction, the size gives how much

Read a product or quotient of integers from a real situation in two parts.

Its sign gives the direction, using the meanings you chose for positive and negative.
Its absolute value (its distance from $0$ ) gives the size, in the unit of the answer.

To find the sign when none of the numbers is $0$ : the result is negative exactly when the count of negative numbers being multiplied or divided is odd, and positive otherwise. If any factor (a number being multiplied) is $0$ , the product is $0$ . Zero divided by a nonzero number (a number that is not $0$ ) is $0$ . Division by $0$ is undefined, so every divisor must be nonzero.

Use the same four moves every time.

Name each quantity, its unit, and what positive and negative mean.
Write the multiplication or division with the signed numbers.
Compute the product or quotient. Before you divide, check that the divisor is not $0$ .
Say the answer in a sentence with a direction, a size and a unit. Pair a direction word with the absolute value, as in 24 meters lower. The word lower already says what the negative sign says.

Worked example

A sinking balloon: looking ahead

A hot-air balloon is sinking at a steady rate of 4 meters each minute, and it will keep sinking at that rate. Compared with its height right now, where will it be 6 minutes from now?

Name the quantities and choose the signs. Measure the change in height from where the balloon is now, with up as positive. Measure time from now, with later as positive. Let r be the rate in meters per minute, t the time in minutes and c the change in height in meters.
Give each known quantity its sign. Sinking is moving down, so r is negative 4. Six minutes from now is later, so t is positive 6.
$r = -4 \text{ meters per minute}, \quad t = 6 \text{ minutes}$
Write the product. The rate stays the same, so each of the 6 minutes brings another change of negative 4 meters. That makes the total change equal to the time multiplied by the rate.
$c = t \times r = 6 \times (-4)$
Compute. Neither factor is zero. Exactly one factor is negative, and 1 is an odd count, so the product is negative. Its size is 6 times 4, which is 24.
$6 \times (-4) = -24$
Say what it means. The sign is negative, and negative means down. The absolute value is 24. The unit is meters, because a number of minutes multiplied by meters per minute gives meters.
$c = -24 \text{ meters}$

Answer

Six minutes from now, the balloon will be 24 meters lower than it is right now.

Worked example

A sinking balloon: looking back

The same balloon has been sinking at a steady rate of 4 meters each minute for the last 10 minutes. Compared with its height right now, where was it 6 minutes ago?

Name the quantities with the same choices as before. The rate r is still negative 4 meters per minute. Six minutes ago is earlier than now, so the time t is negative 6 minutes.
$r = -4 \text{ meters per minute}, \quad t = -6 \text{ minutes}$
Write the product. The rate was the same during those 6 minutes, so the time multiplied by the rate still gives c, the balloon's height at that time compared with now.
$c = t \times r = (-6) \times (-4)$
Compute. Neither factor is zero. Two factors are negative, and 2 is an even count, so the product is positive. Its size is 6 times 4, which is 24.
$(-6) \times (-4) = 24$
Say what it means. Positive means up, so the balloon was 24 meters above where it is now. This fits the story: the balloon loses 24 meters of height in 6 minutes of sinking, so 6 minutes ago it was 24 meters higher.
$c = 24 \text{ meters}$

Answer

Six minutes ago, the balloon was 24 meters higher than it is right now.

Worked example

Club fees: a weekly change, then a number of weeks

A robotics club pays the same fee every week. Over 6 weeks, the fees change the club's account balance by $-90$ dollars. What is the change in the balance each week? At that weekly change, how many weeks of fees make a total change of $-150$ dollars?

Name the quantities. A change in the balance is measured in dollars: positive means the balance goes up and negative means it goes down. The number of weeks is counted forward from the first fee, so it is positive. The weekly change is a rate in dollars per week.
$\text{total change} = -90 \text{ dollars}, \quad \text{number of weeks} = 6$
Find the weekly change. The change is the same every week, so the number of weeks times the weekly change equals the total change. That means the total change divided by the number of weeks is the weekly change. The divisor 6 is not zero. Neither number is zero and exactly one is negative, an odd count, so the quotient is negative. Its size is 90 divided by 6, which is 15.
$\text{weekly change} = -90 \div 6 = -15$
Say what it means. Negative means the balance goes down, the absolute value is 15 and the unit is dollars per week. Check by multiplying the quotient by the nonzero divisor: negative 15 dollars per week for 6 weeks gives negative 90 dollars, the total change you started with.
$(-15) \times 6 = -90$
Find the number of weeks. Now the total change is negative 150 dollars. The number of weeks times the weekly change still equals the total change, so divide the total change by the weekly change. The divisor negative 15 is not zero. Neither number is zero and two are negative, an even count, so the quotient is positive. Its size is 150 divided by 15, which is 10.
$\text{number of weeks} = -150 \div (-15) = 10$
Say what it means. The quotient is a positive count: 10 weeks of fees. That fits, because the total change and the weekly change both point down. Check by multiplying the quotient by the nonzero divisor: 10 weeks at negative 15 dollars per week gives negative 150 dollars.
$10 \times (-15) = -150$

Answer

The balance changes by $-15$ dollars each week, which is a drop of 15 dollars. It takes 10 weeks of fees to make a total change of $-150$ dollars.

Zero has a meaning in these situations too.

$0 \times (-4) = 0$ : in 0 minutes, the balloon's height does not change. Any factor of $0$ gives a product of $0$ .
$0 \div 6 = 0$ : if a balance changes by 0 dollars in total over 6 weeks, with the same change each week, then the change each week is 0 dollars. Zero divided by a nonzero number is $0$ .
$-90 \div 0$ is undefined: you cannot spread a change of $-90$ dollars over 0 weeks. No number multiplied by $0$ gives $-90$ , because any number times $0$ is $0$ .

So a divisor is never $0$ .

Takeaway: In a real situation, the sign and the size of a product or quotient of integers both carry meaning. First choose what positive and negative mean. Then let the sign of your answer give the direction and its absolute value give the size, and say the result in a sentence with its unit.

When none of the numbers is $0$ , the answer is negative exactly when the count of negative numbers is odd. A factor of $0$ gives a product of $0$ , zero divided by a nonzero number is $0$ , and a divisor is never $0$ .

Next, you will explain why the rules for multiplying signed numbers work, and you will find and fix errors in multiplying and dividing integers.