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# Proportional Relationships With Decimals

Lesson 2 of 12

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

#### Introduction

*10 min*

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.

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#### Guided Problem Solving

*15 min*

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.

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#### Independent Problem Solving

*20 min*

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.

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#### Exit Ticket

*5 min*

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.

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- LESSON 1: Proportional Relationships of Whole Numbers
- LESSON 2: Proportional Relationships With Decimals
- LESSON 3: Proportional Relationships With Fractions
- LESSON 4: Finding Distances on Maps
- LESSON 5: Scaling a Recipe
- LESSON 6: Determine Equivalent Ratios - Scale Factor Between Ratios
- LESSON 7: Determine Equivalent Ratios - Scale Factor Between Terms
- LESSON 8: Determine The Graph of a Proportional Relationship
- LESSON 9: Determine Equivalent Ratios - Common Unit Rate
- LESSON 10: Writing The Constant of Proportionality Equation
- LESSON 11: Writing Equations for Proportional Relationships
- LESSON 12: The Distance Formula