Lesson 10On or Off the Line?

Learning Goal

Let’s interpret the meaning of points in a coordinate plane.

Learning Targets

  • I can identify ordered pairs that are solutions to an equation.

  • I can interpret ordered pairs that are solutions to an equation.

Warm Up: Which One Doesn’t Belong: Lines in the Plane

Problem 1

Which one doesn’t belong? Explain your reasoning.

  1. An x y axis with 3 parallel lines with negative slopes
  2. An x y axis with 2 lines intersecting in Quadrant II
  3. An x y axis with 3 lines intersecting at a single point in Quadrant I
  4. An x y axis with 3 lines intersecting in Quadrant I. 2 lines intersect at a point and a third line crosses them.

Activity 1: Pocket Full of Change

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.

Problem 1

number of quarters

number of dimes

Problem 2

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.

A graph with number of quarters vs number of dimes. There is a dotted line from (0,17) to (17,0) and a point A at (8,9).

  1. What does Point represent?

  2. How much money, in dollars, is the combination represented by Point worth?

Problem 3

Is it possible for Jada to have 4 quarters and 13 dimes in her pocket? Explain how you know.

Problem 4

How many quarters and dimes must Jada have? Explain your reasoning.

Activity 2: Making Signs

Problem 1

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.

A graph of time in minutes vs number of completed signs. A line starts at (0,5) with points A, B, D. Another line starts at C (0,15) and cross at point A (40, 25).

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.

Are you ready for more?

Problem 1

  • 4 toothpicks make 1 square

  • 7 toothpicks make 2 squares

  • 10 toothpicks make 3 squares

Three drawings - a square, two connected squares, and three connected squares.

Do you see a pattern? If so, how many toothpicks would you need to make 10 squares according to your pattern? Can you represent your pattern with an expression?

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, if we have pounds of flour that costs $0.80 per pound and pounds of sugar that costs $0.50 per pound, and the total cost is $9.00, then we can write an equation like this to represent the relationship between and

Since 5 pounds of flour costs $4.00 and 10 pounds of sugar costs $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:

A graph of flour (pounds) vs sugar (pounds). A line starts at (0,18) and ends at (11,0) with a point on the line (5,10). Other points (1,14) and (9,16).

Notice that there are two points shown that are not on the line. What do they mean in the context? The point means that there is 1 pound of flour and 14 pounds of sugar. The total cost for this is or $7.80. Since the cost is not $9.00, this point is not on the graph. Likewise, 9 pounds of flour and 16 pounds of sugar costs or $15.20, so the other point is not on the graph either.

Suppose we also know that the flour and sugar 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 two of the three labeled points:

The graph from above with a line going through point (1,14) ending at (15,1).

The point is on the graph of because . Similarly, . But , so is not on the graph of . In general, if we have two lines in the coordinate plane,

  • The coordinates of a point that is on both lines makes both equations true.

  • The coordinates of a point on only one line makes only one equation true.

  • The coordinates of a point on neither line make both equations false.