Unit Rate

Students work in pairs on the Think About It problem.  

Many pairs will incorrectly conclude that Deal 2 is the better deal.  I take the time to hear from many students about their thinking.

• Some pairs will say that Deal 2 is better because for only a 'little more' money, you get three more nuggets.
• Some pairs will have started to make two ratio tables or double number lines to find a comparison point. 

I make sure I ask students why we can't simply say that Deal 1 is better because it costs less money.

Before concluding this warm-up, I frame the lesson by telling students that the better deal is Deal 1 because the cost per chicken nugget is cheaper.  This is known as a special kind of ratio, called a unit rate.  I suggest that we can use unit rates to make valid comparisons.  

During the Intro to New Material, the key idea for students to reflect on is that a unit rate is a ratio of two values in which the second term is one. Because of this, unit rates make it easy to compare values that are originally given in different (non-unit) ratios.

In raising this idea for my students, I guide them through the examples following these steps:

1. Write the given ratio, using a table, double number line, or a fraction.
2. Label the units.
3. In the second ratio, place a 1 for the unit you are finding the rate for (e.g. per dollar, etc.) Place an x in the other term. Find the scale factor.
4. Apply the scale factor to find the unit rate, using division

In this lesson I allow my students to use calculators, if they choose.  My goal is for students to master how to find and use a unit rate to make comparisons.  I don't want division to stand in the way of internalizing the idea of using unit rates to make comparisons efficiently.  In future lessons, once students have a firm grasp of this idea, I will combine unit rate with division practice.  

Students work in pairs on the Partner Practice.  As students are working, I circulate around the classroom.  I am looking for:  

• Are students correctly labeling the double number lines or ratio table?
• Are students correctly identifying the unit in the unit rate?
• Are students correctly dividing the terms?
• Are students showing clear, logical work?
• Are students providing an answer to the specific question?
• Are students checking for the reasonableness of their answer.

I ask students:

• What does the ratio mean in this problem?
• What is the question asking you to find?
• What information are you given?
• What term is going to be your single unit?
• What operation are you performing?
• What does your unit rate mean or represent? 

Students independently complete the check for understanding problem.  I have students clap out their answer choices, and share a student work sample on the document camera.  I then also go through each answer choice and discuss with students its reasonableness.  Choice A in in terms of hours per mile.  Choice B is far too slow, given the distance for 5 hours is over 2,000 miles.  Choice D is more than 2000, which does not make sense for one hour given what the problem tells us.  I want students to analyze the answer choices.  

Students work on the Independent Practice problem set.  

After independent work time, I bring the class back together to discuss problem 4.  Students should recognize that John's thinking is incorrect because he did not find a rate in terms of 1 hour.   I show student work on the document camera, and students give feedback about the work.  A student work sample is here.

Problem 11 is also a good problem to discuss because it requires students make sense of the problem, organize their work, and analyze their answers.  

Students work on the Exit Ticket to end the lesson.  The questions on this exit ticket require students to apply their mastery of unit rates in multiple ways. An exit ticket sample is here.