# 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.

# 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-
http://digitallesson.com/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://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