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Pre-Algebra View details

4. Integer Numbers

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Pre-Algebra /

Chapter 1

Integer Numbers

This lesson provides a comprehensive guide to understanding integer numbers, focusing on key concepts like absolute value, additive inverse, and the number line. These integers are not just positive numbers; they include negatives and zero as well. The lesson explains how to use the number line to visualize and compare these numbers. Absolute value helps in understanding the 'distance' a number is from zero, while additive inverse is about finding a number that, when added to a given number, results in zero. These concepts have practical applications in various fields, from accounting to engineering, and are fundamental in developing strong mathematical skills.

Student Learning Objectives:
Identify integer numbers on a number line Understand absolute value Compare positive and negative numbers

	13 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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Integer Numbers

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Players typically lose and win points based on their actions in video games. The points won can be represented by natural and whole numbers but not the points lost. Integers, however, can represent these losses. This lesson presents attributes and uses of integer numbers used in such situations.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Natural Numbers
Whole Numbers
Number Line

Explore

Measuring Temperatures

Play with the thermometer below to measure different temperatures.

How can temperatures below zero be expressed?

Discussion

Integers

Integers are whole numbers and negative numbers without fractions or decimal parts. An integer can be positive, negative or zero. The set of all integers is denoted by Z.

Z={...- 2, - 1, 0,1,2...}

The dots indicate that the set continues to increase or decrease by one unit at a time indefinitely.

Discussion

Graphing Integer Numbers

Similarly to natural and whole numbers, integer numbers can be displayed on a number line. This process helps to compare numbers and visualize which are greater and smaller.

Concept

Number Line

A number line can display integer numbers by extending the line to the left- and right-hand sides of 0. The positive integers, or natural numbers, are to the right of 0, and the negative integers are to the left of 0.

Recall the thermometer from earlier. A thermometer is also a type of number line. It can be extended down to measure negative temperatures.

Pop Quiz

Integer Numbers on a Number Line

Determine the integer represented by the point on the number line.

Random Generator of Integer Numbers Numbers

Example

The Nature Scavenger Hunt Adventure

Maya is taking part in a nature scavenger hunt at summer camp. In this game, players are divided into teams; each has a map, a list of items to find, and tasks to complete. There are many ways for teams to either earn or lose points.

Action or Task	Points
Find a hidden item (I)	2
Cheating (C)	-7
Complete a physical challenge (P)	4
Solve a puzzle (S)	7
Take a wrong turn (W)	-3
Run out of time before finishing a task (T)	-5

The team that gets the most points will be the winner. Maya wants a better idea of the meaning of these points to help her team win. She thinks that plotting them on a number line should help. Which of the following graphs matches Maya's?

Hint

Check whether the points given by each action or task are negative or positive. When the number is positive, count that number of units going to the right of 0 on the number line. If the number of points is negative, count that number of units to the left of 0.

Solution

The points assigned to each task or action are integers. Let's mark them on a number line. For example, finding a hidden item (I) earns a team 2 points. We will mark it on the number line by moving two units to the right of 0 to display the points earned.

Next, -7 points are assigned to cheating (C), so the team loses 7 points if they cheat. We should move seven units to the left of 0 to show the points given by this action.

Let's follow a similar method to graph the remaining points.

The final graph matches option B. Now that Maya has a better understanding of the game, she can help her team win this wonderful adventure.

Example

Recording Temperatures

Maya measures the temperatures of various things around her summer camp. She created a table with the temperatures she recorded.

Temperatures Measured
-1^(∘) C
9 ^(∘) C
-4^(∘) C
3 ^(∘) C
12^(∘) C

She now wants to know the coldest and warmest temperature she recorded. Help Maya order these temperatures from the coldest to the warmest on a number line.

Hint

On a number line, the smaller numbers are further left and the greater ones are further right. Plot one temperature at a time as a point on a number line. Negative temperatures will be on the left side of 0, and positive temperatures will be on the right side of 0.

Solution

We will use a number line to compare the temperatures. The first recorded temperature is -1^(∘) C. Let's move one unit to the left of 0 and draw a point to plot this temperature.

The second temperature recorded is 9^(∘) C. We plot this by moving 9 units to the right of 0.

Now let's plot the remaining temperatures.

On the final number line, the farthest point to the left is the coldest temperature and the point farthest point to the right is the warmest.

Discussion

Opposite Numbers

Consider the lucky number -7. Or is it lucky number 7? In any case, let's see what happens when 7 is added to -7. -7+7=0 Notice that 7 cancels out -7. This is because 7 is the additive inverse of -7. This property of integer numbers is worth exploring deeper!

Concept

Additive Inverse

The additive inverse of a number is another number such that their sum equals 0. If y is the additive inverse of x, then the following equation holds true.

x+y=0

Given a number, its additive inverse — also called its opposite number — can be found by changing the sign. Some examples of additive inverses are listed in the table.

Number	Additive Inverse	Sum
5	- 5	5+(- 5)=0
- 21	21	- 21+21=0
a	- a	a+(- a)=0
- b	b	- b+b=0

Pop Quiz

Finding the Additive Inverse of Integer Numbers

Find the additive inverse of the indicated number.

An applet asking for the additive inverse of random integer numbers.

Discussion

Comparing Distances Written as Integer Numbers

We often compare quantities that express distances. Knowing which number represents a farther distance is simple if the numbers are positive. What if they are negative? This is where the absolute value comes to the rescue.

Concept

Absolute Value

The absolute value of a number a is the distance between a and 0 on the number line. It is denoted as |a| and is always a non-negative value.

The absolute value of a negative number is its opposite value, while the absolute value of a positive number is equal to itself.

|a| = a &if a≥ 0 - a &if a < 0

For any integer number a, these two properties hold true.

Property	Algebraic Representation	Example
Non-negativity	\|a\| ≥ 0	\|7\| = 7 and 7≥0
Symmetry	\| - a\| = \|a\|	\|-7\| = 7 and \|7\| = 7 \|-7\| = \|7\|

The non-negative property says that the absolute value of any integer must always be greater than or equal than 0. The symmetry property means that the absolute value of a number and the absolute value of its opposite are the same.

Pop Quiz

Comparison of Numbers

Determine whether the given statement is true or false. Some statements involve absolute values.

Example

Measuring Elevation

At a particular animal rehabilitation center, the animals have space to roam. Maya takes photos of the animals and notes their elevation compared to her standing position.

Animal	Elevation (ft)
Kingfisher	10
River otter	-8
Porcupine	7
Sturgeon	-12
Painted turtle	-3

Use Maya's notes to identify which animal is farthest from her standing position in elevation.

What is the animal closest to Maya's standing position?

Hint

Use the absolute values to order the elevations of the animals from Maya's position.

Solution

The elevations of the animals contain both positive and negative integers. We will use the absolute value of the elevations to get the distances the animals are from Maya.

Animal	Elevation (ft)	Distance from Maya (ft)
Kingfisher	10	\|10\| = 10
River otter	-8	\|-8\| = 8
Porcupine	7	\|7\| = 7
Sturgeon	-12	\|12\| = 12
Painted turtle	-3	\|-3\| = 3

Let's put the distances on a vertical number line. The farthest distance is on top, and the closest is on the bottom.

The graph shows that the sturgeon is the farthest from Maya, even though it has a negative elevation! The painted turtle is the closest to her.

Closure

The Bear Sculpture

Maya has $20 in her camp store account. She wants to buy a custom bear sculpture that costs $30. After the purchase, she checks her balance.

Maya's balance is -$10. What does this amount mean? A negative balance in an account usually means that money is owed. The amount owed is the absolute value of the negative balance. Maya's Debt: |-10|=$10 Maya owes $10.

Balance	Meaning
Negative	Debt or money owed
0	No debt and no credit
Positive	Credit

After Maya adds $10 to her account, the balance is 0. This means the sculpture is fully paid for and no money is owed.

Level 1

Level 2

Level 3

Integer Numbers

Exercise 3.1

Point P is on a number line midway between -14 and 4. Point Q is midway between point P and 1. What is the integer value that represents point Q?

Let's begin by drawing two points on a number line to represent -14 and 4.

Next, note that the number of units between -14 and 4 is 18.

Now, consider that half of 18 is 9. This means that point P is 9 units to the right of -14 and 9 units to the left of 4. If we add 9 to -14 or subtract 9 from 4, we will get the number -5.

Therefore, the integer that represents point P is -5.

Now, we know that point Q is midway between point P and 1. Note that there are 6 units between -5 and 1. Since the half of 6 equals 3, point Q is located three units to the right of P and three units to the left of 1.

If we subtract 3 from 1 or add 3 to -5, we will get -2. This is the integer value that represents point Q.

Integer Numbers

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