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Unit 3, Lesson 6

Rational Equations (Part 2)

Algebra 2

6.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR5: Co-Craft Questions

Materials

None

Activity Narrative

In this Warm-up, students have a chance to become familiar with the context (MP1) of the following activity, a boat traveling along a river, before they write equations to represent the context and solve for unknown values.

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

Launch

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context of using a boat traveling on a river. Use Co-Craft Questions to familiarize students with the context and elicit possible mathematical questions.

Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.

Activity

None

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 to go upstream 5 kilometers as it does to travel downstream 10 kilometers.

Student Response

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Building on Student Thinking

Activity Synthesis

Use this discussion to highlight students’ questions and give time for students to share their experiences with watercrafts on moving currents.

Invite several partners to share one question with the class, and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “equation,” “domain,” “distance is rate times time,” or “time is distance divided by rate.”

If any students have experiences with boating on rivers, invite them to share these with the class. For example, students who have kayaked along rivers may have noticed how much faster they moved downstream versus upstream. If not brought up by students, note that if the speed of the river is 8 kilometers per hour or greater, then the boat won’t be able to travel upstream. Also, if the river has no current, then the time it takes to go upstream or downstream would be the same.

6.2

Activity

Instructional Routines

Graph It MLR8: Discussion Supports

Materials

None

Activity Narrative

Continuing their work from the previous lesson and building on the Warm-up, students write a rational equation to model a situation and then use it to answer questions (MP2). 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 .

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).

6.3

Activity

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

None

Activity Narrative

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. Students then solve the equation by graphing, which relates back to work earlier in the unit when students identified a solution to a fifth-degree polynomial using a graph.

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

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

Lesson Synthesis

Arrange students in groups of 2. Tell students that while they have solved this equation using a graph, now they are going to solve it using algebra. Display the equation for all to see and tell students that a first step, rewriting the equation to have a single fraction on each side, has been done for them.

Ask students to identify what they could multiply each side of the equation by to get a new equation with no variables in any denominators. After a brief quiet think time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Select 2–3 students to share what they would multiply by, and record responses for all to see. ( and or just )

Once students are in agreement, add

At this point, tell students they have a choice. is a number, so we can use the distributive property on the left expression, or we could multiply each side by 85. Ask students to use whichever method they choose to rewrite the equation in the form “.” After a brief quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Next, ask students what type of equation this is. Depending on the form students used, it may be more or less clear that they have a quadratic equation, which is a type of equation students learned to solve, in a previous course, using methods such as the quadratic formula. If time allows, ask students to solve this equation using a method of their choice. Students will have more opportunity to solve these types of equations in a future lesson.

Conclude this discussion by telling students that solving rational equations by multiplying strategically in order to rewrite the equation without variables in denominators is sometimes called “clearing the denominators.”

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Student Lesson Summary

A boat travels about 6 kilometers per hour in still water. If the boat is on a river that flows at a constant speed of kilometers per hour, it can travel at a speed of kilometers per hour downstream and kilometers per hour upstream. (And if the river current is the same speed as the boat, the boat wouldn’t be able to travel upstream at all!)

On one particular river, the boat can travel 4 kilometers upstream in the same amount of time it takes to travel 12 kilometers downstream. Since time is equal to distance divided by speed, we can express the travel time as either hours or hours. If we don’t know the travel time, we can make an equation using the fact that these two expressions are equal to one another and figure out the speed of the river.

Substituting this value into the original expressions, we have and , so these two expressions are equal when . This means that when the water flow in the river is about 3 kilometers per hour, it takes the boat 1 hour and 20 minutes to go 4 kilometers upstream and 1 hour and 20 minutes to go 12 kilometers downstream.

Even though we started out with a rational expression on each side of the equation, multiplying each side by the product of the denominators, , resulted in an equation similar to ones we have solved before. Multiplying to get an equation with no variables in denominators is sometimes called “clearing the denominators.”

Launch

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 , , and 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. Display the Task Statement and first 2 questions for all to see. Give quiet work time for students to answer these questions and then to share their work with a partner. Select 2–3 students to share their equation with the class, and record student reasoning for all to see before telling students to work on the last question.

MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the last question. Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive.
Advances: Writing, Speaking, Listening

Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for the first question before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language

Activity

None

Circuits in parallel follow this law: The inverse of the total resistance is the sum of the inverses of each individual resistance. We can write this as: , where there are parallel circuits and is the total resistance. Resistance is measured in ohms.

Two circuits are placed in parallel. The first circuit has a resistance of 40 ohms, and the second circuit has a resistance of 60 ohms. What is the total resistance of the two circuits?
Two circuits are placed in parallel. The second circuit has a resistance of 150 ohms more than the first. Write an equation for this situation showing the relationships between and the resistance of the first circuit.
For this circuit, Clare wants to use graphs to estimate the resistance of the first circuit if is 85 ohms. Describe how she could use a graph to determine the value of , and then follow your instructions to find .

Student Response

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Building on Student Thinking

If students are unsure of how to use graphs to find the solution to , consider asking:

“Can you explain how you wrote your equation.”
“Considering a simpler problem, such as , how could you use a graph to identify the values of that make the equation true?”

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Lesson 6

Rational Equations (Part 2)

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSA-REI.A

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-CED.A.1

HSA-REI.A.1

Building Toward

Standards Alignment

Building On

Addressing

HSA-REI.A.1

HSA-REI.D.11

Building Toward