Lesson 20

Rational Equations (Part 1)

20.1: Notice and Wonder: Denominators and Solutions (5 minutes)

Warm-up

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that if two rational expressions are equal and have the same denominators, then their numerators must also be equal.

Launch

Display the equations and solutions for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

A: \(\frac{2x+5}{x} = \frac{7x-5}{x}\), \(x=2\)

B: \(2x+5 = 7x-5\), \(x=2\)

C: \(\frac{2x^2+3}{x+2} = \frac{4x+1}{x+2}\), \(x=1\)

D: \(2x^2+3=4x+1\), \(x=1\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

Conclude the discussion by displaying the following equation and asking students to identify what value of \(x\) makes it true: \(\frac{2x^2+3}{x-1} = \frac{4x+1}{x-1}\). Students may recognize the numerators and assume \(x=1\) is the solution. Encourage these students to test their answer by substituting it back into the original equation. If no student points it out, remind them that the functions represented by the expressions have an asymptote at the value \(x=1\), so \(x=1\) cannot be a solution, even though it is a value that makes the numerators equal. Tell students they will investigate what is happening in cases like this more in the future. For now, they should remember to always check solutions by substituting them into the original rational equation.

20.2: Rationalizing the Price of T-shirts (15 minutes)

Activity

The purpose of this task is for students to write a rational equation to model a situation and then use it to answer questions. While students worked with average cost functions earlier in the course, this is the first time they are asked to write an equation for one. In addition to writing equations, students also practice solving rational equations for an unknown variable, which is the focus of the discussion.

Monitor for students identifying solutions to the last question in different ways, such as:

  • by guess and check
  • graphing the function and \(y=5\) to find a point of intersection
  • solving algebraically

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Pause students after 5 minutes and select 2–3 students to share their equations and answers to the questions about the average cost to print 25 and 100 shirts and the cheapest the average cost will get. If students have different equations, help them reach consensus on which is correct for the situation.

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 partners to share their response to the question “How many shirts should be printed to have an average cost of \$5 or less per shirt? Explain how you know.” Students should first check to see if they agree with each other about the number of shirts printed. Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “What did you do first?”, “Why did you do _____ ?”, “How does _____ help you here?”, or “Can we have partial shirts?” Next, provide students with 3–4 minutes to revise their initial draft based on feedback from their peers. This will help students explain how to solve rational equations.
Design Principle(s): Optimize output (for explanation)
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success (calculating averages). Sketch examples of shirts and their costs showing that individual shirts cost \$4.25, so 1 shirt with set-up cost would be \$39.25. Calculate the average cost of 5 shirts, and show how the \$35 set-up cost is spread out among the shirts. This means the average cost decreases compared to when we only purchased 1 shirt. Show how when we abstract to purchasing \(x\) shirts, the average cost is \$4.25 plus the \$35 set-up cost divided among the number of T-shirts. Another option would be to physically act out the situation with shirts and money or representations of both.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

The school art club at a large high school is in charge of designing school T-shirts and getting them printed this year. A local business charges \$35 to set up their T-shirt printing machine with the design and \$4.25 in materials per T-shirt to print.

  1. Create an equation to represent the average cost \(C(x)\), in dollars, per T-shirt if \(x\) T-shirts are printed by this business.
  2. What is the average cost per shirt to print 25 shirts? 100 shirts?
  3. What is the cheapest the average cost per T-shirt will get? Explain or show your reasoning.
  4. How many shirts should be printed to have an average cost of $5 or less per shirt? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

For students who are unsure where to start when writing an equation for the average price in terms of the number of shirts \(x\), it may help if they write an unsimplified expression for the average price if the number of shirts is a known value, say 5, and then modify it so that \(x\) represents the unknown number of shirts.

Activity Synthesis

The purpose of this discussion is for students to understand different ways a rational equation can be solved, with an emphasis on algebraic strategies.

Select previously identified students to share, in the order listed in the Activity Narrative, how they determined how many shirts would need to be printed in order for the average cost to be \$5 or less, ending with those who can show clearly how they solved the equation step by step. If no student says it, point out that after multiplying each side of the equation by \(x\), the result is the linear equation \(5x=35+4.25x\), which is something students have quite a bit of practice solving.

20.3: Batting Averages (15 minutes)

Activity

Building on the work from the previous activity, in this activity, students are asked to write an equation to model a situation about a student’s batting average. The third question purposefully echoes the second to give students a second opportunity to solve an equation with a rational expression whose denominator is not a single term.

Monitor for students who have clearly written out how they solved their equation and rounded their answer appropriately given the context to share during the discussion.

Launch

Begin the activity by explaining (or, if possible, having a student explain) how a batting average is calculated as the number of base hits divided by the number of at bats, and then written as a decimal followed by 3 digits.

Reading, Conversing, Writing: MLR5 Co-Craft Questions. To help students make sense of contexts that can be modeled with rational expressions, start by displaying only the context for the question “Tyler is on a school baseball team and he has had 24 base hits out of 110 at bats this year.” Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions involving modeling the situation with rational expressions.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

baseball, glove, and bat on grass

Tyler is on a school baseball team and he has had 24 base hits out of 110 at bats this year.

  1. What is his current batting average?
  2. He wants to raise his batting average to .300. How many of the next consecutive at bats need to be base hits to raise his batting average to .300? Write and solve an equation to describe this situation using \(x\) for the number of consecutive base hits. Be prepared to explain how you wrote your equation and each of your solving steps.
  3. Unfortunately, Tyler gets no base hits in his next three at bats. Revise your equation and then calculate how many of his next consecutive at bats need to be base hits to raise his batting average to .300. Be prepared to explain how you revised your equation and each of your solving steps.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Elena had 24 base hits in 110 at bats. She has done a lot of practice and now thinks that for all of her future at bats in this season, she will have a batting average of 0.350. If she does, then how many more at bats will Elena need so that with this average, she reaches a 0.300 batting average overall for the whole season?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Select 2–3 previously identified students to share their equation and solution steps for the questions about how many base hits are needed, recording responses for all to see. If students do not make the connection, point out how the solving here is similar to the previous activity in which multiplying by the denominator of the rational function results in a linear equation. Make sure students explain any rounding choices they made during the discussion.

Lesson Synthesis

Lesson Synthesis

In this discussion, students return to the T-shirt situation from an earlier activity and consider how the original company compares to a new company when deciding who the Art Club should hire.

Tell students that there is another printing business in town that charges only \$15 to set up and \$4.40 in materials per T-shirt to print. Ask students to write an expression for this business that gives the average cost per T-shirt if \(x\) T-shirts are printed. After work time, select 2–3 students to share their expressions with the class.

Next, ask, “What does the solution to the equation \(\frac{35+4.25x}{x}=\frac{15+4.40x}{x}\) tell us?” (The number of T-shirts for which both companies will charge the same price.) After some quiet think time, select 2–3 students to share their answers. If not brought up by students, note that this equation is similar to those in the warm-up, so the equation can be solved by asking what value of \(x\) makes \(35+4.25x=15+4.40x\) true (133 \(\frac13\)), and then reasoning about the value in context.

20.4: Cool-down - Selling Sweatshirts (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Consider a student on a school softball team who wants to raise her batting average to .200. So far this year, she has 20 base hits out of 120 at bats, making her current batting average .167 since \(\frac{20}{120} = 0.167\).

To increase her batting average, she needs to have more base hits. But each base hit means the number of at bats also increases by 1. Since batting average is the number of base hits divided by the number of at bats, we can use the rational expression \(\frac{20+x}{120+x}\) to model how her batting average changes based on the number of consecutive base hits \(x\) she gets. Her batting average is the value of this expression to 3 decimal places. The value of \(x\) that makes this expression equal to .200 will tell us how many consecutive base hits she needs to get the batting average she wants.

\(\displaystyle \begin{align*} .200&=\tfrac{20+x}{120+x} \\ .200 \boldcdot (120+x) &=\tfrac{20+x}{120+x} \boldcdot (120+x) \\ 0.2(120+x) &= 20 +x \\ 24 + 0.2x &= 20 +x \\ 4 &= 0.8x \\ 5 &= x \\ \end{align*}\)

Even though we started out with a rational expression on the right side of the equation, multiplying each side by \((120+x)\) resulted in an equation similar to ones we have solved before. Checking \(x=5\) in our original expression, \(\frac{20+5}{120+5}=\frac{25}{125}=.2\), so she needs 5 consecutive base hits to have a batting average of .200.