# Proportional Relationships With Decimals

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## Objective

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

#### Big Idea

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

## Introduction

10 minutes

I will open by asking a simple question.  If I earn \$16 for 2 hours of work, how can we determine my hourly rate of pay.  While asking this equation it may be helpful to have double lines displayed with dollars and hours labeled.  I will mark a place for 2 hours on one number line and \$16 on the other.  The values will be vertically aligned.  As the question is answered, I will mark the locations for 1 hour and \$8 on the number lines.

Then I will ask, what type of rate is this - \$8 for 1 hour or \$8 per hour?  Students may recall the term unit rate from 6th grade.

I may ask another question.  If I pay \$1 for 2 cookies, how can I determine the cost of 1 cookie? Again, I can mark these values on a new double number line along with a drawing of 2 cookies with a total price of \$1 labeled.   We will conclude that we can again divide to find the cost per cookie.

Note:  I will allow my students to use a calculator during this lesson.  I do not want to spend time on the details of calculation.  That being said, time must be made for students to practice dividing decimals without a calculator.

Finally, I will present the example problem.  I explain that I will put the value 5 pounds, five intervals to the right of 0 and the corresponding price will go 5 intervals to the right as well.

Before solving for the unit price, I will ask students if they expect to pay more or less for 1 pound of rice? This should be brief but it will serve as a good way to determine if they understand that a lower weight will have a lower price.   (MP2)

The final question is given throughout the work today.  It asks students to recognize how the unit price can be used to find various quantities.  This is a precursor to the work to come several lessons later on the constant of proportionality.

## Guided Problem Solving

15 minutes

The first guided practice problem uses only whole numbers.  It may appear similar to the previous lesson, but here students are asked to find a unit rate.  I also wanted to give students a fairly straightforward problem to help them successfully number the number lines.

The second problem will be a bit more difficult.  Students may struggle with where to place 2.5 hours.  I will explain that we want each interval on the number line to equal the same amount and that we want to leave a place for the unit rate showing 1 hour.  If needed I will encourage students to think about labeling the hours in 0.5 hour increments.  Then we will notice how many intervals there are between 0 and 2.5 and 0 and \$21.25.  This will help make sense of finding the unit rate by dividing 21.25 by 5.

Both problems ask students to explain how to use the unit rate to solve problems.

## Independent Problem Solving

20 minutes

Problem 1 of independent practice is slightly different than any others we have seen so far.  It involves two values that are between 0 and 1 and the terms of the unit rate are greater than the terms of the given rate.  When we review this problem, I will ask students if they see another way to find the unit rate.  They may see that the unit rate is the fourth increment of the given rate.

Problem 2 may cause some problems for students when labeling the double number line.  Students might put \$2 on the 1st interval mark after 0.  If so, I will ask them to suggest where the unit rate (cost per 1 ounce) will go? How can we make sure to show the unit rate on the double number line?  This should help them realize that \$2 should go on the 5th interval along with 5 ounces.

Problem 3 is similar to the exit ticket.  I even give them a number line with increments of 0.5 though only the whole numbers are labeled.  As we review this problem I will ask students to notice the number of intervals between 0 and the given rate.  There are 9.  We can see then that the value of each interval by dividing each term by 9.  I then may ask if they see more than one way to find the unit rate.

Problem 4 is the biggest challenge because no double number line is given.  Also students are given 1.25 hours.  If students struggle I will ask them how they can break up 1.25 hours into equal parts.  It may be necessary to remind students to imagine \$1.25 first.  This should lead students to labeling increments of 0.25 hours.

## Exit Ticket

5 minutes

The exit ticket is similar to problem 3 from independent practice.  Some help has been given by labeling the hours on the number line.

Part C only assess whether students can interpret their work on the double number line though some may solve it based on the unit rate.  Either way is okay.

Part D is again to tie into the essential question of the lesson.

I would make this exit ticket worth 5 points.  One point each for problem A-C.  Problem D will be worth 2 points: 1 point for a correct answer and 1 point for a valid explanation.  A valid explanation.

A successful exit ticket will be worth 4 out of 5 points.