Real-world interpretations of negative integers

We can work with negative integers more easily if we can interpret them using meaningful real-world phenomena, such as temperatures below zero, building floors below ground, or amounts of money owed. For example, –4 could represent a fridge temperature of –4ºC, a parking level 4 floors below ground, or a $4 debt. Some other ideas for negative amounts that lend themselves to story problems are:

• Hot-air balloons with puffs of gas (positive) and sand bags (negative). Adding a puff of gas or taking away a sand bag makes the balloon go up by 1 metre, while taking away a puff of gas or adding a sand bag makes the balloon go down by 1 metre.
• Magic soup with hot cubes (positive) and cold cubes (negative). Adding a hot cube or taking away a cold cube makes the soup temperature go up by 1 degree, while taking away a hot cube or adding a cold cube makes the soup temperature go down by 1 degree.
• People with happy thoughts (positive) and sad thoughts (negative). Adding a happy thought or taking away a sad thought makes the person’s happiness go up by 1 level, while taking away a happy thought or adding a sad thought makes the person’s happiness go down by 1 level.

Adding a negative integer to a positive integer

Add To Problem: Add a negative amount of an object to a positive amount of that same object. For example, add a $4 debt to a $7 credit. Recall from 3b: Adding and subtracting with negative integers that we use the subtraction algorithm to calculate A + (–B) by rewriting it as A – B (if A > B) or –(B – A) (if A < B).

Here’s an example of an “add to” problem involving a negative integer:

• Arithmetic problem: Solve 7 + (–4).
• Word problem: Keenan has $7 in his pocket but he owes his friend $4. How much money does Keenan have to spend?
• 7 + (–4) = 3, so Keenan has $3 to spend.

Subtracting a negative integer from a positive integer

Take From Problem: Taking away something negative has the same effect as adding something positive. For example, taking away a $4 debt has the same effect as adding a $4 credit. Recall from 3b: Adding and subtracting with negative integers that we use the addition algorithm to calculate A – (–B) by rewriting it as A + B.

Here’s an example of a “take from” problem involving a negative integer:

• Arithmetic problem: Solve 7 – (–4).
• Word problem: A hot-air balloon with 4 puffs of gas and 4 sand bags is balanced (neither going up nor going down). What happens to the balloon if we add 7 more puffs of gas and take away the 4 sand bags?
• 7 – (–4) = 11, so the ballon goes up by 11 metres.

Compare Problem: By comparing a positive integer with a negative integer, we can see that the difference between A and –B is the same as the sum A + B. For example:

• Arithmetic problem: Solve 7 – (–4).
• Word problem: At 12 noon it is 7ºC in Vancouver and –4ºC in Whistler. How much warmer is it in Vancouver than in Whistler?
• 7 – (–4) = 11, so it is 11ºC warmer in Vancouver than in Whistler.

The video below works through some examples of word problems for adding and subtracting with negative integers.