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| 13 Theory slides |
| 13 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
A coordinate system is a reference framework used to describe the positions of objects like points, lines, and surfaces in a space. A fixed point called the origin is used as a reference in the coordinate system. The most common types of coordinate systems are one-, two-, or three-dimensional.
A coordinate plane is a two-dimensional coordinate system. It is a grid that results from intersecting a vertical number line with a horizontal number line at their zero points. The horizontal number line is usually named the x-axis and the vertical number line is usually the y-axis.
A coordinate is the position of an object in a coordinate system relative to the corresponding axis. Coordinates are often seen together with other coordinates, which can describe the position of an object in a coordinate system with one, two, three, or even more dimensions.
In a two-dimensional coordinate system, points are usually expressed as a coordinate pair — also called an ordered pair — denoted by (x,y). The first coordinate states the position along the x-axis and the second coordinate states the position along the y-axis.
Identify the coordinates of the given point. Write the coordinates as an ordered pair (x,y), where x represents the x-coordinate and y the y-coordinate.
Dominika plays basketball with her friends every afternoon. She warms up by running and picks up two friends, Jordan and Emily, on the way to the court. Dominika uses ordered pairs to represent the positions of her and her friends' houses.
Position | |
---|---|
Dominika's House | D=(2,1) |
Jordan's House | J=(2,4) |
Emily's House | E=(-1,4) |
Moving one unit on the x-axis or y-axis on the coordinate plane is counted as moving one block. Since Dominika can only stay on the straight roads, her position can only move up, down, left, and right on the graph.
Moving one unit on the x-axis or y-axis on the coordinate plane is the same as Dominika moving one block. Dominika only runs on the straight roads, so her progress on the graph can only move in four directions — up, down, left, and right.
Keeping this in mind, one route she can take is moving five blocks directly to the right. Move 5 units to the right starting her house to show this possible destination. Remember that Dominika's house is located at (2,1), not the origin!
Another spot Dominika might go is one block up and four blocks to the right of her starting point. It does not matter if she moves up on her first, second, third, fourth, or final move. Her destination will always be the same if she moves one block up out of the five total moves.
Follow the same reasoning to find the other places that Dominika can go. Plot each as ordered pairs on the coordinate plane.
Now that all the points are plotted on the graph, connect them with lines to see the overall shape.
The shape is a rotated square.
The places Dominika can go can be determined by identifying all the possible routes she can take. Remember, she can only go five blocks away from her home. Moving left or down indicates a negative direction on the corresponding axis, represented by negative numbers.
Route from Home | Calculations | Destination |
---|---|---|
5 blocks to the right | (2+5,1) | (7,1) |
4 blocks to the right and 1 block up | (2+4,1+1) | (6,2) |
3 blocks to the right and 2 blocks up | (2+3,1+2) | (5,3) |
2 blocks to the right and 3 blocks up | (2+2,1+3) | (4,4) |
1 block to the right and 4 blocks up | (2+1,1+4) | (3,5) |
5 blocks up | (2,1+5) | (2,6) |
1 block to the left and 4 blocks up | (2−1,1+4) | (1,5) |
2 blocks to the left and 3 blocks up | (2−2,1+3) | (0,4) |
3 blocks to the left and 2 blocks up | (2−3,1+2) | (-1,3) |
4 blocks to the left and 1 block up | (2−4,1+1) | (-2,2) |
5 blocks to the left | (2−5,1) | (-3,1) |
4 blocks to the left and 1 block down | (2−4,1−1) | (-2,0) |
3 blocks to the left and 2 blocks down | (2−3,1−2) | (-1,-1) |
2 blocks to the left and 3 blocks down | (2−2,1−3) | (0,-2) |
1 block to the left and 4 blocks down | (2−1,1−4) | (1,-3) |
5 blocks down | (2,1−5) | (2,-4) |
1 block to the right and 4 blocks down | (2+1,1−4) | (3,-3) |
2 blocks to the right and 3 blocks down | (2+2,1−3) | (4,-2) |
3 blocks to the right and 2 blocks down | (2+3,1−2) | (5,-1) |
4 blocks to the right and 1 block down | (2+4,1−1) | (6,0) |
In a coordinate plane, the intersection of the x-axis and the y-axis produces four regions called quadrants. The quadrants are numbered counterclockwise from the top right quadrant as Quadrant I to Quadrant IV in the bottom right.
The signs of the coordinates of a point can be determined based on which quadrant the point lies in.
A coordinate system has four quadrants that provide information about the signs of the x- and y-coordinates of a point. In the following applet, identify in which quadrant the given point lies.
Look at the signs of the coordinates of each point to identify their quadrants.
Point | Signs of Coordinates | Quadrant |
---|---|---|
(-2,-8) | (−,−) | Quadrant III |
(4,-2) | (+,−) | Quadrant IV |
(-11,3) | (−,+) | Quadrant II |
(4,3) | (+,+) | Quadrant I |
Reflecting a point means making a mirror image of that point by flipping it across a certain line or axis. The following rules show how to find reflected points across a specific axis.
Point | Axis of Reflection | Procedure | Reflected Point |
---|---|---|---|
(x1,y1) | x | Change the y-coordinate to its opposite. | (x1,-y1) |
(x1,y1) | y | Change the x-coordinate to its opposite. | (-x1,y1) |
(x1,y1) | x and y | Change the x- and y-coordinates to their opposites. | (-x1,-y1) |
Next, the process for reflecting points across the x-axis, the y-axis, or both, on a coordinate plane will be shown. Consider the points A, B, and C.
Point | Axis of Reflection | Reflected Point |
---|---|---|
A=(5,8) | x | ? |
B=(10,-5) | y | ? |
C=(-4,-7) | x and y | ? |
Once each initial point and its reflection across a given axis are found, plot both points on the coordinate plane.
Dominika's team is only one game away from winning the basketball tournament! Dominika is a shooting guard and wants to use her math skills to improve her game. She starts by figuring out the point on the court from where she usually scores using a coordinate plane. This point is (-4,-3).
Dominika is interested in the mirror point of (-4,-3) because it is a great shooting position. This point is as far away from the basket as the original point but is oriented in a different direction. This can give her an advantage over her opponents and help her score more points during game!
Now that Dominika understands her attack and defense positions better, she can perform much better on the court. Her team is doing great and they keep scoring point after point until before they know it, they win first place in the tournament. They are the champions!
This lesson showed how to locate points with respect to the x- and y-axis on a coordinate plane. A third axis, usually called the z-axis, can be used to provide more information about the position of objects. Think of it like this.
Description | |
---|---|
x-axis | Tells if something is in front of or behind the origin. |
y-axis | Tells if something is to the left or right of the origin |
z-axis | Tells if something is above or below the origin. |
These three axes create a 3D coordinate system that is similar to the coordinate plane but with the addition of an extra dimension. This additional dimension means that points are represented as ordered triples (x,y,z) instead of as ordered pairs, as in the coordinate plane.
Identify the ordered pair that represents each point.
Start by looking at the given graph.
Let's draw a straight line upwards from point P to the x-axis to find its x-coordinate. Then, we will draw another straight line from point P to the y-axis, going horizontally to determine its y-coordinate.
The vertical line intersects the x-axis at 1 and the horizontal line intersects the y-axis at -4. This indicates that the x-coordinate of point P is 1 and the y-coordinate is -4. Now we can express this as an ordered pair. P( 1, -4)
We can use a similar process to determine the coordinates of point Q. Let's start by drawing a vertical line from point Q to the x-axis. We will then draw a horizontal line from point Q to the y-axis.
In this case, the vertical line intersects the x-axis at -7 and the y-axis at 8. This means that the x-coordinate of point Q is -7 and the y-coordinate is 8. Let's write it as an ordered pair! Q( -7, 8)
Let's graph each ordered pair on the coordinate plane one by one to determine the correct option. Remember that we graph an ordered pair as a point on a coordinate plane by moving horizontally from the origin the number of units specified by its x-coordinate and vertically the number of units specified by its y-coordinate.
Now, let's take a look at the first given ordered pair. M( 2, 4) Point M has an x-coordinate of 2 and a y-coordinate of 4. We need to move 2 units to the right of the origin and 4 units up to graph this ordered pair.
We can use the same approach to plot the other ordered pairs. Let's start by identifying the x- and y-coordinates for each point. Once we have these coordinates, we can plot them on the graph.
Ordered Pair | x-coordinate | y-coordinate |
---|---|---|
M( 2, 4) | 2 | 4 |
N( -5, 0) | -5 | 0 |
O( -3, -5) | -3 | -5 |
P( 7, 1) | 7 | 1 |
We can add now these points to the graph.
This graph corresponds to option A.
We can check the signs of the x- and y-coordinates to find out where a point is located.
With this information in mind, let's examine the coordinates of the point ( -5, 14). ( -5, 14) ⇓ ( -, +) This point has a negative x-coordinate and a positive y-coordinate. This means that the point ( -5, 14) is located in Quadrant II. We can use a similar reasoning to find the corresponding quadrant of the remaining points.
Point | Coordinates' Signs | Quadrant |
---|---|---|
( -5, 14) | ( -, +) | Quadrant II |
( 14, -11) | ( +, -) | Quadrant IV |
( -7, -22) | ( -, -) | Quadrant III |
( 21, 19) | ( +, +) | Quadrant I |
We have found the quadrant where each of the given points is located. Let's plot them on a coordinate plane to confirm our matches. For each point, we will move horizontally from the origin according to its x-coordinate and vertically based on its y-coordinate.
Use a coordinate plane to find the reflection of the given point across the indicated axis. Write the answer as an ordered pair.
We are asked to reflect the point ( -5, -1) across the x-axis. This means keeping its x-coordinate the same and changing the y-coordinate to its opposite. The x-coordinate is -5. The opposite of the y-coordinate -1 is 1. Point&Reflection Acrossx-axis ( -5, -1)&( -5, 1) We can plot both points on a coordinate plane to visualize that they are a mirror of each other across the x-axis.
Let's reflect the point ( 4, 15) across the y-axis. In this case, the y-coordinate remains the same and the x-coordinate changes to its opposite. The y-coordinate is 15. The opposite of the x-coordinate 4 is -4. Point&Reflection Acrossy-axis ( 4, 15)&( -4, 15) Again, it is useful to plot these points on a coordinate plane. Let's do it!
Let's reflect the point ( 2, -9) across both the x- and y-axes. To do that, we need to change the x- and y-coordinates both to their opposites. The x-coordinate is 2, so we change it to -2. The y-coordinate is -9, so we change it to 9 to get its reflection. Point&Reflection Across Both Axes ( 2, -9)&( -2, 9) Let's graph both points on a coordinate plane.
Select the statement that best describes the relationship between the given ordered pairs.
Let's review how to reflect a point across each axis.
Axis of Reflection | Procedure |
---|---|
x-axis | Change the y-coordinate to its opposite and keep the x-coordinate the same. |
y-axis | Change the x-coordinate to its opposite and keep the y-coordinate the same. |
x- and y-axes | Change the x- and y-coordinates to their opposites. |
Now let's take a look at the points A and B. A( -8, 2), B( -8, -2) They both have an x-coordinate of -8, but their y-coordinates are opposites of each other. This means that B is a reflection of A across the x-axis.
Let's begin by looking at the points C and D.
C( 4, 12), D( -4, 12)
In this case, C and D both have a y-coordinate of 12, but their x-coordinates are opposites. This fits the rule for a reflection across the y-axis, so D is a reflection of C across the y-axis.
Consider the points E and F.
E( -11, 3), F( 11, -3)
Notice that the x-coordinate of E is the opposite of the x-coordinate of F and the y-coordinate of E is also the opposite of the y-coordinate of F. This means that F is the reflection of E across the x- and y-axes, following the rule for reflecting a point across both axes.
Tell if the given statement is always, sometimes, or never true.
A point that lies on the y-axis has a y-coordinate of 0. |
Points in Quadrant II have negative x-coordinates. |
Consider the given statement.
A point that lies on the y-axis has a y-coordinate of 0.
Let's examine some points on the y-axis to find out if the given statement is always, sometimes, or never true.
Point A is 5 units up and 0 units left or right from the origin, so its x-coordinate is 0 and its y-coordinate is 5. A( 0, 5) Points B and D also have an x-coordinate of 0, but their y-coordinates are not 0. B( 0, 2) D( 0, -3) Does this mean that the y-coordinate of a point on the y-axis is never 0? Not necessarily. Point C, which lies on the origin of the coordinate plane, is on the y-axis and has coordinates ( 0, 0). C( 0, 0) We found examples of points on the y-axis whose y-coordinate is not 0 and an example of a point on the y-axis whose y-coordinate is 0. Therefore, the given statement is sometimes true.
Let's look at the given statement.
Points in Quadrant II have negative x-coordinates.
Recall that quadrants of a coordinate plane are numbered counterclockwise, from Quadrant I in the top right to Quadrant IV in the bottom right.
Quadrant II is the top left part of the coordinate plane. Any point in this area is to the left of the y-axis. Since all points to the left of the y-axis have an x-coordinate less than 0, any point in Quadrant II will have a negative x-coordinate. Therefore, this statement is always true.