Understanding and Predicting Proportional Relationships

Lesson 4.2.1 – Teacher Notes • Standard: • 7.RP.A.2a • Recognize and represent proportional relationships between quantities. • Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. • Full mastery by end of chapter • Lesson Focus: • The focus is for students to understand what a proportional relationship looks like and how growth is shown in a table and graph. (4-25, 4-26, and 4-27) ***It may be a good idea for teachers to have students create a table and/or graph to prove their work in 4-27. • I can translate a proportion into a table to determine equivalency. • Graph ordered pairs on a coordinate plane. • Represent ratios as ordered pairs on the coordinate plane. • Recognize that ordered pairs are proportional if they form a straight line through the origin. • Calculator: Yes • Literacy/Teaching Strategy: Think-Pair-Share (4-21 and 4-27); Walk and Talk (4-26)

Grocery stores often advertise special prices for their foods. You might see a sign that says, “Special Today! Buy 2 lbs. of apples for $1.29!” How would we use that information to predict how much we will pay if you want to buy 6 lbs. of apples? Or just 1 pound of apples? • The way that the cost of apples grows or shrinks allows you to use a variety of different strategies to predict and estimate prices for different amounts of apples. In this section, you will explore different kinds of growth patterns. You will use those patterns to develop strategies for making predictions and deciding if answers are reasonable.

As you work in this section, ask yourself these questions to help you identify different patterns: • How are the entries in the table related? • Can I double the values? • What patterns can I see in a graph?

4-21. COLLEGE FUND Five years ago, R.J’s grandma put some money in a college savings account. The account pays simple interest, and after five years, the account is worth $500. R.J predicts that if he does not deposit or withdraw any money, then the account balance will be $1000 five years from now. a. Do you agree with R.J’s reasoning? Explain why or why not.

4-22. Last week, R.J got his bank statement in the mail. He was surprised (O.O) to see a graph that showed, although his balance was growing at a steady rate, the bank predicted that in 5years his account balance would be only $600. “What is going on?” he wondered. “Why isn’t my money growing the way I thought it would?” R.J decided to look more carefully at his balances for the last few years to see if the bank’s prediction might be a mistake. He put together the table below.

Gustavo’s Money a. How has Gustavo’s bank balance been growing? b. Is the bank’s prediction a mistake? Explain your answer.

4-24. Once R.J saw the balances written in a table, he decided to take a closer look at the graph from the bank to see if he could figure out where he made the mistake in his prediction. There is additional information about R.J’s account that you can tell from the graph. For example, what was his starting balance? • How much does it grow in 5 years? b. Is it possible that R.J’s account could have had $0 in it in Year 0? Why or why not?

4-25. FOR THE BIRDS • When filling her bird feeder, Brianna noticed that she paid $27 for four pounds of bulk birdseed. “Next time, I’m going to buy 8 pounds instead so I can make it through the spring. That should cost $54.” • Does Brianna’s assumption that doubling the amount of birdseed would double the price make sense? Why or why not? • How much would you predict that 2 pounds of birdseed would cost?

4-25. FOR THE BIRDS • When filling her bird feeder, Sonja noticed that she paid $27 for four pounds of bulk birdseed. “Next time, I’m going to buy 8 pounds instead so I can make it through the spring. That should cost $54.” • b. To check her assumption, she found a receipt for 1 pound of birdseed. She decided to make a table, which is started below. Copy and complete her table.

4-25. FOR THE BIRDS • When filling her bird feeder, Brianna noticed that she paid $27 for four pounds of bulk birdseed. “Next time, I’m going to buy 8 pounds instead so I can make it through the spring. That should cost $54.” • c. How do the amounts in the table grow? • d. Does the table confirm Brianna’s doubling relationship? Give two examples from the table that show how doubling the pounds will double the cost.

4-26. Examine the graphs below. • Describe how each graph is the same. • b. Describe what makes each graph different. • c. How do the differences explain why doubling works in one situation and not in the other? Generalize why doubling works in one situation and not in another. Gustavo Sonja

4-26. Examine the graphs below. • d. The pattern of growth in Brianna’s example of buying birdseed is an example of a proportional relationship. In a proportional relationship, if one quantity is multiplied by a scale factor, the other is scaled by the same amount. R.J’s bank account is not proportional, because it grows differently; when the number of years doubled, his balance did not. Gustavo Sonja

4-26. Examine the graphs below. d. Work with your team to list other characteristics of proportional relationships, based on Sonja’s and Gustavo’s examples. Be as specific as possible. Gustavo Sonja

4-27. IS IT PROPORTIONAL? (also on the handout) • When you are making a prediction, it is important to be able to recognize whether a relationship is proportional or not. • Your Task: Work with your team to read each new situation below. Decide whether you think the relationship described is proportional or non-proportional and justify your reasoning. Be prepared to share your decisions and justifications with the class. Make a table for each situation to prove your answer. • Carlos wants to buy some new video games. Each game he buys costs him $36. Is the relationship between the number of games Carlos buys and the total price proportional? • b. A single ticket to a concert costs $56, while buying five tickets costs $250. Is the relationship between the number of tickets bought and the total price proportional?

4-27. IS IT PROPORTIONAL? (also on the handout) • When you are making a prediction, it is important to be able to recognize whether a relationship is proportional or not. • Your Task: Work with your team to read each new situation below. Decide whether you think the relationship described is proportional or non-proportional and justify your reasoning. Be prepared to share your decisions and justifications with the class. Make a table for each situation to prove your answer. • c. Vu is four years older than his sister. Is the relationship between Vu and his sister’s age proportional? • d. Janna runs at a steady pace of 7 minutes per mile. Is the relationship between the number of miles she ran and the distance she covered proportional?

4-27. IS IT PROPORTIONAL? (also on the handout) • When you are making a prediction, it is important to be able to recognize whether a relationship is proportional or not. • Your Task: Work with your team to read each new situation below. Decide whether you think the relationship described is proportional or non-proportional and justify your reasoning. Be prepared to share your decisions and justifications with the class. Make a table for each situation to prove your answer. • e. Carl just bought a music player and plans to load 50 songs each week. Is the relationship between the number of weeks after Carl bought the music player and the number of songs on his player proportional? • f. Anna has a new video game. It takes her five hours of playing the game to master level one. After so much time, Anna understands the game better and it only takes her three hours of playing the game to master level two. Is the number of hours played and the game level proportional?

Practice • Drew is an artist. He paints portraits. The table below shows the number of portraits painted in hours. the Do the numbers in the table represent a proportional relationship? • Daisy made an envelope from sheets of paper. The table below shows the number of envelopes made by the number of sheets. Do the numbers in the table represent a proportional relationship? • Alice went to market and bought comics. The table below shows the price for different numbers of comics. Do the numbers in the table represent a proportional relationship?

Lesson 4.2.1 1) Which graph is proportional? 7.RP.2a 2) Which table is proportional?

Understanding and Predicting Proportional Relationships