Targeted Content Standard(s): Student Friendly Learning...

Grade: 8 Lesson Title: Infinite and No Solution Equations Unit: 2 Time Frame: 1-2 weeks Essential Question:

How do we express a relationship mathematically?

How do we determine the value of an unknown quantity?

Targeted Content Standard(s): Student Friendly Learning Targets 8.EE.7 Solve linear equations in one variable.

a) Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until all equivalent equations of the form , , or result (where and are different numbers).

I can…

Solve equations with linear expressions on either or both sides including equations with one solution, infinitely many solutions, and no solutions. (8.EE.7)

Give examples of and identify equations as having one solution, infinitely many solutions, or no solutions. (8.EE.7)

Targeted Mathematical Practice(s): 1 Make sense of problems and persevere in solving them 2 Reason abstractly and quantitatively 3 Construct viable arguments and critique the reasoning of others 4 Model with mathematics 5 Use appropriate tools strategically 6 Attend to precision 7 Look for and make use of structure 8 Look for an express regularity in repeated reasoning

Supporting Content Standard(s): (optional) 8.EE.7 Solve linear equations in one variable.

b) Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Purpose of Lesson: The purpose of this lesson is for students to understand when linear equations have one solution, infinitely many solutions, or no solution as well as to create equations of that type.

Explanation of Rigor: (Fill in those that are appropriate.) Conceptual: Students will understand when linear equations have one solution, infinitely many solutions, or no solution. (8.EE.7a)

Procedural: Students will be able to create examples of linear equations in one variable with one solution, infinitely many solutions, and no solution. (8.EE.7a)

Application: Students will apply solving linear equations through real-world or “thin” context word problems. (8.EE.7b)

Vocabulary: Distributive property Simplify Like terms Solution Inverse operations

Evidence of Learning (Assessment): Pre-Assessment: 0 Pre-Assessment for prior knowledge*, 1 Linear Equations Pre-Test* Formative Assessment(s): 2 Scavenger Hunt Summative Assessment: 3 Linear Equations Post-Test*, 4 Roller Coaster Design* Self-Assessment: Built into lesson *Assessment is a unit assessment, not lesson assessment




Lesson Procedures: Segment 1 – Always, Sometimes, or Never Taken from:

http://map.mathshell.org/materials/download.php?fileid=1286

Approximate Time Frame:

70 minutes

Lesson Format: Whole Group Small Group Independent

Modeled Guided Collaborative Assessment

Resources:

Per group: Card set (page 14 MARS activity) (answers on page 10) Scissors Poster board Glue stick Markers

MARS Activity found at: http://map.mathshell.org/materials/download.php?fileid=1286

Focus:

The focus of this segment is to develop a conceptual understanding of when equations have one solution, infinite solutions, or no solution. This lesson assumes that students have already solved equations in one variable (8.EE.7b).

Modalities Represented: Concrete/Manipulative Picture/Graph Table/Chart Symbolic Oral/Written Language Real-Life Situation

Math Practice Look For(s):

MP.3 Students must share and critique each other’s posters.

MP.7 Students must use the structure of linear equations to determine how many solutions there are.

Differentiation for Remediation:

Differentiation for English Language Learners:

Differentiation for Enrichment:

Potential Pitfall(s): Independent Practice (Homework): Create equations that are always true, sometimes true, and never true.







Steps:

1. Always, Sometimes, Never: (65 min) This is a MARS task where students will group equations according to their solution of one solution, no solution, or infinitely many solutions and put their answers on a poster board.

a. Begin with the collaborative activity on pages 6 – 8. Student directions are on page 19 of the MARS task.

b. Have student groups share their posters. Student directions are on page 20 and teacher directions on page 8.

c. After sharing posters, have a whole class discussion on what students learned and exploring methods for justification. See pages 8 – 9 for example questions to use for this discussion.

Teacher Notes/Reflections:

2. Self-Assessment: (5-10 min) On a scrap piece of paper, have students write an explanation answering the following questions. Then have students discuss these answers with a partner.

a. What makes a mathematical equation in one variable always true?

b. What makes a mathematical equation in one variable sometimes true?

c. What makes a mathematical equation in one variable never true?


3. Independent Practice: (10 min) Ask students to create one equation in one variable that is always true, one that is sometimes true, and one that is never true. Ask students to create an additional problem of any type that they may have misunderstood based on their self-assessment.





Segment 2 – Infinite and No Solution Equations Created by ISBE

Approximate Time Frame:

170 minutes

Lesson Format: Whole Group Small Group Independent

Modeled Guided Collaborative Assessment

Resources:

Creation and Investigation Chart

Scavenger Hunt Cards

Scavenger Hunt Worksheet

Scavenger Hunt Observational Checklist

Create and Check Chart

Focus:

The focus of this segment is to move from a conceptual understanding of infinite and no solutions to procedural fluency in creating linear equations with infinite solutions, no solution, and one solution.

Modalities Represented: Concrete/Manipulative Picture/Graph Table/Chart Symbolic Oral/Written Language Real-Life Situation

Math Practice Look For(s):

MP 3: While students are investigating equations created by their partner, they need to develop an argument for why they believe the created equations are the correct type or not.

Differentiation for Remediation:

Differentiation for English Language Learners:

Differentiation for Enrichment:

Potential Pitfall(s):

Students may need a review on distributive property as well as when an equation equals zero as opposed to

undefined (

and

is undefined).

Independent Practice (Homework):

Create and Check Chart, Solving Practice




Steps:

1. Opening Activity: (5 min) Have students solve the following equations as bell work. Review answers with students.

a.

b. You started getting an allowance of per week but had to pay your sister back for bubble gum. If you had left, how many weeks had you been getting your allowance?

c.


This should be a review of solving linear equations with one solution.

Answers:

a. Answer:

b. Answer:

c. Answer:

2. Balancing No Solutions: (5-10 min) Using a balance scale, model solving the following equations. Students should see that the scale does not balance no matter how the equation is solved.

a. –

b.


Inquiry Prompt for whole class: What do you notice about the balance after solving the equations? What conclusions can we draw from this?

For more information on using a balance scale, see: http://www.borenson.com

3. No Solutions: (10-15 min) Have students individually solve the following equations. (Note that each equation has no solution.) After a few minutes of work, they should discuss their results with a partner.

a.

b. You and your friend have a race. You run three miles per hour and start six miles behind the actual start line. Your friend runs three miles per hour and gets a two mile head start. After how many hours will you catch up with your friend?

c.


Inquiry Prompt for partners: Did you get the same answers? Did you arrive at that answer the same way? How did your solution method differ from your partner’s?

4. What does it mean? (5 min) As a whole class, discuss what no solution actually means as an answer. Guide students to the concept that no solution means no value for the variable that will make the statement true.


http://www.borenson.com/




5. How do we know? (15-20 min) With a partner, students should discuss how we can tell that an equation has no solution.

a. After time to discuss, have partners write an explanation of their findings.

b. Students should then share their results with another pair of students. The two pairs of students should discuss the explanations and determine an efficient method for determining if a linear equation has no solution.

c. As a whole class, discuss the results from the partner work. Guide students toward a symbolic representation of this which is where and the written form which is that coefficients are the same on both sides of the equation but constants are different.


Inquiry Prompt for partners: Are there any patterns that prove there is no solution so that we don’t have to solve the equation completely?

Note that students may still need to simplify the expression on either side of the equation to a linear binomial.

6. Balancing Infinitely Many Solutions: (5-10 min) Using a balance scale, model solving the following equations, students should see that the scale is empty after manipulating the equation. (If using canisters for variables, you can model that adding the same on each side still keeps it balanced.)

a.

b.


Inquiry Prompt for whole class: What do you notice about the balance after solving the equations?

For more information on using a balance scale, see: http://www.borenson.com

7. Infinitely Many Solutions: (10-15 min) Have students solve the following equations individually and then discuss their results with a new partner. (Note that all answers are infinite solutions.)

a.

b.

c.


Inquiry Prompt for partners: Did you get the same answers? Did you arrive at that answer the same way? How did your solution method differ from your partner’s?

8. What does it mean? (5 min) As a whole class, discuss what having infinitely many solutions actually means as an answer. Guide students to the concept that this means any value for the variable will make the statement true.


http://www.borenson.com/




9. How do we know? (15-20 min) With a partner, students should discuss how we can tell that an equation has infinite solutions.

a. After time to discuss, have partners write an explanation of their findings.

b. Students should then share their results with another pair of students. The two pairs of students should discuss the explanations and determine an efficient method for determining if a linear equation has infinite solutions.

c. As a whole class, discuss the results from the partner work. Guide students toward a symbolic representation of this which is and the written form which is both the coefficients and constants are the same on both sides of the equation.


Inquiry Prompt for partners: Are there any patterns that prove there are infinite solutions so that we don’t have to solve the equation completely?

Note that students may still need to simplify the expression on either side of the equation to a linear binomial.

10. Partner Practice: (25-30 min) Have students create 6 multi-step equations filling out the equation section of the Creation and Investigation Chart.

a. Students should then exchange papers and have their partners determine if their equations are in the correct column on the Creation and Investigation Chart without solving. They should also explain their reasoning.

b. Note: Multi-step equation means it should be at least a two-step equation. So would not be acceptable but would be acceptable. Encourage students to write equations using everything they learned including combining like terms and the distributive property.


Materials: Each student will need a copy of the Creation and Investigation Chart.




11. Formative Assessment: Scavenger Hunt (25-30 min) a. Ahead of time, have cut and placed the 16

scavenger hunt cards around the room (space enough that groups of students can read them at one time).

b. Give each student a Scavenger Hunt Worksheet and have scratch paper to show their work.

c. Students are to work in pairs to complete the scavenger hunt by solving the equations and filling in the worksheet. Gather formative data using the Scavenger Hunt Observational Checklist.

d. Once students have completed their worksheet, they should hand it in with their work attached.


Materials: 16 scavenger hunt cards – cut out. Scavenger Hunt Worksheet for each student. Scavenger Hunt Observational Checklist.

Differentiation: Equations on cards can be modified to meet the levels of your students.

Note that the Scavenger Hunt worksheet can be used as a formative assessment to discuss with students.

12. Formative Assessment: (5 min) Have students write how they know if an equation has one solution, no solution, or infinitely many solutions on an index card and turn it in.


13. Independent Practice: (10 min) Have students create 6 more equations using the Create and Check Chart.

a. Students should create the equations and then prove that they are of the correct type by either writing an explanation or showing the formal symbolic work.


Materials: Create and Check chart per student.

14. Independent Practice: (20 min) Have students apply their knowledge of infinite and no solution equations by solving and creating more equations using the Solving Practice worksheet.


Materials: Solving Practice worksheet.

Students can use this as a self-assessment as long as you make the answers available.

Create and Check Chart Name ____________________________

1) Create 6 multi-step equations. Two of the equations should have one solution, two equations should have no solution and two equations should have infinitely many solutions. Write one equation in each rectangle under the title with that number of solution(s).

2) Prove that your created equations are in the correct column by either writing out a 1-2 sentence explanation or solving algebraically.

3) For the “One Solution” equations, give the solution.

Infinitely Many Solutions No Solution One Solution

Creation and Investigation Chart Name (creator of the equations) ____________________________ Name (investigator of the equations) ____________________________ CREATOR: Create 6 multi-step equations. Two of the equations should have one solution, two equations should have no solution and two equations should have infinitely many solutions. Write one equation in each rectangle under the title with that number of solution(s). INVESTIGATOR: Without solving the equation, determine if the creator has placed the equation in the correct column and write an explanation to justify your position. On the back of this sheet, choose one equation in each column to solve algebraically.

Infinitely Many Solutions No Solution One Solution

8.EE.7

Solving Linear Equations with Infinite and No Solution Practice

Solve the following equations. Some equations will have a single answer, others will have no solution, and still others will have infinite solutions. 1. 2. 3. 4. 5. 6.

7. 8.

9.

10. 11. 12.

13. 14.

15.

16.

17. 18.

19.

20. 21.

22. 23. 24. 25. 26. 27. 28. 29. 30.

8.EE.7

Create multi-step equations with the given number of solutions. 31. A single solution 32. Infinite solutions 33. No solution 34. Infinite solutions 35. No solution 36. A single solution 37. No solution 38. A single solution 39. Infinite solutions 40. A single solution 41. Infinite solutions 42. No solution

8.EE.7

Solving Linear Equations with Infinite and No Solution Practice Answers

Solve the following equations. Some equations will have a single answer, others will have no solution, and still others will have infinite solutions. 1. 2. 3. 4. 5. 6.

7. 8.

9.

10. 11. 12.

13. 14.

15.

16.

17. 18.

19.

20. 21.

22. 23. 24. 25. 26. 27. 28. 29. 30.

8.EE.7

Create multi-step equations with the given number of solutions. 31. A single solution 32. Infinite solutions 33. No solution All answers will vary 34. Infinite solutions 35. No solution 36. A single solution 37. No solution 38. A single solution 39. Infinite solutions 40. A single solution 41. Infinite solutions 42. No solution

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