Common Core Standards: Math

Functions 8.F.A.2

2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Jon's grandmother moved to the U.S. from Italy when she was in her twenties. Over the years, she's done a great job of learning English, but every once in a while she says something that's a little off. For instance, during a production of Into the Woods, she wished him a spirited, "In the mouth of the wolf!" It meant something very different to her than it did to Jon, who played one of the three little pigs.

Functions are kind of like Jon's grandmother and her language situation. Some ideas translate well from one interpretation to another. Other ideas simply don't.

Students should know that there are a number of ways of expressing the idea of a function. A word problem, a graph, a table of values, and an algebraic equation can all be used to express the same idea. Translating these ideas from one form to another will be useful especially when two different representations of functions have to be compared.

In order to convert functions from word problems to tables, equations, and graphs, students need to know what functions are and how to handle them. Thankfully, little mistakes in translation can be easily checked if students remember the definition of a function and use their knowledge of linear equations. When it comes to Jon and his nonna, though, he may just need to brush up on his Italian skills.

Drills

1. Marco and Katie have been asked to tutor in math. Marco charges \$7.75 per hour and Katie charges a flat rate of \$10 plus \$4.25 per hour of tutoring. If a 2-hour tutoring session is needed, who earns more money?

Katie earns \$3.00 more than Marco

What we have here are two functions. The input is the number of hours tutoring h and the number of sessions s, and the output is the price p. We can translate this word problem into two different equations: Marco's price is p = 7.75h while Katie's price is p = 10s + 4.25h. Since we need 2 hours of tutoring and one session, we can substitute h = 2 and s = 1 and calculate the p for each. We end up with p = \$15.5 for Marco and p = \$18.5 for Katie. This translates to a difference of 18.5 – 15.5 = \$3 in Katie's favor.

2. Marco and Katie have been asked to tutor in math. Marco charges \$7.75 per hour and Katie charges a flat rate of \$10 per session plus \$4.25 per hour of tutoring. If one 8-hour session is needed, who earns more money?

Marco earns \$18 more than Katie

Using the hours and sessions as the inputs and the price as the output, we can calculate Marco's price as p = 7.75h = 7.75(8) = \$62 and Katie's price as p = 10s + 4.25h = 10(1) + 4.25(8) = \$44. This means Marco earns 62 – 44 = \$18 more than Katie for a single 8-hour session.

3. Marco and Katie have been asked to tutor in math. Marco charges \$7.75 per hour and Katie charges a flat rate of \$10 per session plus \$4.25 per hour of tutoring. If four 2-hour sessions are needed, who earns more money?

Katie earns \$12 more than Marco

Here, we have four sessions of two hours for a total of eight hours. Our inputs should be s = 4 and h = 8. If we use the function rules to calculate the prices, we end up with Marco earning p = 7.75h = 7.75(8) = \$62 and Katie earning p = 10s + 4.25h = 10(4) + 4.25(8) = \$74. Just by adding more sessions, Katie manages to earn 74 – 62 = \$12 more than Marco.

4. Marco and Katie have been asked to tutor in math. Marco charges \$7.75 per hour and Katie charges a flat rate of \$10 per session plus \$4.25 per hour of tutoring. If both of them plan to work 20 hours this week and both hope to maximize their pay, which is the best option for them?

It doesn't matter for Marco, but Katie should work ten 2-hour sessions

Since Marco only charges per hour, he'll earn the same amount of money (\$155) no matter how many sessions he has. That means (A) and (B) aren't true because Marco will earn the same amount either way. Katie earns \$10 more for each additional session she has, so she'll want to have lots of sessions in order to maximize her pay. The option with the most sessions is (D), which will earn her \$185 instead of the \$135 that (C) would give her.

5. Giovanni and Natalie have different cell phone plans. Gio's plan charges him \$0.20 per text. Natalie's plan gives her 20 texts free per month, but after that she has to pay \$0.25 per text. If both of them send 500 texts in one month, who pays more?

Natalie pays \$20 more than Gio

Of the total 500 texts, Natalie only gets charged for the last 480 texts. Since each cost her \$0.25, we can multiply 480 texts by 0.25 per text, which gives her a total of \$120. Gio, on the other hand, pays \$0.20 for each of the 500 texts he sent, meaning a total of \$100. Natalie pays 120 – 100 = \$20 more than Gio.

6. Giovanni and Natalie have different cell phone plans. Gio's plan charges him \$0.20 per text. Natalie's plan gives her 20 texts free per month, but after that she has to pay \$0.25 per text. Unfortunately, Gio breaks his phone and Natalie loses hers so each is only able to send 10 texts that month. Who pays less for those 10 texts?

Natalie pays \$2.50 less than Gio

Natalie pays nothing for the first 20 texts per month, and the first 10 fall under that umbrella. Gio has to pay \$0.20 for every single text, which means a total of \$0.20(10) = \$2 worth of texts. Natalie pays nothing, which is \$2 less than the amount Gio has to pay. Not that it matters, since their new phones will cost them an arm and a leg, anyway.

7. Theresa has managed to be remarkably steady with her exercise regimen, losing 2.5 pounds every week for the past 6 weeks. Denise tried a fad diet and has tracked her weight loss with the following chart. Which of the following is true?

• Theresa has managed to be remarkably steady with her exercise regimen, losing 2.5 pounds every week for the past 6 weeks. Denise tried a fad diet and has tracked her weight loss with the following chart. Which of the following is true?

Theresa lost 15 pounds total while Denise lost 12 pounds

Theresa lost 2.5 pounds per week for 6 weeks, which means a total of 2.5 × 6 = 15 pounds. According to the chart, Denise lost a total of 12 pounds over the course of her six weeks. This means (B) is right. The other options are incorrect because Theresa lost 3 more pounds than Denise. They both should be very proud of themselves and celebrate the achievement with cheeseburgers.

Happytown has been charting its temperatures over the course of a year. Neighboring Smileyville has kept track too, but chooses to list the low and high temperatures for each month. Which town had a warmer August?

 Month Low Temp (°F) High Temp (°F) January 12 22 February 22 30 March 30 45 April 45 50 May 50 68 June 68 75 July 75 95 August 95 80 September 80 60 October 60 35 November 35 20 December 20 12
• Happytown has been charting its temperatures over the course of a year. Neighboring Smileyville has kept track too, but chooses to list the low and high temperatures for each month. Which town had a warmer August?

Happytown, at 101°F

We can see clearly that the highest temperature in Smileyville during August was 95°F. All it takes is going down to the August row and looking at the high temperature. To find the high August temperature in Happytown, we just need to look at the highest point on the graph in the August section on the x-axis. If we look at the y-axis, it's over 100°F. Since 100 > 95, we know that Happytown had a higher August temperature.

8. Which town had a colder January?

Smileyville, with temperatures in the teens

From the data table, Smileyville's lowest January temperature was 12°F. Looking at the graph of Happytown's temperatures, we can see that its lowest January temperature was above 20°F. This means that Smileyville had lower temperatures than Happytown during January because it was in the teens rather than the twenties.

9. Which town had the largest rate of change and during which month?

Happytown in September