# 1.05 Practical problems with integers

Lesson

There are many situations that require us to compare two or more amounts where the second amount can be above or below the first. The integers are the perfect tool to use in these cases.

For example, if we are on a road trip through the United States we might be interested in our altitude at different parts of our journey. Altitude is a quantity that tells us how far we are from sea level, and in which direction. Let's say we visit Seeley, California, which is about $15$15 m below sea level. Then our altitude at this location would be $-15$15 m. Later we might hike up to the peak of Red Butte, which is about $2228$2228 m above sea level. On the summit, our altitude would be $2228$2228 m.

Altitude can be a positive or negative number, just like an integer.

In both cases, the vertical distance from sea level is given by the magnitude of the altitude (the size of the integer), and the direction from sea level is given by the sign of the altitude (whether the integer is positive or negative).

### Setting up a number line

In general, there are three things we need to keep in mind when applying a number line to a particular real-world situation:

1. What point in the real world shall be represented by the integer $0$0 on the number line?
2. What direction in the real world shall be represented by the positive direction on the number line?
3. What length in the real world shall be represented by $1$1 unit on the number line?

Once we have covered these three things, we can use our knowledge of addition and subtraction on the number line to describe how the real world quantities change.

The key features of a number line.

We are free to choose any point we like as the number $0$0, but there are some common choices for certain situations. Here are some physical points that are often chosen:

• Ground level
• Sea level
• A starting location in space
• A starting event in time

#### Practice questions

##### Question 3

The image below shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line.

1. What integer represents $3$3 m above the surface?

2. What integer represents $4$4 m below the surface?

3. If Nadia is initially $2$2 m above the surface, and descends $6$6 m in the elevator, what integer represents her end point?

4. If Nadia is at a location represented by the integer $-4$4, and ascends $3$3 m, which option describes her new location?

$7$7 m below the surface

A

$1$1 m below the surface

B

$3$3 m above the surface

C

$7$7 m above the surface

D

$7$7 m below the surface

A

$1$1 m below the surface

B

$3$3 m above the surface

C

$7$7 m above the surface

D

##### Question 4

Tara is waiting for the next flight to Los Angeles, which was scheduled to be in $36$36 minutes, but there is a $48$48 minute delay. She takes a nap, and wakes up $24$24 minutes later. How much longer does Tara have to wait before the plane departs?

##### Question 5

Luigi enters an elevator at the $7$7th floor (the ground floor being floor $0$0). The elevator goes down $3$3 floors, then up $9$9 floors and finally it goes down $2$2 floors, where Luigi gets out.

1. On which floor does Luigi end up?

2. When Luigi gets off the elevator, how many floors from his starting point is he?

### Outcomes

#### 7.NS.3

Solve real-world and mathematical problems involving the four operations with rational numbers. (Note: computations with rational numbers extend the rules for manipulating fractions to complex fractions.)