This article covers everything you need to know about how to teach linear equations.
Picture this. You open the teacher’s manual to peek into the next unit: “Linear Equations“.
Ah, yes! Graphing lines, determining slope, changing forms! A subtle smile forms across your face as you delightfully remember learning to graph these equations that seem so simple to you. As you peruse a little further, you wonder if your students will encounter that same joy of learning. It seems simple enough to you, but will it be simple enough to them? Is there a way to make this experience enjoyable, even for the students who lack confidence?
You wonder to yourself, “Where do I begin?”
Wonder no more. Read on as we discuss not only where to begin your linear equations unit but also prerequisite skills, the definition of a linear equation, examples of linear equations, applications of linear equations, math standards on linear equations, tips about slope, common misconceptions, and ideas for you!
What We Review
Where to Begin A Linear Equations Unit?
Start With End in Mind
When we consider beginning a unit, we should start with the end in mind. After all, how will we know where to begin if we have not established our goals?
First, let’s begin with the standards we need to cover. Keep in mind, these standards may differ depending on your specific grade level or state standards. Once you have determined the primary standards to cover, you can create learning goals. Keep reading, and we will further provide a list of some important linear equations common core standards.
Let’s start with this important eighth-grade common core standard:
“Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.”
Learning Goals
We may choose to focus on the interpretation of slope. We can create a learning goal using student-friendly language. Here is an example of a learning goal based on this standard:
| “Students will be able to compare slopes in graphs to slopes in equations and explain their answers in context.” |
For more about comparing slopes, check out this post on parallel and perpendicular lines.
Now that we’ve selected one of the goals for our unit, we know we’ll need to plan specific lessons involving word problems and written responses. We know students must be able to not only identify slopes in equations and graphs but also explain slope in context. Creating goals at the beginning of the unit empowers us to use our class time to effectively prepare students to accomplish the learning goals. The learning goals become the bullseye.
Not only is the bullseye for you, but also for your students. Presenting the learning goals to students at the beginning of the unit provides students a focus and the ability to self-reflect. Before test day, students can ask themselves, “Do I know how to explain slope?” or “Do I know how to explain slope in context?” You can provide students opportunities to reflect on their new skills in class throughout the unit and self-advocate if they need help on specific goals.
A strong unit should only have about three to five learning goals. Creating too many learning goals gives us too much to accomplish in the time typically allotted for a unit. Having too few makes it difficult to address all of the standards. Remember, one learning goal may address the essential components from more than one standard. Be mindful of the academic calendar and how much time is allotted for this unit. It is better to successfully accomplish three goals than to quickly rush through five goals.
Once the learning goals have been created, we can focus on lesson planning. What skills do students need to support the tasks we are asking them to do?
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Prerequisite Skills for Solving Linear Equations
There are several prerequisite skills that will bolster students’ confidence as they embark on the linear equation journey. We will discuss three important prerequisites for teaching linear equations: distributive property, solving equations for a variable, and graphing.
1. Distributive Property
Students will be required to use the distributive property when changing point-slope form to slope-intercept form.
Be sure to take time to evaluate how well students can distribute before you begin the unit or at least early on in the unit. You may be pleasantly surprised by their abilities or shocked by gaps in their knowledge. Either way, using in-class assessments is important to target instruction at the skills where students most need assistance. While you may have time for a review day, it may be best to incorporate a small amount of distributive property practice each day. Consider using class “Warm-Up”/”Do Now” time or incorporating a few distribution problems into your lessons and homework.
2. Solving Equations for a Variable
Students must solve for y when changing standard form to slope-intercept form (sometimes called “literal equations”). For instance, when given 6x+19y=2 and asked to put it into slope-intercept form, students must determine how to isolate the variable y.
Early on in the unit, provide students with ample opportunities to solve for a variable in simple equations, such as solving for b in the equation A=bh. Warm-Ups and formative assessments are great places to put this type of practice problem.
We will discuss this topic further when we address misconceptions.
3. Graphing Basics
There is a large range of graphing abilities. Some students easily latch onto graphing and love expressing equations in graphical form. Other students get bogged down in understanding the new notation and how a coordinate plane works.
When graphing an equation, be sure to articulate important notation items. For example, when graphing an equation in point-slope form, such as y-7=\frac{1}{2}(x+4), be sure to write the point as (-4, 7) and explain to students that the x-coordinate is always written first and the y-coordinate is always written second. Explain how you know where to put the point based on the x-axis and the y-axis.
In teaching, we rarely have enough time to facilitate all of the review we would like to do. With graphing, instead of spending a day reviewing, try to emphasize key graphing notation and review as you teach the new concepts. Emphasize what a y-intercept is and how rise and run relate to the x-axis and y-axis. Even excelling graphing students will benefit from hearing the process articulated repeatedly.
For help seeing all the essential graphing skills that students will need to know, here is a post all about graphing linear equations.
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Linear Equation: Definition
“Linear equations” are a specific type of equation that describes a straight line.
Remember, a linear equation cannot describe just any random graph. The slope of the graph must remain constant because linear equations only describe straight lines (not curved lines).
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3 Examples of Linear Equations
To know how to teach linear equations, let’s start with some examples.
Linear equations can be written in three forms: slope-intercept form, point-slope form, and standard form.
- Slope-intercept form helps us to identify the slope and y-intercept of a line. Visit this post for more on slope-intercept form.
Slope-Intercept Form: y=mx+b
Example: y=7x+\frac{13}{2}
- Point-slope form is determined by the slope of the line and one point on the line. Visit this post for more on point-slope form.
Point-Slope Form: y-y_1=m(x-x_1)
Example: y-7=\frac{1}{2}(x+4)
- Standard form makes it easier to find the x-intercept and is useful for solving systems of equations. Visit this post for more on standard form of linear equations.
Standard From: ax+by=c
Example: 2x+6y=7
For more reviews and examples about all three forms of linear equations, check out this article showing differences and conversions between all the forms.
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3 Real-Life Applications of Linear Equations
Example 1: Gasoline Usage
Gasoline consumption is something students are familiar with. Whether students have smelled the gasoline fumes of a big yellow bus or waited in the car while their parents pumped gasoline, in some for another, the students have experienced gasoline.
We can simply look up a vehicle. For this example, we will use a Chrysler Pacifica. The Chrysler Pacifica has a fuel efficiency of 22 \text{ mpg} when combining city and highway fuel efficiency (source). The Chrysler Pacifica’s gas tank can hold 19 \text{ gallons} of gasoline (source).
Now, we can create an equation to relate the amount of gasoline with the number of miles driven after filling up.
We will let x represent the number of miles traveled and y represent the amount of gas left in the vehicle. We know the y-intercept is 19 because that is the amount of gas in the vehicle before driving it away from the gas station.
Determining slope takes a bit more work. Our graph is describing the number of miles you can travel for every gallon of gasoline. However, fuel efficiency describes the amount of gasoline burned for a certain number of miles. We can discuss the reciprocal. The reciprocal of \frac{ 22 \text{ miles}}{1 \text{ gallon}} is \frac{1 \text{ gallon}}{22 \text{ miles}} . The car will burn one gallon of gas for every 22 miles traveled. Because we are describing the amount of gas left in the vehicle, our slope is negative. When the gas is burned it is no longer in the gas tank. The slope of our equation is \frac{-1}{22}.
Once we have the equation:
y=\frac{-1}{22}x+19
…we can cover all kinds of questions!
- Ask students to determine how many miles the vehicle can travel or how much gasoline will be used for a certain trip.
- Students can create a graph and explain the x and y-intercepts in context.
- Have students research another vehicle and create another equation. Then, there are many comparison questions to be asked!
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Example 2: Cell Phone Bill
Many students are able to enjoy the luxury of a cell phone without the burden of a bill. Having students calculate a cell phone bill based may be a bit of an eye-opener!
Let’s look at a service provider such as T-Mobile. Their Essentials plan is \$30\text{ per month} This is a great place to discuss the connection to “rise over run”. However, we cannot pay \$30\text{ per month} until we have a phone to use. There is such a variety of phone types, but we can use a Galaxy A11 that costs \$180 . This will be our starting cost.
Now, we can create the equation:
y=30x+180
…to represent the amount of money that has been spent on the phone and phone usage. From this point, we can ask a variety of questions such as “How long before we have paid over one thousand dollars for the phone usage?” We could also have students create a graph or compare this cost to the cost of a different plan and phone combination.
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Example 3: Line of Best Fit Connection
The wonderful creator of scaffolded math has shared a project in which students grow their own grass. Students measure the growth each day to create a graph. The line created is not a linear equation. However, the students can create linear equations using two points on the graph. These linear equations can allow students to predict the future growth of grass.
This project can lead to meaningful discussions about real-world data.
In teaching mathematics, sometimes we avoid examples that break patterns. While sometimes this is appropriate, it is also valuable for students to see data that does not fit the pattern we are teaching about. Students can see how the skills they are learning can help them to understand and interpret data even if it does not quite match the current unit’s examples.
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Math Standards on Linear Equations
Earlier, we discussed the importance of standards in how to teach linear equations. Below are a comprehensive list of standards related to linear equations.
Our list begins with three eighth-grade standards and also includes two high school mathematics standards. The hyperlinks direct you to the website of the Common Core State Standards Initiative website where some of the standards are broken into substandard. Note that specific standards for your unit may differ depending on grade level and state standards.
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Solve linear equations in one variable.
Analyze and solve pairs of simultaneous linear equations.
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
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Tips to Help Students Understand Slope
Slope is a concept that will come up over and over again in the mathematics future of our students. This, however, should not give you a reason to gloss over slope. Slope is a foundational piece of understanding that students need to build upon. Understanding slope will help students to internalize more complex functions with changing slopes, to understand concepts like acceleration, and to understand how to determine the derivative of a function.
To build a strong foundation of understanding slope, we must use visual methods, oral methods, and kinesthetic methods. Remember, all students benefit from all methods. It is an educational myth that a teacher should alter their teaching style to reach a particular learning type. Instead, a teacher should always use all three methods, particularly when laying the groundwork for an important concept such as slope.
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Visual Methods
First: visual methods. Clearly show students what slope “looks like”. What is rise over run? Where is “rise” on the graph? Where is “run” on the graph? What does an equation with a slope of \frac{1}{3} look like? What does an equation with a slope of 17 look like?
Read this review article for a great interplay of content and visuals of graphs.
To go even further, show them what the equation is modeling. If you create an equation modeling the speed of an object, show them an object moving at that speed. Make visual connections to what the graph is modeling to help students understand what slope means. Keep reading to view the Turtle Time Trials activity which does this very well.
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Oral Methods
For oral methods, remember to use as much vocabulary as possible every time you explain graphing or calculating slope.
State when you count the rise, when you count the run, when you make a new point, when you graph a line with a slope of zero, etc. It is not sufficient to show students the graph and the image. You must explicitly state all of the connections to the equations.
Additionally, incorporate moments where you talk through identifying the slope, both with and without a visual. Allow students to verbalize how to determine the slope and how to calculate the slope. Students can benefit from listening to one another and students benefit from sharing their solving processes. The act of expressing how to find the slope forces the student to process their method more fully and increases their ability to retain and recall the information.
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Kinesthetic Methods
For kinesthetic methods, remember, you cannot “over practice”. Give students as much variety of problems about slope as you can.
Have them write out their work. Kinesthetic learning does not need to be complicated. Students can go step by step with a partner alternating who writes which step. You can have one student speak the steps while the other student writes what their partner says and then switch roles. Students can use their arms to visually show what different slopes look like. Students can use magnets to create a large graph on a classroom whiteboard. There are so many different ways!
Some teachers love craziness and energy and get their students up and moving and interacting while others want peace and calm. Both teachers can effectively use kinesthetic learning. Wherever you fall on that spectrum, it is imperative to provide your students an opportunity to practice slope themselves and to make them write it or physically demonstrate their understanding.
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Common Student Misconceptions with Linear Equations
Horizontal and Vertical Lines
After a bit of practice, students usually become quite confident using:
y=mx+b
Students beam with pride as they easily identify a slope of 5 and a y-intercept of 2 in the equation y=5x+2.
Then comes an equation that does not exactly fit the pattern. The students are raising their hands and you know they are looking at an equation such as:
x=6
While horizontal and vertical lines may seem easier than graphing a line in point-slope form, these lines do not fit the pattern we have instilled into our students. We must take the time to explain why these equations are different. As stated above, it is imperative to provide visual connections to the equations and state how you know what the graph should look like.
This is a topic that you may feel students master quickly. Then, you may be disappointed when they perform poorly on this topic on the summative assessment. Remember, if students are not rehearsing the information frequently, they will forget. It takes time before information goes into long-term memory storage and even more practice to make retrieving that information easy. Do not forget to include a few horizontal and vertical lines practice questions during in-class practice and during homework practice with high frequency.
For more info, we have an entire post dedicated to horizontal and vertical lines.
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Solving for a Variable
Sometimes as math teachers, we forget the mental hurdles students go through when learning new skills. To us, solving for one variable seems simple enough whether or not there are any additional variables.
To many young students of mathematics, solving for a variable means “getting an answer”. They want a number, and when you beautifully finish changing standard form to slope-intercept form and you write:
y=\frac{-6}{19}+\frac{2}{19}
…and proudly exclaim, “I have solved the equation for y,” some students may appear puzzled. To those students, you could not have solved for y because you do not have “an answer”.
This is a very simple mental hurdle that can be overcome with an explanation, examples, and practice. Tell your students that the solution is not a number but an equation. Explain that their answers will look different than what they are used to. As stated above, identifying solving for a variable as a prerequisite skill can help you make the jump into solving for y more easily. Provide students with ample opportunity to solve for a variable in simple equations.
Of course, the misconceptions for your students may be different. There is a myriad of other reasons your students may have a specific hole in their knowledge. This is why formative assessments are so important. Take the time to get feedback from your students. Ask them questions and have students answer on a whiteboard, on Kahoot, through random calling.
When students are unable to correctly answer questions, investigate what mistakes they are making. Correct the mistake and provide frequent opportunities to practice using the new math skills correctly!
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Ideas for Linear Equation Activities
1. Albert Practice
At Albert.io you will find great information and thousands of practice questions. These practice questions are a great way to ensure all students receive meaningful feedback. Not only does Albert.io explain all solutions to their questions, but they include explanations about why the incorrect answers should not have been chosen. These explanations can help students identify their mistakes and learn from them.
- Check out The Best Algebra 1 Review Guides which includes links to review guides and practice questions.
- You can explore the full Albert Algebra 1 course here.
2. Linear Equations Game from TeachEngineering
If you love incorporating science and STEM education, this activity is excellent!
It creates a unique experience where students are tasked with guiding space shuttles through obstacles. The only caveat is that you do need to have MATLAB, but you can download MATLAB as a 30-day trial for free. The activity provides meaningful practice determining the locations of points, determining slope, and using the slope to solve for the y-intercept in slope-intercept form.
3. Turtle Time Trials (Desmos)
This interactive experience called “Turtle Time Trials” is sure to be a student favorite. Desmos walks students through the process of creating an equation to represent the velocity of a racing turtle. Not only are the turtles wonderfully adorable, but the mathematical skills incorporated help students connect linear equations skills to modeling real situations.
Desmore has additional activities for use in a linear equations unit.
4. The Road Trip Project by Carl Oliver
While this road trip project may be lengthy, the application and explanation skills are meaningful. Students are presented with an opportunity for a $10,000 road trip, but must make critical decisions about companions, transportations, and more.
Throughout the project, students are organizing data and analyzing information to make decisions!
