I will begin by asking the essential question. I will then quickly ask students to provide an example of a rate and its unit rate (constant of proportionality). Throughout the lesson I will use the words together by saying "unit rate or constant of proportionality". The main idea is to again remind students that a unit rate is a rate that compares a quantity to 1 unit of another quantity.
This lesson uses some of the same skills as the first few lessons of the unit, but where those were designed to help develop proportional reasoning and the idea of "for each" and "for every" this lesson has a different purpose. It is to develop students understanding of how to find a unit rate when the denominator is a fraction - sometimes it is easier to multiply. All of the problems are designed to lead students to this conclusion (MP8).
I will model the first problem. Students are expected to watch and then write what I write when I ask them to. My students have been making mistakes with multiples of mixed numbers like 2 1/2 so we have spent time modeling these multiples using bar models. It may be necessary to have a model ready so that students can see the progression more easily.
Question iii and iv are meant to lead students to seeing how multiplying can help us find the unit rate.
This lesson will help tie into the next lesson where the 2nd term of the rate will be a variety of fractions - not just unit fractions.
Now students will work in pairs or trios. I will most likely ask students to do 1 part at a time, so that I can walk the room while they work to check each step along the way. If it looks like a lot of students are stuck on any step, I can immediately bring it to the board. Otherwise, I can help the few individuals who are stuck along the way.
The first thing to look out for is a correctly modeled double number line. It will be important that students label each number line with the appropriate unit and that they keep the hash marks vertically aligned. Also, students should label the first hash mark on the far left of each number line with 0. If they have a hard time finding each multiple correctly ask "For every 1/4 quart, how many ounces?" It may even helped to write and total.
1/4 q = 8 ounces
+1/4 q = 8 oz
2/4 q = 16 oz
+1/4 q = 8 oz
3/4 q = 24 oz
On the second problem students may be tempted to find the rate of hours to buckets. I will ask them to focus on how the unit rate is described - buckets per hour. I will then ask buckets per hour means buckets for every how many hours? By now they should recall that per hour means per 1 hour.
Now it is time for students to work independently and quietly. As the year started, I often allowed students to collaborate during independent problem solving. It started to become apparent that it would be helpful to carve out more time in a lesson for students to work independently outside of the exit ticket.
I expect the first two problems to be fairly routine for most students at this point. I purposely removed number lines from this section so that students are forced to create it for themselves. I want them to build their own tools (MP5) as opposed to always providing a template. I may warn students of this before hand and remind them to draw as neatly as possible - rulers should not be necessary.
I anticipate lots of questions of students for the third and last problem. Rates are given in miles per minute but they are asked to find miles per hour. I still find students in 7th grade who forget that an hour equals 60 minutes. Far too many can not read an analog clock with ease. Other students may realize these minutes represent fractions of an hour by may mistakenly use a denominator of 100. Once reminded that an hour is 60 minutes a bar model or fraction circle may help them see that 15 minutes is 1/4 hour and 20 minutes is 1/3 hour.
Before beginning the exit ticket, we will summarize based on the essential question. I may ask students to discuss how we found our unit rates today. Did we divide? What operation did our models suggest? Why were we able to multiply?
The exit ticket has 3 parts. The first part requires students to use a double number line. The second part requires students to multiply. The third part is a simple question based on the results from the first two parts.
A successful exit ticket will have 3 correct responses out of 3.