Lesson 21

In this warm-up, students have an opportunity to notice and make use of structure (MP7), because the skills they use to solve equations involving fractions also work to solve equations with more complex rational expressions.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

Continuing their work from the previous lesson, the purpose of this activity is for students to write a rational equation to model a situation and then use it to answer questions. The difference between this activity and previous ones in which students solved rational equations is that now students must solve a rational equation in which the denominators of the two expressions on each side are different expressions involving the variable \(r\).

Monitor for students who either graph each expression to find the point of intersection or set up an equation that they solve to share during the class discussion.

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

Arrange students in groups of 2. Tell students to read the activity and answer the first problem. After quiet work time, ask students to compare their responses to their partner’s and reach agreement on the two expressions. Select 2–3 students to share their expressions with the class, recording them for all to see. Once students are in agreement on the expressions, allow them to continue to the next problem.

Students may not know how to use the information that the boat goes 8 kph in still water. Tell them that if a boat is going downstream, then the river is pushing it forward, so it will travel at its speed plus the river’s speed. If the boat is going upstream, then the river will push against it and slow it down.

The purpose of this discussion is for students to share how they solved a rational equation in which the denominators of the rational expressions are not the same. During the discussion, students should make connections to how they solved rational equations in the previous lesson in order to reason that they can strategically multiply by a common denominator in order to get an equation without variables in denominators.

Select previously identified students to share how they answered the last question, starting with any who used graphs. The main focus of the discussion should be on how students identified a solution to the equation \(\frac{5}{8-r}=\frac{10}{8+r}\). Here are some questions for discussion:

• “What are some possible first steps to solving the equation?” (Multiplying by \(8-r\) or \(8+r\).)
• “Is it better to multiply each side by \(8-r\) first or \(8+r\) first?” (No matter the order, multiplying by one then the other results in the linear equation \(5(8+r) = 10(8-r)\).)

If time allows, display a graph of \(y=\frac{5}{8-r}\) and ask, “What is the meaning of the vertical asymptote \(r=8\) in this situation?” (If the boat goes 8 kilometers per hour and the river goes 8 kilometers per hour, then the boat does not go anywhere.)

In this activity, students use the formula for the total resistance of circuits in parallel to write a rational equation for an unknown resistance. The particular equation students write involves the sum of two rational expressions, which students have not seen before. They then solve the equation by graphing, which relates back to work earlier in the unit when students identified a solution to a 5^th degree polynomial using a graph.

Monitor for students writing clear descriptions for how Clare could use graphs to determine the value of \(R\).

Begin the activity by asking students if they know what a circuit is and where it is used. If not suggested, tell students that one example of a circuit is a flashlight, in which the batteries, light, and switch together make a circuit. In a circuit, resistance is like friction—it makes it harder for electricity to flow. Often we want some resistance in a circuit so it can do work. For example, the filament in a light bulb glows because of its high resistance. In this activity, students are going to consider a law about circuits that are run in parallel, like the ones shown in this diagram.

For students who are still unsure about what a circuit is, it may be helpful for students to mentally picture \(R_1\), \(R_2\), and \(R_3\) in this picture as three light bulbs that are connected to the 12-volt battery on the far left. If students have not experienced subscript notation previously, let them know that when talking about the same type of thing, such as 3 light bulbs, the same letter with a different number written smaller and to the bottom right can be used to tell the difference between the objects.

Arrange students in groups of 2. Provide access to graphing technology. Display the task statement and first 2 questions for all to see. Give quiet work time for students to answer these questions, followed by sharing work with a partner. Select 2–3 students to share their equation with the class, recording student reasoning for all to see. 

Students may be unsure how to use graphs to find the solution to \(\frac{1}{85} = \frac{1}{R}+\frac{1}{R+150}\). Ask them to consider a simpler problem, such as \(85=x(x+15)\), and how they could use a graph to identify the value(s) of \(x\) that make(s) the equation true.

Earlier in this unit, students used graphs to identify what value of \(x\) made the equation \(1,\!580=300x^4+500x^3+250x^2+400x\) true. This activity asks students a similar question, but with a rational equation instead of just a polynomial. An important takeaway from this discussion is for students to recognize that even as equations become more complicated and include things like rational expressions added together, everything they have learned about identifying solutions to equations is still true. Specifically, the point of intersection for graphs of the expressions on the left and right side gives the \(R\) value that makes each side equal to \(\frac{1}{85}\).

Select previously identified students to share their graphing directions, and then select one set of directions to follow while displaying the graphs for all to see. If time allows, ask students to use graphing technology to identify the value of \(R\) needed for different values of \(R_T\).