<p>This lesson introduces students to the concept of rates, particularly unit rates. Students will:&nbsp;<br>- calculate unit rates, particularly in situations combining unit pricing and constant speed.&nbsp;<br>- use unit rates to solve problems.&nbsp;</p>

<p>- How are relationships represented mathematically?&nbsp;<br>- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?&nbsp;<br>- How can mathematics help us communicate more effectively?&nbsp;<br>- How may patterns be used to describe mathematical relationships?&nbsp;<br>- How can mathematics help to quantify, compare, depict, and model numbers?&nbsp;<br>- What makes a tool and/or strategy suitable for a certain task?&nbsp;</p>

<p>- Ratio: A comparison of two numbers by division.&nbsp;<br>- Unit Rate: A rate simplified so it has a denominator of 1.</p>

<p>- one copy of the Best Bargains worksheet (M-6-7-2_Best Bargains and KEY) per pair of students&nbsp;<br>- one copy of the Problem Solving with Unit Rates sheet (M-6-7-2_Problem Solving with Unit Rates and KEY) per student&nbsp;<br>- one copy of the Lesson 2 Exit Ticket (M-6-7-2_Lesson 2 Exit Ticket and KEY) per student</p>

<p>- Use the Best Bargains worksheet (M-6-7-2_Best Bargains and KEY) to assess students' comprehension.&nbsp;<br>- Examine students' responses on the Problem Solving with Unit Rates worksheet (M-6-7-2_Problem Solving with Unit Rates and KEY) to assess their ability to solve real-world situations using rates.&nbsp;<br>- The Lesson 2 Exit Ticket (M-6-7-2_Lesson 2 Exit Ticket and KEY) can be used to quickly evaluate student mastery before the end of the lesson.</p>

<p>Scaffolding, Active Engagement, Explicit Instruction, and Formative Assessment&nbsp;<br>W: Students will learn about and practice finding unit rates, comparing quantities using unit rates, as well as applying rates to solve real-world situations.&nbsp;<br>H: The lesson will engage students by examining a rate that is essential to their life - their heart rate!&nbsp;<br>E: Students will learn about the concept of rates through real-life examples and compare them using various methods (e.g., "doubling" or "halving" rates, unit rates, etc.).&nbsp;<br>R: Students will have several opportunities to review and refresh their grasp of the concept of rates. The three activities are progressively more challenging, but each uses concepts from the previous activity. Each activity will allow students to further expand their understanding of what they learnt in the prior activity.&nbsp;<br>E: Students' progress can be assessed using the Best Bargains and Problem Solving with Unit Rates worksheets. An exit ticket is also available to help measure the level of student understanding.&nbsp;<br>T: Use the Extension section to personalize the lesson to students' specific requirements. The Routine section includes strategies for reviewing lesson concepts throughout the year. The Small Group section contains exercises for students who could benefit from more learning opportunities. The Expansion section includes a more difficult practice for students who are ready to go beyond the requirements of the standard.&nbsp;<br>O: The lesson begins with a hands-on activity, followed by a teacher-led introduction to the idea. Students work with you to answer questions about rates before working independently to assess their comprehension of the concepts.&nbsp;</p>

<p><strong>Activity 1</strong><br><br>Have students check their pulse (by holding a finger against the inside of their wrist). Allow students time to find it and practice counting each pulse. Then, instruct students to count their pulse for 30 seconds. After they've recorded their pulses on paper (so they don't forget), have them calculate their heart rate for 1 minute (by doubling the number they recorded).<br><br><strong>"The number you recorded, beats per minute, represents a rate. It compares two values: the number of your heartbeats and the amount of time that has passed. Rates exist all around us. When you ride in an automobile, it travels at a certain rate, which is commonly expressed in miles per hour. What two quantities do a car's rate compare to?"</strong> (<i>Miles traveled and time passed.</i>)&nbsp;<br><br><strong>"The number you recorded is a sort of rate known as a </strong><i><strong>unit rate</strong></i><strong>. We already discussed why it's called a rate. It compares two different quantities. What does the word </strong><i><strong>unit</strong></i><strong> mean?"</strong> Help students understand that the word <i>unit</i> implies, simply, one. If necessary, compare it to the terms unicorn (one horn), unicycle (one wheel), and even union (one country).<br><br><strong>"So a rate may tell us, for example, how much it will cost us to buy six ounces of juice or how many miles we'd travel for every four hours, but those aren't unit rates. A unit rate will tell us how much it will cost for one ounce of juice, how far we will travel in one hour of driving, and how many times our heart will beat in one minute."</strong><br><br><strong>Activity 2</strong><br><br><strong>"Suppose you go to the supermarket to get some soda and you have two options. You can get 6 cans of soda for $2.25 or 12 cans of soda for $3.00. Which is a better deal?"</strong> Write the two options on the board and ask the class which is the better offer. They may conclude that if 12 cans cost $3, 6 cans should only cost $1.50, or use another approach to compare the two.&nbsp;<br><br><strong>"If you buy 12 cans for $3.00, how much are you paying for each can of soda?"</strong> (<i>$0.25</i>) Ask students to explain how they arrived at their answer (<i>by diving 3 by 12</i>). Emphasize that, to get the dollars per can, we should divide the dollars by the number of cans.<br><br><strong>"If you buy 6 cans for $2.25, how much are you paying for each can of soda?"</strong> (<i>$0.375, or about 38 cents a can.</i>)&nbsp;<br><br><strong>"So, this clearly supports the conclusion that buying 12 cans is cheaper. We saved $0.13 each can."</strong>&nbsp;<br><br>Write <i>$2.99 for 32 ounces</i> and <i>$4.75 for 48 ounces</i> on the board. <strong>"Assume orange juice is offered for these two amounts. Which container of orange juice is the best deal? To discover out, we may calculate the </strong><i><strong>cost per ounce</strong></i><strong>—how much one ounce costs in each package. To calculate the cost per ounce, we simply divide the overall cost, like $2.99, by the number of ounces, 32."</strong> Allow students time to determine the cost per ounce for each package. (The answers are around 9.3¢ and 9.9¢ per ounce, respectively.)<br><br><strong>"Which container should we purchase if we want to get the best deal?"</strong> (<i>The 32-ounce container</i>) <strong>"And how much do we save per ounce?"</strong> (<i>About 0.6¢</i>) Explain to students that we're saving 6 tenths of a cent per ounce—less than a penny per ounce. <strong>"Even though the savings aren't huge, the smaller container is definitely cheaper in this case.</strong>"<br><br><strong>"Here's one more example: In 6.5 hours, Clark drives at a constant speed for a total of 273 miles."</strong> Write these figures on the board. <strong>"If we need to write a rate, we can simply state his rate is 273 miles per 6.5 hours. However, in order to calculate his unit rate, we must first determine how many miles he travels every hour. We divide the number of miles by the number of hours. What is Clark's unit rate?"</strong> (<i>42 miles per hour</i>)<br><br><strong>"Notice that I said Clark drives at a constant speed, which means he drives 42 miles per hour throughout the entire trip. If he doesn't move at a constant speed, he may drive slightly faster or slower at times, but it all averages out to 42 miles per hour."</strong>&nbsp;<br><br>Students should work in pairs to complete the Best Bargains worksheet (M-6-7-2_Best Bargains and KEY).<br><br><strong>Activity 3</strong><br><br><strong>"Suppose a store has an offer that you can get 4 apples for $1.20. At this rate, how much would it cost to buy 8 apples?"</strong> (<i>$2.40</i>) Have students explain how they calculated the cost of 8 apples.<br><br><strong>"At this rate, how much would it cost for 7 apples?"</strong> If students are struggling, ask them why this problem with 7 apples is more difficult than the problem with 8 apples. They will see that 8 is a multiple of 4, whereas 7 is not. As an intermediate problem, ask how much 1 apple would cost. Keep in mind that this is a unit rate—the cost of one apple. They should determine that one apple costs $0.30.&nbsp;<br><br><strong>"Now that you know the unit rate, is it easier to find the cost for 7 apples?"</strong> (<i>Yes</i>) <strong>"It's easier because we can just multiply our cost for 1 apple by 7. How much do 7 apples cost?"</strong> (<i>$2.10</i>)<br><br>Note that the answer makes sense in the context of the fact that it costs $2.40 to buy 8 apples and each apple costs 30¢. To get the cost of 7 apples, simply subtract 30¢ from the cost of 8 apples.&nbsp;<br><br><strong>"So, we can use unit rates as kind of a bridge to solve problems about different quantities."</strong> Write the rate of <i>150 miles in 6 hours</i> on the board.&nbsp;<br><br><strong>"Traveling at this rate, how far can someone travel in 12 hours?"</strong> (<i>300 miles</i>)<br><br><strong>"At this rate, how far can someone travel in 3 hours?"</strong> (<i>75 miles</i>) Students may use a unit rate, but emphasize that if you travel for half the time, you would go half the distance.<br><br><strong>"At this rate, how far can someone travel in 7 hours?"</strong> Remind students that they should first identify the unit rate—how far someone can travel in one hour—and then use it to calculate how far they can travel in 7 hours. (The answer is 175 miles.)&nbsp;<br><br><strong>"At this rate, how far can someone travel in two-and-a-half hours?"</strong> (<i>62.5 miles</i>)&nbsp;<br><br>Allow students to work individually on the Problem Solving with Unit Rates worksheet (M-6-7-2_Problem Solving with Unit Rates and KEY).&nbsp;<br><br><strong>Extension:&nbsp;</strong><br><br>Use the strategies listed below to adjust the lesson to your students' needs throughout the year.<br><br><strong>Routine:</strong> Students can continue to investigate rates provided in real life by bringing in newspaper articles (or articles printed from the Internet) that include rates, such as gasoline prices (dollars per gallon), airplane or spaceship speeds, birth rates, literacy rates, etc. Suggest that students make a daily/weekly notebook of all the rates they encounter in the real world.&nbsp;<br><br><strong>Small Group:</strong> Ask students to design their own pricing strategies for various objects (oranges, apples, etc.). Students' pricing plans should not be shown in unit rates. Given unit rates might help students plan their pricing strategies. (That is, you want to charge more than $0.30 but less than $0.40 each apple; how much can you charge for 8 apples?) Students can compare their pricing plans to those of others in their group, selecting which "store" they would use to buy each of various items. The websites listed in the Related Resources section below might be used for extra instruction or practice.&nbsp;<br><br><strong>Expansion:</strong> This lesson can be developed to include percentages and rates (or unit rates). Inflation rates, for example, are a unit rate (percent change per year). The lesson can also be broadened to look at an object's acceleration, which is a rate that measures how fast another rate changes.</p>