Comparing Unit Rates Lesson Plan: Unit Rates

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.

Lesson Plan Common Core State Standards Alignments

Grade: 04

CCSS.Math.Content.4.NBT.A.3

Use place value understanding to round multi-digit whole numbers to any place.

Grade: 05

CCSS.Math.Content.5.MD.B.2

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Grade: 05

CCSS.Math.Content.5.NBT.B.5

Fluently multiply multi-digit whole numbers using the standard algorithm.

Grade: 05

CCSS.Math.Content.5.NF.B.3

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Grade: 05

CCSS.Math.Content.5.NF.B.6

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Grade: 05

CCSS.Math.Content.5.NF.B.7c

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Grade: 06

CCSS.Math.Content.6.RP.A.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

Grade: 06

CCSS.Math.Content.6.RP.A.2

Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1

Grade: 06

CCSS.Math.Content.6.RP.A.3

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Grade: 07

CCSS.Math.Content.7.EE.A.2

Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

Grade: 07

CCSS.Math.Content.7.RP.A.2

Recognize and represent proportional relationships between quantities.

Grade: K, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12

CCSS.Math.Practice.MP3

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Grade: K, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12

CCSS.Math.Practice.MP7

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Grade: K, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12

CCSS.Math.Practice.MP8

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

Lesson Plan Next Generation Science Standards Alignments

Students will:

Identify relationships between quantities
Find missing values in a ratio table by making use of multiplicative patterns
Calculate unit rates
Generate equivalent ratios
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:

Play one or more of the BrainPOP movies suggested in the Preparation section for background.
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.
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?
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.
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).
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.
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.
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.
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
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?”
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.
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