Introduction to Rates

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Objective

SWBAT: • Define rate as a comparison of two quantities using different units. • Define a unit rate as a rate where one of the measurements is one. • Use a rate to determine a unit rate. • Use a unit rate to determine a rate.

Big Idea

Who is the fastest at making posters? Who is the slowest? Students learn about rates and unit rates and apply their knowledge to solve word problems.

Do Now

7 minutes

See my Do Now video for an explanation of my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today we are going to work with rates.  I want to see what students think about who is fastest and slowest at making posters.  Some students may recognize that Andrea is fastest, since she makes 3 posters in 6 minutes or 1 poster in 2 minutes.  Other students may be confused, and pick Kai because he makes the most posters.  It is okay if students are confused or not able to use math to support their answer, we will return to this example later in the lesson.

I ask students to share who they think is the fastest and slowest.  Students participate in a Think Pair Share. I quickly ask students to call out who they think is the fastest and slowest so I can gauge student thinking.

Class Notes: Introducing Rates

3 minutes

I want students to recognize that a ratio shows a relationship between two quantities, while a rate is a comparison of two quantities using different types of units.  Common ways of expressing rates are using words, fractions, or tables.  I stress the importance of labeling units when using rates.

We read over the rate examples.  I ask students how we could write “3 miles in 30 minutes” as a fraction.  I explain that it doesn’t matter which unit is in the numerator or denominator.  It is important that we label our units correctly.

Ratey the Math Cat

10 minutes

We watch the Ratey the Math Cat video at http://mathsnacks.com/ratey.php .   I pause the video frequently so we can talk about the rates and use the rates to answer the follow up questions.  Students work with their partner to answer and compare work.

I want students to be able to turn information into a rate and use that rate to create equivalent rates.  I make the connection between equivalent rates and equivalent fractions.  I do not teach students to cross-multiply and divide.  When I have done this in the past, I have found that students memorize the procedure and lose focus on what is actually going on in the problem.  Instead, I have students set up a fraction with the rate that they know and then create an equivalent rate with what they want to know.  Students determine whether they need to use multiplication or division to create the equivalent rate.  I stress to students that their units must match.  The units used in the numerator of the first fraction must match the units used in the numerator of the second fraction.  I explain that if students fail to write their units they are more likely to make mistakes.

Students may initially struggle with the last question.  I have them set up the rate of 50 miles/ 1 hour.  What can we multiply 1 hour to get 7.5 hours?  What must we multiply 50 by?  Students should be able to apply their knowledge of decimals to find the answer.

Rate Practice

8 minutes

I have students work in partners to answer these questions.  I walk around and monitor student progress.  I am looking to see that students are correctly setting up their equivalent rates and writing the units.

After 5 minutes we come back together.  I call on students to come up and show and explain their thinking for problem (b) and (d).  I ask students if they agree or disagree with their classmates.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

Unit Rates

5 minutes

I want students to understand that a unit rate is when one part of the rate is compared to a single unit of the other part of the rate.  Students should realize that traveling 75 miles per hour is equivalent to traveling 75 miles in one hour.  We review the examples and I ask for a student to create their own unit rate.

Students work in partners to write the guided practice examples as rates.  I ask students how they could show how long it takes Jeremy to run 1 mile.  Students should set up the beginning rate as a fraction.  To create an equivalent rate, they must divide 3 miles by 3 to get 1 mile.  If they divided the numerator by 3, they must divide the denominator by 3.

Making Posters

10 minutes

Students work with their partner on the Making Poster questions.  I walk around and monitor student progress.  I am looking to see that students are labeling their rates and correctly calculating unit rates.  My goal is that students complete problems 1-4. Students are engaging in MP:1 Make sense of problems and persevere in solving them and MP5: Attend to precision.

If students are struggling, I may ask one or more of the following questions:

  • What information do you know?  What information are you trying to find out?
  • How can you set up equivalent rates?
  • How can we move from 4 posters to 1 poster? 
  • If you divide the numerator by 4, what must you do to the denominator?
  • What does it mean to be the fastest?  What information should you compare?

I Post A KeyAs students complete 1-3 questions, they raise their hand.  If they are on track, I send them to check their answers.  Then they can move on to question 4, which is more complex.  Students must recognize that they need to divide 20 minutes by 8 to find the amount of time it takes Jason to make one poster.

If students successfully complete problem 4, they can move on to “More Shopping with Ratey the Cat”.  I am interested to see how students work through part (d).  

Closure and Ticket to Go

7 minutes

I ask students who they think the fastest and slowest students are at creating posters.  The important part is that students are able to express why they think the way they do.  I push students to use rates and units to defend their answer.  I declare that I think Kai is the fastest, because he completed the most posters.  I want students to explain that we can’t compare rates when they have different amounts of time.  I am looking for students to explain that we can compare the students’ times for completing one poster.  Students are engaging in MP2: Reason abstractly and quantitatively and MP3: Construct viable arguments and critique the reasoning of others.

If I have time, I ask students to explain how they worked with Jason’s information in problem 4.

With a few minutes left, I pass out the Ticket to Go and the HW Introduction to Rates.