Introduction to Linear Relationships

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Create a Graph of Proportional Relationships.

We have two stacks of styrofoam cups as below.

  • One stack has 6 cups, and its height is 15 cm.
  • The other stack has 12 cups, and its height is 23 cm.

(1). How many cups are needed for a stack with a height of 50 cm? To answer this question, create a graph of the situation below.

(1). What are some ways you can tell that the number of cups is not proportional to the height of the stack?

(2). What is the slope of the line in your graph? What does the slope mean in this situation?

(3). At what point does your line intersect the vertical axis? What do the coordinates of this point tell you about the cups?

(4). How much height does each cup after the first add to the stack?


Use a Graph of Linear Relationships to Solve Real World Problems.

To paint a house, a painting company charges a flat rate of $500 for supplies, plus $50 for each hour of labor.

(1). How much would the painting company charge to paint a house that needs 20 hours of labor?

(2). How much would the painting company charge to paint a house that needs 50 hours of labor?

(3). Draw a line representing the relationship between x, the number of hours it takes the painting company to finish the house, and y, the total cost of painting the house. Label the two points from the earlier questions on your graph.

(4). Find the slope of the line. What is the meaning of the slope in this context?


Compare Situations From Graphs of Linear Relationships.

Tyler and Elena are on the cross country team. Tyler’s distances and times for a training run are shown on the graph.

Elena’s distances and times for a training run are given by the equation= 8.5 , where represents distance in miles and represents time in minutes.

(1). Who ran farther in 10 minutes? 

(2). How much farther?

(3). Explain how you know.

(4). Calculate each runner’s pace in minutes per mile.

(5). Who ran faster during the training run?

(6). Explain or show your reasoning using the applet below.



A restaurant offers delivery for their pizzas. The total cost is a delivery fee added to the price of the pizzas. One customer pays $25 to have 2 pizzas delivered. Another customer pays $58 for 5 pizzas. How many pizzas are delivered to a customer who pays $80?


Write an equation for the line that passes through (2, 5) and (6, 7).