Lesson 20

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.