Learning Module: Linear Equations
Multiple Representations of Linear Equations
Learning Module
- Creator(s):
Pre-Algebra is one of the foundational courses of Middle School Mathematics. However, it can be pretty challenging for some students. In fact, there has been a long debate as to whether Algebra concepts should even be taught in 8th grade, or if they should be withheld until students are more capable of dealing with the ideas and rigors of the course (Clotfelter et al., 2012). It is one of the first times that middle school students are asked to work with more abstract mathematical concepts on a larger scale. Linear relationships are one of the first concepts where students begin to see the abstract representation of mathematical concepts, starting with graphs and tables. Although they have had some experience making graphs, during 8th grade they make a more formal connection between the various representations of linear relationships and learn to use those connections to create shortcuts and deepen their understanding.
The focus of this Learning Module is to help students be able to comfortably switch between multiple representations of linear relationships, seeing the connections between each representation. It is intended for students who have already gone through the class but struggled to master the concepts during the unit and need additional help, review, or even just a different, asynchronous opportunity to view the material in a multi-modal presentation. The benefit of this Learning Module is that students can work at their own pace, re-watch videos, or redo worksheets until they feel comfortable moving on to the next module.
Students will have been introduced to linear relationships, slope, and y-intercept prior to this Learning Module, but not in an extensive and thorough manner. The hope is that this module will be used to deepen understanding and mastery before moving on to further concepts that build on linear relationships.
Having taught 8th-grade math for over 15 years, I am always on the lookout for new or fresh ways to present the material to reach a diverse population of students. That is the overarching goal of this Learning Module--to make challenging Algebraic concepts, such as working with linear equations, easily and comfortably accessible to all types of learners.
The following are the mathematics learning objectives that will be covered in this unit. Please note that in order to fully grasp the concept of working with linear relationships, we will need to review some foundational skills first.
Learning Objectives:
Upon completing these modules, students will be able to:
These goals align with the following Common Core State Standards (CCSS):
TIMELINE:
This unit will be comprised of 6 lessons, but will also require one-three days (at teacher discretion) for review and practice of skills prior to the unit assessment.
Learning Objective: State characteristics of linear relationships
Hello everyone. Let's get started. You have been working on linear equations this chapter, but I know some of the concepts have been challenging. First, we need to see how much you know. Please take the following Pre-Assessment and submit your work on Google Classroom. Don't worry if you don't know how to do everything. Just try your best, and know that this is what you will be working on. We do this so you can see how much you learned over these lessons.
How did that go? Did you know how to do all of those problems? If not, that's okay. Remember, the goal of these lessons is to help clear up any confusion and make sure you are solid with the material. As you do these lessons, make sure you pay attention to what things are similar to those we mentioned in class, and what things are different or unique. Sometimes things are presented in different ways, and that can help make our understanding stronger.
The first thing we are going to do is start with the basics. What is a linear function or equation? Watch the following video and make sure you take notes. You will be using the information to answer some questions.
Fig 2. Basic Facts about linear functions. https://www.youtube.com/watch?v=MXV65i9g1Xg
I know that was a lot of information, but hopefully, that made sense. We will be going more in-depth into this information in the next few days. For now, let's see what you remember (by the way, you are welcome to go back and watch some or all of the video again if you need to).
WORK FOR TODAY'S MODULE
Post a Comment: Write about one concept, idea, or fact that stood out to you from the video. Why did that stick out? Was it something that you had heard in class, or was it new to you?
Each comment should consist of at least 5 full sentences. Your comment should demonstrate that you watched the video and should offer real insight into your thinking. Please use complete sentences.
Update: Think about an example of a linear equation in real life. What would the "starting value" be? What would the "slope" or "growth" be? How do you know it is linear? If you graphed it, what would it look like? Make a sketch of the graph, and make sure you label the "x" and "y" variables.
Your update should be at least 3 paragraphs (5 or more sentences in each paragraph). It should also include some type of illustration, which could be a picture off of the internet or even a picture of your own work. Anyone should be able to read your work and understand what you are doing and showing. Please be creative and show your effort in your work.
Replies: Reply to two of your classmates' updates. Do you agree that it is a linear equation? What might make the situation NOT be linear? What would have to be different?
Your replies should include at least 5 sentences and demonstrate that you have read and/or viewed the content posted in the update and have put forth an effort to understand it. Your reply should also directly answer the questions listed above.
Learning Target: The student will be able to define slope and y-intercept and explain the basic characteristics of linear relationships.
Purpose of this update: In order to do this unit, it is essential that we cover the basics. That is the goal of this lesson. Students will take a Pre-Assessment to demonstrate their knowledge prior to the unit, and then take the same assessment at the end. The video then goes over the basics of linear equations, particularly slope and y-intercept.
Teaching tips: This lesson is pretty foundational for the section. The video describes a lot of the information concerning linear equations, particularly slope and y-intercept, but it goes pretty quickly. Students may need to watch and re-watch the video several times (or at least parts of it) in order to feel confident with the material. It is not essential for kids to master this material at this point. This is just an introduction to the unit, but students should already have a pretty good grasp of these concepts from the unit when it was covered in class. This is intended for students who struggled the first time around.
The pre-assessment may be pretty ugly. Some of the problems are pretty straightforward, but others can be tricky. It is okay if they do not know how to do them at this point. Encourage them to do their best, but not to get frustrated.
Extensions: An appropriate extension for this activity would be for students to create their own video in which they share how they would define linear relationships. What examples would they include to teach fellow students about linear relationships?
Learning Objective: Be able to graph linear equations correc
Hello again. Today we are going to do some graphing! You might feel very comfortable making graphs already, but are you a master at it? That is what we will find out. First, watch the video, and then you will get a chance to prove how comfortable you are with graphing.
Fig 3. Process of graphing linear equations. https://www.youtube.com/watch?v=94OsUWGZ5K8
What do you think? Do you feel the video covered everything you need to make a graph? Isn't it amazing how much there is to correctly graph a linear equation? Perhaps it would be helpful to see things a different way. Here is a graph showing the steps to graphing linear equations:
Wow, that's a lot of information! Which one do you like best? It is important to understand that ALL of these methods is beneficial. It just depends on what situation and information you have. Do not just rely on one method, but know this information so you can use the best method for whatever problem you have.
Now, let's have a little fun. Go to Desmos graphing calculator and play around a little bit with different linear relationships. Follow these steps:
1. Go to: www.desmos.com
2. Type in one of the equations that were listed in the video and see if it matches the graph given.
3. Type in several equations that have the same y-intercept, but different slopes. What do you notice?
4. Type in several equations that have the same slope, but a different y-intercept. What do you notice?
5. Type in several equations that have the same numbers, but different signs (for example, 5 and -5 as slopes or 3 and -3 as y-intercepts). How does that change your lines?
6. Play around a little and explore different equations. Try fractional slopes or y-intercepts. Try slopes of 0. Does anything stand out?
Work for Today's Module
Comment: Today we discussed how to graph linear equations. Which of the methods do you prefer when you make a graph and why?
Each comment should consist of at least 5 full sentences. Your comment should demonstrate that you watched the video and should offer real insight into your thinking. Please use complete sentences.
Update: Pick one of the methods for graphing linear equations mentioned today. Come up with a real-world problem that you could graph using that method. Create a graph and describe the process you followed to make the graph. Show all your work and upload a picture of your work into your update.
Your update should be at least 3 paragraphs (5 or more sentences in each paragraph). It should also include some type of illustration, which could be a picture off of the internet or even a picture of your own work. Anyone should be able to read your work and understand what you are doing and showing. Please be creative and show your effort in your work.
Replies: Reply to two of your classmates' updates. What did you think of their example? Provide another real-world problem that could be graphed using this method.
Your replies should include at least 5 sentences and demonstrate that you have read and/or viewed the content posted in the update and have put forth an effort to understand it. Your reply should also directly answer the questions listed above.
Purpose of this update: This lesson is focused on making sure students understand how to graph linear equations. There are various different techniques to graph linear equations and students need to be familiar with them all. Understanding the methods and the appropriate situation for when to use them is important as we move forward.
Teaching tips: Please make sure that the students are graphing the points correctly from the y = mx + b form. All the students should be able to graph linear equations, but there may be several that forgot how to do it and will need this refresher. Keep an eye out for those struggling with this concept--particularly those switching the slope and y-intercept.
There are a lot of different methods presented here. It might be easy for students to get overwhelmed. Try to ease their concerns, but also stress that it is important that they understand the different methods so they can successfully graph in a variety of situations. Perhaps have a discussion with students about the different methods and have a group conversation about when to use each.
Extensions: An appropriate extension for this activity would be for students to create a handout or poster for one of the methods presented in which students list the steps for graphing and include a description of when this method would be preferred.
Learning Objective: Be able to write the linear equation from data on a table
Okay, everyone. Now it's time to start writing some actual equations from different representations. We will begin with tables. Watch the following video about how to do this. Be careful, not all problems will be as straightforward as the first problem.
Fig 5. How to write equations from a table. https://www.youtube.com/watch?v=xXxfDT_GLJ0
Did that make sense? Although it was only two examples, they are of very different situations. In the first problem, all the "x" values increased by 1, which makes finding the equation pretty easy. However, that is not always the case. Please make sure you check how "x" is changing before finding the equation, which is why the video presented the idea of "change in y, over change in x." Be careful as you do your work.
Here is your chance to prove you paid attention. Do the following worksheet and submit your work through Google Classroom.
How did that go? Were they easy or challenging? We will talk about them in class tomorrow, so make sure you bring in any questions you may have.
Work for Today's Module
Comment: Today we discussed how to write equations from linear situations. Respond to this module with a description of when it would be helpful to be able to write a linear equation from a table. Why is it important?
Update: Writing equations from tables can be very simple in certain situations. Create your own example of a "simple" table to write an equation and then create an example of a challenging table. Also, include a table that is NOT linear, along with an explanation of how you know it is not linear.
Replies: Reply to two of your classmates' updates. Why was their "challenging" table challenging? Describe the characteristics that made it challenging.
Purpose of this update: One of the main goals of our linear equations unit is to be able to switch back and forth between different representations. Students should be comfortable creating tables from equations, but often struggle to reverse the process. This lesson is intended to solidify their knowledge of writing linear equations from a variety of different tables, including situations where "x" does not increase by 1 each time.
Teaching tips: It might be easy for students to be overconfident going into this lesson. Writing equations from tables may seem pretty easy, but that is because most cases only include tables where "x" increases by 1 each time. The video clearly shows examples where "x" increases by a different value and shows how to write equations from those tables. Please make sure students don't shy away from these more challenging tables and instead, help them embrace the "challenge" of doing tougher tables. Also stress that as long as they learn the correct process, finding equations from ANY type of table is the same.
The worksheet includes a variety of different situations and tables. Students will have to be very careful as they are doing their work, as it will be very easy for them to make silly mistakes. You may need to go over the directions with the students to make sure they understand what they need to do. It might also be tough for some students to identify the y-intercept when the "x" does not equal zero. That part will take additional work.
Extensions: An appropriate extension for this activity would be for students to create a short video in which they present one or two "challenging" tables and walk classmates through how to write their equations, paying particular attention to the process needed.
Learning Objective: Be able to write a linear equation from a graph
How have you done so far? Has it been easy? Tough? Hopefully, you feel pretty confident with the skills we have worked on. We are now moving on to my favorite--writing linear equations from graphs. You will need to show some work, but it is fascinating how easy writing equations can be once you master the process. Watch the following video and get ready to show off your skills!
Fig 6. Writing equations from graphs. https://www.youtube.com/watch?v=NvZHnDGLePw
How was that? Do not overcomplicate the process. Find the lattice points, make your slope triangle, and use that to find your slope. Then, find your y-intercept on the graph and use it to finish your equation. Here is a visual of the steps to write the equations:
However, it is not always this easy. Beware of the two following situations--the biggest mistakes students always make with writing equations from graphs:
1. Students will confuse positive and negative slopes. Make sure you always follow the graph from left to right. As you are doing that, does the line go up or down? If it goes up, it is positive; if it goes down, it is negative.
2. Students will assume all graphs have a scale of 1. Many graphs will have a scale of 1 for each square, but that is not always the case. Not only could the scale be going by 2 or more, but sometimes the scales will go by different quantities! Be careful not to just count the number of squares between points, but to find the value they are changing by. It is very easy to miss this step.
Now is your chance to prove you can't be tricked! Find the equations of the following graphs and be careful that you don't fall for any traps!!
Although there were only four problems, be careful! These problems are pretty challenging!
Work for Today's Module
Comment: Today we discussed writing linear equations from graphs. What do you think will be the issue that would confuse students the most? Why?
Update: One of the toughest things to do is to create examples that can challenge others. Create 4 graphs that would challenge your peers. Make sure each one has a different "challenge" and have some fun with it (don't make them impossible but do push yourself!).
Replies: Reply to two of your classmates' updates. Which of their examples was the toughest? Were any of their examples too easy? Find the equation for at least two of the graphs given.
Purpose of this update: Graphs are one of the more common types of representations of equations, both linear and nonlinear. Future years will require students to be able to graph a variety of different equations quickly and accurately. Today's lesson challenges students to write equations from a graph, paying special attention to situations when the scale is not one.
Teaching tips: Today's lesson may seem very simple, but when the scale is not one, there are often a lot of mistakes. Another common mistake is switching the rise and the run. You will need to make sure you are circulating as students are working on their worksheets and verify that they are doing the first couple of problems correctly. You may also need to discuss the steps for finding the slope with the students, as lattice points could be something that trips students up.
Be sure you start class tomorrow by going over the questions your students will surely have. However, make sure you first open it up to peers to see if someone can explain it.
Extensions: An appropriate extension for this activity would be for students to create a short video answer key to the homework yesterday. Students should show the correct answers, along with their work, and then be able to explain to their classmates how they found it. You could assign other students to watch it as they check their answers and give students feedback.
Learning Objective: Be able to write an equation from two coordinate points
We are almost there! Hang in there. Today is probably the toughest day, as we have a somewhat abstract concept to study. Our focus now is on finding the equation of a line when you are only given two points. No graph. No table. Just two points. No, you may not graph them or make a table (actually, you could always do that, but in many cases, it would take you a ridiculous amount of time!). Instead, we will learn a method to use Algebra to solve this rather quickly and painlessly! Here is a video walking you through the process with two coordinate points: (4, -1) and (0, 5)
Fig 8. Finding the equation from two points. https://www.youtube.com/watch?v=un5tmpTiNkU
There, that seemed easy enough, right? I will say, the one thing I did not like about this video is that they picked two very easy points, didn't they? In fact, did you catch that you didn't even have to do the second step in this situation to find the y-intercept? Why is that? Well, because one of the coordinate points they chose already told you the y-intercept--point (0, 5). However, the process is what is important. In fact, let's see the process using different coordinate points:
Fig 9. Finding the equation with two points. https://www.youtube.com/watch?v=4yy94rZCT-4
As you see in this case, we still use the same process, just the numbers are different. Please be careful as you do this work and make sure that you identify which is your first and which is your second coordinate point. It is very easy to make a silly mistake using this method.
Okay, now it's time to practice. Please do the following worksheet and submit it through Google Classroom. Although the worksheet says to just find the slope, I want you to find the y-intercept as well and write the whole equation.
As you see, it does have the solutions to the slope on the last two pages. Please use that to check your work and make sure you are doing it correctly before finding the y-intercept. We will discuss it tomorrow in class if you have questions.
WORK FOR TODAY'S MODULE
Post a Comment: What is the toughest part of this process? Why? You can also talk about what you think will be the two easiest mistakes to make in this process and talk about how you could avoid making those mistakes.
Update: Think of a real-world situation that would require you to use find the equation from two coordinate points. Describe the situation and provide an example. Show your work for finding the equation and then talk about how the equation would help solve the problem.
Replies: Reply to two of your classmates' updates. What do you think of their example? Can you expand on their work? Where might their example fall short? Why?
Learning Target: The student will be able to find the equation of a line from two points.
Purpose of this update: Finding an equation from two points is a necessary skill in Algebra 1. Students are learning to manipulate equations to find missing values and applying algebraic skills to problem-solving.
Teaching tips: This is probably the most complicated process in this learning module. Although not necessarily difficult, it is a complex set of steps and there are a lot of things that could go wrong. You may want to go over the process with the students in the class a few times just to make sure they completely grasp the concept before having them do it on their own. They are also welcome to watch one or both of the videos repeatedly in order to follow along.
Two of the most common mistakes made in this process are either switching the change in "y" over the change in "x" or switching which "y" and "x" come first between the two coordinate points. It may be wise to push students to pick a point and label it "1" and the other "2" before starting to do the actual equation.
Extensions: An appropriate extension for this activity would be for students to have students create a poster in which they clearly show the steps to finding the equation between two points. The work should be clear and complete and include challenging points.
Learning Objective: Be able to correctly find the slope and y-intercept in a variety of different situations
So there it is. You have learned how to work with linear equations in various situations. You have also learned how to write a linear equation from a variety of starting points. Now it is time to see how well you can repeat the process and how deep your understanding was. This will be a two-part process. First, you will retake the assessment that we took at the beginning of this unit. My hope is that you will find it a lot easier than the first time around! Let's give it a shot:
When you are done with the quiz, upload it into Google Classroom.
For your final work in this unit, you are going to create an end-of-the-unit assessment for your class. It must include at least 12 problems, with the following distribution of each topic we have gone over in this unit:
2 problems of Graphing from an equation
3 problems of Writing an equation from a table
3 problems of Writing an equation from a graph
2 problems of Writing an equation from two coordinate points
2 problems of writing and solving linear equations around a real-world problem
Problems should be challenging, but reasonable. They should also be presented in a way that challenges your classmates to repeat the appropriate method. Think of all that we have covered to this point and try to make sure that your assessment includes a good representation of each topic.
You must also include an answer key to your assessment, showing all your work, a clear answer, and even a rubric of how you would grade the assessment.
The most effective assessment created will be the one used for the class, INCLUDING yourself!!
The assessment will be graded on the following rubric:
Please take your time and make sure that you make an excellent test. You will get your rubric score back within 2 days. My hope is that you were able to understand the process of writing linear equations and that your test would challenge anyone who takes it (within reason). Make sure you don't make it too ridiculously challenging, as your peers will be taking it as well as yourself.
The goal of this Peer Reviewed Work is to gauge the understanding and depth of knowledge attained by the students in this module/unit. The test will hopefully show their ability to determine key facts, create real-world problems related to the material, and challenge them to think beyond rote memorization or concept repetition. The added bonus is that the chosen test will be used to assess the class. This also makes sure that the student is able to do the work they included in their own test and that they did not get help creating the test in a way that was beyond their understanding or abilities.
Baker, Charles. (2020, March 31). Writing Linear Equations from a Table. YouTube. https://www.youtube.com/watch?v=xXxfDT_GLJ0
Clotfelter, Charles T., et al. “Algebra for 8th Graders: Evidence on Its Effects from 10 North Carolina Districts.” NBER. www.nber.org/papers/w18649.
Graphing Linear Equations (solutions, examples, videos). (n.d.). www.onlinemathlearning.com. https://www.onlinemathlearning.com/graphing-linear-equations.html
Mathantics. (2017, August 29). Basic Linear Functions - Math Antics [Video]. YouTube. https://www.youtube.com/watch?v=MXV65i9g1Xg&feature=youtu.be
Mathwithmrbarnes. (2015, May 9). Finding the Equation of a Line from a Graph (y = mx+b) [Video]. YouTube. https://www.youtube.com/watch?v=NvZHnDGLePw
McLogan, Brian. (2012, October 11). Writing the equation of a line given two points [Video]. YouTube. Retrieved October 7, 2022, from https://www.youtube.com/watch?v=4yy94rZCT-4
MrsALovesMath. (2017, April 14). Equation of a Line Given Two Points [Video]. YouTube. Retrieved October 7, 2022, from https://www.youtube.com/watch?v=un5tmpTiNkU
Spring, W. (2020, May 13). Linear Relations - Middle School Math Alberta [Video]. YouTube. https://www.youtube.com/watch?v=94OsUWGZ5K8
Staff, S. (2022, June 23). Teaching Linear Equations in Math. Houghton Mifflin Harcourt. https://www.hmhco.com/blog/teaching-linear-equations-in-math