2.1 Number Lines and Coordinate Grids

Learning Objectives:

You will locate positive and negative rational numbers on a number line.
You will plot positive and negative numbers on a coordinate grid.
You will use adding and subtracting with integers.
You will demonstrate an understanding of the absolute value by using the symbol and a number line.
You will solve problems in various contexts using inverse relationships between addition and subtraction.

Introduction

The Sunken Ship

Eleni and Yuri are very excited because they are going to go on a dive to see a sunken ship. The dive is quite shallow which is unusual because most sunken ship dives are found at depths that are too deep for two junior divers. However, this one is at 40 feet, so the two divers can go to see it.

They have the following map to chart their course. Yuri wants to figure out exactly how far the boat will be from the sunken ship. Each square represents 160 feet of water.

First, he makes a note of the coordinates. Then he can use the map to calculate the distance.

We use coordinate grids like this one all the time. Use the information in this lesson to help Yuri figure out the coordinates of his boat and the sunken ship. Then you will be able to estimate the distance between them.

Guided Learning

The first step to identify absolute value and opposites of given integers is to realize that zero is neither positive nor negative and can be the starting point to find where other numbers exist on a number line.

Sometimes, when we look at an integer, we aren’t concerned with whether it is positive or negative, but we are interested in how far that number is from zero. Think about water. You might not be concerned about whether the depth of a treasure chest is positive or negative simply how far it is from the surface.

This is where absolute value comes in.

What is absolute value?

The absolute value of a number is its distance from zero on the number line.

We use symbols to represent the absolute value of a number. For example, we write the absolute value of 3 as |3| .

Writing an absolute value is very simple you just leave off the positive or negative sign and simply count the number of units that an integer is from zero.

Example A

Find the absolute value of 3. Then determine what other integer has an absolute value equal to |3| .

Look at the positive integer, 3, on the number line. It is 3 units from zero on the number line, so it has an absolute value of 3.

Now that you have found the absolute value of 3, we can find another integer with the same absolute value. Remember that with absolute value you are concerned with the distance an integer is from zero and not with the sign.

Here is how we find another integer that is exactly 3 units from 0 on the number line. The negative integer, -3, is also 3 units from zero on the number line, so it has an absolute value of 3 also.

So, |3|=|-3|=3 .

This example shows that the positive integer, 3, and its opposite, -3, have the same absolute value. On a number line, opposites are found on opposite sides of zero. They are each the same distance from zero on the number line. Because of this, any integer and its opposite will always have the same absolute value. To find the opposite of an integer, change the sign of the integer.

Just like we can find the absolute value of a number, we can also find the opposite of a number.

Example B

Find the opposite of each of these numbers: -16 and 900.

-16 is a negative integer. We can change the negative sign to a positive sign to find its opposite. The opposite of -16 is +16 or 16.

900 is the same thing as +900. We can change the positive sign to a negative sign to find its opposite. So, the opposite of 900 is -900.

Find the absolute value of each number.

Find the opposite of -18.

Check your answers with a peer.

Compare and Order Integers on a Number Line

Now that you know about absolute value, opposites, zero, and integers, we can work on learning how to compare and order integers. Often, the easiest way to do this is by using a number line.

Remember, a number line shows numbers ordered from least to greatest. So, on a number line, the further a number is to the right, the greater its value.

This can be a little tricky because when you look at a number line -222, you might think that it is larger number, but it isn’t.

How can this be?

Consider an example. If you were thinking about how much money you had, a positive amount could be a bank account balance. However, if you spent too much money, you would owe more than you had. This could be considered a negative number. Therefore, if your bank account balance was -$222 this means that you have even less money than if your bank account balance was -$22.The best way to think about it is the farther that a negative number is on the left side of the number line, the smaller that number is.

Example C

Order these numbers on a number line. Then determine which one is the greatest and which is the least: -12, 5, -6, -9, 10, 7

First, let’s draw a number line and plot the numbers on it.

Now look at the values on this number line and where they are located.

You can see that the number furthest to the right is 10. That is the largest number. The number furthest to the left is -12. That is the smallest number.

Example D

Order these integers from least to greatest: -6, 0, 5, -1.

To help you order these integers, draw a number line from -6 to 6. Then plot points for -6, 0, 5, and -1.

The numbers, ordered from least to greatest, are -6, -1, 0, 5.

Once we have an idea about which numbers are smaller or larger, we can work on comparing them.

Try a few sample problems: use a number line and write these numbers in order from least to greatest.

-4, 2, 8, 9, -11, -5
6, -16, 7, -22, 1, 4
-3, -2, -7, -12, -1

Find Sums of Integers on a Number Line

In this lesson, we will extend our understanding of integers by exploring different strategies for adding them.

What is an integer?

An integer is the set of whole numbers and their opposites. Said another way, integers are positive and negative whole numbers.

For example, imagine a person standing at the zero mark and facing the positive numbers. If that person moved 4 units forward (in a positive direction), that person would end up at the point representing 4. That is because 4 is 4 units to the right of zero on a number line.

How do we add integers?

There are strategies to help us do this. The first strategy we will explore for adding integers involves using a number line. To model addition of integers on a number line, imagine a person standing at zero, facing the positive numbers. To represent a positive integer, the person moves forward. To represent a negative integer, the person moves backward.

Let’s look at an example that shows how we can use a number line to model the addition of two positive integers.

Example E

Use a number line to find the sum of 4+6 .

You are adding two positive numbers. The positive numbers are to the right of zero on a number line.

So, to model 4+6 , imagine the person moving 4 units forward and then 6 more units forward. In other words, the person will move 4 units to the right of zero, and then moving 6 more units to the right.

All in all, the person moved 10 units to the right of zero and ended up at the tic mark representing 10. So, 4+6=10 .

Now, let's examine how to find the sum of two negative integers on a number line.

Example F

Take notes on using a number line to solve addition of integers.

Use a number line to find the sum of -4+(-6) .

Imagine the person starting at zero on the number line. You are adding two negative numbers. The negative numbers are to the left of zero on a number line.

So, to model -4+(-6) , imagine the person moving 4 units backward (in a negative direction), and then moving 6 more units backward.

All in all, the person moved 10 units to the left of zero and ended up at the tic mark representing -10. So, -4+(-6)=-10 .

Finally, let's explore how to use a number line to find the sum of two integers, each with a different sign. This may seem a little tricky, but if you think it through step by step you can come up with the correct sum.

Example G

Use a number line to find the sum of 4+(-6) .

Imagine starting at zero on the number line. You are adding a positive number, 4, to a negative number, -6.

To model 4+(-6) , imagine the person moving 4 units forward and to the right of zero. To model adding -6 to that integer, imagine the person moving 6 units backward and to the left.

The person moved 4 units to the right of zero and then 6 more units to the left from that point until the person reached the tic mark representing -2. So 4+(-6)=-2 .

Example H

Use a number line to find the sum of -4+6 .

Imagine starting at zero on the number line. You are adding a negative number, -4, to a positive number, 6.

So, to model -4+6 , first represent the -4 by moving the person 4 units backward and to the left of zero. To model adding 6 to that integer, imagine the person then moving 6 units forward and to the right.

The person moved 4 units to the left of zero and then 6 more units to the right until reaching the tic mark representing 2. So, -4+6=2 .

Now let’s practice a few on your own.

Add by using a number line.

Check your answers with a peer.

Add Two or More Integers Using Absolute Value

Using a number line is one strategy for adding integers now let’s look at another strategy.

Another strategy for adding integers involves using absolute values. An absolute value is the distance or the number of units that a number is from zero. Remember, with absolute value, the sign doesn’t matter. You will see the symbol $| \ |$ with an integer in the middle when absolute value is being represented.

Here are the steps to the absolute value strategy.

To add two integers with the same sign, add their absolute values. Then give the answer the same sign as the two original integers.
To add two integers with different signs, subtract the lesser absolute value from the greater absolute value. Then give the answer the same sign as the integer with the greater absolute value.

How do we apply these steps?

Let’s look at an example and see what this looks like in action.

Example I

Find the sum of -13+(-12) .

Both integers being added have the same sign––a negative sign. So, add their absolute values.

Since |-13|=13 and |-12|=12 , add those values.

13+12=25 .

Give that answer, 25, the same sign as the original two integers––a negative sign.

So -13+(-12)=-25 .

Now you have seen this strategy with two negative numbers. Next, let’s see how it applies with a negative and a positive number.

Example J

Find the sum of 13+(-12) .

The two integers being added have different signs. So, subtract their absolute values.

|13|=13 and |-12|=12 , subtract the lesser absolute value from the greater absolute value :

13-12=1 .

Give that answer, 1, the same sign as the integer with the greater absolute value. 13>12, so 13 has a greater absolute value than 12. Give the answer the sign of the number with the greater absolute value, in this case the 13, so the answer will have a positive sign.

So, 13+(-12)=1 .

What happens if we are finding the sum of more than two integers?

You can use this same strategy to add three or more integers. When adding three or more integers, remember that the associative property of addition states that the grouping of numbers being added does not matter.

Example K

Find the sum of 7+2+(-10) .

According to the associative property of addition, the integers being added can be grouped in any way. Here is one way to group numbers. Notice that we used brackets because parentheses are helpful when separating a negative sign and an addition sign. Brackets can mean the same thing as parentheses in these examples.

[7+2]+(-10)

If you group the numbers this way, you will add 7+2 first. Then you will add (-10) to that sum.

To add 7+2 , first notice that both integers have the same sign––a positive sign. So, add their absolute values.

|7|+|2|=7+2=9

Since the two original integers both had positive signs, give the sum a positive sign.

[7+2]+(-10)=9+(-10)

Now, add 9+(-10) . Since both integers have different signs, find the absolute value of each integer.

|9|=9 and |-10|=10 , so subtract the lesser absolute value from the greater absolute value.

10-9=1

Give that answer, 1, the same sign as the integer with the greater absolute value. 10>9 , so -10 has a greater absolute value than 9. Give the answer a negative sign.

So, 7+2+(-10)=9+(-10)=-1 .

Could I use a number line too?

Sure. A number line would have worked too. It just would have involved drawing a number line and then working through the math. Either way, you would still end up with the same answer.

Let’s practice using this method of adding integers.

Find Differences of Integers on a Number Line

We can subtract integers by using a strategy. Using a strategy will allow us to find the difference between two integers. Remember that the word “difference” is a key word that means the answer in a subtraction problem.

Do you remember what an integer is?

An integer is the set of whole numbers and their opposites. Essentially, we can think of integers as positive and negative whole numbers.

You may recall that one way to add integers is to use a number line. A similar strategy can be used to subtract integers as well. Let’s look at how to do this.

In the first strategy, we will explore how to subtract integers using a number line. When using a number line to model subtraction, imagine a person standing at 0, facing the positive numbers on the line. The person will then move forward or backward to show the first quantity in the problem. Then, the person will turn around and move forward or backward again to show the second quantity.

To subtract a positive number, the person moves forward. To subtract a negative number, the person moves backward.

This first example shows how we can use a number line to model the subtraction of two positive integers.

Try this:

Take notes on using a number line to solve subtraction of integers.

Use a number line to find the difference: 4-3 .

You are subtracting two positive integers.

So, to model 4-3 , imagine the person moving 4 units to the right of zero. This shows the quantity, 4.

Next, imagine the person turning around.

Since you are subtracting a positive integer, 3, the person moves forward. After turning around, imagine that person moving 3 units forward, or to the left.

The person ends up at 1. So, 4-3=1 .

Our answer is 1.

Now, let's imagine subtracting a negative integer from a positive integer.

Example L

Use a number line to find the difference of 4-(-3) .

To model 4-(-3) , imagine the person moving 4 units forward and to the right of zero. Then imagine the person turning around. Since a negative integer, -3, is being subtracted, imagine the person moving 3 units backward. That means the person will be moving to the left.

The person ends up at 7. So, 4-(-3)=7 .

Our answer is 7.

Example M

Use a number line to find the difference of -4-3 .

To model -4-3 , imagine the person moving 4 units backward and to the left. The person moves backward because the initial quantity, -4, is a negative integer.

Next, imagine the person turning around. Since a positive integer, 3, is being subtracted, imagine the person moving 3 units forward and to the left.

The person ends up at -7. So, -4-3=-7 .

Our answer is -7.

Now, let's imagine subtracting a negative integer from a negative integer.

Example N

Use a number line to find the difference of -4-(-3) .

To model -4-(-3) , imagine the person moving 4 units backward and to the left. Then imagine the person turning around. Since a negative integer, -3, is being subtracted, imagine the person moving 3 units backward and to the right.

The person ends up at -1. So, -4-(-3)=-1 .

Our answer is -1.

Subtract the following integers. Use a number line to help you.

Check your answers with a peer.

Name Ordered Pairs of Integer Coordinates Representing Points in a Coordinate Plane

In working with integers in previous lessons, we used both horizontal (left-to-right) and vertical (up-and-down) number lines. Imagine putting a horizontal and a vertical number line together. In doing this, you would create a coordinate plane.

In a coordinate plane like the one shown, the horizontal number line is called the axis. The vertical number line is called the axis. The point at which both of these axes meet is called the origin.

We can use coordinate planes to represent points, two-dimensional geometric figures, or even real-world locations. If you look at a map, you will realize that you often see a coordinate plane (or lines of longitude and latitude) on a map. You use the coordinates to find different locations. Let’s look at how we can use a coordinate plane.

How do we name points on a coordinate plane?

Each point on a coordinate plane can be named by an ordered pair of numbers, in the form (x, y) .

The first number in an ordered pair identifies the coordinate. That coordinate describes the point's position in relation to the axis.
The second number in an ordered pair identifies the coordinate. That coordinate describes the point's position in relation to the axis.

You can remember that the coordinate is listed before the coordinate in an ordered pair (x, y) , because comes before in the alphabet.

Identifying the coordinates of a point is similar to locating a point on a number line. The main difference is that you need to find the point that corresponds to both of the given coordinates.

Name the ordered pair that represents the location of point below.

Here are the steps to naming the coordinates.

To start, place your finger at the origin.
Then move your finger to the right along the axis until your finger is lined up under point . You will need to move 4 units to the right to do that. Moving to the right along a number line means you are moving in a positive direction. So, the coordinate is a positive integer 4.
Now, move your finger up from the axis until your finger reaches point . You will need to move 5 units up to do that. Moving up along the axis means you are moving in a positive direction. So, the coordinate is a positive integer 5.

The arrows below show how you should have moved your finger to determine the coordinates.

To name the ordered pair, write the coordinate first and the coordinate second. Separate the coordinates with a comma and put parentheses around them, like this (4, 5).

So, the ordered pair (4, 5) names the location of point .

Example O

This coordinate grid shows locations in Jimmy's city. Name the ordered pair that represents the location of the city park.

Here are the steps to figuring out the coordinates of the city park.

Place your finger at the origin.
Next, move your finger to the left along the axis until your finger is lined up above the point representing the city park. You will need to move 2 units to the left to do that. Moving to the left along the axis means that you are moving in a negative direction. Your finger will point to a negative integer, -2, so that is the coordinate.
Now, move your finger down from the axis until your finger reaches the point for the city park. You will need to move 6 units down to do that. Moving down parallel to the axis means that you are moving in a negative direction. Your finger will be aligned with the negative integer, -6, on the axis, so, that is the coordinate.

The arrows below show how you should have moved your finger to find the coordinates.

So, the ordered pair (-2, -6) names the location of the city park.

Now that you know how to name points using an ordered pair, it is time learn how to graph them from an ordered pair.

Graph Ordered Pairs of Integer Coordinates as Points in a Coordinate Plane

Graphing points on a coordinate plane is similar to naming them. Given an ordered pair, you can move your finger left or right along the axis and then up or down parallel to the axis until you find the location named by the ordered pair. Then you can plot a point at that location.

There are a few key points to remember.

If the coordinate is positive, move to the right of the origin. If the coordinate is negative, move to the left.
If the coordinate is positive, move up parallel to the axis. If the coordinate is negative, move down.

Example P

Plot the ordered pair (-5, 3) as a point on the coordinate plane.

Here are the steps:

The coordinate is a negative integer, -5, so move your finger 5 units to the left along the axis. Your finger should be pointing to the integer -5 on the axis.
The coordinate is a positive integer, 3, so move your finger 3 units up from the axis.

Plot a point at that location. That point represents the ordered pair (-5, 3).

Sometimes, the points you plot on a coordinate grid will form the vertices of a geometric figure, such as a triangle.

Example Q

Triangle ABC has vertices $A (-2, -5), \ B(0, 3)$ , and C(6, -3) . Graph triangle ABC on a coordinate plane. Label the coordinates of its vertices.

Here are the steps to graphing the triangle.

To plot vertex at (-2, -5), start at the origin. Move 2 units to the left and then 5 units down. Plot and label point .
To plot vertex at (0, 3), start at the origin. The coordinate is zero, so do not move to the left or right. From the origin, simply move 3 units up. Plot and label point .
To plot vertex at (6, -3), start at the origin. Move 6 units to the right and then 3 units down. Plot and label point .

Connect the vertices with line segments to show the sides of the triangle, as shown.

There are other figures that can be graphed on the coordinate plane as well. When you graph a rectangle, you can also look at length, width and area of the rectangle. Let’s look at how this works using the coordinate plane.

Real-Life Example Completed

The Sunken Ship

Here is the original problem once again. Use this information to help Cameron with the coordinates and the distance.

Gina and Cameron are very excited because they are going to go on a dive to see a sunken ship. The dive is quite shallow which is unusual because most sunken ship dives are found at depths that are too deep for two junior divers. However, this one is at 40 feet, so the two divers can go to see it.

They have the following map to chart their course. Cameron wants to figure out exactly how far the boat will be from the sunken ship. Each square represents 160 feet of water.

First, he makes a note of the coordinates. Then he can use the map to calculate the distance.

First, here are the coordinates of each item on the map.

The sunken ship is marked at (4, 8).

The dive boat is marked at (-3, 7).

Notice the arrows. Once they get to the sunken ship, Gina and Cameron will swim up 1 unit and over 6 units.

1 + 6 = 7

If each unit = 160 feet, then we can multiply $160 \times 7$

Gina and Cameron will swim through 1120 feet of water from the boat to the sunken ship.

Review

When finding the absolute value, find the distance from zero.
Remember to plot points using the first coordinate, x, in the horizontal direction and the second coordinate, y, in the vertical direction.
Use inverse operations to help determine accurate calculations when adding and subtracting with positive and negative numbers.