Objective
Understand that equations can have no solutions, infinite solutions, or a unique solution; classify equations by their solution.
Common Core Standards
Core Standards
The core standards covered in this lesson
8.EE.C.7.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 an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
Expressions and Equations
8.EE.C.7.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 an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
Foundational Standards
The foundational standards covered in this lesson
7.EE.B.4
Expressions and Equations
7.EE.B.4— Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Criteria for Success
The essential concepts students need to demonstrate or understand to achieve the lesson objective
- Understand that if an equation has no solution, then there is no value for $$x$$that will make the equation true; for example$${ x+2=x+3}$$ has no solution.
- Understand that if an equation has infinite solutions, then $$x$$can represent any value and it will make the equation true; for example$$ x+1=x+1$$ has infinite solutions.
- Understand that if an equation has a unique solution, then there is one value for $$x$$that will make the equation true; for example$$ x+1=3$$ has a unique solution of $$x=2$$.
Tips for Teachers
Suggestions for teachers to help them teach this lesson
Lessons 8 and 9 introduce students to the three types of solutions that an equation can have. This lesson focuses on students understanding what each solution means about the equation, and on becoming familiar with arriving at solutions that look like $${4=2}$$or$${-8=-8}$$(MP.2). In the next lesson, students will reason with all three types of solutions to further internalize what they mean. The concept of no solution, infinite solutions, and unique solution will re-appear when students study systems of linear equations in Unit 6.
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Anchor Problems
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Problem 1
These scales are all currently balanced. You must choose onenumber to fill into the boxes in each problem that will keep them balanced. In each individual box you may only use one number, and it must be the same number in each box for that problem.
Guiding Questions
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References
Everybody is a Genius Solving Special Case Equations
Solving Special Case Equations is made available on Everybody is a Genius under the CC BY-NC-SA 3.0 US license. Accessed Aug. 31, 2017, 2:41 p.m..
Sort the equations below into the three categories.
a. $${3x=0}$$
b. $${3x=2x}$$
c. $${3x=2x}+x$$
d. $${2x+1=2x+2}$$
e. $${3x-1=2x-1}$$
f. $${x+2=2+x}$$
g. $${x+2=x-2}$$
h. $${2x-3=3-2x}$$
| Infinite Solutions | Unique Solution | No Solution |
Guiding Questions
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Problem 3
Solve the equations below and explain what each solution means.
a.$${5x-2\left ({1\over2}+x \right ) + 8 = - {1\over4}(16-12x)}$$
b.$${1.6(x+4)-3(0.2x+1)=2x-{1\over2}(18.6)}$$
c.$${{{{1\over2}x +4}\over-1} = {{-x-8}\over2}}$$
Guiding Questions
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Problem Set
A set of suggested resources or problem types that teachers can turn into a problem set
Fishtank Plus Content
Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.
Target Task
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Todd and Jason both solved an equation and ended up with this final line in their work:$${{-2}x=4x}$$.
- Todd says, “This equation has no solution because $${{-2}\neq 4}$$.”
- Jason says, “The solution is $${-2}$$because $${-2}({-2})=4$$.”
Do you agree with either Todd or Jason? Explain your reasoning.
Student Response
An example response to the Target Task at the level of detail expected of the students.
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Additional Practice
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
- MARS Formative Assessment Lesson for Grade 8 Solving Linear Equations in One Variable—This lesson has a great sorting activity, as well as problems where students examine and critique the reasoning of others.
- EngageNY Mathematics Grade 8 Mathematics > Module 4 > Topic A > Lesson 7—Exercises and Problem Set
Lesson 7
Lesson 9
