Proportional Relationships With Fractions

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SWBAT create proportional relationships of fraction ratios using a double number line

Big Idea

Students expand and solve ratio problems involving fractions using unit rates and a double number line.


10 minutes

In this lesson we continue to explore proportional relationships.  This lesson focuses on ratios of fractions.

I will start by asking the essential question.  We have already answered this question in the last two lessons by discussing a common scale factor between ratios, but now we will apply it to fractions.  I will give students a chance to turn-and-talk before discussing their thoughts.

Next we will read the example problem and model it using a double number line.  This example problem could also be modeled using pattern blocks.  The yellow hexagon is the whole, the red trapezoid is the half, and the blue parallelogram is the third.  I will model this rate using the blocks while filling in the numbers.

I will allow students to think through each question before revealing an answer.  


Guided Problem Solving

15 minutes

This section has two problems.  I am looking for similar things in each problem:

1) Can students correctly use the double number line?

2) Are students able to determine the unit rates?

3) Are students able to explain how the unit rate can be used to solve problems?


With both problems it is important to emphasize that each interval per number line represents the same value.  It is also important to show the relationship between a few of the equivalent rates by drawing arrows between the rates and show the scale factor.  Using GP2 as an example, I could draw arrows from the unit rate of $8 per hour to the rate $20 for 2 1/2 hours showing a scale factor of 2 1/2.

Independent Problem Solving

20 minutes

The first two problems of independent practice follow the structure of the problems already explored.  The third problem has students compare rates in a problem.  There are many ways to model these problems but I am focusing on double number lines because I think it emphasizes the idea of a rate being "for each" or "for every".  Bar models, circle models, or a physical manipulative are encouraged if this helps students to see the relationships.

In part iv of problem 3, students are asked to find the unit rate in miles per hour.  I expect many students to mistakenly find miles per minutes, especially for Lilo.  They may have less problem with this for Marcus as this value should appear on their number line at 60 minutes.  That being said, I expect to have to ask a few students if they know the number of minutes in an hour.


For finding Lilo's rate in miles per hour, I look forward to seeing and discussing several strategies at play.  Some students may find the unit rate in miles per minute and multiply by 60.  Others may use 100 minutes or 150 minutes to scale down to 10 minutes or 15 minutes and then scale up to 60 minutes.  This is an opportunity for students to explain their own reasoning and make sense of the reasoning of others (MP3).

Exit Ticket

5 minutes

Before beginning the exit ticket we will summarize.  What is the relationship we have found between equivalent rates?  

This exit ticket will be worth 4 points.  The measure of success will be a score of at least 3 correct answers.