Engaging students: Defining the terms perpendicular and parallel

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 Diana Calderon. Her topic, from Geometry: defining the terms perpendicular and parallel.

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

– This topic of parallel and perpendicular appears in art in the early 1900’s, late 1910’-1930’s. The movement was widely known as De Stijl, which in Dutch means “the style”. This movement had characteristics of “abstract, pared-down aesthetic centered in basic visual elements such as geometric forms and primary colors.” , the two main artists of this artistic movement were Theo can Doesburg and Piet Mondrian. The artistic movement started because of a reaction to the end of World War I, “Partly a reaction against the decorative excesses of Art Deco, the reduced quality of De Stijl art was envisioned by its creators as a universal visual language appropriate to the modern era, a time of a new, spiritualized world order”. As seen below, there are multiple lines, all of which are either perpendicular to each other or parallel, “De Stijl artists espoused a visual language consisting of precisely rendered geometric forms – usually straight lines, squares, and rectangles–and primary colors.”.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

– When we think of geometry a lot of people instantly think of triangles, SOHCAHTOA, and other 2D or 3D shapes. But when I think of geometry I think of the Greeks and Euclid, the literal father of geometry, only because I learned about him in Dr. Cherry’s class. With that being said, Euclid was one of the first mathematicians to define the term parallel, in specific, parallel lines. In 300 BCE Euclid stated some definitions for the basics of geometry, then give five postulates, “The postulates (or axioms) are the assumptions used to define what we now call Euclidean geometry.” The fifth postulate is what we want to focus on because it is called the parallel postulate, “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” He also states how to construct a perpendicular in Proposition 12, “To draw a straight line perpendicular to a given infinite straight line from a given point not on it.”, this construction states that by a given line AB and a point C not on the line then it is possible to construct a perpendicular on line AB.

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

– A good group project for the topic of parallel and perpendicular lines would be to allow the students to create a town. The requirements would be for the student’s town to be no bigger than 100 square inches, the students would have the liberty to create any quadrilateral shape as long as it meets the 100 square feet requirement. Another requirement that the project would have is that there must be at least 4 parallel streets, one perpendicular street that is only perpendicular for one of the parallel streets and finally one diagonal street that intersects 3 parallel streets. A town obviously needs to have shops so the students would be required to put shops within the town but must have an explanation as to why the shops were chosen. Finally the students must bring a physical final product, the shops must be in 3D form, the town area may be made with cardboard, cardstock or any material that would sustain the shops on top of it, the streets and corners of streets must be labeled with the corresponding angles. Finally, when they bring their final piece as a class we will walk around and allow the groups to present their product. As an exit ticket for presentation day the students must turn in the definitions of parallel and perpendicular in their owns words and how it was shown in their project product.

Citations:

o Mondrian Returns to France (Figure 1)
https://worldhistoryproject.org/1919/mondrian-returns-to-france

o De Stijl
https://www.theartstory.org/movement/de-stijl/

o The Three Geometries
https://mathstat.slu.edu/escher/index.php/The_Three_Geometries

o Euclid’s Elements I-XIII
https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html#posts

Engaging students: Using the undefined terms of points, line and plane

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 Alec Bui. His topic, from Geometry: using the undefined terms of points, line and plane.

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

This topic appears in a game that I play called League of Legends. To give context, in the game there are a total of 10 players with 5 players on each team. Within each team, they must work together to achieve the goal of destroying the enemy team’s Nexus, which is their base. Most games are not usually this straightforward. You must work with your team to take objectives such as towers, special buffs, and secure kills. This video provides a great summary as to what this game is https://www.youtube.com/watch?v=BGtROJeMPeE.

This topic appears in the game in a very dynamic way, specifically with the champion’s abilities. Each champion’s abilities have different interactions with the game and how it is used. It depends on your cursor placement, which decides what point in the plane you’re aiming at. Also the distance and position of your cursor from your champion can dictate the line in which the ability is casted towards. All of this is conducted on a plane in which the game is played on. These terms are very basic and is inherently understood when playing the game. These inherent concepts depends on the champions and their abilities. Basically the position of the cursor dictates the path and location in which the ability is used on the plane. In terms of the game, this is how you aim and move depending on your character.One specific champion that comes to mind is Thresh. This video (https://www.youtube.com/watch?v=Sv95nBi7ulQ) goes over the champions abilities. Death Sentence, Dark Passage, and Flay all clearly makes use of points (the position in which the cursor is related to the plane) and lines (in relation to where the cursor is respective to the champion, what path & direction will the ability follows).

My background with League of Legends:

●Played the game for about 3-4 years
●I played for our school at UNT (Esports Club)
●What role I played – ADC (Attack Damage Carry)
●In the U.S, I was ranked top 1.87% (in season 8)
●In the world top 2.1% (in season 8)
●Favorite champion: Lee Sin

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D5. How have different cultures throughout time used this topic in their society?

This question and topic reminded me of a YouTube channel which I think uses the undefined terms without explicitly saying so. The channel is called “Primitive Tool” which you can search on YouTube. The channel has a multitude of construction videos of different structures such as pools and houses built back with primitive tools. It makes you think how basic knowledge of lines, points, and plane were naturally used without a mathematical explanation or background. It was very natural to consider this vocabulary in normal day to day life which was continuously used. You can see the progression and advancement of this simple vocabulary in our architecture over human history. Different cultures have used this topic expressed through art, architecture, etc. One that sticks out in my head are the Egyptians. It’s clear that basic mastery of the topic is needed to construct the phenomenal pyramids that stand today.

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

Geometer’s sketchpad would be a great way to engage this topic. It would be a great way for students to explore the different tools with little to no explanation needed. Student’s can play around with the tools up until guided practice is needed up to the discretion of the teacher. It would be far more engaging than simply explaining what these terms are and going through examples. It provides a dynamic way for the students to interact with all of the terms. This allows them to see the relationship between the terms. They can further their exploration by creating shapes and different polygons with the tools. Overall it’s great dynamic way to learn what the terms are instead of a static manner.

Engaging students: Defining the words acute, right, and obtuse

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: how to engage geometry students when defining the words acute, right, and obtuse.

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A1.) How could you as a teacher create an activity or project that involves your topic?

To have the students get engaged with the topic of Defining the terms acute, right, and obtuse, I will begin with having the classroom set up into groups of 4-5. Within their group they will create 10 examples of where each acute, right, and obtuse angles or triangles can be found in the classroom or in the real world in general. For example, the letter Y, end of a sharpened pencil, and the angle under a ladder can be used. They will be given about 10-15 minutes depending on how fast they can all finish. This is a great activity as the students can work together to try to come up with these examples and can familiarize themselves with amount of ways these terms are used in life. I will tell them before I begin the activity that the group that comes up with the most examples will be given extra credit in the next exam or quiz. This will give them extra incentive to stay on task as I am well aware that some groups may finish earlier than the rest and may take that extra time to cause disruptions.

 

 

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

In previous courses, students have learned had some exposure to these types of angles. Most students have been familiar with the use of right triangles and have learned methods like the Pythagorean theorem. When we extend the terms acute, right, and obtuse in geometry, it begins to be more intensified. These angles then extend in terms of triangle that will then have many uses. Students will then be expected to not only find missing side lengths but also angles. Students will then be exposed to methods later like, law of sines and cosines, special right triangles, triangle inequality theorem and triangle congruency in. This topic essentially is the stepping stone for a large part of what is soon to be learned. Other courses will use a variety of other was to incorporate the terms acute, right, and obtuse. Geometry, precalculus and trigonometry will essentially have a great deal of uses for these terms for starters and can then also be extended in many higher-level math courses in universities.

 

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

An effective way to teach this topic using technology and the terms acute, right, and obtuse would be games. There is a magnitude of game that involve angles and be beneficial in the understanding of these angles. I have found this one game called Alien Angles. In this game, you are given the angle of where the friendly alien at and you have to launch your rocket to rescue them. the purpose of the game is for students to be familiar with angles and how to find them. after you launch the rocket, you are given a protractor that shows the angles and I believe this is beneficial for students as they can also be more familiar with the application of protractors. I can post this on the promethean board and have students identify what the angle I need to rescue the aliens. I can then call for volunteers to go on the board and try to find the correct angle to launch the rocket.

https://www.mathplayground.com/alienangles.html

 

High-flying functions: Marching Jayhawks’ star twirler sees math in her pastime

From the YouTube description:

As an internationally competitive baton twirler in Toronto, Canada, Nicole Johnson impressed audiences with the degree of difficulty found in her acrobatic routines. As a Jayhawk, she found a way to express that complexity using another talent — math. Nicole Johnson received an Undergraduate Research Award for “Baton Twirling into the Fourth Dimension.” Working with associate professor Estela Gavosto, she translated her astonishing performances into equally beautiful graphs. “Just being able to show that there’s this different side of math than what you normally see — that is one of the really cool outcomes of this project,” says Johnson, who graduated in May 2019 with honors in both mathematics and engineering physics.

https://www.youtube.com/watch?v=peGl6N7SqjY

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.

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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.

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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.

 

 

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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: 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.

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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.

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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.

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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

https://study.com/academy/lesson/production-function-in-economics-definition-formula-example.html

 

 

Engaging students: Solving linear systems of equations with matrices

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 Andrew Sansom. His topic, from Algebra II: solving linear systems of equations with matrices.

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

The Square in Downtown Denton is a popular place to visit and hang out. A new business owner needs to decide which road he should put an advertisement so that the most people will see it as they drive by. He does not have enough resources to traffic every block and street, but he knows that he can use algebra to solve for the ones he missed. In the above map, he put a blue box that contains the number of people that walked on each street during one hour. Use a system of linear equations to determine how much traffic is on every street/block on this map.

HINT: Remember that in every intersection, the same number of people have to walk in and walk out each hour, so write an equation for each intersection that has the sum of people walking in is equal to the number of people walking out.
HINT: Remember that the same people enter and exit the entire map every hour. Write an equation that has the sum of each street going into the map equal to the sum of each street going out of the map.

Solution:

1. Build each equation, as suggested by the hints.

2. Rewrite the system of simultaneous linear equations in standard form.

3. Rewrite the system as an augmented matrix

4. Reduce the system to Reduced Row Echelon Form (using a calculator)

 


5. Use this reduced matrix to find solutions for each variable

 

This gives us a completed map:

 


Clearly, the business owner should advertise on Hickory Street between Elm and Locust St (Possibly in front of Beth Marie’s).

 

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

Systems of Simultaneous Linear Equations appear frequently in most problems that involve modelling more than one thing at a time. In high school, the ability to use matrices to solve such systems (especially large ones) simply many problems that would appear in AP or IB Physics exams. Circuit Analysis (including Kirchhof’s and Ohm’s laws) frequently amounts to setting up large systems of simultaneous equations similar to the above network traffic problem. Similarly, there are kinematics problems where multiple forces/torques acting on an object that naturally lend themselves to large systems of equations.

In chemistry, mixture problems can be solved using systems of equations. If more than substance is being mixed, then the system can become too large to efficiently solve except by Gaussian Elimination and matrix operations. (DeFreese, n.d.)

At the university level, learning to solve systems using matrices prepares the student for Linear Algebra, which is useful in almost every math class taken thereafter.

 

 

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

Simultaneous linear equations were featured in Ancient China in a text called Jiuzhang Suanshu or Nine Chapters of the Mathematical Art to solve problems involving weights and quantities of grains. The method prescribed involves listing the coefficients of terms in an array is exceptionally similar to Gaussian Elimination.

Later, in early modern Europe, the methods of elimination were known, but not taught in textbooks until Newton published such an English text in 1720, though he did not use matrices in that text. Gauss provided an even more systematic approach to solving simultaneous linear equations involving least squares by 1794, which was used in 1801 to find Ceres when it was sighted and then lost. During Gauss’s lifetime and in the century that followed, Gauss’s method of elimination because a standard way of solving large systems for human computers. Furthermore, by adopting brackets, “Gauss relieved computers of the tedium of having to rewrite equations, and in so doing, he enabled them to consider how to best organize their work.” (Grcar J. F., 2011).

The use of matrices in elimination appeared in 1895 with Wilhelm Jordan and 1888 by B.I. Clasen. Since then, the method we use today has become commonly attributed to Jordan and commemorated with the name “Gauss-Jordan Method”.
References:
DeFreese, C. (n.d.). Mixture Problems. Retrieved from University of Missouri-St. Louis–Department of Mathematics and Computer Science: http://www.umsl.edu/~defreeseca/intalg/ch8extra/mixture.htm
Grcar, J. F. (2011, May). Historia Mathematica–How ordinary elimination became Gaussian elimination. Retrieved from ScienceDirect: https://www.sciencedirect.com/science/article/pii/S0315086010000376
Grcar, J. F. (n.d.). Mathematics of Gaussian Elimination. Retrieved from American Mathematical Society: https://www.ams.org/notices/201106/rtx110600782p.pdf

 

 

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.

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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.

 

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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.

 

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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: Completing the square

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 Haley Higginbotham. Her topic, from Algebra: completing the square.

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

To start the activity, I think I would do some examples of how to complete the square and see if anybody notices a pattern in how it is done. If not, I would give them some hints and some time to think about it more deeply, and maybe give them a few more examples to do depending on time and number of previous examples. After they have figured out the pattern, I would ask them if they knew why it worked to add (b/2)^2, and why they need to both add and subtract it. Then, we would go into the second part of the activity, which would require manipulatives. They would get into partners and model different completing the square problems with algebra tiles, and explain both verbally and in writing why adding (and subtracting) (b/2)^2 works to complete the square. I would probably also ask if you could “complete the cube,” and have them justify their answer as an elaborate. green line

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

Completing the square is a fairly nifty trick that pops up a decent bit in Calculus 2, particularly in taking integrals of trig functions. Since they need to be in the specific form of (x+a)^2, or some variation thereof. If a student didn’t know how to complete the square, they would get stuck on how to integrate that type of problem. In addition, completing the square is useful when you want to transform a quadratic equation into the vertex form of the equation. It also could have applications in partial fraction decomposition if you are trying to simplify before doing the partial fraction decomposition, and has applications in Laplace transforms through partial fraction decomposition. It is also helpful in solving quadratic equations if it’s not obviously factorable and the quadratic equation is useful but can be tedious to use, especially if you don’t remember how to simplify radicals.

 

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

Students typically learn, or at least have heard of, the quadratic formula before they have learned completing the square. Completing the square can be used to derive the quadratic formula, so they get more of an idea of why it works as opposed to just memorizing the formula. Also, if a student is having trouble remembering what exactly the quadratic formula is, they can use completing the square to re-derive it fairly quickly. Also, it ties the concepts of what they are learning together more so they are more likely to remember what they learned and less likely to see the quadratic formula and completing the square as two random pieces of mathematical information. Depending on the grade level, completing the square can also extend the idea of rewriting equations. They might have been familiar with turning point-slope form into slope intercept form, but not turning what is sometimes the standard form (the quadratic form) into the vertex form of the equation.

 

Engaging students: Solving absolute value 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 Biviana Esparza. Her topic, from Algebra: solving absolute value equations.

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

One of the things that I love about math is that it all builds up on itself. Absolute value is first introduced in sixth grade, where they just have to determine the absolute value of a number. Given |-4|, the answer is 4, |5|=5, |-16|=16, and so on. In seventh grade, students are expected to be able to use the operations on numbers, such as multiply, add, subtract, and divide. In eighth grade, students should be able to write one variable equations; all lead up to learning how to solve absolute value equations in algebra 2.

 

 

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

How I Met Your Mother is a TV show that aired from 2005 to 2014 on CBS. It is a very popular show to watch on Netflix. In the show’s second to last episode, titled “Last Forever, Part 1,” Marshall Eriksen is asked about his new job, and all of his responses are positive but sound slightly awkward. His wife Lily then explains that Marshall decided to only say positive things about his new job now that he is back in corporate law.

This scene could be used to engage students before a lesson on absolute value equations because the two are sort of related in that with every input, there is a positive output. After watching the scene, the teacher could explain how absolute value equations usually require you to break them up into a positive and negative solution and continue to solve. The positive answer is more straight forward to solve for, and the negative answer probably requires more thought and steps, similar to Marshall having to answer cautiously and slowly when trying to answer in a positive way in the scene.

 

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

If the students have access to laptops or tablets or the teacher has access to a class set, Desmos has a nice teacher program and one of the lessons on the site scaffolds student knowledge on distances on number lines all the way up to solving absolute value functions using number lines. The link is provided below. This lesson would be engaging for students because many of them are usually drawn to projects or lessons involving technology. Also, the virtual, interactive lesson does a good job of scaffolding, starting from basic number line knowledge which the students should all be starting with.

https://teacher.desmos.com/activitybuilder/custom/59a6c80e7620f30615d2b566