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7.03 Solving contextual problems with integers

Lesson

Introduction

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 Australia 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.

A vertical number line showing altitudes of negative 15 metres and 2228 metres. Ask your teacher for more information.

Let's say we visit Lake Eyre, which is about 15\text{ m} below sea level. Then our altitude at this location would be -15\text{ m.} Later we might hike up to the peak of Mt Kosciuszko, which is about 2228\text{ m} above sea level. On the summit, our altitude would be 2228\text{ m.}

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).

Set up of 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 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 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.

A number line showing one tick labeled as 1 unit and an arrow going right labeled as positive direction.

The point that we identify with the integer 0 is called the datum.

We are free to choose any point we like as the number 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
  • An account balance of \$0
  • 0\degree \text{C}, the freezing point of water
PositiveNegative
RightLeft
UpDown
AboveBelow
NorthSouth
EastWest
AfterBefore
CreditDebit
HotterColder

With the zero selected, we can then orient the number line by choosing a positive direction. Again, we are free to work with any orientation we like, but quite often the words we use to describe the physical situation suggest the most sensible orientation. Here are some physical directions that are common in situations with integers.

Finally, we need to relate the magnitude of a real world quantity with the distance between each integer on the number line. We can think of this as scaling the number line to suit the physical problem. In our earlier example, we were measuring altitude in metres, so in that case, it would make sense to let 1 unit on the number line represent 1\text{ m} of altitude in the real world. Here are some quantities we could identify with 1 unit on the number line:

  • Length - cm, m, km, etc.
  • Time - seconds, minutes, hours, days, years, etc.
  • Temperature - degrees Celsius (or Centigrade), degrees Fahrenheit, Kelvin, etc.
  • Money - cents, dollars, etc.

Examples

Example 1

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

An image showing a miner's location on the right, and a number line on the left. Ask your teacher for more information.
a

What integer represents 3\text{ m} above the surface?

Worked Solution
Create a strategy

Find the height 3\text{ m} above the surface using the vertical scale on the left of the image.

Apply the idea
\displaystyle \text{Integer}\displaystyle =\displaystyle 3
b

What integer represents 4\text{ m} below the surface?

Worked Solution
Create a strategy

Find the depth 4\text{ m} below the surface using the vertical scale on the left of the image.

Apply the idea
\displaystyle \text{Integer}\displaystyle =\displaystyle -4
Idea summary

In setting up the number line, note these when applying a number line to a particular real world situation:

  • The point in the real world shall be represented by the integer 0 on the number line, such as:

    • Ground level

    • Sea level

    • A starting location in space

    • A starting event in time

    • An account balance of \$0

    • 0\degreeC, the freezing point of water

  • The direction in the real world shall be represented by the positive direction on the number line, such as:

    • Positive: Right, Up, Above, North, East, After, Credit, Hotter

    • Negative: Left, Down, Below, South, West, Before, Debit, Colder

  • The length in the real world shall be represented by 1 unit on the number line, such as:

    • Length - cm, m, km, etc.

    • Time - seconds, minutes, hours, days, years, etc.

    • Temperature - degrees Celsius (or Centigrade), degrees Fahrenheit, Kelvin, etc.

    • Money - cents, dollars, etc.

Movement on the number line

After setting up a number line, we can talk about changes in the quantity we are representing using integer arithmetic. Given a starting temperature, and some change in a certain direction, what is the final temperature? Given a starting balance and an ending balance of money in an account, what has been the magnitude and sign of the change?

Answers that are integers can be positive or negative:

Question: What is the balance of your account?

Answer: \text{Balance }=-\$31

When describing this situation in words it is more natural to combine a positive number with a directional word:

Question: How much do you owe the bank?

Answer: I owe the bank \$31.

Often we will start to think about a problem using directional words like "above/below" or "east/west" or "credit/debit". Then we will set up a number line and use integers to make calculations. It is tempting to combine the integer that is the result of this calculation with the directional word we first used to describe the situation, but this can lead to confusion.

Examples

Example 2

Tara is waiting for the next flight to Los Angeles, which was scheduled to be in 64 minutes, but there is a 34 minute delay. She takes a nap, and wakes up 23 minutes later. How much longer does Tara have to wait before the plane departs?

Worked Solution
Create a strategy

Add the delay and subtract her sleep time from her wait time.

Apply the idea
\displaystyle \text{Waiting time}\displaystyle =\displaystyle 64 + 34 - 23Set up the equation
\displaystyle \text{ }\displaystyle =\displaystyle 98 - 23Perform 64 + 34
\displaystyle \text{ }\displaystyle =\displaystyle 75 \text{ minutes}Subtract the values

Example 3

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

a

On which floor does Luigi end up?

Worked Solution
Create a strategy

We start at floor 7. Whenever the elevator goes down, we subtract the number. Whenever the elevator goes up, we add the number.

Apply the idea
\displaystyle \text{Floor}\displaystyle =\displaystyle 7 - 3 + 9 - 2Write the values
\displaystyle \text{ }\displaystyle =\displaystyle 4 + 9 - 2Perform 7 - 3
\displaystyle \text{ }\displaystyle =\displaystyle 13 - 2Perform 4 + 9
\displaystyle \text{ }\displaystyle =\displaystyle 11Evaluate
b

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

Worked Solution
Create a strategy

Subtract the starting point from the end point.

Apply the idea

From the previous problem, Luigi starts on floor 7, and ends up on floor 11.

\displaystyle \text{Number of floors}\displaystyle =\displaystyle 11 - 7Substitute the values
\displaystyle \text{ }\displaystyle =\displaystyle 4Evaluate
Idea summary

Answers that are integers can be positive or negative. The sign of the integer determines the location of a thing or person, or whether we have a profit or loss, or savings or debt.

Outcomes

MA4-4NA

compares, orders and calculates with integers, applying a range of strategies to aid computation

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