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 and reasoning about that solution in context (MP2), which is the focus of the discussion.

Monitor for students who identify solutions to the last question in different ways. Here are some strategies they might use, ordered from less efficient to more efficient:

• Guessing and checking
• Graphing the function and to find a point of intersection
• Solving algebraically

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

If students are unsure where to start when writing an equation for the average price in terms of the number of shirts , consider saying:

• “Tell me more about what you notice about the situation.”
• “How could calculating the cost for a known number of T-shirts, say 5, help you create an equation to represent the average cost?”

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

Invite previously selected students to share how they determined how many shirts would need to be printed in order for the average cost to be $5 or less. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “When is guess and check a good strategy to use? When is it more challenging?”
• “Why does the intersection of and tell us the solution?”
• “If you solved the equation algebraically, what was your first step?”

If no student says it, point out that after multiplying each side of the equation by , the result is the linear equation , which is something students have quite a bit of practice solving. In a future lesson, students will focus on problems that can arise when multiplying each side of an equation by a value that can be 0, like . For now, it suffices to note that it doesn't make sense to print 0 shirts.

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 an additional opportunity to solve an equation with a rational expression whose denominator is not a single term and reason about the solution in context (MP2).

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.

The purpose of this discussion is for students to share their process for solving the rational equation. 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 resulted in a linear equation. Make sure students explain any rounding choices they made during the discussion.