standard_description: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^(2) + y^(2))^(2) = (x^(2 )– y^(2))^(2) + (2xy)^(2 )can be used to generate Pythagorean triples.
problem_activity: Activity
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).
Launch
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.
Conversing: MLR8 Discussion Supports.
Use this routine to help students explain their expression to their partner. Students should take turns sharing their expressions and explaining their reasoning to their partner. Display the following sentence frames for all to see: “_____ represents _____”, “Where does _____ show . . . ?”, and “That could (or couldn’t) be true because . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about modeling situations with rational expressions.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Noah likes to go for boat rides along a river with his family. In still water, the boat travels about 8 kilometers per hour. In the river, it takes them the same amount of time
\(t\)
to go upstream 5 kilometers as it does to travel downstream 10 kilometers.
[IMG]
If the speed of the river is
\(r\)
, write an expression for the time it takes to travel 5 kilometers upstream and an expression for the time it takes to travel 10 kilometers downstream.
Use your expressions to calculate the speed of the river. Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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.)


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