# Writing Equations to Solve Rate Problems - Ratios of Fractions

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## Objective

SWBAT write equations to represent and solve problems involving ratios of fractions

#### Big Idea

Writing and solving equations is a useful tool for solving rate problems.

## Introduction

10 minutes

I will begin with the essential question:  How can we use equations to solve rate problems?

I will present the equation for proportional relationships:  y = mx

I will ask students to discuss and share in a turn-and-talk what each variable in the equation means.  This is putting MP2 to practice. I may jar their memory by asking them to identify the dependent, independent, and constant of proportionality.  A few example problems may be helpful.

Oranges are 1.99 per pound.  How much are 2, 3, 4 pounds.

This can be shown below the equation y = mx.

1.99 = 1.99 * 1

3.98 = 1.99 * 2

5.97 = 1.99 * 3

etc...

It may be helpful to color code each variable to help students visually see the structure.  Perhaps red for the y, green for the m, and blue for the x.  Use black for the operators.  Do this with the variables and the actual values.

Once we've reminded ourselves of the parts of the equation, it may be helpful to give one more example where the constant of proportionality is not explicitly given.  It can be a simple problem:  Marley earned \$27 for 3 hours of work.  How can we write an equation to determine how much Marley would be paid for any amount of work?  This is all pretty much a review of work we've already done.  Perhaps to show how useful the equation can be, the problem could be changed.  Marley earned \$45. How many hours did she work?  Again, these values should be modeled next to y = mx.

Next we will work through a problem.  I will give my students about 3 minutes to see if they can solve this problem first.  I may have to ask students:  What value do we need to find in order to write the equation?  The answer is m, the constant of proportionality, in this case report per hour.

In part ii, students should substitute a value into the equation.  This may be tricky for students.  I'll ask, what is a complete report?  How much was completed in 4/5 of an hour? Answer: 3/10.  If 3/10 is part of the report what value would be the entire report?  The answer: 10/10 or 1.

Part iii is presented so that students see there are two ways to answer part ii.  It would be writing the equation m = y/x where we are finding hours per report.

Students then have another problem to solve.  I will use this as a check for understanding.  It will let me know how much support will be necessary in the next section of the lesson.  If a large majority (75-80%+) of the class is successful, I'll know just to focus on the remaining group of students.

## Guided Problem Solving

15 minutes

There are three problems each with 3 parts.  I have attempted to vary what students are asked to find, but the equation still is the main tool for modeling the problem.

GP1 is similar to the two problems from the previous section.

GP2 allows students to work with mixed numbers.

GP3 presents time in minutes but I want the rates to be solved in hours.  Again, some students will need to be reminded to turn 30 minutes into hours.

## Independent Problem Solving

20 minutes

Now nearly all students should be ready to work independently.  The primary objective is for students to learn to use an equation to solve rate problems.  That being said, I have varied some of the tasks so that students have a chance to bring together several skills learned throughout the unit (and to prepare them for the unit final).  In addition to being asked to write an equation for every problem, problem 2 asks students to model the rate with a double number line.  Problem 3 again brings in the need to convert minutes to hours.  Problem 4 asks students to graph the relationship on a coordinate plane and to describe where the unit rate can be found as an ordered pair.

## Exit Ticket

5 minutes

The exit ticket is in 4 parts.  Students should be able to solve at least 3 of the 4.  A student who stayed engaged throughout the lesson should have no problem doing at least this well.  The first 3 problems echo what students have done several times throughout the lesson.  These all deal with one rate.

I also wanted to assess students ability to convert minutes to hours in a rate.  We already solved this rate problem earlier, the only difference was that instead of 15 minutes I gave students 1/4 of an hour.  For this fourth part, students will only need to write a valid equation.