Systems of Linear Equations
Learning Module
- Creator(s):
This Learning Module is meant to help students learn the mathematical concept of systems of linear equations. This concept is covered within the Common Core class known as “Algebra I”, which is taken by grade 9 students, or freshmen in high school. The updates in this module will include videos and screenshots from Khan Academy, a great way to learn mathematics. If you are interested in learning more than just how to solve a system of equations, then Khan Academy is a great place to start.
By the end of this Learning Module, students will be able to:
These abilities correspond with the following Common Core Standards:
In order to be successful at solving a system of equations, there are some mathematical ideas that need to have been grasped prior to being introduced to this concept. They are:
These ideas are inscribed within the methodology of solving a system of equations. Before you start using this learning module, please make sure you have a solid understanding of the three ideas pointed out above.
Assignment: On your own, answer the following problems in your notebook.
Purpose: If students don’t understand the prior knowledge needed to solve a system of equations, they will easily fall behind at some point within this learning module. This portion of it is meant to help the students refresh their mind on what a linear equation looks like, and also act as a “warm up.”
Method: The students will work on the set of 4 problems. You will then collect the problems and grade them, not to be returned to the students. The problems should be gone over with the class as a whole after the assignments are collected and graded. Grading this assignment shouldn't have any sort of point value attached to it, but instead be a gauge of how much prior knowledge is available amongst the students. The teacher should then pair up the students in groups of 3-4. These groups will be used at certain points throughout the module. Each pairing will contain students who have a great understanding of linear equations already, and those who don’t. This way, multiple assignments and the project in this course should run smoothly as they will be group-based works. Try to make the groupings are even as possible so no one group is too strong or too weak quantitatively. The grades on this prior knowledge assignment can be a help in creating these pairings.
Tips: You must navigate the classroom, watching for students that seem to be struggling, as well as those you finish early. This will give you a quick idea of who may struggle and who may not, even if all the grades turn out fairly well. Just because a student gets a decent grade doesn’t mean that they can’t struggle. They may not realize how to solve a linear equation immediately, resulting in a potential future problem.
Imagine you are the treasurer for a new movie theater selling two types of tickets, adult and children. In order to make a profit on opening night, a total of 600 tickets must be sold. And, 4 times as many adult tickets must be sold as child tickets. How many adult tickets must be sold? What about child tickets?
A problem like this may be hard to solve if you don’t realize that this is a system of linear equations. This system can be described exactly as it sounds: it’s a system of two equations, which have some sort of relation, with two variables to solve for. The solutions to both variables must satisfy both equations.
The video below is a great introduction to what systems of equations look like. This video shows how to test a solution set:
After watching the video, please read through the link provided on systems of equations:
Comment: Email your teacher one or two questions you have about systems of equations. The link on systems of equations can give you a basis for your questions.
Purpose: To give students an idea of what a linear system of equations is, and how a real life problem can be solved using this system. The emails from the students are meant to get them thinking about what they are going to be learning, and the link provided is meant to support this curiosity.
Method: Talk through the problem given with your students, attempting to not write down any mathematical formulas. Encourage them to try and make up formulas for themselves. If they aren’t able to do this, talk about the problem conceptually. Don’t solve the problem for them as this will take away from the remaining modules.
Tips: I think that watching the video and looking through the link provided would help students get an idea on how to generate the formulas for the problem given. Therefore, they may be watched and read through first before the Method section is performed. Express to your students that reading the description of the module is necessary, but the video and link should be watched and read before attempting to solve the problem.
There are multiple ways of solving a linear system of equations. The three main ways are the elimination method, the substitution method, and by graphing. We will start with the elimination method. Below are a couple of links, and a video, explaining and describing the elimination method:
Assignment: In a notebook, come up with your own two examples of systems of equations that can be solved using the elimination method. Potentially have a word problem go along with each problem. As a challenge, attempt to create a third example that includes three variables (this would mean that there would need to be three equations).
Comment: At the bottom of the assignment, write a rating from 0-3 of how you felt about this assignment. Use the following guidelines to help you:
0 – “I don’t have any idea of what is going on.”
1 – “I can come up with my own problem if I look at an example.”
2 – “I can come up with my own algebraic problems, but I don’t know how to create a word problem out of it.”
3 – “This is too easy. I need more of a challenge.”
Purpose: To give students a start at how to solve systems of equations. Also, if a student really wants to solve the problem in the last module, this will help them get away from the “guess and check” method, promoting the idea that a system can be solved efficiently; this method may be thought to be useful because of the video provided previously.
Method: Have the students get in the groups you have set up to work on the assignment. Provide an example to help guide them through the assignment if needed. Then, collect the assignment. Setting a point value for an assignment like this is up to you. But, half the points should come from the first example, and half should come from the second. If a group decides to challenge themselves with a three-variable system, then extra credit should be rewarded.
Tips: Please take the student ratings seriously. Also, promote your students to be honest about the difficulty of this method. Even though the pairings are meant to get the assignment finished quickly, some students may just “tag along”, watch the quantitative students do the work, and not gain an understanding of what is going on.
This method is usually more difficult than the elimination method not because of its process, but because of the more advanced algebraic work that must be done. It involves getting one variable by itself on a side of the equation, and then replacing that variable in the other equation within the system. Then, using the equation where the variable was substituted, the remaining variable must be solved for. Once it is solved for, it is easy to find the other, missing variable. Below is a video explaining this process in more detail:
Assignment: With your group members, solve the problems below in a notebook. Feel free to use either the substitution or elimination method. Discuss with your group about which method you like the most for each problem and why.
Purpose: To allow students to have another way for solving a system of equations. As all teachers already know, there is most likely more than one way to solve a math problem. This also gives students a choice for which algebraic method they prefer the most. Not every learner is created equal, so a variation in solution methods is ideal.
Method: Watch the video with your class and discuss any questions that arise. Then, once the assignment is completed, collect it. As before, setting a point value for this assignment is up to you. But, a third of the points should come from each problem. Partial credit may be awarded if desired.
Tips: Promote students to not only be group learners, but independent academics. Each person learns in a different way, and express this to your students. Show them that no one method is correct.
I have once heard a mathematics teacher say, “If you don’t see what is going on, graph it.” That is exactly what we are going to be doing in this module.
First, you must make each linear equation in the system into slope-intercept form, also known as y = mx + b, where m is the slope and b is the y-intercept. After that, you are able to graph each equation. Please take a look at the examples below.
Because the two lines never cross, there is no solution to this system. If you haven’t already noticed, they won’t cross because they are parallel lines. When two lines are parallel, they have the same slope. This can be a quick check before you take the time to graph what is happening. The system is also known to be inconsistent, as inconsistent systems have no solution to them.
Since we are only dealing with linear equations, two lines may only cross once at most (if they don’t have the same slope), resulting in one solution to the system. The point at which they cross is the coordinate that satisfies both equations. The system above is known as a consistent system because consistent systems have at least one solution.
There is also ways to describe systems as either independent or dependent. A system is independent if the two lines “do their own thing.” That is, they will intersect at one point only. A system is dependent if any point that satisfies one equation will satisfy the other. These concepts, and the concepts discussed above can all be proven algebraically, but they are much easier to see graphically. Hence, this is why they are discussed within this section.
Assignment: Take a look at the graph below. The two line graphs are the exact same. After reviewing the graph, answer the following questions in your notebook: How many solutions do you think there are to the system of equations? Are the equations independent or dependent? Consistent or inconsistent?
Purpose: To give students even another way to solve systems of equations. This may be meant for more visual learners, as graphing can be very visual to see a relationship between two lines.
Method: After completing the last two assignments in groups, the students should complete this assignment on their own. This also goes along with the independent learning tip from the last module. The assignment is also meant to have the learners to think about a “new” problem on their own. They have the basis to solve the problem from the information provided, but the problem is something they haven’t seen before, and this will challenge them. Encourage them to provide a "because" response to each question on this assignment. Collect assignment when finished. The point value is again up to you.
Tips: Show the students how to change a linear equation into slope-intercept form, as a system of equations problem almost never has the equations written in this form.
Get in your pre-assigned groups. Take a look at the videos of word problems provided. Start each video, pause it quickly, read the problem, and attempt to solve it with your group members. Once you get an answer, watch the remainder of the video to see if your answer was correct.
Discussion: Talk with your teacher about what your group believes makes each word problem harder than being given a system of equations algebraically. Then, come up with ways that allow your group, and your classmates, to solve word problems easily and efficiently.
Purpose: To reiterate the idea given in the second module: that mathematics is word problems generated into numbers and variables. It also helps students see that math is qualitative, and this property must be understood just as much as the quantitative side of math.
Method: It is fairly straightforward. Just read the discussion portion of the student side of the module, and then lead a classroom discussion after all students are done watching the videos.
Tips: You may make a list of strategies on solving word problems and put them up in your classroom for students for future reference.
“How can mathematics be used in the occupations?” – Almost every student has said this at one point. Take a look at the links provided to see jobs that use mathematics, and linear equations specifically:
Project: With your groups, pick an occupation from the links and create a slideshow presentation to present to the class. Your group must go out and interview someone of this occupation and question them on the mathematical components of their job.
The guidelines for the slide elements are as follows:
For each of the first 4 guidelines above, so you should have a slide covering each, your classmates will be grading you according to the following rubric:
1 – “The slide doesn’t look clean, has too many words, and isn’t related to the content necessary. No media contained either.”
2 – “There are 1-2 quick points that are relevant, but no media for support. Too many words on the slide.”
3 – “There are 3-4 quick points, as well as media. The wording is very precise, and the group has done most of the talking to describe their project. They still sometimes read word-for-word what is on each slide.”
4 – “There are at least 5 points, and media such as links and images to make the slide stand out. The students has great knowledge on their project, and show this by expanding on the facts and details in each slide. Very little, if none at all, reading word-for-word from each slide.”
When you are grading, please write the score for each slide, and then total them up, out of 16.
The average of your classmates’ scores will account for 50% of your project grade, and the other half will come from the teacher-designed rubric.
Purpose: To have students see what they are learning is very applicable to everyday life. The lack of application of math content in the classroom hurts the field, as students don’t get an understanding of why it is necessary for their futures.
Method: The project should take about two weeks, with the first taking the time to gather the data and conduct the interview, and the second putting it all together. The point value of the project should be worth the total of the point values from the previous assignments as it should be at least half of the class grade.
Tips: The interview meeting should be facilitated to make sure that it happens in a timely manner, as it may be hard for both parties to find a time to conduct the interview. You may have meetings halfway through the project with students to track their progress. Some meetings might even have to be more often to make sure students are going to finish on time. The meetings are also meant to address any student concerns, such as certain group members not contributing equally to the project.
Below is the rubric you will use to grade the student projects. The student rubric is listed as points because they are just meant to total up the scores. It also is meant to only cover the objective of making the slides media-heavy, and not word-heavy. Your rubric is more based on the fluidity of the presentation.
Please print out and complete the survey attached below. Once completed, give to your teacher so they can be returned to me.
There is also a survey for you, the educator, to complete on what you thought about this module.
I have included this segment after the module survey as it is extra information for those who want to get a head start on learning the next topic: Systems of Inequalities. Below are three videos. The first discusses how to test a solution to a system of inequalities, the next is an intro to graphing a system of inequalities, and the third solves a two-variable system of inequalities. Please watch these videos:
Discussion: If you decide to complete this segment, let your teacher know. Then, talk with your teacher about what extracurricular activities/assignments can be done to gain more understanding on this topic.
Purpose: To let the students who have a great curiosity of mathematics continue their learning.
Method: Prepare questions to be answered for the students who decide to complete this segment. I am leaving this preparation up to you as you can decide how in-depth you want to take it because you will be the teacher of this new topic, without the basis of a module.
Tips: Promote this extra module even to the students who seemed to struggle with learning systems of equations. Getting students started early on a new topic may help them gain a better understanding as they have more time to grasp what is being taught.
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“Linear Equations.” Maths Careers, www.mathscareers.org.uk/article/linear-equations/.
“Solving Equations | Algebra I | Math.” Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/one-variable-linear-equations.
“Systems of Equations | Algebra I | Math.” Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/systems-of-linear-equations.
“Systems of Linear Equations.” Math Is Fun, www.mathsisfun.com/algebra/systems-linear-equations.html.
“The Elimination Method for Solving Linear Systems (Algebra 1, Systems of Linear Equations and Inequalities) – Mathplanet.” Mathplanet, www.mathplanet.com/education/algebra-1/systems-of-linear-equations-and-inequalities/the-elimination-method-for-solving-linear-systems.