Minds On
Cartesian Plane
What do you notice about the following values?
What is similar and what is different? Record your ideas using a method of your choice.
- (2, 2)
- (-2, 2)
- (-3, -2)
- (3, -2)
- (-2, -3)
- (2, -3)
- (0, -4)
Press ‘Reveal’ to access information about this type of graph.
This type of graph is called a Cartesian plane. There are four quadrants, that start at the top right and continue counterclockwise.
- Quadrant 1 has both x and y values that are positive.
- Quadrant 2 has negative x values and positive y values.
- Quadrant 3 has both x and y values that are negative.
- Quadrant 4 has positive x values and negative y values.
What might you use it for? Let’s list some possibilities…
- Example: A board game where you guess positions to try to find the other people’s locations.
Action
Task 1: Graphing inequalities on a Cartesian plane
When there is more than one variable, a number line would not be useful. Instead, we graph inequalities on a Cartesian plane.
You may notice that there are 4 quadrants made by 2 intersecting lines. The line that goes up and down, or vertical, is called the y-axis. The line that goes side to side, or horizontal, is called the x-axis. The point where the x-axis and y-axis intersect is called the origin and is marked with a 0. Everything above and to the right of that 0 is a positive number, while everything below and to the left of that 0 is a negative number.
Let’s plot an inequality with 2 variables together. For example, y ≤ x + 2. The +2 indicates what point to plot on the y-axis (0,2). Our y-axis includes values from –1 to 6. Our x-axis includes values from –1 to 6.
Make a straight diagonal line from that point by going up and over by 1 towards the positive quadrant. To continue the line in the opposite direction but decrease by 1 in each direction.
The solution set, the numbers that solve the inequality, will either be all of the numbers above or below the line, depending on the inequality symbol. Since the symbol indicates that y is less than, the solution set will be below the line created. If the symbol described y as greater than, the solution set would be above the line. We shade the area above or below the line to show the solution set, in this case, we’ll shade below the line.
The symbol also describes something about the actual line. If the inequality is greater/less than OR equal to, the line is solid because the number that the line lands on is part of the solution set. If the symbol is greater than/less than, the line should be dotted because the number that the line lands on is not part of the solution set.
The possible solutions are within the lower part of the graph. You can choose any point in that section and determine the value of x and y. For example, coordinate (3, 1) in this range. That means that x can be 3 if y is 1. The x coordinate or value, is always listed first followed by the y.
Determine if the following points are part of the solution set. How do you know? You could see where they fall by plotting them on the graph. Or you could substitute them into the inequality. Record your responses using a method of your choice. Remember, we’re looking at the inequality y≤x + 2.
- (2, 2)
- (1, 5)
- (6, 2)
- (4, 1)
- (2, 0)
Press ‘Answers’ to access the solutions to check your work.
Point 1: (2, 2)
- y < x + 2
- 2 < 2 + 2
- 2 < 4
- Yes!
Point 2: (1, 5)
- y < x + 2
- 5 < 1 + 2
- 5 < 3
- No!
Point 3: (6, 2)
- y < x + 2
- 2 < 6 + 2
- 2 < 8
- Yes!
Point 4: (4, 1)
- y < x + 2
- 1 < 4 + 2
- 1 < 6
- Yes!
Point 5: (2, 0)
- y < x + 2
- 0 < 2 + 2
- 0 < 4
- Yes!
Task 2: Practice graphing inequalities
For the following inequalities, graph them on a Cartesian plane, or create a table of values, and list at least possible values for x and y. Describe your ideas using a method of your choice.
Press ‘Answers’ to reveal a possible solution to check your work.
x + y < 4
| Term number | Term value |
|---|---|
| 1 | 2 |
| 2 | 1 |
| 3 | 0 |
| 4 | -1 |
| 5 | -2 |
y > x – 1
| Term number | Term value |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
y ≥ x + 3
| Term number | Term value |
|---|---|
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
| 4 | 7 |
| 5 | 8 |
Consolidation
Task 1: Plotting inequalities for real-life scenarios
A baker has a certain number of sprinkles, represented by y. She can use these sprinkles on the cookies and brownies she is baking. She will make 7 brownies and an unknown number of cookies (x). Create an inequality to show how many sprinkles the baker will use on x number of cookies and 7 brownies. She must have some sprinkles leftover.
Graph, create a table of values for the inequality and list at least 3 possible solutions for x and y. Record your solution using a method of your choice.
Task 2: Think about your learning
Student Success
Think-Pair-Share
Reflect on the following questions, and record your ideas using a method of your choice. If possible, share your thoughts with a partner.
- How is plotting inequalities on a number line different than plotting inequalities on a Cartesian plane? What is similar?
- How do you know when to use a Cartesian plane to plot inequalities?
Note to teachers: See your teacher guide for collaboration tools, ideas and suggestions.
Reflection
As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.
I feel...
Now, record your ideas using a voice recorder, speech-to-text, or writing tool.
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