Transcript Coordinate Plane

Module 4
Test Review
Now is a chance to review all of the great
stuff you have been learning in Module 4!
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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 xcoordinate is negative and
the y-coordinate is positive
in an ordered pair: (−x, y).
For example: (−3, 5)
In this quadrant, the xcoordinate is negative
and the y-coordinate is
negative in an ordered
pair: (−x, −y). For
example: (−3, −5)
In this quadrant, the xcoordinate is positive and the
y-coordinate is positive in an
ordered pair: (x, y). For
example: (3, 5)
In this quadrant, the xcoordinate 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
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.
When you reflect a point across the yaxis, the y-coordinates of the two
ordered pairs are the same and the xcoordinates are opposite. So
reflecting across the y-axis is the
same as taking the opposite of the xcoordinate
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 ycoordinate will be positive.
The first number of the ordered pair is the xcoordinate. 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 xaxis 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
Because all of the ratios are equivalent,
only one pair is needed. So for every 3
hours of time, this person traveled 6 miles.
Determine the Unit Rate
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.