Lesson 5 Getting Your Share of the Pie Develop Understanding

Learning Focus

Identify and create inverse variation functions, tables, and graphs to model real-world situations.

How do I model situations where some fixed amount has to be shared equally among more and more people, such as sharing my birthday cake with more and more of my family and friends?

Open Up the Math: Launch, Explore, Discuss

Tehani, Taska, and Trevor are planning a picnic to celebrate the completion of their engineering project. Tehani plans to bring three $12 -inch$ submarine sandwiches, Taska plans to bring three $20 -ounce$ beverages, and Trevor plans to bring $2$ cherry pies. They have chosen a location for the picnic that is equidistant from each of their homes, so they will each need to travel $10 miles$ to get to the picnic site: Tehani will travel by car, Taska by bus, and Trevor plans to bike to the picnic balancing the pies on his handlebars.

Tehani, Taska, and Trevor each think there will be too much food for just the three of them, so unbeknown to the others, they have each invited some of their friends.

This picnic scenario contains at least four important questions that can be explored:

Question 1: How many inches of submarine sandwich will each person get, depending upon the number of people that attend the picnic? Include the possibility that Tehani shows up with the subs, but no one else, not even Taska and Trevor, make it to the picnic.

Question 2: How many ounces of beverage will each person get, depending upon the number of people that attend the picnic? Include the possibility that Taska shows up with the drinks, but no one else, not even Tehani and Trevor, make it to the picnic.

Question 3: How much of a whole pie will each person get, depending upon the number of people that attend the picnic? Include the possibility that Trevor shows up with the pies, but no one else, not even Taska and Tehani, make it to the picnic.

Question 4: What is the average speed that each of the three friends who live $10 miles$ from the picnic might travel to get from their home to the picnic site? Assume that Tehani, traveling by car, might need $10 – 20 minutes$ , depending on traffic; Taska, traveling by bus, might need $20 – 40 minutes$ , depending upon the time of day; and Trevor, traveling by bike and balancing pies on his handlebars, might need $40 – 60 minutes$ , depending upon the obstacles he needs to navigate.

1.

Analyze two or more of these questions using various representations, including tables, graphs, and equations.

2.

Based on your explorations, what do all of the four contexts described in the questions have in common?

3.

What would be similar in the tables representing each of these four contexts? What would be different?

4.

What would be similar in the graphs representing each of these four contexts? What would be different?

5.

What would be similar in the equations representing each of these four contexts? What would be different?

6.

Each of these contexts describes an inverse variation. What are the defining features of an inverse variation function?

Ready for More?

Because the scenarios in the task didn’t make sense for input values between $0$ and $1$ , we haven’t explored what happens in an inverse variation function as $x$ approaches $0$ . Ignoring the restrictions imposed by the context, examine what happens when one or more of your inverse variation function equations is evaluated as $x$ gets smaller and smaller. What behavior do you predict for the graph as $x$ approaches $0$ and why?

Takeaways

Key features of inverse variation functions:

Vocabulary

inverse variation
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about inverse variations, functions in which the quantities are said to vary inversely because doubling one quantity cuts the other in half, tripling the quantity cuts the other quantity in thirds, etc.

Retrieval

1.

Given: $\overset{―}{J K} ⊥ \overset{―}{K L}$ and $\overset{―}{M N} ⊥ \overset{―}{K L}$

$J K = 30 ft$

$K L = 72 ft$

$M L = 13 ft$

Find the length of $J L$ , $M N$ , and $N L$ .

Begin by proving the relationship between $△ J K L$ and $△ M N L$ .

Show your work.

2.

Solve. $3 x^{2} + 5 x = 4$ .