# Engaging students: Equations of two variables

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 again comes from my former student Julie Thompson. Her topic, from Algebra: equations of two variables.

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

4000 years ago the Babylonians knew how to solve systems of two linear equations with two variables. The most basic notion in all of math, most of us would say, is the equation- you manipulate both sides until you obtain what you were solving for. It sounds very simple to us these days, but it actually took a LONG time to develop as a concrete concept. Not until the 16th century was the concept of an equation taken as its own mathematical entity!

Early text from Egypt, specifically in the Rhind Papyrus, 1650 BC, shows that Egyptians were able to solve linear equations of one variable. Then as late as 300 BC evidence shows that the Egyptians also know how to solve equations with TWO unknowns!

In their society, equations of two variables could be used for very useful things such as finding the length and width of their field given the area and perimeter. There were no symbols at the time, so all calculation was done mentally and verbally. Linear equations with two unknowns have been discovered and used for a very long time!

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

A popular question among students is, “How does this relate to the real world?” The unit on equations of two variables is one of the best ways to show students how equations really do help you solve real life problems. We solve them in our heads all the time without even realizing it! For example, if we are at a store shopping for clothes, and shirts cost \$5 and pants cost \$10, but we only have a \$50 budget, we mentally set up 5x+10y=50 in our minds and try to find how many of each item we could buy that satisfies our equation. When introduced in a math class, it may not look so interesting. That is why as a teacher I would want to have a project where students can have fun setting up equations of two variables modeling a real life situation (maybe something they could even save and use for the future)! The idea is having students plan their ideal vacation! There are many variables that are in play when planning a trip. The students must consider gas, lodging, food, transportation, activity cost, etc. The idea is to have students do research and actually plan a trip they would want to take, while considering their budget and making decisions based on what their calculations allow them to do. An example of an equation they would write involves a rental car and gas. Let’s say that a rental car costs \$50 per day and gas costs 2.33/gallon. Then the total cost, y, for their transportation after arriving at their destination PER DAY will be modeled by y=50+2.33x, where x is the number of gallons of gas they use in a day. Hopefully by the end of the project they will be able to make the connection between the topic and the real world, and even have a trip planned that they can take one day!

How has this topic appeared in the news?

As I was reading the news regarding hurricanes, I ran across this article that is comparing the European model with the American model in regards to which model is tracking the hurricane more accurately. “The European model collects data continuously over several hours at various observation points, measured in 4D. It does this before making a prediction for the next 10 days. This is updated two times per day.” Contrastingly, the American model only collects data 4 times per day in 3D. The part that especially caught my attention and relates to my topic is, “Additionally, ensembles are used to test different variables in forecasting equations. Since the European model uses more than twice the number of ensembles than the GFS, it is able to plug-in more numbers, thus generating more outcomes for potential hurricane paths.” The actual process may be a lot more complicated than what students are introduced to in Algebra I or II, but when reading this brief news article, students are exposed to content that they have learned in class and may be able to make a real-world connection. ‘Testing different variables in equations’ is exactly what students are doing when writing and solving equations with two variables. They are trying to come up with an equation to model a situation and find possible solutions. This is relatable to what the European model is doing with the hurricane- testing different variables in the equation and coming up with possible paths for Florence. Although it is more complicated, it is still the same concept in action!

References:

https://www.britannica.com/science/algebra#ref761896