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Title of Lesson: Solving Literal Equations

UFTeach Students’ Names: Cassaundra Haggard and Xavier Monroe

Teaching Date and Time: 10/25/12, 12:20-2:40pm

Length of Lesson: 100 minutes

Grade / Topic: Math for College Readiness

Source of the Lesson: Cassaundra Haggard and Xavier Monroe

Appropriateness for High School Students:

The teaching strategies in this lesson are appropriate for high school students, because the course dialogue focused on developing and solving equations based on real-life situations provide the students with a way to remain engaged while developing their mathematical skills to create and solve equations.

Concepts

Variables are usually denoted by letters and represent a number, which is not constant, but symbols may also be used as a variable. Variables allow situations to be expressed mathematically when one element is not known or changes. When numbers form a mathematical statement with at least one operation and an equal sign, the statement is called an equation. Mathematical equations may consist of only constant numbers, only variables, or both constant numbers and variables. Equations can be translated to and from words and can often be simplified. Equations can also be made to model real-world situations, particularly the sciences. Similarly, a real-world situation can often be quantified using an equation. When a scientific expression is known to be equivalent to another expression or value, it can be written as an equation with an equals sign. An equation can then be solved to find what value(s) of the variable make the equation true. When a number solution is not possible, equations can be solved in terms of a variable. These equations are known as literal equations, which is the topic of this lesson.

Florida State Standards (NGSSS):

· Grade 9-12, MA.912.A.3.3, Standard 3: Linear Equations and Inequalities - Solve literal equations for a specified variable. Cognitive Complexity: Level 2.

· Grade 9-12, MA.912.A.3.2, Standard 3: Linear Equations and Inequalities - Identify and apply the distributive, associative, and commutative properties of real numbers and the properties of equality. Cognitive Complexity: Level 2.

Performance Objectives

1. Students will be able to solve a given equation for the specified variable.

2. Students will be able to use order of operations and other properties to isolate the specified variable.

Materials List and Student Handouts

· 5 copies of ordering activity (1 of each).

· 15 copies of the assessment.

· 15 copies of Literal Equations through Science.

Advance Preparations

· Cut out ordering activity.

· Make ActivInspire presentation.

Safety

· There are no significant safety concerns.

ENGAGEMENT Time: 5 minutes

What the Teacher Will Do

Teacher Directions and Probing Questions

Student Responses and Potential Misconceptions

Teachers will introduce themselves.

Hello, I’m Ms. Haggard. I’m going to be teaching the lesson today until you go to lunch. Then Mr. Monroe will finish the lesson for the remainder of the class period.

Show video and ask students about what they saw. http://www.youtube.com/watch?v=oIxxqztQz3Y

What did you think of this video?

What did you learn?

Today we are going to be working with literal equations. Literal equations are equations with multiple variables. What are some examples of literal equations?

How does the video apply to solving literal equations?

Cool, not as good as the other one, etc.

Answers will vary.

Formulas, specific answers will vary.

It doesn’t. [You do the same thing.]

EXPLORATION Time: 25 minutes


Teacher Directions and Probing/Eliciting Questions

Student Responses and Misconceptions

Put students in groups of three. Pass out ordering activity. Instruct students on how to complete the activity.

I am now going to assign you to groups of three. Each group is getting a packet with an equation and steps to solve it. The blue equation is the initial equation that you start with. The red letter is the variable you need to solve for. Put the steps you are given in order. There are different ways to solve each equation so there are different steps you can use depending on the method you choose. I would encourage you to solve the equation multiple ways. There are also incorrect steps. Be prepared to explain what happened between steps as well as why some steps are incorrect. [Call on student] Repeat the directions.

Put the steps in order. Start with the blue one. Solve for the red letter. There are different ways to do it. Some steps are wrong. We will have to explain it.

As students work, ask them probing questions to check for understanding.

How did you know this step was next?

What happened between these steps?

Answers will vary depending on problem and step. Correct responses could include Order of operations, distributive property; you have to do the opposite operation to the other side. Responses such as I guessed or it is closest to the one before it should be followed by additional questioning.

Answers will vary depending on problem and step. Correct responses could include: It was distributed, divided, multiplied, added, or subtracted. Responses such as I don’t know should be followed by additional questioning.

What is a different way you could solve this?

Will you get a different answer if you solve the equation in a different way? Why or why not?

How did you know this step was incorrect?

How is this related to solving the types of equations you have worked with before?

Answers will vary depending on problem and step. Correct responses should lead the student to find correct alternative solution methods. Incorrect responses may lead to attempting to use the incorrect steps.

Yes, [No]. If you do it differently, it works out to something different. [As long as you do it correctly, you will get the right answer.]

It did not have a step after it so I knew it was not supposed to be there. [You were supposed to (add, subtract, multiply, divide, distribute, etc.), but it (subtracts, adds, divides, multiplies, doesn’t distribute, etc.).]

[You do the same stuff only this has letters instead of numbers.]

EXPLANATION

Time: 20 minutes




Ask students to come to the board and drag the steps into the order they put for their solution. The entire group should be called to the board to take turns explaining their reasoning. They should be told to explain any alternative solutions and incorrect responses.

When your equation is displayed on the board, your group should come up and drag the steps into the right order. You should each take turns dragging the next step and explain what happened between steps. Show different ways that the equation could be solved and explain why some steps were incorrect. If anyone else has any questions about the problem being worked, they may ask the group at the board.

(Each group should be asked the following questions)

How did you know to put the steps in that order?

Does order always matter?

What is an example of a time order doesn’t matter?

What is an example of a time it does?

How would you tell your classmates to avoiding using the incorrect solutions?

Someone else did the work. [You have to do this step before this step for this reason.]

Yes, [No].

See correct examples below for incorrect examples for this question. [When you are adding and subtracting or multiplying and dividing multiple things, it does not matter which goes first.]

See correct examples above for incorrect examples for this question.

[You have to distribute before you can add or subtract things inside the parentheses.]

Make sure you do everything correctly. When you have, something added on one side you subtract it from both sides. When you have a number and then parentheses and then two numbers added or subtracted, you have to distribute to both. Etc.

EXPLORATION Time: 20 minutes




Teacher will pass out the Literal Equations through Science worksheet and ask the students to continue working in pairs using the knowledge they gained in the first half of class.

Now we are going to build on what you learned during the first part of the class.

For this worksheet you and your partner need to work together to solve each problem for the stated variable. You should discuss each equation presented and individually explain how each equation pertains to your daily life.

What questions do you have?

What are you expected to do for this activity?

Now that you all know what to do for this activity, let’s complete number 1 on the activity so you know what I expect.

Problem 1 on the activity is, where we are asked to find t.

How should we approach this problem?

What variable? Why V? Does someone else have a suggestion on how to approach the problem?

We need to isolate t and therefore make sure the remaining variables are on the other side of the equation.

How do we isolate t? What happens if you multiply by v?

Did that isolate the variable t?

How should we solve the problem?

On your activity, you are given the names of each variable. In this equation, what variable did we solve for?

How does this equation apply to your daily lives?

[None.]

[Solve each equation for the given variable, discuss the importance of the equation, and individually explain how the equation pertains to us individually.]

[Isolate the variable.]

V, because it is on the same side as t [Yes, We should isolate t as that is the variable we are asked to find.]

Multiply by v [You get dv=v2t]

[No.]

[Divide v on both sides, which gives d/v=t.]

[Time.]

Answers will vary. [Possible answers include: this could tell me how long it takes to drive somewhere.]

Teacher will allow students to complete the assigned activity and circle the room with probing questions to gage their understanding.

Now, I want you to work on the activity with your partner. Be sure to read the directions on the activity again.


[None.]

What number are you working on?

How did you approach the problem?

Was that the best way to begin the problem?

What variable are you solving for?

Have you seen this equation before?

How does this equation apply to your life on a daily basis?

Answers will vary.

Answers will vary. Possible answers include: [moving the other variables to the other side of the equation by division, multiplication, addition or subtraction.]

[Yes, there may have been another way.]

Answers will vary.

Answers will vary.

Answers will vary. [Possible answers include: Driving, determining my weight, determining how much power or energy I use, etc.]

Teacher will transition into the explanation portion of the lesson.

Now we are going to review what you completed during the activity.

EXPLANATION

Time: 20 minutes




Pull up a copy of the worksheet on the ActivInspire. Ask students to come to the board and write their answer in the blank.

(Call on student) Come write the answer you and your partner got for the second question on the board. (Repeat for third, fifth, and seventh question with different students.)

Teacher will allow each student to explain how he or she solved the particular equations while standing at the board. The following questions will be asked to solicit input from all students.

How did you approach this problem?

How did you do isolate the desired variable?

Did anyone else get a different answer?

Did you approach the problem differently?

Why was this answer correct?

Why is there more than way to approach these types of problems?

Can you have more than one answer when solving literal equations?

Why or why not?

How does this equation apply to you on a daily basis?

Repeat the above equations for each problem presented to the class.

[First, I looked for the variable I needed to solve for and isolated that variable by moving the other variables to the other side of the equation.]

Answers will vary. [Possible answers include: I needed to first subtract, add, multiply or divide certain variables.]

Answers will be given.

Answers will be given and students will be allowed to explain their procedure.

[The variable asked for has been solved for.] A more thorough answer is [You can solve for another variable or reverse the above steps to get back to the original equation.]

There is only one way to solve the problem.[There are a number of variables in each problem and you can move each variable step by step or you can move variables with like operators at one time.]

[No.

[Because there is a variable being solved for.]

Answers will vary.

Transition to Elaboration

Now we are going to take what we have learned to another level!

ELABORATION

Time: 10 minutes


Probing/Eliciting Questions


Explain the activity.

Now that you know how to solve literal equations and have reasons for why these equations apply to you in your daily life, you are going to develop word problems using the literal equations.

Really, why?

The purpose of this activity is for you to take on the role of a teacher writing a test, much like your ROPES problems. You need to incorporate percents, decimals, fractions and use all operators (addition, subtraction, multiplication and division). These are all skills that you have developed over the past few weeks.

You are expected to write two word problems and are expected to provide the correct solution.

Provide students with an example.

Here is an example of what I expect. Using d=vt, here is an example word problem.

Mark is expected to travel to Miami for Christmas. As Mark likes to speed, he knows that his timing will be 20% less than expected and his velocity will be 1.15 times the posted speed limit. He also expects to take a number of back roads that will increase his distance by 5/4. Develop an equation that finds Marks travel time.

(5/4)d=(1.15v)(0.80t)

T=(5/4d)/(0.80*1.15v)

Allow students to create and solve their word problems with a partner.

Now create and solve two written word problems based on one of the science based literal equations discussed earlier. Your word problem should be original and unique.


EVALUATION

Time: 10 minutes


Assessment

Student Responses

Pass out the Variables and Expressions Assessment. Ask students to complete it individually and without their graphic organizers.

Now you are going to take an assessment that will show how well you grasped the concept.

Make sure you pack up, so you are ready to leave once the bell rings and you finish your assessment.

This assessment should take ten minutes. You will not need to use a calculator. All you need is a pencil.

Once you have finished the assessment, raise your hand and I will pick up your paper. If you finish ahead of time, please quietly remain in your seat until you are dismissed.

Name: _____________________________________________Date: _____________________

Literal Equations through Science

Math for College Readiness

Below are a number of literal equations. Please solve each equation for the stated variable. You should clearly write one step per line. When finished, explain how each equation applies to you on a daily basis.

1) , Find t

d=distance, v=velocity, t=time

2) , Find R

H=heat, I=electric charge, R=resistance, t=time

3) , Find G

F=force between two objects, G=gravity, M=1st mass, m=2nd mass, r=distance between centers

4) , Find vf

A=acceleration, vf=final velocity, vi=initial velocity, t=time

5) , Find v

F=force, m=mass, v=velocity, r=distance

6) , Find h

V=volume, R=radios, h=height

7) , Find w

P=perimeter, l=length, w=width

8) , Find F

C=Celsius, F=Fahrenheit

9) , Find a

S=position, V=velocity, a=acceleration, t=time

10) , Find

L=Inductance, M=mass, P=power, Q=charge

Name: __________________________________________Date: ____________________

Assessment


For the following problem, number the steps in the order that you would complete them if solving the equation. Place an X next to any incorrect steps. In the right column, explain your reasoning.

1) , Find h

For the following problems, solve for the stated variable. Be sure to show all steps and write a brief statement as to how the equation applies to you on a daily basis.

2) , Find R


3) , Find V


4) , Find C


Name: __________________________________________Date: _____________________

Explore #2 Activity


Below are a number of literal equations. Please solve each equation for the stated variable. You should clearly write one step per line. When finished, explain how each equation applies to you on a daily basis.

1) , Find t

t=d/v

d=distance, v=velocity, t=time

2) , Find R

R=H/(I2t)

H=heat, I=electric charge, R=resistance, t=time

3) , Find G

G=(Fr2)/(Mm)

F=force between two objects, G=gravity, M=1st mass, m=2nd mass, r=distance between centers

4) , Find vf

vf=at+vi

A=acceleration, vf=final velocity, vi=initial velocity, t=time

5) , Find v

v=

F=force, m=mass, v=velocity, r=distance

6) , Find h


7) , Find w

W=

P=perimeter, l=length, w=width

8) , Find F


9) , Find a


10) , Find

L=Inductance, M=mass, P=power, Q=charge

Name: ____________________________________________________Date: _____________________

Assessment


For the following problem, number the steps in the order that you would complete them if solving the equation. Place an X next to any incorrect steps. In the right column, explain your reasoning.

1) , Find h

3

Third, you factor out the h on the right hand side.

X

You cannot cancel out the 2’s because the S doesn’t have one.

1

First, you distribute the 2l.

2

Second, you subtract the 2lw because it doesn’t have an h.

4

Finally, you divide by the 2l+2w to isolate the h.

X

You must distribute the 2l to both the w and the h.

For the following problems, solve for the stated variable. Be sure to show all steps and write a brief statement as to how the equation applies to you on a daily basis.

2) , Find R


3) , Find V


4) , Find C


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