Unit Plan

## Solving Equations and Inequalities

• Assessment Anchors
• Eligible Content
• Big Ideas
• Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
• Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
• Numerical measures describe the center and spread of numerical data.
• Patterns exhibit relationships that can be extended, described, and generalized.
• Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
• Some questions can be answered by collecting, representing, and analyzing data, and the question to be answered determines the data to be collected, how best to collect it, and how best to represent it.
• The set of real numbers has infinite subsets including the sets of whole numbers, integers, rational, and irrational numbers.
• There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
• Concepts
• Distance, Pythagorean Theorem
• Linear equations and inequalities
• Linear functions
• Polygons and Polyhedra
• Rate of change
• Representations
• Competencies
• Understand and apply the Pythagorean Theorem to find distances between points in a coordinate plane and to analyze polygons and polyhedra.
• Use fundamental facts about distances and angles to describe and analyze figures and situations in 2- and 3-dimensional spaces and to solve problems including those with multiple steps.
• Use linear functions, linear equations, and linear inequalities to represent, analyze, and solve a variety of problems.
• Use the appropriate graphical data representation and extend understanding of the influence of scale in data interpretation.

### Objectives

In this unit, students will learn to use equations and inequalities to model real-world situations and to solve basic linear equations and inequalities, including the following forms/techniques. Students will:

• isolate a variable using inverse operations.
• combine like terms.
• balance variables on both sides of an equation or inequality.
• use the distributive property.
• write an equation in slope-intercept form.

#### Essential Questions

• How do we recognize when it is appropriate to use a linear model to represent a real-world situation and what are the benefits of using a linear model to answer questions about the situation?

### Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

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# Multiple -Choice Items:

1. Solve the following for x: 2x + 3 = 7
1. 4
2. −2
3. −4
4. 2

1. Which of the following was used to solve the equation in problem 1?
1. the distributive property
2. the subtraction property of equality
3. the multiplication property of inequality
4. the lowest common denominator

1. Solve the following for x: −4x − 3 > 9
1. x > −3
2. x < −3

1. Use the equation to answer the question that follows:

Which of the following situations could most likely be represented by the equation?

1. Breanne is calculating how much money she will make after 4 weeks of lawn mowing.
2. Aaron wants to find out what score he needs to get on his fourth quiz for an average quiz score greater than 75.
3. Breanne wants to know how many hours she needs to work in the fourth week to average exactly 80 hours per week.
4. Aaron wants to find out what score he needs to get on his fourth quiz for an average quiz score of at least 80.
1. Which of the following is a false number sentence?
1. 2x + 7 = 3x − 1
2. 3x + 7 < x − 2 + 2x
3. −4y −4 = 0

1. Which of the following is a possible solution of ?
1. (6, −2)
2. (1, 2)
3. (−2, 2)
4. (−2, 6)

1. Solve the following for y:
1. y = 10
2. y = −10
3. y = −6
4. y = 6
1. Which inequality is represented by the graph?

1. y ≥ 3x - 5
2. y ≤ -3x - 5
3. y ≤ 3x - 5
4. y ≥ -3x - 5

1. Use the inequality to answer the question that follows:

Which of the following would have the same solution as the inequality?

1. −3x > 36
2. −12 x > 36
3. −3 x > −36
4. −12 x > −36

1. D
2. B
3. C
4. D
5. C
6. D
7. B
8. C
9. A

1. Solve each of the following for x. Check your answers by substituting a value from your solution into the original statement. Compare and contrast your solutions.

3 x − 5 = 10     and      3 x − 5 > 10

1. Is the process below correct? If so, state a reason for each step. If not, state and correct the error.

1. Solve the following inequality for x. Show each step and graph the solution on a number line.

# Short-Answer Key and Scoring Rubrics:

10. Solve each of the following for x. Check your answers by substituting a value from your solution into the original statement. Compare and contrast your solutions.

3 x − 5 = 10     and      3 x − 5 > 10

x = 5 and x > 5

3(5) − 5 = 10              and                  3(6) − 5 > 10

15 − 5 = 10                                         18 − 5 > 10

10 = 10                                               13 > 10

For the equation, when we substitute 5 into the original statement it holds true. For the inequality, it will only hold true if we substitute numbers greater than 5.

 Points Description 2 Both correct solutions given Understanding of the different solutions demonstrated Correct substitution of 5 into the equation and any number greater than 5 into the inequality Explanation of the contrast correct and complete 1 Both correct solutions given Partial understanding of the different solutions demonstrated Correct substitution of solution in one statement or attempted substitution in both statements Explanation of the contrast correct but may be incomplete 0 Solutions incorrect or missing No understanding of the different solutions demonstrated No attempt at substitution or substitution completely incorrect or missing Explanation completely incorrect or missing

11. Is the process below correct? If so, state a reason for each step. If not, state and correct the error.

correct; given

correct; subtracting 2x from either side

incorrect; should flip the sign of the inequality because both sides need to be divided by a −3;

 Points Description 2 Written explanation and/or justification for each step correct and complete. 1 Written explanation and/or justification for two steps correct, but incorrect for one step. 0 Written explanation and/or justification for each step incorrect or missing (for two or more steps).

1. Solve the following inequality for x. Show each step and graph the solution on a number line.

x < 8

Multiply both sides by the LCD of 12

−9x + 84  >  8x − 36 − 2x           Distributive property

−9x + 84  >  6x − 36                    Combine like terms

−9x + 84 − 84  >  6x − 36 − 84  Subtract 84 from both sides or subtraction property of inequality

(Note: If the student adds −84 to both sides, this is acceptable. S/he would then need the explanation of Add −84 to both sides or addition property of inequality)

−9x  >  6x − 120                          Simplify or combine like terms

−9x − 6x  >  6x − 120 − 6x         Subtract 6x from both sides or subtraction property of inequality (adding −6x also acceptable, see previous note)

−15x  >  −120                              Simplify or combine like terms

Divide both sides by −15 or division property of inequality (Multiplying by −1/15 also acceptable, with explanation of Multiply both sides by −1/15 or multiplication property of inequality)

(Also note: The student must reverse the inequality sign to get full credit.)

x  <  8                                            Simplify

 Points Description 2 Correctly and completely solves the inequality Demonstrates thorough understanding of finding the LCD, the distributive property, the addition/subtraction property of inequalities, and the multiplication/division property of inequalities Supports each step with an explanation or identification of the correct property. 1 Correctly solves the inequality, but the answer may be incomplete and does not show all the steps or shows all the correct procedures but includes one calculation error Demonstrates partial understanding of finding the LCD, the distributive property, the addition/subtraction property of inequalities, or the multiplication/division property of inequalities Attempts to support each step with an explanation or identification of the correct property 0 Makes no attempt at solving or incorrectly attempts to solve the inequality Demonstrates no understanding of finding the LCD, the distributive property, the addition/subtraction property of inequalities, or the multiplication/division property of inequalities Does not give any explanation or property identification

# Performance Assessment:

The Eighth Grade Student Council in your school has been given permission to use the gymnasium to hold a Talent Show to raise funds for the local food shelf. As part of the planning group, you have the chance to help.

1. Set a goal of how much money you would like to raise. Think of different ways to make this money at the event, including ticket sales and refreshments. Do you want to charge the same for all ages or have separate prices? If you are using different prices, estimate the fraction of the total that will be in the different age brackets.
2. After deciding upon the ticket pricing and other items to be sold, list these different amounts. Use p for the number of people attending, and if some amounts involve only some of the attendees (for example, a portion of the attendees will purchase a soda, a portion of the attendees will be under 10 years old, etc.), estimate these amounts and represent them as fractions of the total p.
3. Research and determine any expenses involved with the event. These could include making programs, purchasing refreshments to sell, and producing posters and flyers to advertise the event.
4. Using the amounts from problem 2, write an equation that can be used to determine the number of people attending (p) needed to make your goal amount of money. Be sure to include expenses in this equation.
5. Does your answer in problem 4 sound reasonable? If it does not seem possible to get that many attendees, what other solution(s) could be used? Show how a new solution would change your equation.
6. Write a plan that your group will submit to the principal. Include the estimates and equation that you wrote.

# Performance Assessment Scoring Rubric:

 Points Description 4 Responds completely with detailed explanation Contains no math/calculation errors Demonstrates complete understanding of how to model a real-world situation in mathematical terms Shows complete understanding of the questions, mathematical ideas, and processes Goes beyond what is required by the problem, shows creativity 3 Responds completely with clear explanation Contains no major math errors or conceptual/procedural errors Demonstrates understanding of how to model a real-world situation in mathematical terms Shows substantial understanding of the questions, mathematical ideas, and processes Meets the problem requirements 2 Responds unclearly or has some parts missing. Contains several minor errors or one or more serious math errors or conceptual/procedural errors Demonstrates some understanding of how to model real-world situations Shows some understanding of the problem Partially meets the problem requirements 1 Misses key points and/or sections Contains major math errors or serious conceptual/procedural errors Demonstrates lack of understanding of how to model real-world situations Shows lack of understanding of the problem Does not meet the problem requirements 0 Fails to complete or incorrectly completes most sections Contains major math errors or serious conceptual/procedural errors Demonstrates complete lack of understanding of how to model real-world situations Shows complete lack of understanding of the problem Does not meet the problem requirements
Final 04/12/13