# Clowns and Graphing Rational Functions

I thought I had heard every silly mnemonic device for remembering mathematical formulas, but I recently heard a new one: the clowns BOBO, BOTU, and BETC for remembering how to graph rational functions.

• BOB0: bigger (exponent) on bottom, $x = 0$
• BOTU: bigger on top, undefined
• BETC: bottom equals top eponent, coefficients (i.e., the ratio of coefficients)

Which naturally leads to this pearl of wisdom:

# 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

# Graphs can be made to show anything

It’s 2016, which means it’s another election year. Here’s a very nice article from the last campaign cycle about the bipartisan abuse of statistics to mislead: http://www.washingtonpost.com/blogs/the-fix/wp/2014/10/30/graphs-can-be-made-to-show-anything-campaign-ads-edition/?tid=sm_fb

# 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: Graphing parabolas

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 Tiffany Wilhoit. Her topic, from Algebra: graphing parabolas.

How did people’s conception of this topic change over time?

The parabola has been around for a long time! Menaechmus (380 BC-320 BC) was likely the first person to have found the parabola. Therefore, the parabola has been around since the ancient Greek times. However, it wasn’t until around a century later that Apollonius gave the parabola its name. Pappus (290-350) is the mathematician who discovered the focus and directrix of the parabola, and their given relation. One of the most famous mathematicians to contribute to the study of parabolas was Galileo. He determined that objects falling due to gravity fall in parabolic pathways, since gravity has a constant acceleration. Later, in the 17th century, many mathematicians studied properties of the parabola. Gregory and Newton discovered that parabolas cause rays of light to meet at a focus. While Newton opted out of using parabolic mirrors for his first telescope, most modern reflecting telescopes use them. Mathematicians have been studying parabolas for thousands of years, and have discovered many interesting properties of the parabola.

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

A fun activity to set up for your students will include several boxes and balls, for a smaller set up, you can use solo cups and ping pong balls. Divide the class into groups, and give each group a set of boxes and balls. First, have the students set up a tower(s) with the boxes. The students will now attempt to knock the boxes down using the balls. The students can map out the parabolic curve showing the path they want to take. By changing the distance from the student throwing the ball and the boxes, the students will be able to see how the curve changes. If students have the tendency to throw the ball straight instead of in the shape of a parabola, have a member of the group stand between the thrower and the boxes. This will force the ball to be thrown over the student’s head, resulting in the parabolic curve. The students can also see what happens to the curve depending on where the student stands between the thrower and boxes. In order for the students to make a positive parabolic curve, have them throw the ball underhanded. This activity will engage the students by getting them involved and active, plus they will have some fun too! (To start off with, you can show the video from part E1, since the students are playing a real life version of Angry Birds!)

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

A great video to show students before studying parabolas can be found on YouTube:

The video uses the popular game Angry Birds to introduce parabolic graphs. First, the video shows the bird flying a parabolic path, but the bird misses the pig. The video goes on to explain why the pig can’t be hit. It does a good job of explaining what a parabola is, why the first parabolic curve would not allow the bird to hit the pig, and how to change the curve to line up the path of the bird to the pig. This video would be interesting to the students, because a majority of the class (if not all) will know the game, and most have played the game! The video goes even further by encouraging students to look for parabolas in their lives. It even gives other examples such as arches and basketball. This will get the students thinking about parabolas outside of the classroom. (This video would be perfect to show before the students try their own version of Angry Birds discussed in part A2)

Resources:

Youtube.com/watch?v=bsYLPIXI7VQ

Parabolaonline.tripod.com/history.html

http://www-history.mcs.st-and.ac.uk/Curves/Parabola.html

# Drought in California

Source: http://www.xkcd.com/1410/

# Importance of labeling axes

Source: http://www.xkcd.com/833/

# Graphing a function by plugging in points

I love showing this engaging example to my students to emphasize the importance of the various curve-sketching techniques that are taught in Precalculus and Calculus.

Problem. Sketch the graph of $f(x) = x^5 - 5x^3 + 4x + 6$.

“Solution”. Let’s plug in some convenient points, graph the points, and then connect the dots to produce the graph.

• $f(-2) = (-32) - 5(-8) + 4(-2) + 6 = 6$
• $f(-1) = (-1) - 5(-1) + 4(-1) + 6 = 6$
• $f(0) = (0) - 5(0) + 4(0) + 6 = 6$
• $f(1) = (1) - 5(1) + 4(1) + 6 = 6$
• $f(2) = (32) - 5(8) + 4(2) + 6 = 6$

That’s five points (shown in red), and surely that’s good enough for drawing the picture. Therefore, we can obtain the graph by connecting the dots (shown in blue). So we conclude the graph is as follows.

Of course, the above picture is not the graph of $f(x) = x^5 - 5x^3 + 4x + 6$, even though the five points in red are correct. I love using this example to illustrate to students that there’s a lot more to sketching a curve accurately than finding a few points and then connecting the dots.

# Engaging students: Finding x- and y-intercepts

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 Maranda Edmonson. Her topic, from Algebra: finding $x-$ and $y-$intercepts. Unlike most student submissions, Maranda’s idea answers three different questions at once.

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

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

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

This link is to a reflection by a mathematics teacher who used the popular TV show “The Big Bang Theory” to teach linear functions. She taught this lesson prior to teaching students about finding $y-$intercepts of linear functions, but it can be adapted in order to teach how to find the intercepts themselves.

ENGAGE:

One thing I would not change would be to show the students the above clip of the show where Howard and Sheldon are heatedly discussing crickets at the beginning of the activity. By showing the video at the beginning, students will be engaged and want to figure out what will be done throughout the lesson. Being a clip of a popular show that many probably watch during the week, students will be even more engaged and interested since they are able to watch something that they are already familiar with. Being something that they are already familiar with or can relate to, students have a tendency to remember the material or at least the topic longer than they would remember something that they were unfamiliar with or could not relate.

In the clip, Sheldon argues that the cricket the guys hear while eating dinner is a snowy tree cricket based on the temperature of the room and the frequency of chirps; Howard argues that it is an ordinary field cricket.  The beginning of their discussion is as follows:

Sheldon: “Based on the number of chirps per minute, and the ambient temperature in this room, it is a snowy tree cricket.”

Howard: “Oh, give me a frickin’ break. How could you possibly know that?”

Sheldon: “In 1890, Amos Dolbear determined that there was a fixed relationship between the number of chirps per minute of the snowy tree cricket and the ambient temperature – a precise relationship that is not present with ordinary field crickets.”

The whole episode revolves around the guys finding the exact genus and species of the cricket, but that is not the importance here. The importance of this clip is the linear relationship between the temperature and the number of chirps per minute of the cricket, which the activity should then be centered around.

EXPLORE:

After showing the short clip, it could be beneficial to show students the Wikipedia link that discusses Dolbear’s Law. Toward the bottom of the page, the relationship is written out in several formats, but there is a basic linear function that students could focus on for the activity.

Assuming students know how to graph linear functions (as stated above, the link is for a lesson the teacher taught before teaching students about $y-$intercepts), I would have students graph Dolbear’s Law on a piece of graph paper. The challenge would be for students to find out what happens when there are variations to the number of chirps of the cricket, the temperature or both to see how the graph changes – specifically where the graph crosses each axis.

EXPLAIN/ELABORATE/EVALUATE:

At this point, students should be able to state what changes they noticed with the graph – specifically where the graph crossed the axes as changes are made to the function. After they have explained what they found, fill in any gaps and correct vocabulary as needed. Basically, teach what little there is left for the lesson. Follow-up by providing extra examples or a worksheet for students to practice before giving them a quiz or test to assess their performance.