# Engaging students: Solving linear systems of equations with matrices

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Alyssa Dalling. Her topic, from Algebra II: finding the area of a square or rectangle.

A. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

A fun way to engage students on the topic of solving systems of equations using matrices is by using real world problems they can actually understand. Below are some such problems that students can relate to and understand a purpose in finding the result.

• The owner of Campbell Florist is assembling flower arrangements for Valentine’s Day. This morning, she assembled one large flower arrangement and found it took her 8 minutes. After lunch, she arranged 2 small arrangements and 15 large arrangements which took 130 minutes. She wants to know how long it takes her to complete each type of arrangement.
• The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point  and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make?

(Idea and solution on http://www.algebra-class.com/system-of-equations-word-problems.html )

A. How could you as a teacher create an activity or project that involves your topic?

• For this topic, creating a fun activity would be one of the best ways to help students learn and explore solving systems of equations using matrices. One way in which this could be done is by creating a fun engaging activity that allows the students to use matrices while completing a fun task. The type of activity I would create would be a sort of “treasure hunt.” Students would have a question they are trying to find the solution for using matrices. They would solve the system of equations and use that solution to count to the letter in the alphabet that corresponds to the number they found. In the end, the solution would create different blocks of letters that the student would have to unscramble.

For Example: The top of the page would start a joke such as “What did the Zero say to the Eight?…

Solve $x+y=26$ and $4x+12y=90$ using matrices.

To solve this, the student would put this information into a matrix and find the solution came out to be $x=12$ and $y=14$. They would count in the alphabet and see that the 12th letter was L and the 14th letter was N. Then at the bottom of their page, they would find where it said to write the letters for x and y such as below-

N  __  __  __     __  __    L  __! (Nice Belt!)

x     a    c    z       d    z     y    w

E. How can technology be used to effectively engage students with this topic?

This activity would be used after students have learned the basics of putting a matrix into their calculator to solve. The class would be separated into small groups (>5 or more if possible with 2-3 kids per group) The rules are as follows: a group can work together to set up the equation, but each individual in the group had to come up to the board and write out their groups matrices and solution. The teacher would hand out a paper of 8-12 problems and tell the students they can begin. The first group to finish all the problems correctly on the board wins. There would be problems ranging from 2 variables to 4.

Ex: One of the problems could be  and . The groups would have to first solve this on their paper using their calculator then the first person would come up to the board to write how they solved it-

Written on the board:

The technology of calculators allows this to be a fun and fast paced game. It will allow students to understand how to use their calculator better while allowing them to have fun while learning.

# Engaging students: Solving linear systems of equations by either substitution or graphing

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Angel Pacheco. His topic, from Algebra II: solving linear systems of equations by either substitution or graphing.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Westerville South High School (WSHS) is located in Westerville, Ohio. In 2010, the math department of WSHS worked together with their students to create parodies of popular rap songs about particular mathematical topics. They have made a Facebook page as well as their own account for YouTube. This is a great idea because it uses websites that are popular among the students. In one of their recent videos, it is called All I Do is Solve, which is the parody of ‘All I Do is Win’ by DJ Khaled. This video has been constructed really well. It contains three ways to solve systems of equations, which are graphing, substitution, and elimination.

This video will be a great tool for an Engagement as well as right before the Evaluation. The sound of it being a famous rap song will certainly grab the interest of all students. I, personally, am not a big fan of rap but when I saw this video I could not stop watching it. It was really entertaining. A lot of teachers can gain a lot of ideas from this type of teaching.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

There are a lot things to say. There are a lot of different cultures that had their own procedure or different perspective to this topic. I found a website called History of Math. In early 200 BC, there are sections in an ancient China text called the ‘Jiuzhang suanshu’ that contains examples of linear equations. This is a selection from the text:

One pint of good wine costs 50 gold pieces, while one pint of poor wine costs 10. Two pints of wine are bought for 30 gold pieces. How much of each kind of wine was bought?

The solution of this problem is used by using systems of linear equations. I can use this example as well as other examples from the different cultures. I will primarily use this as an Engagement. I will begin to ask the class, “Do any of you know how long solving systems of equations has been around?” “Do you know who discovered this concept?” Using these questions to get them interested, I will use the website to inform the different contributions that each culture made.

How could you as a teacher create an activity or project that involves your topic?

I will create a project based activity that requires the students to work in groups of 3-4. The students will each have their own role: Gate Keeper, Focus Keeper, Analyst, and Encourager. The link below will be to the website that describes the same roles and the same project. Each students will have to learn the material to complete the project on their own, but they will not do it by themselves. The group complete it by itself.

The project consists of the real life scenario that their parent(s) have decided not to pay for their cell phone bill so they have a \$50 limit per month so they must research the different options they have with different service providers. They will create a system of linear equations and they must be able to solve the systems of linear equations by the three methods: graphing, substitution, and elimination. This will allow for students to work together as well by themselves on an activity that is exciting. The students will be required to present their results at the end of the project. The project will turn to be an interdisciplinary lesson with systems of equations.

Source(s): The image below is a copy of the layout of the roles and project.