Module 4Test Review

Now is a chance to review all of the great stuff you have been learning in Module 4!

• Ordered Pairs• Plotting on the Coordinate Plane• Ratios and the Coordinate Plane• Applications with Coordinates

Key Terms

• Number Line– The number line goes on forever in the negative

direction and in the positive direction.– The number line is a one-dimensional graph because it

maps only from left to right (or up and down)• Coordinate Plane

– two-dimensional. It maps left to right, as well as up and down

• Ordered Pair– Ordered list of two numbers that describes the location

of a point on the coordinate plane; takes the form (x-coordinate, y-coordinate).

Coordinate Plane

Notice the center of the map is where the bold horizontal line and the bold vertical line cross over. This location is numbered (0, 0)

coordinate points are written inside parentheses, separated by a comma.

The first number in an ordered pair is the x-coordinate, which tells you where the point is along the x-axis.

The second number in an ordered pair is the y-coordinate, which tells you where the point is along the y-axis.

The location of a point (left, right, above, and below) in relation to zero will determine the signs of coordinates

example

Point A is located at the coordinates (−2, 4). Are the x- and y-coordinates positive or negative?

The x-coordinate is the first number of the ordered pair, and the y-coordinate is the second number in the ordered pair.

You are given that Point A is located at (−2, 4). In this case, the x-coordinate is negative and the y-coordinate is positive.

Coordinate Grid

Coordinate Grid

Coordinate Grid

Coordinate Grid

Coordinate Plane

Quadrants

In this quadrant, the x-coordinate is positive and the y-coordinate is positive in an ordered pair: (x, y). For example: (3, 5)

In this quadrant, the x-coordinate is negative and the y-coordinate is positive in an ordered pair: (−x, y). For example: (−3, 5)

In this quadrant, the x-coordinate is negative and the y-coordinate is negative in an ordered pair: (−x, −y). For example: (−3, −5)

In this quadrant, the x-coordinate is positive and the y-coordinate is negative in an ordered pair: (x, −y). For example: (3, −5)

Reflecting a Point

You can reflect points on a coordinate plane as if there was a mirror on it. The two most common ways are reflecting across the x-axis or the y-axis.

When you reflect a point across the y-axis, the y-coordinates of the two ordered pairs are the same and the x-coordinates are opposite. So reflecting across the y-axis is the same as taking the opposite of the x-coordinate

When you reflect a point across the x-axis, the x-coordinates of the two ordered pairs are the same and the y-coordinates are opposite. So reflecting across the x-axis is the same as taking the opposite of the y-coordinate.

Examples

Point (−5, 3) is a reflection of (−5, −3). Across which axis is the reflection occurring?

Plotting Points

In order to understand how to plot points on a coordinate plane, we must first recall how to locate integers on horizontal and vertical number line diagrams

Point R is on –7. The negative shows that the point is 7 spaces to the left of 0.

Point T is on 9. This is positive, so the point is 9 spaces to the right of 0.

Plotting Points

Let’s plot point S at 4 and point P at –5 on a vertical number line.

Point P is on –5. The negative shows that the point is 5 spaces below 0.

Point S is on 4. This is positive, so the point is 4 spaces above 0.

Plotting a Point

Plot the point (3,1)

Start at the origin. (Remember the first number tells you how far to go left or right.) Because the first coordinate is positive three, you should travel three units to the right.

The second number tells you up or down.The second number is positive so you will move up 1 unit

Place a dot here to mark your point.

Example

Richard’s ticket says that he is sitting at (–3, 4). Let’s find his location!

Rational Numbers

Rational numbers can be integers, decimals, or fractions.

Plotting decimals and fractions on a number line is as easy as plotting any integer; you just have to think about the values in between integers.

Let’s Plot the point (3.5, 0.5)

Plotting decimals is similar to plotting integers. Follow the same steps:

1. Start at the origin.

2. Move 3.5 units to the right. The 3.5 is in the middle of 3 and 4.

3. Then go up 0.5. The 0.5 is in the middle of 0 and 1.

4. Plot the point.

Example

Doris wants to buy some cupcakes. The cupcake stand is located at (−5, 2 3/4). Let’s help Doris figure out where it is by plotting the cupcake stand on the coordinate plane!

Determining the Coordinates of a Point

We need to find the coordinates of Point P

It is in the second quadrant. So we already know that the x-coordinate will be negative and the y-coordinate will be positive.

The first number of the ordered pair is the x-coordinate. Count along the x-axis to find how many units left of the origin the point lies. This point is three units left of the origin. So the first number in the ordered pair is −3.

The second number in the ordered pair is the y-coordinate. You must find out how many units the point is above the x-axis. Because Point P is 2 units above the x-axis, the y-coordinate is 2.

That means the coordinates of Point P are (−3, 2)

Ratios as Ordered Pairs

Let’s see how a coordinate plane can show the equivalent ratios

Given the ratio: for every 3 cans of fruit, there are 2 cans of soup.

make a table of equivalent ratios

Fruit Soup

3 2

6 4

9 6

12 8

15 10

Let’s choose the fruit cans to be on the x-axis and the soup cans to be on the y-axis.

Now we can use the table to set the equivalent ratios up as coordinate points.There are 5 points (3, 2), (6, 4), (9, 6), (12, 8), and (15, 10)

Now you can use the points to plot them on the coordinate plane.

One thing you should notice is that only the first quadrant is needed, as there are no negative cans of soup or frui

The Graph

Example

Melissa is creating designs on her gift boxes for her group members, as they all worked so hard. Because there is only room for 12 designs, for every 7 diamonds, she places 5 stars. Let’s create an equivalent ratio table and plot those ratios to see how many stars are needed if there are 28 diamonds.

1. Create the table of equivalent ratios 2. Plot the points

Applying Rates

You can determine the unit rate in a relationship by the points plotted on a coordinate plane

The coordinate plane shows the distance traveled by a person. Use the coordinates to determine the miles per hour.

Applying Rates

Determine the coordinates

From the graph, you can see that there are three ordered pairs, (3, 6), (5, 10), and (7, 14). In order to determine the unit rate, you must use the ordered pairs to determine the ratio for the two quantities.

Create a table

Determine the Unit Rate

Because all of the ratios are equivalent, only one pair is needed. So for every 3 hours of time, this person traveled 6 miles.

Try It!

Sarah bought 3 game controllers at Games Plus for $27. Determine the unit price for a game controller. Then plot the relationship on a coordinate plane to show other prices, depending on the number of controllers purchased.

Using the Coordinate Plane to solve a real world problem

We can use the coordinate plane to help solve real world problems

• Let’s look at some examples

Example 1

Cooper's parents just bought a new swing set for the backyard. His dad decided to put four garden wind spinners around it. He drew a coordinate plane to help him see where to place the spinners.

Help Cooper's dad to place the spinners at the following points: (−7, 5), (7, 5), (7, −3), and (−7, −3).

Example 2

The local park is creating a new walking trail. The town mayor asks the town architect to design the trail for the next meeting. The trail needs to start and end near the park entrance. The architect plots the points on the coordinate grid to offer his proposal to the mayor.

Example 2 plotting the points

Finding Measurements Quadrant 1

We can use the coordinate plane to find the measurement of line segments.

Method 1: Counting spaces

By counting the spaces between the points, you can determine how far two points are from each other. You will see that the door is 4 units. Start on the ordered pair (6, 15), and count the number of spaces (or units) from that point to (10, 15).

Finding Measurements – Quadrant 1

Method 2 - Subtraction With the subtraction method, you just need the coordinates that are different. Identify the first and last coordinates of the line segment.

First point: (6, 15)

Second point: (10, 15)

Notice that the x-coordinates are different and the y-coordinates are the same. Take the values of the x-coordinates and subtract them (remember to put the number with the larger value first).

10 − 6 = 4 units

This shows that the door is 4 units long on the grid.

Finding Measurements – Quadrant 2

Using the coordinate plane, find the measurements of the TV

Method 1: Counting Spaces

From (−13, 12) to (−12, 12), there is 1 unit.

From (−12, 12) to (−12, 3), there are 9 units.

Because it is a rectangle, you know the other two sides will be the same length.

Finding Measurement – Quadrant 2

Method 2: Subtracting In the second quadrant, some of the values are negative. So how would you subtract?

Since you are finding the distance, remember distance is always positive. This means you have to take the absolute value of each negative value before subtracting.

Review absolute value facts if you need to.

From (−13, 12) to (−12, 12), |−13| − |−12| = 13 − 12 = 1 unit

From (−12, 12) to (−12, 3), 12 − 3 = 9 units.

Because it is a rectangle, you know the other two sides will be the same length

Finding Measurements: Multiple Quadrants

What about the foosball table? Notice it's in more than one quadrant.

Finding Measurements: Multiple Quadrants

Method 1: counting spaces

You can still count the spaces to find the lengths.

From (−4, −2) to (2, −2), there are 6 units.

From (2, −2) to (2, −4), there are 2 units.

Finding Measurements: Multiple Quadrants

Method 2: Subtraction

When a line segment crosses over an quadrant, you have to find the distance the endpoints are from the axis that is crossed. Since the axes have a value of 0, find the absolute value of the coordinates that are different in the two ordered pairs.

Finally, instead of subtracting, you add the two distances.

From (−4, −2) to (2, −2), notice this goes from Quadrant 3 to Quadrant 4. This goes across the axis, so you add the absolute value of the coordinates that are different.

Distance would be |−4| + |2| = 4 + 2 = 6 units.

The distance from (−4, −4) to (−4, −2) is different. No axis is crossed, so you subtract.

|−4| − |−2| = 4 − 2 = 2 units.

You have now had a chance to review all of the great stuff you learned in Module 4!

• Ordered Pairs• Plotting on the Coordinate Plane• Ratios and the Coordinate Plane• Applications with Coordinates

Have you completed all assessments in module 4? Have you completed your Module 4 DBA?

Now you are ready to move forward and complete your module 4 test. Please make sure you are ready to complete your test before you enter the test session.