# 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.

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.

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.

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.

References:

(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)

https://bim.easyaccessmaterials.com/index.php?location_user=cchs

https://www.ncdc.noaa.gov/news/recent-global-surface-warming-hiatus

https://www.climate.gov/news-features/climate-qa/did-global-warming-stop-1998

# Engaging students: Slope-intercept form of a line

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 Fidel Gonzales. His topic, from Algebra I: the point-slope intercept form of a line.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Technology is always advancing right in front of us. Using it in the classroom can be a tool that allows students to have a more hands on experience in the classroom. When I was in middle school, the only tool that we had to learn slope intercept form of a line was using a ti-inspire calculator. However, schools are receiving more funding and can provide students with tablets or computers to assist in their academic career. Gizmos is a website that contains many user-friendly programs that a student can use to learn a concept, or an educator can present to reinforce a skill. For the topic of slope intercept form of a line, the gizmo has two sliding parts that allows the user to change the values of the equation. One for the slope and one for the y- intercept. The student can adjust the values of both and observe the changes that occur to the line. This experience is more user friendly since it only allows the person to change those two aspects compared to having to input the equation each time into the graphing calculator. The reason that students would be more likely to be engaged is because they are already used to technology and there is still a need to incorporate technology into the classroom. So, students would prefer using a computer compared to the traditional paper and pencil. Imagine them having to graph by hand each graph to compare differences!

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

Slope intercept form is a way that data can be displayed. The data is usually continuously decreasing or continuously increasing. There is a magnitude of activities that can be used to help students gather a better understanding of the topic. As an educator, I would create a scavenger hunt that displayed either a word problem or a graph. Both will ask for the student to represent the information as slope intercept form. For each problem, there will be 4 answer choices that the student could choose for their answer. On their worksheet, there will be fill in the blanks that will be filled up from the letter that is in front of the correct answer. As the student progresses to the next problem, they will be filling out the letter blanks in a random order. So, if the person does the activity correctly, they should end up with the correct word phrase. The word phrase will be a math pun to add to the magic. This activity will allow students to switch from graph and word problems to slope intercept form.

How does this topic extend what your students should have learned in previous courses?

As educators, we want to ensure that our students have the proper foundation to continue advancing their mathematic skills. Slope intercept form is an algebra base lesson. The skills that students used to reach this topic is addition. At a young age, students learn to count numbers in repeated increments. An example of this is when a student keeps adding 5 until they reach a certain number. Displaying this as slope intercept would be a line with no y intercept and a slope of 5. We have even used y intercepts in context to adding in past classes. An example of this would be a person wanting to sell 200 dollars’ worth of tickets that are worth 5 dollars each and they already started with 57 dollars. If they were to solve the problem using slope intercept form, they would put 200 as the y value and 57 as the y intercept of the problem. The slope would be 5. In the past, they would add 5 to 57 until they reach their goal. Slope intercept form is a way for students to display data with a constant increasing or decreasing value. It is more convenient for students to use slope intercept form compared to how they displayed the pattern in the past. They use it now since they learned why it works before they reach algebra.

References:

# Engaging students: Finding the slope of a line

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 Bri Del Pozzo. Her topic, from Algebra: finding the slope of a line. How could you as a teacher create an activity or project that involves your topic? As a teacher I would likely introduce a very popular and well-received project to my students, the project where students draw an angular image on a graph and then calculate the slope of 20 lines from their image. I love this project because it allows students to connect mathematics to art and encourages them to express themselves creatively. As a precursor to the project, I would introduce students to the types of slopes and their characteristics using a tool that I learned in my Algebra Class, Mr. Slope Guy (pictured below).
In the image the positive slope is indicated by the left eyebrow and above the plus-sign eye, the negative slope is the right eyebrow by the negative sign eye, the nose represents the undefined slope and is denoted by a vertical line and a “u” for undefined, and the mouth represents the zero-slope shown by a horizontal line and two zeros. The students could use this resource while completing their projects to serve as a reminder of the types of slopes. The main focus of the project, however, would focus on the process of how to find slope given two points on a line. (This project is based on an example from: https://kidcourses.com/slope/ #5) How does this topic extend what your students should have learned in previous courses?             In the grade 7 TEKS for mathematics, students are expected to “represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form .” This creates a foundation for finding the slope of a line by introducing students to multiple representations of what slope looks like. When discussing how to find the slope of a line, I think that the tabular representation is a great tool for students to visualize the meaning behind slope. In seventh grade math, students were able to conceptualize slope without using the formula. When finding slope in early algebra, I would encourage students to look at graphs from a new lens, noticing features such as the sign of the line, the steepness of the line, the difference in x’s and y’s at different points on the line, and the slope itself. When looking at a table, I would ask students to calculate the difference in x’s and y’s as they go down the rows of the table and have them compare those numbers to those that they saw in the graph. How has this topic appeared in the news?             As many of us know, over the past 18 months or so, the number of Covid-19 Cases in the United States has been on the rise. For a long time, the total number of cases in the United States was growing exponentially and very quickly. As more research has been done by the Centers for Disease Control and Prevention, we have learned that there is a way to flatten the curve and reduce the number of daily cases. This initiative to flatten the curve has resulted in the growth of cases to resemble liner growth rather than exponential growth. As mathematicians, we can calculate the slope of the line that represents the (linear) growth of Covid-19 cases per day. We can make comparisons between growth rates in different states and use that data to make predictions about effectiveness of Covid-19 prevention procedures.

# Engaging students: Finding the slope of a line

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 Austin Stone. His topic, from Algebra: finding the slope of a line.

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

Using “pull back” toy cars, you can create a fun little activity that students can compete in to see who wins. Students can be put into groups or do it individually depending on how many cars you have available. The idea of the activity would have students pull back the cars a small amount and record how far they took it back and how far the car went. After doing this from three or four different distances, the students would then graph their data with x=how far they took it back and y=how far the car went. Then the teacher would tell the students to find how far back they would need to pull for the car to go a specified distance by finding the slope of their line (or rate of change in this example). After students have done their calculations, they would then pull back their cars however far they calculated and the closest team to the distance gets a prize.

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

Students will continually use slope throughout their future math and science classes. In math courses, slope is used to graph data and predict what will happen if certain numbers are used. It is also used to notice observations about the graph such as steepness (how quickly it changes) and if the rate of change is increasing or decreasing. It is also used in science for very similar reasons. In physics, slope is used commonly to calculate velocity and force. In chemistry labs, slope is used to predict how much of a certain substance needs to be added to find observational differences. In calculus, when taking the first derivative of a function, if the slope is negative, then the function is decreasing during that interval and vice versa if it is positive. Slope is also widely used in Algebra II, so learning how to find the slope is very important for future math and science classes whether it be in high school or college.

How does this topic extend what your students should have learned in previous courses?

Students should have already learned how to graph points on the coordinate plane. They can take this knowledge and now not only plot seemingly random points, but now see the relationship between these points. Plotting points is a skill usually learned around 6th grade and is used regularly after that. Also, finding the x and y axis can be used when finding the slope of a line. If you have a function with no points, finding the x and y axis can let you find the slope. Finding the x and y axis is learned in Algebra I so this would be fresh on students’ minds. Finding the slope of a line can be scaffolded with finding the x and y axis in lectures or in PBL experience. Also refreshing students on how to graph not only in the first quadrant, but in all four quadrants could be a quick little activity at the beginning of the PBL experience.

Reference:

http://www.andrewbusch.us/home/racing-day-algebra-2

# Engaging students: Solving systems of linear inequalities

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 Angelica Albarracin. Her topic, from Algebra: solving linear systems of inequalities. What interesting (i.e., uncontrived) word problems using this topic can your students do now? One example of an interesting word problem students can do using this topic is based on a technique astronomers use to learn about celestial bodies. Being able to assess the number of craters a body has on its surface can reveal information about the body’s age, as well as its history of impacts. In comparing the number of craters two bodies have experienced over time, astronomers are able to compare their lifetimes and hypothesize reasons for differences and/or similarities. Another example of an interesting word problem pertains to determining whether a specific phone plan is best for you. When choosing between certain plans, individuals may have to decide between a higher flat fee and a lower rate per minute or a lower flat fee and a higher rate per minute. In many cases, the answer may not be so obvious so to be able to figure out which is the best deal can prove to be a very helpful money saver. Of course, the answer to this question depends on how many minutes an individual plans to use a month, but we can use linear systems of equations to find out at which point do the plans differ, and thus finding a starting point to the solution. Taken from https://students.ga.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/1798/12938/Algebra_ReasoningwithEquationsandInequalities12.html How does this topic extend what your students should have learned in previous courses? In previous courses, students should have learned about x and y intercepts and solving linear equations. Solving linear systems of equations is and extension of x and y intercepts because one of the major components in this topic is finding the exact point at which two different linear functions meet. We can think of a typical problem of finding the x or y intercept of a linear function in terms of a system. For example, we can let our first equation be y = 3x + 2 and the second be y = 0. From this we can clearly see that our second equation is the x-axis, and as we are trying to find the point of intersection between a linear function, we end up calculating the x-intercept of our first function. It is also not difficult to see that solving linear systems of equations serves as an extension to solving linear equations. When employing the method of substitution, you must solve for one variable, in terms of the other. This process requires the student to know how to solve singular linear equations, and to apply their solutions through substitution. We can also see an extension regarding graphing linear equations. When solving linear systems of equations by graphing, one must graph each individual linear equation. Once the two individual equations are graphed, the solution can be found by observing the point at which the two equations intersect if at all. How can technology be used to effectively engage students with this topic? Desmos is widely regarded for its creative lessons that integrate mathematical topics in fun and engaging ways. For the topic of solving systems of linear equations with graphing and substitution, one such Desmos activity is titled Playing Catch-Up. The first two slides set up an engaging premise where a video compares the running speed of an average person and a professional runner. Further along the activity, the student can see a graphical representation of their speeds and is able to make a prediction as to whether they think one person will pass the other. Aside from being able to see an animated graph that corresponds to the information given in the video, there is also an interesting short answer feature on the first slide. This feature allows the student to ask a question regarding the situation they are presented with in the video. The most helpful part of this feature is that not only can the teacher view the student responses, but also the students can see each other’s responses. This allows for students to communicate with each other in a controlled environment and lead the way for further elaboration on some of the most asked questions. This specific Desmos activity places much of its emphasis on solving systems of linear equations through graphing, however substitution can still have a place in technology. Typically, when students are introduced to this concept, they are taught the graphing method first as its visual component aids in understanding. Graphing isn’t always reasonable however as it is time consuming and you may be faced with equations that are difficult to graph. By using technology such as the Desmos graphing calculator, the teacher can show the student of an example of a linear system of equations that would be unreasonable to solve by graphing. This gives the students reasoning as to why learning another method such as substitution is necessary while also making them consider a possibility that they might not have thought of before. References: https://spacemath.gsfc.nasa.gov/algebra2.html https://students.ga.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/1798/12938/Algebra_ReasoningwithEquationsandInequalities12.html https://teacher.desmos.com/activitybuilder/custom/5818fb314e762b653c3bf0f3

# Engaging students: Using the point-slope equation of a line

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 Johnny Aviles. His topic, from Algebra: using the point-slope equation of a line.

A2 How could you as a teacher create an activity or project that involves your topic?

On the 1987 NBA Dunk Contest, Michael Jordan won by dunking all the way from the free throw line. (I will play them a clip). Now can anyone tell me how high the hoop is from the ground? And how far is the free throw line from the base of the hoop? So, in total he went 10 feet in the air while jumping 15 feet! This is incredibly difficult and was why he won the contest. Now lets just compute that slope. With rise/ run we get that the slope was 2/3. Another example I can use is the time I took to get to school. I live 30 miles away and it took me 40 minutes to get to school. would anyone be able to find the average speed? (45 MPH) Then I will make it more complex and say I went 60 miles an hour for the first 20 minutes, how fast was I going the last 20 minutes?(30 MPH) Then I will have a round robin activity where I will give 5 min for my students to discuss amongst their groups where they can create a scenario where they can use point-slope equation of a line.

C3 How has this topic appeared in the news?

We all have many factors that interest us and the news’ job is to keep us updated. For many people, the stock market is a very serious subject of interest. Everything is shown in charts and done on points and percentages for simplicity reasons. This uses the concept of point-slope equation of a line to create this data. The news also covers may other topics like the rise of current temperature from given years to see if factors like global warming may have played a role to create the next leading story. The data from previous years can create point-slope equation that can predict the rain and snow fall amount for a given city or town. The weather initially can use point-slope equation of a line to predict all factors all data collected over decades. There is a copious amount of data that the news has to be used in all aspects of the news, one that has been shown is the rise of mass shootings. This is a very controversial matter as many people seek reform of the second amendment. Overall, point-slope equation of a line is widely used in many platforms of our news programs.

D4 What are the contributions of various cultures to this topic?

Architecture has been the biggest contribution that point-slope equation of a line and has to be applied. Various cultures have their own specific style of how they have their cities, towns and neighborhoods but all will apply the basics of point-slope equation of a line. For example, when creating a building, they use materials with large mass and need to be supported. If the slope of a beam is even slightly off, it can generally cause the building to collapse under its own weight causing the lives of many. Every aspect of the building needs to be measured in a precise way to create a solid structure. Styles then range from all cultures and can have tilted and rounded with elaborate beams to add more diversity. Overall, all cultures have their own specific style of houses that all require the same point-slope equation of a lines that contributes them to remain standing.

# Engaging students: Slope-intercept form of a line

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 Michael Garcia. His topic, from Algebra I: the point-slope intercept form of a line.

How does this topic extend what your students should have learned in previous courses?

Writing linear function in slope-intercept form is an important topic in Algebra I. The slope-intercept form of a line consists of an independent and dependent variable, a slope, and a y-intercept. In previous courses, the students should have learned slope. They may not have learned specifically about the word “slope,” but they should have been introduced to the topic of rate of change. The students also should have been introduced to the topic of graphing, specifically graphing a point on a Cartesian coordinate plane. Finally, the students should have learned about independent and dependent variables.

The slope-intercept form of a line extends the concept of rate of change, graphing, and independent/dependent variables by “putting it all together.” Students now have their first glimpse into the world of graphing equations. They can now visually see the representation of the rate of change (or slope) between the different points of a line. The students can see how the slope is a constant rate that goes through our points of data.

How has this topic appeared in the news?

On September 12, 2018, Apple held its annual news conference. They announced plenty of new gear and updates to their IOS, but everyone tuned in to hear about the new iPhone. The world freaked out when the new iPhone XS, iPhone XS Max, and iPhone XR were announced. So, how does this announcement relate to the slope-intercept form of a line? If we wanted to purchase a new iPhone and have a cell service plan with it, we can write a linear equation.

According to the Apple website, you can purchase the iPhone XS for \$999, while you can purchase the iPhone XS Max for as little as \$1,099. However, those price points are for the 64 GB model. If we are going to purchase an iPhone, we are going to buy the biggest and flashiest model. Since the iPhone XR is not currently taking pre-orders, we are going to purchase an iPhone XS Max with 512 GB of storage for \$1,449. Since most people cannot afford to spend \$1,449 on a single item, we are going to have monthly payments of \$68.66.

According to an Apple sales representative I spoke on the phone with, there would not be a down payment on this iPhone model.* Also, according to my mom’s phone bill, it would cost \$40 a month for one cell phone line that has unlimited talking, but 0.05 cents per text (my mom doesn’t text) . Our linear equation would be y= 0.05x +68.66 + 40, which is the same as y= 0.05x + 108.66.

This is a great way of viewing linear functions in slope-intercept form because it makes the problem more personable to the student.

*given that the customer signs up for the Apple iPhone Upgrade program

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

With new technology coming out every day, there are plenty of resources available for teachers. One tool that can be used to engage students with the slope-intercept form of a line is an application called GeoGebra. GeoGebra is “dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package.” There are online resources already created (very similar to Desmos), but I would mainly use their external app. I would use GeoGebra because of the many possibilities that are provided within the app.

The beauty of GeoGebra is student can utilize this app in their studying time as well. When you plot two points, the application automatically writes the equation associated with the line. This is a great way for the student to check their work when graphing/writing linear equation in slope-intercept form.

References:

# Engaging students: Solving systems of linear inequalities

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 Austin Carter. His topic, from Algebra: solving linear systems of inequalities.

How does this topic extend what your students should have learned in previous courses?

System of equations can be solved in several ways. Changing from linear systems to systems of inequalities only means there is a range of viable answers, but the processes for solving them remain the same; graphing, elimination, substitution, or matrices. Learning how to deal with inequalities will also give us access to more interesting real world problems, because we don’t always need an exact value; sometimes we need at least this much or no more than a certain amount. For example:

• In order to get a bonus this month, Leon must sell at least 120 newspaper subscriptions. He sold 85 subscriptions in the first three weeks of the month. How many subscriptions must Leon sell in the last week of the month?
• Virenas Scout troop is trying to raise at least \$650 this spring. How many boxes of cookies must they sell at \$4.50 per box in order to reach their goal?
• The width of a rectangle is 20 inches. What must the length be if the perimeter is at least 180 inches?

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

Systems of inequalities are most easily understood with visual aid. Different colors for each equation, dotted line vs. solid line, and shading are all major components of inequalities and being able to see how each shaded region overlaps is invaluable to understanding the answer. In my experience, the easiest tool to visualize all these components is the desmos online calculator. Desmos is very user friendly and will accept equations in any form. Also, it assigns different colors to each equation entered, allows students to zoom in and out to see detail on any scale, and allows students to “click and drag” and equation line to see the (x,y) components at that location. Desmos could be used to have students create their own graphs and explain the limiting factors of their picture.

Application/Technology

Sensors are how our electronics interact with the real world. Just think about a car, and how many things are being measured and monitored constantly. Every one of those sensors is responsible for measuring something specific and making sure that measurement stays within an acceptable range. What happens if your car gets too hot? What happens if you don’t buckle your seatbelt? As autonomous vehicles come online, what happens if that vehicle gets too close to another object? All of these things are measured by sensors, and those measurements are constantly being run through software to make sure those measurements stay within an acceptable range. But how does the software determine what an acceptable range is? The software uses system of inequalities to make sure every single measurement stays within an acceptable range, and if it doesn’t it alerts the driver. The world as we know it would come crumbling down without the sensors we rely on daily, but the information those sensors collect would be useless if we didn’t have systems of inequalities to make the data meaningful.

References:

Solve Real-World Problems Using Inequalities. (2015, July 7). Retrieved September 14, 2018, from https://students.ga.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/1798/12938/Algebra_ReasoningwithEquationsandInequalities7.html

# 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 Saundra Francis. Her topic, from Algebra: graphs of linear equations.

B1. How can this topic be used in your students’ future courses in mathematics and science?

Learning how to graph linear equations is the basis for many topics that students will learn later in Algebra and future mathematics and science courses. Students will now be able to solve word problems using graphs to model the situation describe in the problem. Being able to graph linear equations will help students graph non-linear equations since they will be able to apply the steps they learn on how to graph to different types of equations, Students will also be able to graph inequalities to find solutions for an equation since graphing equations is the first step in graphing inequalities. Another application of graphing linear equations is when students need to make graphs when completing science labs, many times students need to graph their data collected and find an equation that represents the data.

C3. How has this topic appeared in the news?

Graphs of linear equations are displayed in the markets sections on The New York Times. Segments of different linear equations can be put together match the graphs that display the rise and fall of different markets and stocks. Time is displayed on the x-axis while the y-axis list the price of the stock. The slope of the line is the percent change in the price of the stock and can be positive or negative depending if the price rose or fell. The y-intercept would be the price that the stock or market was at before the percent change. This will engage students because it is an example of how graphs of linear equations is displayed in the real world and they get a chance to see how they can use this concept in the future. This could also be made into an activity where students discover the linear equations that are combined to make a certain market or stock graph.

D1. What interesting things can you say about the people who contributed to the discovery and/or development of this topic?

René Descartes was born in 1596 and was a French scientist, philosopher, and mathematician. He is thought to be the father of modern philosophy. Descartes started his education at age nine and by the time he was twenty-two he had earned a degree in law. Then Descartes tried to understand the natural world using mathematics and logic, which is when he discovered how to visually represent algebraic equations. Descartes was the first to use a coordinate system to display algebraic equations. In 1637 Descartes published La Géométrie, which was where he first showed how to graph equations. He linked geometry and algebra in order to represent equations visually. While thinking about the nature of knowledge and existence Descartes stated, “I think; therefore I am”, which is one of his most famous thoughts. Students will gain interest in graphing equations when they are told about Descartes since he was an interesting person and he discovered things not only in the field of mathematics but philosophy too.

# Engaging students: Graphing inequalities

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 Lyndi Mays. Her topic, from Algebra: graphing inequalities.

A1. Once students get to the point where they’re graphing inequalities, they should have a pretty good understanding of how to graph regular functions. I’ve noticed that where students have issues graphing inequalities is knowing which side of the graph should be shaded. Students get confused thinking that the graph should be shaded depending on the direction of the line instead of checking specific points. One activity that I would like to try in the classroom is giving them a worksheet where they graph inequalities on a small graph and when all the little graphs are graphed and shaded it creates a mosaic picture. I feel like there needs to be some sort of pattern or picture so that the students are sure that they’re doing the questions correctly. Another reason I like this activity is because it reaches the intelligence of artistic students. It’s not often that a math lesson can reach artistic intelligences.

C1. One thing the students might find interesting about linear inequalities is that they appeared in the popular TV series, Numbers. In this particular episode, there is a blackout from attacks on an electrical substation. In order to figure out where the attack was located they mapped out where the blackouts were happening. Once they filled in all the different places that were blacking out, they realized it was one big section. Then they drew lines as if the map was on the coordinate plane. From there they are able to target the location where the attack happened.
Students also might be interested in knowing that this is also the way that policeman use to locate a cell phone. They mark the three closest cell towers that the cell phone pinged off of and are then able to draw a section and use linear functions to find the cell phone.

This video shows students how to solve for a variable and graph with inequalities. I liked the way it was set up because it was a word problem set up like a story and then solved. I know that students can become intimidated by having to learn new material and then having to apply it to a word problem. But this video kind of walks them through it which I believe could be helpful. Another thing was that the thing we were solving for was very realistic and might help students see why they would need to know how to graph linear equations in the future. The video also showed what x represented (cookies) and what y represented (lemonade). This lets the students know that x and y actually mean something instead of just being an arbitrary variable. I also liked that the video checked for specific points for the shading portion since many students forget that that’s a possibility and end up guessing where to shade.

References:
Sayfan, Sayfan. Graphing Linear Inequalities. https://us.sofatutor.com/mathematics/videos/graphing-linear-inequalities.