Unit 4, Lesson 10
On or Off the Line?
Grade 8
10.1
Warm-up
Which three go together? Why do they go together?
<p>A graph in the x y plane. Three lines that do not intersect. One line crosses the x axis to the left of the origin and the y axis below the origin. Another line crosses the y axis above the origin and the x axis to the right of the origin. The third line crosses the y axis above the origin and the x axis to the right of the origin. </p>
<p>A graph in the x y plane. Two intersecting lines. One line crosses the x axis to the left of the origin and the y axis below the origin. Another line crosses the y axis above the origin. </p>
<p>A graph in the x y plane. Three lines that intersect at a single point. One line crosses the y axis above the origin. Another line crosses the x axis to the right of the origin and the y axis below the origin. The third line crosses the x axis to the right of the origin.</p>
<p>A graph in the x y plane. Three lines. There are 3 points of intersection between two lines each. One line crosses the y axis above the origin. Another line crosses the x axis to the left of the origin and the y axis above the origin. The third line crosses the x axis to the right of the origin.</p>
10.2
Activity
Jada told Noah that she has \$2 worth of quarters and dimes in her pocket and 17 coins all together. She asked him to guess how many of each type of coin she has.
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Here is a table that shows some combinations of quarters and dimes that are worth \$2. Complete the table.
number of quarters number of dimes 0 20 4 0 5 -
Here is a graph of the relationship between the number of quarters and the number of dimes when there are a total of 17 coins.
- What does Point represent?
- How much money, in dollars, is the combination represented by Point worth?
<p>Graph of points in the x y plane, origin O, with grid. Horizontal axis, number of quarters, scale 0 to 24 by 1s. Vertical axis, number of dimes, scale 0 to 24 by 1s. Points plotted are 0 comma 17, 1 comma 16, 2 comma 15, 3 comma 14, 4 comma 13, 5 comma 12, 6 comma 11, 7 comma 10, 8 comma 9 labeled A, 9 comma 8, 10 comma 7, 11 comma 6, 12 comma 5, 13 comma 4, 14 comma 3, 15 comma 2, 16 comma 1, and 17 comma 0.</p> - Is it possible for Jada to have 4 quarters and 13 dimes in her pocket? Explain how you know.
- How many quarters and dimes must Jada have? Explain your reasoning.
10.3
Activity
Clare and Andre are making signs for all the lockers as part of the decorations for the upcoming spirit week. Yesterday, Andre made 15 signs and Clare made 5 signs. Today, they need to make more signs. Each person's progress today is shown in the coordinate plane.
<p>Graph of two lines in the x y plane, origin 0, with grid. Horizontal axis, time in minutes, scale 0 to 110, by 5s. Vertical axis, number of completed signs, scale 0 to 70 by 5s. A line crosses the y axis at 5 and passes through the points A and B. Another line crosses the y axis at 15 and passes through the points C and A. Starting at the origin, point A is 8 unit to the right and 5 units up. Starting at the origin, point B is 15 units to the right and 8 point 5 units up. Starting at the origin, point C is 0 units to the right, and 3 units up. Starting at the origin, point D is 20 units to the right and 12 units up.</p>
- Which line represents Andre and which represents Clare?
- Based on the lines, mark the statements as true or false for each person.
| point | what it says | Clare | Andre |
|---|---|---|---|
| At 40 minutes, I have 25 signs completed. | |||
| At 75 minutes, I have 42 and a half signs completed. | |||
| At 0 minutes, I have 15 signs completed. | |||
| At 100 minutes, I have 60 signs completed. |
Student Lesson Summary
We studied linear relationships in an earlier unit. We learned that values of and that make an equation true correspond to points on the graph.
For example, let’s plan the base rocks for a terrarium. We have pounds of river rocks that cost \$0.80 per pound and pounds of unpolished rocks that cost \$0.50 per pound, and the total cost is \$9.00, so we can write an equation like this to represent the relationship between and
Because 5 pounds of river rocks cost \$4.00 and 10 pounds of unpolished rocks cost \$5.00, we know that , is a solution to the equation, and the point is a point on the graph.
The line shown is the graph of the equation. Notice that there are 2 points shown that are not on the line. What do they mean in the context?
<p>The graph of a line in the x y plane. The line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. Two additional points, 9 comma 16 and 1 comma 14, are labeled on the graph.</p>
The point means that there is 1 pound of river rock and 14 pounds of unpolished rocks. The total cost for this is or \$7.80. Because the cost is not \$9.00, this point is not on the line. Likewise, 9 pounds of river rocks and 16 pounds of unpolished rocks cost or \$15.20, so the other point is not on the line either.
Suppose we also know that the river rocks and unpolished rocks together weigh 15 pounds. That means that .
If we draw the graph of this equation on the same coordinate plane, we see it passes through 2 of the 3 labeled points:
<p>The graph of two intersecting lines in the x y plane. The first line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. The second line slants downward and to the right and passes through the points 1 comma 14 and 5 comma 10. An additional point, 9 comma 16, is labeled on the graph.</p>
The point is on the graph of because . Similarly, . But , so is not on the graph of .
In general, if we have 2 lines in the coordinate plane and we have their corresponding equations,
- The coordinates of a point on a line make that equation true.
- The coordinates of a point off of a line make that equation false.
- The coordinates of a point that is the intersection of the 2 lines make both equations true.
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