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.