# Engaging students: Fitting data to a quadratic function

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 Eduardo Torres Manzanarez. His topic, from Algebra: fitting data to a quadratic function.

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

One interesting project that could be done to invoke quadratic modeling is for students to develop a model that fits a business’ data of labor and output. The basic model of labor and output for a given company can be modeled by a quadratic function and it can be used to determine important figures such as the maximum output, minimum output, maximum labor, and minimum labor. The following image is an example of such a relationship.

In general, people would think that the more labor and resources used at the exact same time results in more product. If you have more product produced, then you accumulate more profit. These ideas are not wrong to be thought of but a key aspect that is missed in the thought process is that of land or otherwise known as workspace. The more employees you hire, the more space required so that these individuals can produce but space is limited just like any other resource. Lack of space inhibits production flow and therefore decreases product, decreases profits, and increases cost through increased wages. All of this does not occur until you pass the maximum of the model. So, both of these behaviors are shown and exhibited by a quadratic function. Students can realize these notions of labor and production by analyzing data of various companies. An activity that could show such a relationship in action is having one student create a small particular product such as a card with a particular design and produce as many as they can in a certain amount of time, with certain resources, and a workspace. Record the number of cards produced. Next, have two students create cards with the exact same time, resources, and workspace and record the amount produced. As more students are involved, the behavior of labor and production will be shown to be direct and then inverse to each other. The final piece for this activity would be for students to find realize what function seems to have the same shape as the data on a graph and for them to manipulate the function so that it fits on the data. Turns out the function will have to be a quadratic function.

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

Fitting data onto a quadratic function is useful in analyzing behavior between variables. In various mathematical courses, data is provided but in science usually one must come up with data through an experiment. Particularly there are many situations in physics where this is the case and relationships have to be modeled by fitting data onto various functions. Doing quadratic modeling and even linear modeling early on is a good introduction into other models that are used in the many fields of science. Not every experiment is recorded perfectly and hence there can never be a perfect model. Through analytical skills presented in this topic, it scaffolds students to find a model for bacteria growth, a model for velocity, a model for the position of an object, and a model for nuclear decay in the future and what to expect the behavior of these models to be. This topic in combination with limits from calculus builds onto piece-wise models for probability and statistics.

E1) How can technology be used to effectively engage students with this topic?

Technology such as graphing calculators, Excel, Desmos, and TI-Nspires can be used to create the best model possible based on least-squares regression. This technology is engaging in developing models, not because of the lack of convoluted math that deals with squaring differences but rather the focus on analyzing particular models such as a quadratic model. They could be engaging for students when students can input particular sets of data they find interesting and need a way to model it. Furthermore, students can use technology to develop beautiful graphs that can be easily interpreted than rough sketches of these models. TI-Nspire software can be used by a teacher to send a particular data set to students and their own TI-Nspires. Students can then insert a quadratic function on the graphing application and manipulate the function by changing its overall shape by the mouse cursor. This allows students to dictate their own particular models and allows for comparison between models as to which is more accurate for particular data.

References

# 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 Cody Luttrell. His topic, from Algebra: graphing parabolas.

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

Understanding the graph of a parabola will be very important in an Algebra 1 students future math and science classes. When a student enters Algebra II, they will be dealing with more complicated uses dealing with quadratic functions. An example would be complex numbers. When dealing with a parabola that does not cross the x-axis, you will end up with an imaginary solution, but if the student does not understand the graph of a parabola they may not understand this topic. When the student reaches pre-calculus, understanding the transformations of a parabola will aid when dealing with transformations of other functions such as cubic, square root, and absolute value.
Understanding the graph of a parabola will benefit a student in Physics when they deal with equations of projectiles. Knowing that there is symmetry in a parabola can aid in knowing the position of the projectile at a certain time if they know the time the projectile is at its maximum height.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

The shape of the parabola is used constantly in art and even architecture. A quick engage that I can have for the students would be a powerpoint of photos of parabolas in the real world. Examples would include arches in bridges, roller coasters, water fountains, etc. Ideally, I would want my students to see the pattern that I am getting at and see the parabola in all of these objects. I could then ask the students to brainstorm where else they can find this shape. I would expect to hear answers such as the St. Louis Arch, the sign at McDonalds, or even a rainbow.
After learning about quadratics, we could come back to the topic of architecture and parabolas. After they have learned about the transformations of parabolas, we can discuss how to make arch longer or shorter in bridges(if it follows the parabolic shape). We could also discuss how if we wanted to make a bridge taller, how it would affect the distance between the legs of the bridge.

Using Technology.

A great video from Youtube to show the students to introduce them to graphing parabola: https://www.youtube.com/watch?v=E_0AHIaK48A

In the video, it shows how parabolas are even used in famous videogames such as Mario Bros. In the video, you see a few clips of Mario and Luigi jumping over enemies. The video outlines the path that he jumped and you can notice that it is in the shape of a parabola. The video then goes into explanation that Mario if following the path of y=-x^2. After this explanation, the video switches to Luigi. When Luigi jumps, he also follows the form of a parabola, but slightly different then the way Mario jumps. Luigi can jump higher than Mario, but not as far. The video then states that Luigi is following the path of y=-1.5x^2. This can introduce the idea of compression and stretches. The video than continues on with other examples of how parabolas are used within the game such as vertical shifts.

# Engaging students: Finding the focus and directrix of a parabola

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 Brittnee Lein. Her topic, from Precalculus: finding the focus and directrix of a parabola.

What are the contributions of various cultures to this topic?

Parabolas (as we know them) were first written about in Apollonius’s Conics. Apollonius stated that parabolas were the result of a plane cutting a double right circular cone at an angle parallel to the vertical angle (α). So, what does that actually mean?

Well, if we take a vertical line and intersect it with a straight line at a fixed point, and then rotate that straight line around the fixed point we form the shape below:

If the plane slices the cone at the angle β and β=α, a parabola is formed. This is still how we define parabolas today although you may not think about it that way. When you think of a parabola, you think of the equation $y = ax^2 +bx + c$. This equation is derived using the focus and the directrix. This video shows how to do so:

Understanding how the focus and directrix affect the equation of a parabola is crucial to understanding what each word means. According to mathwords.com, “For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.” The directrix is a line perpendicular to the axis of symmetry and the focus falls on the line of the axis of symmetry.

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

This desmos activity can be used to show students how changing the focus, directrix, and vertex of the parabola affects the graph. https://www.desmos.com/calculator/y90ffrzmco

From this, students can shift values of the vertex and see that the directrix stays constant when the x-value is changed and that the focus remains constant when the y-value is shifted. If students change the value of the focus, they can see how it stretches and contracts the width of the parabola and how the directrix shifts. They can also see that when the focus is negative, the parabola opens downward and the directrix is positive. This website: https://www.intmath.com/plane-analytic-geometry/parabola-interactive.php Is also very helpful in showing the relationships between the focus, directrix and the graph of the parabolas because students can clearly see that the distance between a point on the parabola and the focus and the distance between that same point and the directrix are equal.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

The website http://www.purplemath.com/modules/parabola4.htm has a lot of great real-world word problems involving finding the focus and the directrix of a parabola. For example, one of the questions is:

(This is a graph I made using desmos to model the situation at hand)

This problem requires a lot of prior knowledge of parabolas and really tests students’ ability to interpret information. From the question alone, the students can find the x-intercepts (-15,0) and (15,0) from the information “the base has a width of 30 feet”. They are also able to infer that the slope of the parabola will be negative because of the shape of an arch. The student must also know how to find the slope of the parabola using the x-intercepts, solving for the equation of the parabola using the x-intercepts and vertex and the equations for finding the focus and directrix from the given information. There are a few problems as involved as this one on the listed website above.

# Engaging students: 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 comes from my former student Lisa Sun. Her topic, from Algebra II: parabolas.

How has this topic appeared in high culture?

Parabola is a special curve, shaped like an arch. Any point on a parabola is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). Today, I will be presenting the parabolas’ unique shape to the class. Parabolas are everywhere in our society today. Students just don’t know it yet because no one has informed them. Parabolic structures can be seen in buildings, mosaic art, bridges, and many more. One that I’m going to share with the class is going to be roller coasters. Similar to this image below:

This specific roller coaster is The Behemoth. It is a steel coaster located in Canada’s Wonderland in Vaughan, Ontario, Canada. I will first present this photo to the class and ask the following:

• What do you notice that’s repeating in this roller coaster?
• Do you think you’ve seen this similar structure anywhere else? Where?

–Present definition of Parabola–

• Does this roller coaster have any parabolic structure? Where?

With these guiding questions, I want the students to be familiar with how a parabola looks like and that we can see them in our real world other than school.

How has this topic appeared in the news?

https://www.sciencedaily.com/releases/2016/07/160713143146.htm

This link above is a recent article from Science News on how an engineer from the University of Warwick discovered how to build bridges and buildings to enhance the safety and long durability without the need for repair or restructuring by the use of inverted parabolas. Using inverted parabolas and a design process called “form finding”, engineers will be able to take away the main points of weakness in structures. I believe this is a remarkable discovery that must be shared with students. Math is truly used in our everyday life and can definitely benefit the society today by how fast our technology is advancing.

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

Prezi favors visual learning and works similar to a graphic organizer or a mind map. It helps students to explore a canvas of small ideas then turning it into a bigger picture or vice versa. Prezi is a great tool to maintain an interactive classroom and creates stunning visual impact on students keeping them engaged in the lecture.

http://www.bbcactive.com/BBCActiveIdeasandResources/UsingPreziInEducation.aspx

Above is a link of a Prezi presentation of parabolas in roller coasters. This is a great example as to what I would create for my students to provide them the information of a parabola.

http://www.rollercoasterking.com/article/behemoth/

https://www.mathsisfun.com/definitions/parabola.html

https://www.sciencedaily.com/releases/2016/07/160713143146.htm

https://prezi.com/pwkzfddbu4bu/parabolas-in-roller-coasters/

http://www.bbcactive.com/BBCActiveIdeasandResources/UsingPreziInEducation.aspx

# 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 Irene Ogeto. Her topic, from Algebra: graphing parabolas.

B2. How does this topic extend what your students should have learned in previous courses?

In previous courses, students should have learned about linear functions of the form y = mx + b. Parabolas are functions of the form y = a(x-h) + k. Graphing parabolas extends their thinking because it allows to students to see the graph of a function that is different from the graph of a line. Students can explore the similarities and differences between linear functions and quadratic functions. Students can apply the same logic they used when graphing linear functions by making a table and use the points to plot the graph. Students can use the graph of parabolas to determine the equation of the quadratic function. Students can apply transformations of graphs such as reflecting, stretching or compressing to parabolic functions as well. Graphing parabolas allows students to explore concepts they previously learned such as parent functions, y-intercepts, x-intercepts, and symmetry.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Parabolic curves are all around us in buildings, churches, restaurants, homes, schools and other places. Parabolas are apparent in numerous places in architecture. One example where parabolic curves can be found in architecture is in suspension bridges such as the Brooklyn Bridge in New York, the Golden Gate Bridge in California, or the George Washington Bridge in New Jersey. Suspension bridges are mainly used to carry loads over a long distance and most suspension bridges are lengthy in distance. In suspension bridges, cables, ropes or chains are suspended throughout the road. The cables under tension form the parabolic curve. The towers and hangers are used to support the cables throughout the bridge. Seeing how parabolas appear in high culture will allow students to make a connection between math and the things that may see around them. Hopefully the students can see that math, specifically parabolas in this case are not only found in the classroom.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], VI Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This YouTube video, “Water Slide Stunt,” is a great way to introduce students to graphing parabolas. It allows students to see the curve that parabolic functions make. In addition, it gives students an example of a real-world situation where projectile motion and parabolic functions can be seen. This video can be used at the beginning of a lesson on graphing parabolas. This video is engaging because it gets the students thinking about projectile motion and it shows how math can be related to different things in our society. In addition, students can also look up this video on YouTube on their own time and share with others.

References:

http://science.howstuffworks.com/engineering/civil/bridge6.htm

# Engaging students: Fitting data to a quadratic function

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 Loc Nguyen. His topic, from Algebra: fitting data to a quadratic function.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

To engage students on this topic, I will provide them the word problems in the real life so they can see the usefulness of quadratic regression in predictive purposes. The question to the problem is about the estimated numbers of AIDS cases that can be diagnosed in 2006. The data only show from 1999 to 2003. This will be students’ job to figure out the prediction. I will provide the instructions for this task and I will also walk them through the process of finding the best curve that fit the given data. The best fit to the curve will give us the estimation. Here is how the instruction looks like:

In the end, students will be able to acquire the parabola curve which fit the given data. By letting students work through the real life problems, they will be able to understand why mathematics is important and see how this concept is useful in their lives.

B2. How does this topic extend what your students should have learned in previous courses?

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

At the beginning of the class, I would like to show students the short video of football incident.

This incident was really interesting. The Titans punt went so high so that it hit the scoreboard in Cowboys stadium. Surprisingly, this was Cowboy’s new stadium. There were many questions about what was going on when the architecture built this stadium. It was supposed to be great. This incident revealed the errors in predicting the height of the scoreboard. The data they collected in past year may have been incorrect. I want to incorporate this incident into the concept of quadratic regression. I will pose several questions such as:

Was Titan football punter really that powerful? What was really wrong in this situation?

When the architectures built this stadium, did they ever think that the ball would reach the ceiling?

How come did the architectures fail to measure the height of the ceiling? Did they just assume the height of the stadium tall enough?

What was the path of the ball?

Students will eagerly respond to these questions, and I will slowly bring in the important of quadratic regression. I will then explain how quadratic regression helps us to predict the height based on collected data from past years.

References:

# High-pointing a football?

Today is one of the high points of the American sports calendar: the AFC and NFC championship games to determine who plays in the Super Bowl.

A major pet peeve of mine while watching sports on TV: football announcers who “explain” that a receiver made a great reception because “he caught the ball at its highest point.”

Ignoring the effects of air resistance, the trajectory of a thrown football is parabolic, and the ball is the essentially the same height above the ground when it is either thrown or caught. (Yes, there might be a difference of at most three feet, but that’s negligible compared to the distance that a football is typically thrown.) Therefore, a football reaches the highest point of its trajectory approximately halfway between the quarterback and the receiver.

And anyone who can catch the ball that far above the ground should be immediately tested for steroids.

# 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:

Parabolaonline.tripod.com/history.html

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

# Engaging students: 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 comes from my former student Banner Tuerck. His topic, from Precalculus: finding the equation of a parabola from the focus and directrix.

An interesting way to present the mathematics behind parabolas and their focus points is through the applications it has in science present in our everyday lives. http://spacemath.gsfc.nasa.gov/IRAD/IRAD-4.pdf

The above link includes a great engagement activity for students to do as a group activity. The first exercise presented involves the students in the design of a parabolic dish after observing the properties of a satellite dish with a radio receiver (located at the focus). Once the students have completed the design of the parabolic dish the instructor could then use the second half of the pdf from the link as an elaboration activity. The instructor could either keep the students in the groups or have them work the problems individually. Nevertheless, the second activity would be for the students to work problems one and two, which deal with aiding a bird watcher and a hobbiest in determining the focus points in order to design their parabolically shaped tools. The last problems are excellent real world examples of why one would need to know and apply the mathematics for parabolas. This will encourage students to view everyday objects with a more mathematical respect.

The understanding of the relationship between the focus point and the directrix of a standard parabola is fundamental when students extend their mathematical and science education in post-secondary courses. For example, when students reach multivariable calculus they will graph and study the properties of conic sections on a three dimensional scale. With respect to this topic the students can apply their preexisting knowledge of two-dimensional parabolas to the paraboloids presented in this course. Furthermore, if students from a pre-calculus high school course were to not keep with the theoretical study of mathematics they could benefit greatly from this topic in careers such as architecture, art, or graphic design.

As an instructor of a pre-calculus course one has many technological resources to use in order to construct an elaborate lesson on the directrix and focus of a parabola. For example, modern graphing calculators allow instructors to link their calculator to a projector and show the entire class various parabolas in order to further visualize the changing distances to these specific points. Furthermore, I believe a unique homework assignment would be for students to graph given quadratic equations with an online resource such as http://www.wolframalpha.com/. This assignment would also be a great review of how to apply the distance formula. I recommend having the students check that the points on the parabola are equidistant apart from the focus and the directrix they have already found after graphing and computing. Another idea is requesting (for full credit of the assignment) the students use the following link: https://www.khanacademy.org/math/algebra2/conics_precalc/parabolas_precalc/v/parabola-focus-and-directrix-1

to facilitate their understanding of the definition of a parabola as well as the importance of the focus point and directrix line. This is a way to involve technology while simultaneously ensuring that students review key aspects of the lesson after it was given by the instructor during class time.

References