 # A Synthesis of Representations of Equivalent Ratio Collections

Video solutions to help Grade 6 students learn how to associate with the ratio A:B with the ordered pair (A, B) and plot the line in the xy - coordinate plane.

Lesson 15 Student Outcomes

• Students associate with each ratio A:B the ordered pair (A, B) and plot it in the xy - coordinate plane.
• Given a ratio table, students plot the ratios in the plane and observe that they lie on a line through the origin.
• Students conclude that the coordinates in the line satisfy y = ky where k is the value of an associated ratio.

Lesson 15 Summary

There are several ways that we can represent the same collection of equivalent ratios. These include ratio tables, tape diagrams, double number line diagrams, equations, and graphs on coordinate planes.

Lesson 15 Classwork

Exploratory Challenge
At the end of this morning’s news segment, the local television station highlighted area pets that need to be adopted. The station posted a specific website on the screen for viewers to find more information on the pets shown and the adoption process. The station producer checked the website two hours after the end of the broadcast and saw that the website had 24 views. One hour after that, the website had 36 views.

Exercise 1
Create a table to determine how many views the website probably had one hour after the end of the broadcast based on how many views it had two and three hours after the end of the broadcast. Using this relationship, predict how many views the website will have 4, 5, and 6 hours after the end of the broadcast.

Exercise 2
What is the constant number, c, that makes these ratios equivalent?
Using an equation, represent the relationship between the number of views (v) the website received and the number of hours (h) after this morning's news broadcast.

Exercise 3
Use the table created in Exercise 1 to identify sets of ordered pairs that can be graphed.

Exercise 4
Use the ordered pairs you created to depict the relationship between hours and number of views on a coordinate plane.
Label your axes and create a title for the graph. Do the points you plotted lie on a line? If so, draw the line through the points.

Exercise 5
Predict how many views the website will have after twelve hours. Use at least two representations (e.g., tape diagram, table, double number line diagram) to justify your answer.

Exercise 6
Also on the news broadcast, a chef from a local Italian restaurant demonstrated how he makes fresh pasta daily for his restaurant. The recipe for his pasta is below:
3 eggs, beaten
1 teaspoon salt
2 cups all-purpose flour
2 tablespoons water
2 tablespoons vegetable oil
Determine the ratio of the number of tablespoons of water to the number of eggs.
Provided the information in the table below, complete the table to determine ordered pairs. Use the ordered pairs to graph the relationship of the number of tablespoons of water to the number of eggs.
What would you have to do to the graph in order to find how many eggs would be needed if the recipe was larger and called for 16 tablespoons of water?
How many eggs would be needed if the recipe called for 16 tablespoons of water?

Exercise 7
Determine how many tablespoons of water will be needed if the chef is making a large batch of pasta and the recipe increases to 36 eggs. Support your reasoning using at least one diagram you find applies best to the situation, and explain why that tool is the best to use.

Problem Set
1. The producer of the news station posted an article about the high school’s football championship ceremony on a new website. The website had 500 views after four hours. Create a table to show how many views the website would have had after the first, second, and third hours after posting, if the website receives views at the same rate. How many views would the website receive after 5 hours?

2. Write an equation that represents the relationship from question 1. Do you see any connections between the equations you wrote and the ratio of the number of views to the number of hours?

3. Use the table in question 1 to make a list of ordered pairs that you could plot on a coordinate plane.

4. Graph the ordered pairs on a coordinate plane. Label your axes and create a title for the graph. Connect the ordered pairs.

5. Use multiple tools to predict how many views the website would have after 12 hours.

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