Engaging students: Graphs of linear equations

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 Morgan Mayfield. His topic, from Algebra: graphs of linear equations.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Given a rather vague statement such as ”Graphs of Linear Equations”, I was unsure if it meant only the technique of analyzing graphs or being able to have the ability to build a graph of a linear equation. In A1, I attempt to rely on analysis. Here are the problems I encountered on Space Math @ NASA:

  • Problem 1 – Calculate the Rate corresponding to the speed of the galaxies in the Hubble Diagram. (Called the Hubble Constant, it is a measure of how fast the universe is expanding).
  • Problem 2 – Calculate the rate of sunspot number change between the indicated years.

Space Math has these problems listed as “Finding the slope of a linear graph”, the two key phrases being “Finding the slope” and “linear graph”. The students must be able to do both. Students are given three sets of graphical data to analyze (shown below). I am not an expert in any of these fields, but I suspect these graphs were made using real data scientists collected. The Space Math team gave students two points on the data to aid in calculations. What makes these graphs interesting is the fact that they are messy, but real compared to a graph of a linear equation in a classroom. These graphs can be analyzed further than the problems Space Math offered. Students could see how that data can be collected and put into a scatterplot, like in the case of graph 2, and have an approximately linear correlation. Sadly, most things don’t follow a neat model of what we see in our math class, yet we can still derive meaning from real-world phenomena because of what we learn in math class. Scientists use their understanding of graphs of linear equations and linear models to analyze data and come to conclusions about our real-world environment. In graph 2, a scientist would clearly see that there is a linear proportional relationship between the speed and distance from the Hubble space telescope of a galaxy, or more meaningfully understood as a rate, 76 km/sec/mpc. Now, if a scientist encountered a new galaxy, they could determine an approximate speed or distance of the galaxy given the other variable.

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How can this topic be used in your students’ future courses in mathematics or science?

Students will formalize learning about graphing linear function in Algebra I. Graphs of linear equations are important in solving linear inequalities in two variables, solving systems of linear inequalities, solving systems of linear equations, and solving systems of equations involving linear and nonlinear equations which are all topics in Algebra I and II. Solving systems can be done algebraically, but graphing systems give students a more concrete way in finding a solution and is an excellent way of conveying information to others. If a student ever found themself in a business class, they may be asked to make “business decisions” on a product to buy. If I were the student explaining my decision to my teacher and potential “investors”, I would be making a graph of linear systems to help explain my “business decisions”. Generally, a business class would also introduce “Supply and Demand” graphs, where the solution is called the “equilibrium”. Many graphs in an intro class depict supply and demand as a system of linear equations.

In the high school sciences, a student will come across many linear equations. Students in a regular physics course and an AP physics course will come across simplified distance vs. time graphs to represent velocity, velocity vs. time graphs to represent acceleration, and force vs. distance graphs to represent work and energy (khan academy link included below). Note, just because many of the examples used in a physics class are graphs of linear equations, real life rarely works out like this.

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

We are shown data daily that our climate is changing fast through infographics on social media, posters set up by environmentalists, and news broadcasting. Climate change is one of the most important issues that society faces today and is on the collective consciousness of my generation as we have already seen the early consequences of climate change. Climate change, like most real-world data collecting does not always follow a good linear fit or any other specific fit with 100% accurately. However, a way scientists and media want to convey a message to us is to overlay a “trend line” or a “line of best fit” over the graphed data. Looking at the examples below, we can clearly understand that average global temperatures have been on the rise since 1880 despite fluctuations year-to-year and comparisons to the expected average global temperature. The same graph also gives insight on how the same data can also be cherrypicked to fit another person’s agenda. From 1998 – 2012, the rate of change, represented by a line, is lower than both 1970 – 1984 and 1984 – 1998. In fact, the rate is dramatically lower, thus climate change is no more! Not so, this period of slowing down doesn’t immediately refute the notion of climate change but could be construed as so. Actually, in the NOAA article linked below and its corresponding graph actually finds that we were using dated techniques that led to underestimates and concluded that the IPCC was wrong in it’s original analysis of 1998-2012 and that the trend was actually getting worse, indicated by the trend line on the second graph, as the global temperature departed from the long-term average.

Look at how much information could be construed by a few linear functions represented on a graph and some given rate of changes.




(or Problem 226 from https://spacemath.gsfc.nasa.gov/algebra1.html)

https://d1yqpar94jqbqm.cloudfront.net/documents/Gateway5A1VAChart.pdf (or grade 5 – Algebra II Vertical Alignment https://www.texasgateway.org/resource/vertical-alignment-charts-revised-mathematics-teks)





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