### Submitted by: Ian Whitacre, Sebnem Tekin

In this middle school lesson plan students use an interactive simulation called Unit Rates to  compare prices by identifying multiplicative relationships.

### Students will:

1. Identify relationships between quantities
2. Find missing values in a ratio table by making use of multiplicative patterns
3. Calculate unit rates
4. Generate equivalent ratios
5. Solve problems by finding and using unit rates

### Materials:

• Computers or other devices with internet access

### Preparation:

This lesson plan features an interactive simulation titled Unit Rates, developed by our award-winning partner PhET through the University of Colorado Boulder. The simulation invites students interact with the unit rate in the contexts of price per item, price per pound, and speed.

Review the Unit Rates Simulation Overview to learn about the sim controls, model simplifications, and insights into student thinking.

Preview and play with the Unit Rates sim to plan how you will adapt it to your students’ needs.

Build background knowledge or reinforce topics with these BrainPOP movies: Acceleration; Distance, Rate, and Time; Customary Units; Comparing Prices; Money; Decimals; and Multiplying Decimals.

### Lesson Procedure:

1. Play one or more of the BrainPOP movies suggested in the Preparation section for background.
2. Ask students how much apples cost. Have them write down what they know about the price of apples. In a brief, whole class discussions, call on a few students to share what they wrote. Then emphasize that prices are rates (e.g., cost per apple or cost per pound), and not just amounts of money.
3. Pose a quick task based on students responses. For example, if a student says an apple costs 25 cents, ask how much for four apples, eight apples, and twelve?
4. Display the Unit Rates sim on an interactive whiteboard. At their own computers, instruct students to go to Unit Rates: Shopping and play for 3–5 minutes. As they play, circulate and interact. Look at students’ screens and listen in on their conversations. See what they are noticing about the sim and what mathematical ideas are coming up. Feel free to voice important discoveries that some students make. Ask individual students (or small groups of students) open-ended questions about how the sim works and what they think about the relationships between the numbers on the double number line.
5. Lead a brief discussion in which students can share their discoveries or questions.
• Illustrate students’ discoveries or questions on the projected sim, or have students come up to the board to input values or to illustrate their ideas.
• Make sure that everyone knows how to use the relevant controls in the sim (e.g., how do we switch from pears to potatoes?).
• Introduce and define the term unit rate. Encourage students to say “unit rate” or “unit price” when discussing their strategies.
• Begin to focus the discussion on what the unit rate looks like on the double number line (by noticing the differences in prices and corresponding differences in numbers of items).
6. Challenge the class to use the sim to compare the prices of apples and pears. Ask them to identify which are less expensive. Offline, have them record the unit rate of apples and the unit rate of pears. Have them do the same for lemons and oranges.
7. As they work,circulate and notice strategies students are using to find the missing values. Listen especially for efficient strategies that involve reasoning about rates (e.g., using division to find the unit rate or noticing differences in prices and comparing those to differences in the number of items). Possible strategies include:
• Removing all but one item, so that the sim tells them the price per item.
• Dividing the total cost (for a bag) by the number of items
• Noticing the difference in price when one item is removed
Select one or more student to share their solutions.
8. Now ask students which are less expensive: carrots or cucumbers. Instruct them use the sim to compare prices, and to record the unit rate of each offline. Repeat with potatoes and tomatoes.
9. Circulate the room, listening  in on students’ conversations and look for written work. How are they finding the unit rates? Possible strategies include:
• Guess-and-check with repeated addition or multiplication
• Dividing total cost by the number of items
10. Bring the class together and have volunteers share how they found the unit rate. Focus on the reasoning associated with the strategies. For example, ask “Why was it helpful to divide?” and “How can we tell this answer makes sense?”
11. Finally, present the following challenge: A store offers a package of 16 mushrooms for \$2. You buy all of the packages on the shelf, for a total of 112 mushrooms. How much will it cost? Again, have them use the sim to figure out the answer. Have them record their answer offline. Encourage them to use two or more strategies to solve the problem.
12. Circulate the classroom one last time, again listening to students’ discussions and looking at their written work. After everyone is done, identify one or more students to present their work to the class. Look for these two specific methods:
• Find the unit rate and apply it
• Use equivalent rates
Filed as:  3-5-ETS1-2, 3-5-ETS1-3, CCSS.Math.Content.4.NBT.A.3, CCSS.Math.Content.5.MD.B.2, CCSS.Math.Content.5.NBT.B.5, CCSS.Math.Content.5.NF.B.3, CCSS.Math.Content.5.NF.B.6, CCSS.Math.Content.5.NF.B.7c, CCSS.Math.Content.6.RP.A.1, CCSS.Math.Content.6.RP.A.2