Unit 3, Lesson 5
Rational Equations (Part 1)
Algebra 2
5.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-CED.A.1
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
HSA-REI.A
Understand solving equations as a process of reasoning and explain the reasoning.
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to get students thinking critically about the structure of rational equations, which will be useful when students solve rational equations in future activities. While students may notice and wonder many things about these equations, the common denominators are the important discussion points.
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.
Launch
Arrange students in groups of 2. Display the equations 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 with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
- ,
- ,
- ,
- ,
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the equation. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the similarities between the first pair of equations and the second pair of equations does not come up during the conversation, ask students to discuss this idea.
Conclude the discussion by displaying the following equation and asking students to identify what value of makes it true: . Students may recognize the numerators and assume is the solution. Encourage these students to test their answer by substituting it back into the original equation. If no students points it out, remind them that the functions represented by the expressions have an asymptote at the value , so 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.
5.2
Activity
Instructional Routines
5 PracticesMaterials
NoneActivity Narrative
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).
5.3
Activity
Instructional Routines
MLR5: Co-Craft QuestionsMaterials
NoneActivity Narrative
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.
Launch
Begin the activity by explaining (or, if possible, asking a student to 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.
MLR5 Co-Craft Questions. Keep books or devices closed. Display only the problem stem, without revealing the questions, and ask students to record possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the task. Ask, “What do these questions have in common? How are they different?” Reveal the intended questions for this task and invite additional connections.
Advances: Reading, Writing
Advances: Reading, Writing
Lesson Synthesis
The goal of this discussion is for students to consider the context to reason about the solution to a rational equation in which each side has a variable in the denominator. Returning to the T-shirt situation from an earlier activity, students now consider how the original company compares to a new company when deciding where the Art Club should get the T-shirts printed.
Tell students that there is another printing business in town that charges only \$15 to set up and \$4.40 in materials per T-shirt to print. Ask students to write an expression that gives the average cost per T-shirt if T-shirts are printed by this business. After a brief work time, select 1–2 students to share their expressions with the class.
Next, ask, “What does the solution to the equation tell us?” (the number of T-shirts for which both companies will charge the same price) After some quiet think time, select 2–3 students to share their answers. If not brought up by students, note that this equation is similar to those in the Warm-up, so the equation can be solved by asking what value of makes true (133 ), and then reasoning about the value in context.
Student Lesson Summary
Consider a student on a school softball team who wants to raise her batting average to .200. So far this year, she has 20 base hits out of 120 at bats, making her current batting average .167 since .
To increase her batting average, she needs to have more base hits. But each base hit means the number of at bats also increases by 1. Since batting average is the number of base hits divided by the number of at bats, we can use the rational expression to model how her batting average changes on the basis of the number of consecutive base hits she gets. Her batting average is the value of this expression to 3 decimal places. The value of that makes this expression equal to .200 will tell us how many consecutive base hits she needs to get the batting average she wants.
Even though we started out with a rational expression on the right side of the equation, multiplying each side by resulted in an equation similar to ones we have solved before. Checking in our original expression, , so she needs 5 consecutive base hits to have a batting average of .200.