# 8.2.4A Representations of Linear Equations

8
Subject:
Math
Strand:
Algebra
Standard 8.2.4

Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context.

Benchmark: 8.2.4.1 Linear Equations to Represent Relationships

Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships.

For example: For a cylinder with fixed radius of length 5, the surface area  A = 2π(5)h + 2π(5)2 = 10πh + 50π, is a linear function of the height h, but the surface area is not proportional to the height.

Benchmark: 8.2.4.3 Linear Equations & Lines

Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line.

For example: Determine an equation of the line through the points (-1,6) and ($\frac{2}{3}$,-$\frac{3}{4}$).

Benchmark: 8.2.4.4 Linear Inequalities to Represent Relationships

Use linear inequalities to represent relationships in various contexts.

For example: A gas station charges $0.10 less per gallon of gasoline if a customer also gets a car wash. Without the car wash, gas costs$2.79 per gallon. The car wash is $8.95. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most$35?

## Overview

Big Ideas and Essential Understandings

#### Standard 8.2.4 Essential Understandings

At this point in their learning, students are familiar with proportional relationships.  They have used tables, graphs and equations to represent and solve problems that involve proportional relationships.  In this standard, students use this background to move into representing and solving linear equations and inequalities.  It is essential for students to make connections to real situations in order to make sense out of this representation. Students need to be able to use information to find the rate of change and the y-intercept and then write the equation describing the relationship represented in the situation.  Once students can write the equations and inequalities, the focus is on solving.  Students will explore solving linear equations, inequalities, equations containing absolute value, and equations with the variables being squared.  It is important for students to continue looking at tables and graphs as well as symbolic representations to find their solutions.  Sense-making comes when they can see the solutions in more than one representation.  It is crucial for students to be comfortable using the graph and table to find solutions as they move into solving systems of equations and inequalities.  The graph paints a picture of the situation and solution that is more of a visual representation for students.  When working with just the symbolic method, the meaning of the solution is often lost.  Knowing if their answer makes sense is an essential part of students' understanding when solving equations and inequalities.

#### All Standard Benchmarks

8.2.4.1
Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships.
For example: For a cylinder with fixed radius of length 5, the surface area  A = 2π(5)h + 2π(5)2 = 10πh + 50π, is a linear function of the height h, but the surface area is not proportional to the height.

8.2.4.2
Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used.
For example: The equation 10x + 17 = 3x can be changed to 7x + 17 = 0, and then to 7x = -17 by adding/subtracting the same quantities to both sides. These changes do not change the solution of the equation.
Another example: Using the formula for the perimeter of a rectangle, solve for the base in terms of the height and perimeter.

8.2.4.3
Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line. For example: Determine an equation of the line through the points (-1,6) and ([math[\frac{2}{3}[/math], [math[\frac{-3}{4}[/math]).

• Items must not have context

8.2.4.4
Use linear inequalities to represent relationships in various contexts.For example: A gas station charges $0.10 less per gallon of gasoline if a customer also gets a car wash. Without the car wash, gas costs$2.79 per gallon. The car wash is $8.95. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most$35?

• Inequalities contain no more than 1 variable

8.2.4.5
Solve linear inequalities using properties of inequalities.  Graph the solutions on a number line.  For example: The inequality -3x < 6 is equivalent to x > -2, which can be represented on the number line by shading in the interval to the right of -2.

8.2.4.6
Represent relationships in various contexts with equations and inequalities involving the absolute value of a linear expression. Solve such equations and inequalities and graph the solutions on a number line. For example: A cylindrical machine part is manufactured with a radius of 2.1 cm, with a tolerance of 1/100 cm. The radius r satisfies the inequality   |r - 2.1| ≤ .01.

8.2.4.7
Represent relationships in various contexts using systems of linear equations. Solve systems of linear equations in two variables symbolically, graphically and numerically. For example: Marty's cell phone company charges $15 per month plus$0.04 per minute for each call. Jeannine's company charges $0.25 per minute. Use a system of equations to determine the advantages of each plan based on the number of minutes used. 8.2.4.8 Understand that a system of linear equations may have no solution, one solution, or an infinite number of solutions. Relate the number of solutions to pairs of lines that are intersecting, parallel or identical. Check whether a pair of numbers satisfies a system of two linear equations in two unknowns by substituting the numbers into both equations. 8.2.4.9 Use the relationship between square roots and squares of a number to solve problems. For example: If πx2 = 5, then |x| = $\sqrt{\frac{5}{\pi }}$, or equivalently, x = $\sqrt{\frac{5}{\pi }}$ or x = $-\sqrt{\frac{5}{\pi }}$. If x is understood as the radius of a circle in this example, then the negative solution should be discarded and x = $\sqrt{\frac{5}{\pi }}$. • Allowable notation: ±3 • Items may assess the interpretation of square roots based on the context of the item. Benchmark Cluster Benchmark Group A - Representations of Linear Equations 8.2.4.1 Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships. For example: For a cylinder with fixed radius of length 5, the surface area A = 2π(5)h + 2π(5)2 = 10πh + 50π, is a linear function of the height h, but the surface area is not proportional to the height. 8.2.4.3 Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line. For example: Determine an equation of the line through the points (-1,6) and ([math[\frac{2}{3}[/math], [math[\frac{-3}{4}[/math]). • Items must not have context 8.2.4.4 Use linear inequalities to represent relationships in various contexts.For example: A gas station charges$0.10 less per gallon of gasoline if a customer also gets a car wash. Without the car wash, gas costs $2.79 per gallon. The car wash is$8.95. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $35? • Inequalities contain no more than 1 variable What students should know and be able to do [at a mastery level] related to these benchmarks • Students can write an equation or inequality to represent proportional and non-proportional linear relationships. • Students can identify the rate of change from a situation and the y-intercept and use that information to write an equation or inequality. • Students can convert between slope-intercept form, point-slope form and standard form and recognize that each is a form of linear. • Given a situation students can write an equation or inequality in standard form. • Given two points on a line, students can find the slope and y-intercept and write the equation. Work from previous grades that supports this new learning includes: • use/write inequalities when values are described using phrases like, at least, at most, greater than, less than, or more, or less • identify when situations, tables, graphs, and equations are proportional • identify when situations, tables, graphs, and equations are linear Correlations NCTM Standards: Algebra: Understand patterns, relations, and functions Relate and compare different forms of representation for a relationship; Represent and analyze mathematical situations and structures using algebraic symbols develop an initial conceptual understanding of different uses of variables; explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope; use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships; recognize and generate equivalent forms for simple algebraic expressions and solve linear equations #### Common Core State Standards (CCSS) 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.EE.7 Solve linear equations in one variable 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 7.EE.4b Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid$50 per week plus $3 per sale. This week you want your pay to be at least$100. Write an inequality for the number of sales you need to make, and describe the solutions.

8.EE.8  Analyze and solve pairs of simultaneous linear equations.

8.EE.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations.  Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

8.EE.8c  Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x2= p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that $\sqrt{2}$ is irrational.

## Misconceptions

Student Misconceptions

#### Student Misconceptions and Common Errors

• Students often have difficulty interpreting phrases that indicate the use of inequalities.
• Students may not understand the difference between proportional and non-proportional linear relationships.
• Students think that standard form is not linear because they are so familiar with y = mx+b.
• Writing equations in standard form is difficult because students get stuck thinking in slope intercept form.
• Students don't realize the coefficients in standard form (Ax + By = C) represented by A, B, and C different meaning than m (slope) and b (y-intercept) in the slope-intercept form (y = mx + b).
• Students will sometimes mistakenly give the wrong slope when given standard form.  They are familiar with slope-intercept form where the slope is automatically in the equation.  When given standard form they often state the slope as the coefficient of x
• Students may still have difficult finding slope.  They can still mix up the relationship between change of y and change of x.  For example:

dividing the change in x by the change in y
subtracting the x coordinate from the y coordinate
subtracting the x and y coordinates in different order

## Vignette

In the Classroom

Day 1: Focus on writing equations in standard form. (Ax + By = C) because students are usually most familiar with slope intercept form.

Teacher:  As a school we raise a ton of money for Pennies for Patients.  One of the events that brings in the most money is our walk-a-thon.  This year, Costco donated bottles of water and energy bars that we could sell to raise extra money.  We can make a profit of $2 for each water bottle and$4 for each energy bar that we sell.  Let's set a goal.  How much money should we shoot for?

Student:  Let's try and raise $600. Teacher: How are we going to reach our goal? Student: Well, we need to sell the water and energy bars. We'd have to figure out how many waters and how many bars we would need to sell to make$600.

Teacher:  Let's try it out.  As a group, come up with three ways we could raise $600. Student: We could sell 100 bottles of water and 100 energy bars. Because$2(100) is $200 and$4(100) is $400 so that adds up to be$600.

Teacher:  Did anyone come up with another combination of water bottles and energy bars?

Student:  Our group thought that we could sell 200 bottles and 50 energy bars.  We did the same thing $2(200) is$400 and then there is $200 left to go so it has to be 50 because$4(50) is $200. Teacher: Any others? Student: Well, yeah there are a lot...We could sell only one bottle of water or even no bottles...Same with energy bars. Teacher: As a group, come up with at least three more combinations of bottles and energy bars that will get the student council to reach the goal of$600.

(Students work in groups and come up with possible solutions.)

Teacher:  Combine with another group.  You should have 6 different possible solutions. If you have don't have 6 different solutions, find as many more as you need to get 6 different solutions.  Graph these solutions.  What do you notice?

Student:  Hey, it's a line.

Teacher:  How could you write an equation to represent all the possible solutions?

Student:  We would have to find the slope and y-intercept.

Teacher:  Ok. In each of your groups, come up with the slope and y-intercept for this situation.  Once you find the slope and y-intercept, write a sentence describing what each means in this situation.

Student:  I found the gaps and then used the gaps to find the change in energy bars over change in water bottles and got a slope of -.5.  Then I already knew the y-intercept. So in the equation x is the number of water bottles and y is the number of energy bars we need to sell.

Teacher:  Your sentence tells us what to do with the slope and y-intercept, but what does -.5 mean as a slope?

Student: Does it mean for every bottle of water we sell, we have to sell half less of  an energy bar?  That doesn't really make sense.  Or I suppose we could say every two water bottles we sell we have to sell 1 less energy bar.  Energy bars depend on water bottle sales?  That doesn't make sense. There aren't really independent and dependent variables.

Teacher:  Did anyone make energy bars the independent variable?

Student: Yeah. I found the slope by finding the change in the water bottles compared to the change in energy bars.  So we got a different equation.  We got y= -2x + 300.  x is the number of energy bars we sell and y is the number of water bottles.

So we could sell 300 water bottles and then for every energy bar sell 2 less water bottles?  That doesn't make sense either in this situation.  We want to sell as much as possible, not sell less.

Teacher:  Let's stop for a minute.  It sounds like you are having a tough time making sense of this situation in slope intercept form.  Both equations work to find the number of each item we need to sell, but we don't know which is independent or dependent.  Do you think that matters?

Student:  Well, it does if you are finding the slope and y-intercept.

Student:  Yeah, but it doesn't really as long as you label the variables correctly.  It's not like we are talking about selling something over a certain amount of time, like how many energy bars we need to sell per minute. Then independent and dependent would matter more.

Teacher:  So I'm hearing you say that slope intercept form might not be the best for this situation.  Did anyone think of another way to write this equation for this line?  Can we come up with something better?

(No one responds, students unsure.)

Teacher:  Let's look for some patterns.  You all came up with a list of solutions.  Let's organize those solutions on a table and look for a general rule to describe the pattern.

First, let's record your solutions in a table.  Some of you already did that.  If you did, be ready to extend your table.  Otherwise organize your solutions on a table

Next, let's add a few columns to look at the profit for each item.  Let's start with the profit from the water.  The first solution is 100 bottles.  So how much profit did we get?

Student:  We got $2 per bottle so its$2(100) = $200 Teacher: Okay. Let's write that in the profit column on your paper. Let's do the same thing for the profit from energy bars. Fill in the rest of your table. Make sure you record how you got the profit for each item. (Students record in the table and continue for the rest of the solutions.) What do you notice about the patterns in each column? Student: For the profit for the water bottles we always multiply by 2 and for the profit for the energy bars we always multiply by 4. Teacher: How could we write a rule for each item's profit if we sold any amount of water bottles or energy bars? Student: I would write 2b for water bottle profits and 4e for energy bar profits Teacher: Any other ideas? Student: That makes sense because when you look at the pattern, you take the number of bottles times 2 and the number of energy bars times 4. Teacher: So if our goal is for the profit to be$600, what rule would represent this situation?

Student:  Well, if we add the profits we always get $600, so the profit of water plus the profit of energy bars is$600.  So 2b + 4e = 600.

Teacher:  Guess what....That equation is also a linear equation.  It's just in a different form.  Mathematicians call this standard form.    In this situation, it makes more sense to write it this way because we don't have a specific slope and y-intercept that is given to us, and when you wrote it in slope intercept form the meaning of the slope isn't very relevant.  Does this equation make sense?

Student:  Yeah.  You can see that we get $2 for each bottle and$4 for each energy bar and you can also see we are trying to raise $600. That makes more sense than having to sell one less energy bar for every two waters we sell. Teacher: The relationship is still linear, as you saw on the graph. It's just a different way to express the equation of that line. Student: Kind of like a foreign language...I can say the words thank you in many different forms, but it means the same thing. Teacher: Interesting analogy! So now we have an equation of the line to represent all the possible combinations of water and energy bars student council can sell to earn exactly$600. How would the equation change if they wanted to make $400 or$1000?

(Students work to come up with new equations.  They compare to other groups.)

Student:  The profit per bottle and per bar stays the same; we just have to change the value it adds up to, so the 600 was the only thing that changed.

So it would be 2b + 4e = 400 or 2b + 4e = 1000

Teacher:  What if we made $3 per water bottle and$2 per energy bar and we went back to wanting to make $600 total? How would that change our equation? Student: It would just change the number in front of the variable. It would change our profit because we would multiply by a different cost. So the coefficient of x and y would change from$2 to $3 and$4 to $2 like this: it was 2b + 4e = 600 and it would change to 3b + 2e = 600. Teacher: What if we change our goal to raise at least$600? How would that change our mathematical statement?   How would it change our graph?  Think about it tonight and we can discuss it tomorrow in class.

## Resources

Instructional Notes
##### Teacher Notes

Students may need support with further development of previously studied concepts and skills:

• Give students ample opportunities to work with interpreting and writing simple inequalities before writing linear inequalities.
• Build off students prior knowledge of proportional relationships and use the vocabulary as you continue to move into linear relationships that are not proportional. Continue to compare the two.
• Create opportunities for students to compare proportional to non-proportional linear relationships.
• When writing equations in standard form, start with the situation and have students come up with possible solutions. Use the patterns in all the possible solutions to write the equation. (see vignette)
• When finding solutions for equations in standard form, it is important for students to graph the solutions so they can make the connection that standard form is still a linear equation.
• Standard form (Ax + By = C) cannot be written with fractions/decimals. The A, B and C must be integers.
• Traditionally in standard form (Ax + By = C) the coefficient of the x term (A) is a non-negative integer.
• There are multiple ways for students to find the y-intercept of line when it is not given. It is important that students are exposed to these multiple ways: they can extend the graph to see where the line crosses the y-axis, work forward or backward with the table to find the value of y when x = 0, substitute values of an (x, y) coordinate pair and the slope into y = mx + b and solve for b, or use point-slope form and solve for y.
• Continue to give students opportunities to make sense of the idea of slope as rate of change by using contextual problems where students can connect the idea of rate of change directly to slope.
Instructional Resources
• This lesson is like a Linear Battleship Game.  Students write equations to attack other ships.  This is practice for slope intercept form.
##### Additional Instructional Resources
New Vocabulary

slope-intercept form: The equation of a line in the form y = mx + b where m is the slope of the line and b is its y-intercept.

standard form: The standard form of a linear equation is Ax + By = C where A is traditionally non-negative and A, B, and C are integers with a greatest common factor of 1.

point slope form: The equation of a straight line in the form y-y1=m(x - x1), where m is the slope of the line and (x1,y1) are the coordinates of a given point on the line in a Cartesian coordinate system.

linear inequality: A Linear Inequality involves linear expressions on both sides of one of the relational symbols <, >, ≤ or ≥.

Professional Learning Communities
##### Reflection - Critical Questions regarding the teaching and learning of these benchmarks
• In what ways can you help students make connections between the three different forms of equations?
• What other mathematics do students know that would be useful when solving these problems?
• How are you connecting the mathematics to the students' reality?
• What questions could you ask students in order to better understand their thinking about the relationships they see? (see site below)
• What probing questions could you ask to help students expand their understanding of graphical and symbolic representations and the relationship between them?

NCTM Principals and Standards for School Mathematics, 7.5 Exploring Linear Functions: Representational Relationships

##### Materials
• This link has video of teachers preparing lessons with a math coach, the actual lesson, student work, students' debriefing, and follow up.  It would be a valuable tool to help facilitate a PLC session.
References

Comparing linear functions: Lesson planning (part A). (n.d.). The Inside Mathematics website. Retrieved June 12, 2011, from http://www.insidemathematics.org/index.php/classroom-video-visits/public...

Focus in grade 8:  Teaching with curriculum focal points. (2010). Reston, VA: National Council of Teachers of Mathematics.

Hendrickson, K. (n.d.). Equations of attack. Illuminations. Retrieved June 13, 2011, from  http://illuminations.nctm.org/LessonDetail.aspx?id=L782

Insights into Algebra 1 workshop 1. (n.d.). Retrieved June 12, 2011, from http://learner.org/workshops/algebra/workshop1/index.html?pop=yes&pid=2096#

Intoduction-Algebra. (n.d.). California Department of Education. Retrieved June 13, 2011, from  http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqalgebra.pdf

Math graphic organizers: A teachers' page. Retrieved June 12, 2011, from http://www.dgelman.com/graphicorganizers/#ALGI

McShea, B., Vogel, J., & Yarnevich, M. (2005, April). Harry Potter and the Magic of Mathematics. Mathematics Teaching in the Middle School, 10, 408.

Principles and standards for school mathematics. (2000). Reston, VA: National Council of Teachers of Mathematics.

Regents Exam Questions A.A.23: Transforming Formulas 2. (n.d.). Jefferson Math Project. Retrieved June 13, 2011, from www.jmap.org/JMAP/RegentsExamsandQuestions/3-AdobePDFs/WorksheetsByPI-To...

## Assessment

• Mary is ordering t-shirts for spirit week at her school.  U-Make It Tshirt company charges a $100 set up fee and$5 per t-shirt.  Write an equation to represent the cost y for any number of t-shirts x.

Correct answer:  y = 100 + 5x

• Which equation is equivalent to 3x + 4y = 15?

• A doughnut shop charges $0.70 for each doughnut and$0.30 for the carryout box. How many donuts can Shirley buy if she can spend at most \$5.00.   Write the inequality to represent this situation and solve.

Correct answer:  6 doughnuts
.7x  + .3 ≤ 5

• What is the equation of the line that has a slope of 4 and passes through the point (3,-10)?

Taken from California Algebra I Standards Test released questions.
http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqalgebra.pdf

• The equation of line l is 6x + 5y = 3 and the equation of line q is 5x - 6y = 0.  Which statement about the two lines is true?

Taken from California Algebra I Standards Test released questions
http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqalgebra.pdf

Taken from MN MCA III 8th grade item sampler.

Taken from MN MCA III 8th grade item sampler.

## Differentiation

Struggling Learners
• Do an activity with matching the different representations of a linear function.  Ask students to create their own equations that could also be included in the matching problems.
• Graphic organizers help students have a place to start and have a process for solving the problem.  This website has a variety of graphic organizers including How to Find the Equation of a Line.
English Language Learners
• Understanding vocabulary is very important for ELL students.  This website has a game called Vocabulary Toss that students could play to reinforce mathematics vocabulary.
• Graphic organizers help students have a place to start and have a process for solving the problem.  This website has a variety of graphic organizers including How to Find the Equation of a Line.
Extending the Learning
• Convert standard form Ax + By = C to slope intercept form using the variables A, B, and C.  Students explore the relationship between the values and come up with a way to find slope and y-intercept when in standard form based on this relationship without actually solving for y