# 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 Jessica Williams. Her topic, from Algebra I: the point-slope intercept form of a line.

A.2 How could you as a teacher create an activity or project that involves your topic?

In order to teach a lesson regarding slope intercept form of a line, I believe it is crucial to use visual learning to really open the student’s minds to the concept. Prior to this lesson, students should know how to find the slope of a line. I would provide each student with a piece of graph paper and small square deli sheet paper. I would have them fold their deli sheet paper into half corner to corner/triangle way). I would ask each student to put the triangle anywhere on the graph so that it passes through the x and the y-axis. Then I will ask the students to trace the side of the triangle and to find two points that are on that line. For the next step, each student will find the slope of the line they created. Once the students have discovered their slope, I will ask each of them to continue their line further using the slope they found. I will ask a few students to show theirs as an example (picking the one who went through the origin and one who did not). I will scaffold the students into asking what the difference would look like in a formula if you go through the origin or if you go through (0,4) or (0,-3) and so on. Eventually the students will come to the conclusion how the place where their line crosses the y-axis is their y intercept. Lastly, each student will be able to write their equation of the line they specifically created. I will then introduce the y=mx+b formula to them and show how the discovery they found is that exact formula. This is a great way to allow the students to work hands on with the material and have their own individual accountability for the concept. They will have the pride of knowing that they learned the slope intercept formula of a line on their own.

E.1 How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Graphing calculators are a very important aspect of teaching slope-intercept form of a line. It allows the students to visually see where the y-intercept is and what the slope is. Also, another good program to use is desmos. It allows the students to see the graph on the big screen and you can put multiple graphs on the screen at one time to see the affects that the different slopes and y intercept have on the graph. This leads students into learning about transformations of linear functions. Also, the teacher can provide the students with a graph, with no points labeled, and ask them to find the equation of the line on the screen. This could lead into a fun group activity/relay race of who can write the formula of the graph in the quickest time. Also, khan academy has a graphing program where the students are asked to create the graph for a specific equation. This allows the students to practice their graphing abilities and truly master the concept at home. To engage the students, you could also use Kahoot to practice vocabulary. For Kahoot quizzes, you can set the time for any amount up to 2 minutes, so you could throw a few formula questions in their as well. It is an engaging way to have each student actively involved and practicing his or her vocabulary.

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

Learning slope intercept form is very important for the success of their future courses and real world problems. Linear equations are found all over the world in different jobs, art, etc. By mastering this concept, it is easier for students to visualize what the graph of a specific equation will look like, without actually having to graph it. The students will understand that the b in y=mx+b is the y-intercept and they will know how steep the graph will be depending on the value of m. Mastering this concept will better prepare them to lead into quadratic equations and eventually cubic. Slope intercept form is the beginning of what is to come in the graphing world. Once you grasp the concept of how to identify what the graph will look like, it is easier to introduce the students to a graph with a higher degree. It will be easier to explain how y=mx+b is for linear graphs because it is increases or decreases at a constant rate. You could start by asking,
1.What about if we raise the degree of the graph to x^2?
2.What will happen to the graph?
3.Why do you think this will happen, can you explain?
4.What does squaring the x value mean?
It really just prepares the students for real world applications as well. When they are presented a problem in real life, for example, the student is throwing a bday party and has $100 dollars to go to the skating rink. If they have to spend$20 on pizza and each friend costs $10 to take, how many friends can you take? Linear equations are used every day, and it truly helps each one of the students. # 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 Rachel Delflache. Her topic, from Algebra: using the point-slope equation of a line. A2: How could you as a teacher create an activity that involves the topic? An adaptation of the stained-glass window project could be used to practice the point-slope formula (picture beside). Start by giving the students a piece of graph paper that is shaped like a traditional stained-glass window and then let they students create a window of their choosing using straight lines only. Once they are done creating their window, ask them to solve for and label the equations of the lines used in their design. While this project involves the point slope formula in a rather obvious way, giving the students the freedom to create a stained-glass window that they like helps to engage the students more than a normal worksheet. Also, by having them solve for the equations of the lines they created it is very probable that the numbers they must use for the equation will not be “pretty numbers” which would add an addition level of difficulty to the assignment. B2: How does this topic extend what your students should have learned in previous courses? The point-slope formula extends from the students’ knowledge of the slope formula m = (y2-y1)/(x2-x1) (x2-x1)m = y2-y1 y-y1 = m(x-x1). This means that the students could solve for the point-slope formula given the proper information and prompts. By allowing students to solve for the point-slope formula given the previous knowledge of the formula for slope, it gives the students a deeper understanding of how and why the point-slope formula works the way it does. Allowing the students to solve for the point-slope formula also increases the retention rate among the students. C1&3: How has this topic appeared in pop culture and the news? Graphs are everywhere in the news, like the first graph below. While they are often time line charts, each section of the line has its own equation that could be solved for given the information found on the graph. One of the simplest way to solve for each section of the line graph would be to use point slope formula. The benefit of using point slope formula to solve for the equations of these graphs is that there is very minimal information needed—assuming that two coordinates can be located on the graph, the linear equation can be solved for. Another place where graphs appear is in pop culture. It is becoming more common to find graphs like the second one below. These graphs are often time linear equation for which the formula could be solved for using the point slope formula. These kinds of graphs could be used to create an activity where the students use the point slope formula to solve to the equations shown in either the real world or comical graph. References: Stained glass window- Stained Glass Window Graphing Project # 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 Deanna Cravens. Her topic, from Algebra: finding the slope of a line. C3. How has this topic appeared in high culture (art/sports)? While one might not think of ski jumping as an art but more of a sport, there is definitely an artistic way about doing the jumping. The winter Olympics is one of the most popular sporting events, besides the summer Olympics that the world watches. This is a perfect engage for the beginning of class, not only is it extremely humorous but it is extremely engaging. It will instantly get a class interested in the topic of the day. I would first ask the students what the hill the skiers going down is called. Of course the answer that I would be looking for is the “ski slope.” This draws on prior knowledge to help students make a meaningful connection to the mathematical term of slope. Then I would ask students to interpret the meaning of slope in the context of the skiers. This allows for an easy transition into the topic for finding the slope of a line. C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)? Look at this scene from Transformers, it shows a perfect example of a linear line on the edge of the pyramid that the Decepticon is destroying. This video easily catches the attention of students because it is from the very popular Transformer movie. I would play the short twenty second clip and then have some student discussion at the beginning of class. This could be done as an introduction to the topic where students could be asked “how can we find the steepness of that edge of the pyrmaid?” Then the students can discuss with a partner and then group discussion can ensue. It could also be done as a quick review, where students are asked to recall how to find the slope of a line and what it determines. The students would be asked to draw on their knowledge of slope and produce a formula that would calculate it. How can this topic be used in your students’ future courses in mathematics or science? Finding the slope of a line is an essential part of mathematics. It is used in statistics, algebra, calculus, and so much more. One could say it is an integral part of calculus (pun intended). Not only is it used in mathematics classes, but it is also very relevant to science. One specific example is chemistry. There are specific reaction rates of solutions. These rates are expressed in terms of change in concentration divided by the change in time. This is exactly the formula that is used in math classes to find the slope. However, it is usually expressed in terms of change in y divided by change in x. Slope is also used in physics when working with velocity and acceleration of objects. While one could think of slope in the standard way of ‘rise over run,’ in these advanced classes whether math or science, it usually better thought of as ∆y/∆x. References: # 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 Anna Park. Her topic, from Algebra: graphs of linear equations. How could you as a teacher create an activity or project that involves your topic? • Have the students enter the room with all of the desks and chairs to the wall, to create a clear floor. On the floor, put 2 long pieces of duct tape that represent the x and y-axis. Have the students get into groups of 3 or 4 and on the board put up a linear equation. One of the students will stand on the Y-axis and will represent the point of the Y-Intercept. The rest of the students have to represent the slope of the line. The students will be able to see if they are graphing the equation right based on how they form the line. This way the students will be able to participate with each other and get immediate feedback. Have the remaining groups of students, those not participating in the current equation, graph the line on a piece of paper that the other group is representing for them. By the end of the engage, students will have a full paper of linear equation examples. The teacher can make it harder by telling the students to make adjustments like changing the y intercept but keeping the slope the same. Or have two groups race at once to see who can physically graph the equation the fastest. Because there is only one “graph” on the floor, have each group go separately and time each group. • Have the students put their desks into rows of even numbers. Each group should have between 4 and 5 students. On the wall or white board the teacher has an empty, laminated graph. The teacher will have one group go at a time. The teacher will give the group a linear equation and the student’s have to finish graphing the equation as fast as possible. Each group is given one marker, once the equation is given the first student runs up to the graph and will graph ONLY ONE point. The first student runs back to the second student and hands the marker off to them. That student runs up to the board and marks another point for that graph. The graph is completed once all points are on the graph, the x and y intercepts being the most important. If there are two laminated graphs on the board two groups can go at one time to compete against the other. Similar to the first engage, students will have multiple empty graphs on a sheet of paper that they need to fill out during the whole engage. This activity also gives the students immediate feedback. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? Sir William Rowan Hamilton was an Irish mathematician who lived to be 60 years old. Hamilton invented linear equations in 1843. At age 13 he could already speak 13 languages and at the age of 22 he was a professor at the University of Dublin. He also invented quaternions, which are equations that help extend complex numbers. A complex number of the form w + xi + yj + zk, where wxyz are real numbers and ijk are imaginary units that satisfy certain conditions. Hamilton was an Irish physicist, mathematician and astronomer. Hamilton has a paper written over fluctuating functions and solving equations of the 5th degree. He is celebrated in Ireland for being their leading scientist, and through the years he has been celebrated even more because of Ireland’s appreciation of their scientific heritage. Culture: How has this topic appeared in pop culture? An online video game called “Rescue the Zogs” is a fun game for anyone to play. In order for the player to rescue the zogs, they have to identify the linear equation that the zogs are on. This video game is found on mathplayground.com. References https://www.teachingchannel.org/videos/graphing-linear-equations-lesson https://www.reference.com/math/invented-linear-equations-ad360b1f0e2b43b8# https://en.wikipedia.org/wiki/William_Rowan_Hamilton http://www.mathplayground.com/SaveTheZogs/SaveTheZogs.html # 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 Heidee Nicoll. Her topic, from Algebra: solving linear systems of inequalities. How could you as a teacher create an activity or project that involves your topic? I found a fun activity on a high school math teacher’s blog that makes solving systems of linear inequalities rather exciting. The students are given a map of the U.S. with a grid and axes over the top, and their goal is to find where the treasure is hidden. At the bottom of the page there are six possible places the treasure has been buried, marked by points on the map. The students identify the six coordinate points, and then use the given system of inequalities to find the buried treasure. This teacher’s worksheet has six equations, and once the students have graphed all of them, the solution contains only one of the six possible burial points. I think this activity would be very engaging and interesting for the students. Using the map of the U.S. is a good idea, since it gives them a bit of geography as well, but you could also create a map of a fictional island or continent, and use that as well. To make it even more interesting, you could have each student create their own map and system of equations, and then trade with a partner to solve. How does this topic extend what your students should have learned in previous courses? If students have a firm understanding of inequalities as well as linear systems of equations, then they have all the pieces they need to understand linear systems of inequalities quite easily and effectively. They know how to write an inequality, how to graph it on the coordinate plane, and how to shade in the correct region. They also know the different processes whereby they can solve linear systems of equations, whether by graphing or by algebra. The main difference they would need to see is that when solving a linear system of equations, their solution is a point, whereas with a linear system of inequalities, it is a region with many, possibly infinitely many, points that fit the parameters of the system. It would be very easy to remind them of what they have learned before, possibly do a little review if need be, and then make the connection to systems of inequalities and show them that it is not something completely different, but is simply an extension of what they have learned before. How can technology be used effectively to engage students with this topic? Graphing calculators are sufficiently effective when working with linear systems of equations, but when working with inequalities, they are rather limited in what they can help students visualize. They can only do ≥, not just >, and have the same problem with <. It is also difficult to see the regions if you have multiple inequalities because the screen has no color. This link is an online graphing calculator that has several options for inequalities: https://www.desmos.com/calculator. You can choose any inequality, <, >, ≤, or ≥, type in several equations or inequalities, and the regions show up on the graph in different colors, making it easier to find the solution region. Another feature of the graphing calculator is that the equations or inequalities do not have to be in the form of y=. You can type in something like 3x+2y<7 or solve for y and then type it in. I would use this graphing calculator to help students visualize the systems of inequalities, and see the solution. When working with more than two inequalities, I would add just one region at a time to the graph, which you can do in this graphing calculator by clicking the equation on or off, so the students could keep track of what was going on. References Live.Love.Laugh.Teach. Blog by Mrs. Graves. https://livelovelaughteach.wordpress.com/category/linear-inequalities/ Graphing calculator https://www.desmos.com/calculator # 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 Brianna Horwedel. Her topic, from Algebra: finding the slope 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? Algebra vs. the Cockroaches is a great way to get students engaged in learning about slopes. The object of the game is to kill the cockroaches by figuring out what the equation of the line that they are walking on is. It progresses from simple lines such as y=5 to more complicated equations such as y=(-2/3)x+7. It allows the students to quickly recognize y-intercepts and slopes. Once finished, you can print out a “report” that tells you how many the student got correct and how many tries it took them to complete a level. This game could even be used as a formative assessment for the teacher. http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf How could you as a teacher create an activity or project that involves your topic? Last year, I was placed in an eighth grade classroom that was learning about slope. One of the things that really stuck out to me was that the teacher gave a ski illustration to get the students talking about slope. The illustration starts off with the teacher going skiing. She talks about how when she is going up the ski lift she is really excited and having a “positive” experience which correlates to the slope being positive. Once she gets off of the ski lift, she isn’t going up or down, but in a straight line. She talks about how she doesn’t really feel either excited or nervous because she is on flat ground. This corresponds to lines that have a slope of 0. She then proceeds to talk about how when she starts actually going down the ski slope, she hates it! This relates to the negative slope of a line. She also mentions how she went over the side of a cliff and fell straight down. She was so scared she couldn’t even think or “define” her thoughts. This is tied to slopes that are undefined. I thought that this illustration was a great way of explaining the concept of slope from a real world example. After sharing the illustration, the students could work on problems involving calculating the slope of ski hills. How can this topic be used in your students’ future courses in mathematics or science? Understanding how to find the slope of a line is crucial for mathematics courses beyond Algebra I and Algebra II. Particularly, knowing how to find the slope of a line is essential for finding tangent lines of curves. This comes in handy for Calculus when you have to use limits to determine the slope. If a student does not have a strong grasp of what slope means and what its relationship is with the graph and the equation in Algebra I, then they will have a difficult time understanding slopes of lines that are not straight. # 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 Jason Trejo. His topic, from Algebra: finding the slope of a line. A2) How could you as a teacher create an activity or project that involves your topic? I have to start off by giving some credit to my 5th grade math teacher for giving me the idea on how I could create an activity involving this topic. You see, back in my 5th grade math class, we were to plot points given to us on a Cartesian plane and then connect the dots to create a picture (which turned out to be a caveman). Once we created the picture, we were to add more to it and the best drawing would win a prize. My idea is to split the class up into groups and give them an assortment of lines on separate pieces of transparent graphing sheets. They would then find the slopes and trace over the line in a predetermined color (e.g. all lines with m=2 will be blue, when m=1/3 then red, etc.). Next they stack each line with matching slopes above the other to create pictures like this: Of course, what I have them create would be more intricate and colorful, but this is the idea for now. It is also possible to have the students fine the slope of lines at certain points to create a picture like I did back in 5th grade and then have them color their drawing. They would end up with pictures such as: C1) How has this topic appeared in pop culture (movies, TV, current music, videogames, etc.)? Sure there aren’t many places where finding the slope of a line will be the topic that everyone goes on and one about on TV or on the hottest blog or all over Vine (whatever that is), but take a look around and you will be able to see a slope maybe on a building or from the top of Tom Hank’s head to the end of his shadow. Think about it, with enough effort, anyone could imagine a coordinate plane “behind” anything and try to find the slop from one point to another. The example I came up with goes along with this picture I edited: *Picture not accurately to scale This is the infamous, first double backflip ever landed in a major competition. The athlete: Travis Pastrana; the competition: the 2006 X-Games. I would first show the video (found here: https://www.youtube.com/watch?v=rLKERGvwBQ8), then show them the picture above to have them solve for each of the different slopes seen. In reality this is a parabola, but we can break up his motion to certain points in the trick (like when Travis is on the ground or when Travis is upside down for the first backflip). When the students go over parabolas at a later time, we could then come back to this picture. B2) How does this topic extend what your students should have learned in previous courses? It has been many years since I was first introduced to finding the slope of the line so I’m not sure exactly when I learned it, but I do know that I at least saw what a line was in 5th grade based on the drawing project I stated earlier. At that point, all I knew was to plot points on a graph and “connect the dots”, so this builds on that by actually being able to give a formula for those lines that connected the dots. Other than that, finding slopes on a Cartesian plane can give more insight on what negative numbers are and how they relate to positive numbers. Finally, students should have already learned about speed and time, so by creating a representation how those two relate, a line can be drawn. The students would see the rate of change based on speed and time. References: Minimalistic Landscape: http://imgur.com/a/44DNn Minimalistic Flowers: http://imgur.com/Kwk0tW0 Double Backflip Image: http://cdn.motocross.transworld.net/files/2010/03/tp_doubleback_final.jpg Double Backflip Video: : https://www.youtube.com/watch?v=rLKERGvwBQ8 # 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 Nada Al-Ghussain. Her topic, from Algebra: graphs of linear equations. How could you as a teacher create an activity or project that involves your topic? Positive slope, negative slope, no slope, and undefined, are four lines that cross over the coordinate plane. Boring. So how can I engage my students during the topic of graphs of linear equations, when all they can think of is the four images of slope? Simple, I assign a project that brings out the Individuality and creativity of each student. Something to wake up their minds! An individualized image-graphing project. I would give each student a large coordinate plane, where they will graph their picture using straight lines only. I would ask them to use only points at intersections, but this can change to half points if needed. Then each student will receive an Equation sheet where they will find and write 2 equations for each different type of slope. So a student will have equations for two horizontal lines, vertical lines, positive slope, and negative slope. The best part is the project can be tailored to each class weakness or strength. I can also ask them to write the slop-intercept form, point slope form, or to even compare slopes that are parallel or perpendicular. When they are done, students would have practiced graphing and writing linear equations many times using their drawn images. Some students would be able to recognize slopes easier when they recall this project and their specific work on it. Example of a project template: Examples of student work: How has this topic appeared in the news? Millions of people tune in to watch the news daily. Information is poured into our ears and images through our eyes. We cannot absorb it all, so the news makes it easy for us to understand and uses graphs of linear equations. Plus, the Whoa! Factor of the slopping lines is really the attention grabber. News comes in many forms either through, TV, Internet, or newspaper. Students can learn to quickly understand the meaning of graphs with the different slopes the few seconds they are exposed to them. On television, FOX news shows a positive slope of increasing number of job losses through a few years. (Beware for misrepresented data!) A journal article contains the cost of college increase between public and private colleges showing the negative slope of private costs decreasing. Most importantly line graphs can help muggles, half bloods, witches, and wizards to better understand the rise and decline of attractive characters through the Harry Potter series. How can this topic be used in your students’ future courses in mathematics or science? Students are introduced to simple graphs of linear equations where they should be able to name and find the equation of the slope. In a student’s future course with computers or tablets, I would use the Desmos graphing calculator online. This tool gives the students the ability to work backwards. I would ask a class to make certain lines, and they will have to come up with the equation with only their knowledge from previous class. It would really help the students understand the reason behind a negative slope and positive slope plus the difference between zero slope and undefined. After checking their previous knowledge, students can make visual representations of graphing linear inequalities and apply them to real-world problems. References: http://www.hoppeninjamath.com/teacherblog/?p=217 http://walkinginmathland.weebly.com/teaching-math-blog/animal-project-graphing-linear-lines-and-stating-equations http://mediamatters.org/research/2012/10/01/a-history-of-dishonest-fox-charts/190225 http://money.cnn.com/2010/10/28/pf/college/college_tuition/ http://dailyfig.figment.com/2011/07/13/harry-potter-in-charts/ https://www.desmos.com/calculator # 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 again comes from my former student Kelley Nguyen. Her topic, from Algebra: slope-intercept form of a line. How has this topic appeared in high culture (art, classical music, theatre, etc.)? The slope-intercept form of a line is a linear function. Linear functions are dealt with in many ways in everyday life, some of which you probably don’t even notice. One example where the slope-intercept form of a line appears in high culture is through music and arts. Suppose a band wants to book an auditorium for their upcoming concert. As most bands do, they meet with the manager of the location, book a date, and determine a payment. Let’s say it costs$1,500 to rent the building for 2 hours. In addition to this fee, the band earns 20% of each $30 ticket sold. Write an equation that determines whether the band made profit or lost money due to the number of tickets sold – the equation would be y = 0.2(30)x – 1500, where y is the amount gained or lost and x is the number of tickets sold that night. This can also help the band determine their goal on how many tickets to sell. If they want to make a profit of$2,000, they would have to sell x-many tickets to accomplish that.

In reality, most arts performances make a profit from their shows or concerts. Not only do mathematicians and scientists use slope-intercept of a line, but with this example, it shows up in many types of arts and real-world situations. Not only does the form work for calculating cost or profit, it can relate to the number of seats in a theatre, such as x rows of 30 seats and a VIP section of 20 seats. The equation to find how many seats are available in the theatre is y = 30x + 20, where x is the number of rows.

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

A great way to engage students when learning about slope-intercept form of a line is to use Geometer’s Sketchpad. After opening a graph with an x- and y-axis, use the tools to create a line. From there, you can drag the line up or down and notice that the slope increases as you move upward and decreases as you move downward. Students can also find the equation of the line by selecting the line, clicking “Measure” in the menu bar, and selecting “Equation” in the drop-down list. This gives the students an accurate equation of the line they selected in slope-intercept form. Geometer’s Sketchpad allows students to experiment and explore directions of lines, determine whether or not it has an increasing slope, and help create a visual image for positive and negative slopes.

Also, with this program, students can play a matching game with slope-intercept equations and lines. You will instruct the student to create five random lines that move in any direction. Next, they will select all of the lines, go to “Measure” in the menu bar, and click “Equation.” From there, it’ll give them the equation of each line. Then, the student will go back and select the lines once again, go to “Edit” on the menu bar, hover over “Action Buttons,” and select “Hide/Show.” Once a box comes up, they will click the “Label” tab and type Scramble Lines in the text line. Next, the lines will scramble and stop when clicked on. Once the lines are done scrambling, the student could then match the equations with their lines. This activity gives the students the chance to look at equations and determine whether the slope is increasing and decreasing and where the line hits the y-axis.

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

With this topic, I could definitely do a project that consists of slope-intercept equations, their graphs, and word problems that involve computations. For example, growing up, some students had to earn money by doing chores around the house. Parents give allowance on daily duties that their children did.

The project will give the daily amount of allowance that each student earned. With that, say the student needed to reach a certain amount of money before purchasing the iPad Air. In part one of the project, the student will create an equation that reflects their daily allowing of $5 and the amount of money they have at the moment. In part two, the student will construct a graph that shows the rate of their earnings, supposing that they don’t skip a day of chores. In part three, the students will answer a series of questions, such as, • What will you earn after a week? • What is your total amount of money after that week? • When will you have enough money to buy that iPad Air at$540 after tax?

This would be a short project, but it’s definitely something that the students can do outside of class as a fun activity. It can also help them reach their goals of owning something they want and making a financial plan on how to accomplish that.

References

# Finding the equation of a line between two points

Here’s a standard problem that could be found in any Algebra I textbook.

Find the equation of the line between $(-1,-2)$ and $(4,2)$.

The first step is clear: the slope of the line is

$m = \displaystyle \frac{2-(-2)}{4-(-1)} = \frac{4}{5}$

At this point, there are two reasonable approaches for finding the equation of the line.

Method #1. This is the method that was hammered into my head when I took Algebra I. We use the point-slope form of the line:

$y - y_1 = m (x - x_1)$

$y - 2 = \displaystyle \frac{4}{5} (x-4)$

$y - 2 = \displaystyle \frac{4}{5}x - \frac{16}{5}$

$y = \displaystyle \frac{4}{5}x - \frac{6}{5}$

For what it’s worth, the point-slope form of the line relies on the fact that the slope between $(x,y)$ and $(x_1,y_1)$ is also equal to $m$.

Method #2. I can honestly say that I never saw this second method until I became a college professor and I saw it on my students’ homework. In fact, I was so taken aback that I almost marked the solution incorrect until I took a minute to think through the logic of my students’ solution. Let’s set up the slope-intercept form of a line:

$y= \displaystyle \frac{4}{5}x + b$

Then we plug in one of the points for $x$ and $y$ to solve for $b$.

$2 = \displaystyle \frac{4}{5}(4) + b$

$\displaystyle -\frac{6}{5} = b$

Therefore, the line is $y = \displaystyle \frac{4}{5}x - \frac{6}{5}$.

My experience is that most college students prefer Method #2, and I can’t say that I blame them. The slope-intercept form of a line is far easier to use than the point-slope form, and it’s one less formula to memorize.

Still, I’d like to point out that there are instances in courses above Algebra I that the point-slope form is really helpful, and so the point-slope form should continue to be taught in Algebra I so that students are prepared for these applications later in life.

Topic #1. In calculus, if $f$ is differentiable, then the tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ has slope $f'(a)$. Therefore, the equation of the tangent line (or the linearization) has the form

$y = f(a) + f'(a) \cdot (x-a)$

This linearization is immediately obtained from the point-slope form of a line. It also can be obtained using Method #2 above, so it takes a little bit of extra work.

This linearization is used to derive Newton’s method for approximating the roots of functions, and it is a precursor to Taylor series.

Topic #2. In statistics, a common topic is finding the least-squares fit to a set of points $(x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)$. The solution is called the regression line, which has the form

$y - \overline{y} = r \displaystyle \frac{s_y}{s_x} (x - \overline{x})$

In this equation,

• $\overline{x}$ and $\overline{y}$ are the means of the $x-$ and $y-$values, respectively.
• $s_x$ and $s_y$ are the sample standard deviations of the $x-$ and $y-$values, respectively.
• $r$ is the correlation coefficient between the $x-$ and $y-$values.

The formula of the regression line is decidedly easier to write in point-slope form than in slope-intercept form. Also, the point-slope form makes the interpretation of the regression line clear: it must pass through the point of averages $(\overline{x}, \overline{y})$.