Engaging students: The area of a circle

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 Deetria Bowser. Her topic, from Geometry: the area of a circle.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

An example of a helpful and engaging website for students is aaamath.com. On the left side of the webpage, there are a list of subjects. To find the Area of a circle lesson, select geometry and then area of a circle. The lesson is color coded with green being the “learn” part of the lesson, and blue being the “practice.”In the “learn” part of the lesson it explains briefly how to find the area of a circle. While I believe that and actually lesson should be taught before using this website, I think that the “learn” part provided by this lesson would be a great way to quickly review how to find the area of a circle. The next section (“practice”) gives a radius and the student is expected to calculate the area of the circle using said radius. I think this aspect of the lesson will help students gain speed and accuracy in computing the area of a circle. Although I do not think that this website can be used as a complete lesson on finding the area of a circle, on its own, I do believe that it could serve as a great review tool for students.

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

Hands on activities are easier to find for geometry topics, and finding the area of a circle is no exception. An example activity can be found in the YouTube video “Proof Without Words: The Circle.” In this video, the area of a circle is proved using beads and a ruler. The demonstrator creates a circle with silver beads, and shows that the radius of the circle can be measured using the ruler, and the circumference of the circle can be measured by unraveling the outermost part of the circle and measuring it (or by plugging the radius into the equation 2πr). The demonstrator then deconstructs the circle and traces the triangle created by it. From this he shows that $A=0.5bh = 0.5(2\pir)r = \pi r^2$ . Instead of just using symbols to show this idea, I would create a guided explore activity where the students need to actually measure the radius and circumference of the circle they created as well at the base and height of the triangle created by deconstructing the circle they created. I would ask how the circumference and radius of the circle relate to the base and the height of the triangle. Once students recognize that the base of the triangle correlates with the circumference of the circle, and the radius correlates with the height, it will be easier to see why the area of a circle is calculated using the formula $A=\pi r^2$

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.)

Practical uses for finding the area of a circle proved to be quite difficult. For example, most questions contain unrealistic examples such as “making a card with three semi-circles” (Glencoe). Although, many of these impractical exist, I found two example problems that could actually be used in the real world. The first example states “The Cole family owns an above-ground circular
swimming pool that has walls made of aluminum. Find the length of aluminum surrounding the pool as shown if the radius is 15 feet. Round to the nearest tenth” (Glencoe). This example is practical because when constructing a pool, one needs to know the surface area which can be found by using $\pi r^2$ . The final example states “A rug is made up of a quadrant and two semicircles. Find the area of the rug. Use 3.14 for $\pi$ and round to the nearest tenth!” (Glencoe). Although this seems less practical than the pool example, it is still related to real life because finding the area of a rug will help when deciding which rug to choose for a room.

References
M. (2012, May 29). Proof Without Words: The Circle. Retrieved October 06, 2017, from

(n.d.). Retrieved October 06, 2017, from http://www.aaamath.com/geo612x2.htm#pgtp
(n.d.). Retrieved October 06, 2017, from
http://www.glencoe.com/sites/washington/support_student/additional_lessons/Course_1/
584-588_WA_Gr6_AdlLsn_Onln.pdf

Engaging students: Finding the area of a right triangle

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Deanna Cravens. Her topic, from Geometry: finding the area of a right triangle.

How could you as a teacher create an activity or project that involves your topic?
One of the most common questions students ask when working with the area of a triangle is: “Why do I multiply by ½ in the formula?” It is a rather simple explanation for working with right triangles. Students could either do an explore activity where they discover the formula for the area of a right triangle, or a teacher could show this short two minute video in class.

So why do we multiply by ½? If we look at the formula ignoring the ½, you will see that it is the same formula for the area of a rectangle. Each angle in a rectangle forms 90 degrees and if we cut the rectangle along one of the diagonals, we will see that it creates a right triangle. Not only that, but it is exactly one half of the area of the rectangle since it was cut along the diagonal. Another way of showing this is doing the opposite by taking two congruent right triangles and rearranging them to create a rectangle. Either way shows how the ½ in the formula for the area of a right triangle appears and would be a great conceptual explore for students to complete.

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

Students are first introduced to finding the area of right triangle in their sixth grade mathematics class. One way that the topic is advanced in a high school geometry class is by throwing the Pythagorean Theorem into the mix. Students will know that formula for the area of a right triangle is A=½ bh. The way the topic is advanced is by giving the students the length of the hypotenuse and either the length of the base or the height, but not both. Students must use a^2 +b^2=c^2 in order to solve for the missing side length. The side lengths will not always be an integer, so students should be comfortable with working with square roots. Once students utilize the Pythagorean Theorem, they can then continue to solve for the area of the right triangle as they previously learned in sixth grade.

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

In this short music video young students at Builth Wells High School did a parody of Meghan Trainor’s “All About that Bass.” They take the chorus and put the lyrics in “multiply the base, by the height, then half it.” This music video can help several different types of learners in the classroom. Some need a visual aid which is done by specific dance movements by the students in the video. Others will remember it by having the catchy chorus stuck in their head. The parody lyrics are also put on the video to help students who might struggle with English, such as ELL students. Plus, it is a good visual cue to have the lyrics on the screen so it makes it easier to learn. No doubt with this catchy song, students will leave the classroom humming the song to themselves and have connected it to finding the area of a triangle.

References: https://www.youtube.com/watch?v=PHKaqXlki6w and https://www.youtube.com/watch?v=-8HK9H9gNxs

Engaging students: Midpoint

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 Danielle Pope. Her topic, from Geometry: deriving the term midpoint.

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

Introducing the definition of a midpoint in the classroom will take using class time to let students explore for themselves. The activity that I would make my students do is have the entire class stand up and have 2 students stand at opposite sides of the room. I would then ask my students to line up shoulder to shoulder. Once they were in a straight line I would ask “who is perfectly in the middle of this line?” This is where I would give my students 10 minutes initially to come up with various ways of how they would prove a student was in the middle of the line. Various “proofs” that they could tell me would be that there is exactly the same number of people on each side of the middle person. If that answer was given I would make an odd number of students stand in line and ask the same question of “Who is in the middle”? They would have to reconsider this answer because they couldn’t cut the student in half but I would hope that they would come to the conclusion that they would have to half the person in order to find the perfect center. Another “proof” that they may give me is measuring the distance from one end to the other and half that distance to find the person in the middle. This can also start that same conversation of how we would find the exact “midpoint” without cutting the person into pieces.

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

To get just a basic definition of the midpoint, we can look at the lingo used in all sporting events. All sports have some form of a season that lasts for a certain amount of time. For this example specifically, I will be looking at the football season. Towards the middle of the season teams will know what to expect by the end. Most of the stats and predictions for teams are made already by the middle or midpoint of a season. In this article about football it relates to what changes various teams needed to make by the middle of their season. Just in the article itself, it says that “we’re now at the midpoint of the NFL season, and while some things are beginning to take shape, there’s still plenty of football left to be played.” In this context, students can understand that midpoint is being used to describe the middle of a football season. With this knowledge, they can use those context clues and just add the numbers given to them.

One of the most important people in mathematics to date would have to be Euclid. Euclid’s book, The Elements, is still the backbone of all mathematics taught from kindergarten to college. One artist took this book or manual to mathematics and put it in the form of artwork. Crockett Johnson is an artist who bases his work off of mathematics. He takes the complicated proofs, lemmas, and theorem that have been proved and puts those in a form that we see as beautiful. One piece that uses mostly all midpoints titled “Bouquet of Triangle Theorems”. This piece is based off of the many of Euclid’s propositions about triangle just used together in one piece of art. For example “the midpoints of the sides of the large triangle in the painting are joined to form a smaller one.” Giving students a copy of this picture they can find various characteristics given a ruler and other tools that can help them possibly come to this conclusion that Euclid already proved. Crockett’s pieces can also be seen at the Smithsonian so that could show kids that math really does show up everywhere in our world even in unexpected places.

References
http://americanhistory.si.edu/collections/search/object/nmah_694643

http://www.foxsports.com/nfl/gallery/every-nfl-teams-biggest-weakness-at-the-midpoint-of-the-2016-season-110116

Engaging students: Defining the term segment bisector

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 Caroline Wick. Her topic, from Geometry: defining the term segment bisector.

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

A segment bisector is a point, segment, line, or plane that divides a line segment into two equal parts, according to the math dictionary “intermath” (A1). This geometric term has been used throughout history to create art, even before the term was eloquently defined. The people of ancient Greece would use all sorts of geometric ideas to build their vast architectural structures and sculptures, and almost all of these structures would require segment bisectors. According to the Metropolitan Museum of Art page on Greek Architecture, “the vertical structure of [their] temple conformed to an order, a fixed arrangement of forms unified by principles of symmetry and harmony” (A2). The Ancient Greeks prided themselves on their beautiful structures that were pleasing to the eyes because of their symmetry and balance.

Take this picture above, for instance. The columns on the right are perfectly symmetrical to the top beam, and the middle column perfectly divides the top beam. It would be considered a Segment bisector.

Other examples of segment bisectors in high art can be seen in renowned artists work like Picasso who used geometry to paint/express the world in a way that one might not normally see, and other painters have used this geometrical interpretation in their works as well.

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

Segment bisectors could be used in a number of projects or activities. One activity could be showing the use of segment bisectors in origami, or the art of paper folding. Origami requires multiple strategic folds of paper that must be perfect if the shape is to come to fruition. One often has to fold the paper in half perfectly many times, which is the definition of a segment bisector. Students could learn how to use geometric concepts in a concrete and fun way that is applicable in the real world.

The picture above shows just how much segment bisectors are used in the art of paper folding.

Another project could be using the information above on ancient Greek architecture to create their own little architectural temples or structures. The students would use basic materials found around the house, and their knowledge of geometric definitions to create these structures. Not only would this project apply to geometry, but it would also help students see how geometry plays a role in architecture; another real-world application of school knowledge.

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

Segment bisectors do not really sound like the most exciting topic for students to cover. Sure, they can be used in a lot of different applications, but when a student hears that they will be working with the definition of a segment bisector, they likely will not get terribly enthusiastic. However, if students learn these ancient concepts in the context of new technology, it might stick in their brains as a more interesting topic. Geogebra is a website that allows you to construct geometrical shapes and objects as if you were using a ruler and compass. Students could very easily spend hours on the site just finding different ways to construct a geometric shape. They could use the site to create and define multiple geometric concepts, including segment bisectors, so that they discover the words’ meanings for themselves.

The picture above was taken from a youtube video that shows you how to construct perpendicular segment bisectors using a ruler and compass. And though it may seem like it a more advanced subject, students will be able to see the reasoning behind the definition, and might be able to use this website and knowledge for later geometric use.

References:

http://intermath.coe.uga.edu/dictnary/descript.asp?termID=323
https://www.metmuseum.org/toah/hd/grarc/hd_grarc.htm
http://www.crystalinks.com/greekarchitecture.html

https://www.youtube.com/watch?v=cZTEGgsy9W0&ab_channel=Ri chardPieper

Engaging students: Perimeters of polygons

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 Geometry: perimeters of polygons.

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

Finding the perimeter of a polygon has been a necessity since the implementation of architecture and engineering in society. Every advanced society has used this topic to their benefit. For example, if a person wanted to build a fence around their rectangular garden, but wanted to use the least amount of building materials possible, they could find the perimeter around the garden and then calculate how many planks of wood they would need to use before buying that wood. This is a much more effective method than buying the wood and then finding out how much one would need to use.

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

A few interesting word problems I found online involve real world applications of decorating. Both of the problems I found are off of this website: http://www.bbc.co.uk/skillswise/worksheet/ma31peri-e3-w-perimeter-problems

Problem 1 is good for enforcing student understanding because it challenges them to think more abstractly than if the problem were merely stated as “what is the perimeter of the cake?”. This problem tests student understanding of both the meaning of a square and the meaning of perimeter.

Problem 5 is beneficial to students because it includes both a closed and an open question. The closed question allows students to practice what they have learned about finding the perimeter of a polygon and the open-ended question is worded in a way that challenges the student’s conceptual understanding. The student must not only compute the perimeter but also must explain his/her thinking. This problem also forces students to visualize the problem in their head.

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

An engaging activity that a teacher could create to reinforce the topic of finding the perimeter of a polygon is a game where students set out to stop a criminal from entering an a given area/robbing a bank. The students would have to find the perimeter of a building from a simple blueprint mapping the building’s structure (in this case it would just be the outline of the shape of the building with given dimensions). The student would be informed of how much area each officer can cover and they would then be expected to “secure the perimeter” by allotting a certain amount of police officers to the building and placing them along the perimeter (denoted by a given symbol). To ramp up the difficulty of the game, you could set a time limit for each building and have students compete against the clock to stop the robber and you could also increase the variety of officers in the game where each type has a specialty and can cover a different amount of area.

The idea for this activity is found on the website: https://www.teacherspayteachers.com/Product/Secure-the-Perimeter-Cover-the-Area-Hands-on-police-trainee-activity-2770696

References

“Secure the Perimeter! Cover the Area! Hands-on ‘Police Trainee’ Activity.” Teachers Pay Teachers, http://www.teacherspayteachers.com/Product/Secure-the-Perimeter-Cover-the-Area-Hands-on-police-trainee-activity-2770696.

Perimeter Problems.” BBC News, BBC, http://www.bbc.co.uk/skillswise/worksheet/ma31peri-e3-w-perimeter-problems.

Engaging students: Finding the volume and surface area of spheres

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Austin DeLoach. His topic, from Geometry: finding the volume and surface area of spheres..

What interesting word problems using this topic can your students do now?

I found an interesting word problem that has to do with finding the size and density of Pluto using satellite images and data at https://spacemath.gsfc.nasa.gov/Geometry/6Page143.pdf that would be a good way for students to practice finding the volume of a sphere among other things. This problem could not be used at the very beginning of the section, but it is definitely interesting and could be very engaging for some students. There are multiple parts to the problem, but the third part has students calculate the volume of Pluto using the scale of measurement that they discovered in an earlier part. Students would then use their calculated volume to determine the density of the planet and compare it to other common things by using the given mass of the planet. Not only is this practice for the students to be able to calculate volume of spheres, but it helps them by showing further applications and how their calculated volume can be used to make more scientific discoveries. Problems like this are very good for students to see so that they can recognize real-world application for what they are learning in school, even if it is simplified for the sake of the class.

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

One idea that I think is interesting and engaging for students is taking an orange, measuring the diameter, then seeing how many circles of the same diameter the removed orange peel can fit in. There is a short demonstration video at https://www.youtube.com/watch?v=FB-acn7d0zU to see what I mean. This is a good activity because it is very hands-on for students to be actively engaged, and it also helps students recognize that mathematical formulas are not just thrown together, but there is reasoning behind all of them. This will also help the students remember the formula for the surface area of a sphere, as they will be able to think back to this activity and remember the time that they discovered the formula on their own. There is potential to be messy with this activity, but because it is such a memorable activity and will genuinely engage the students and let their curiosity about mathematics come to life, it is worth it if you can set aside the time for clean-up afterwards.

How has this topic appeared in pop culture?

A big place for volume and surface area of spheres to come up in pop culture is in sports. One recent situation can be seen at http://www.espn.com/espn/wire/_/section/ncw/id/18605942 where the Charleston women’s basketball team had to forfeit two victories because their basketballs for those games were not regulation size. The team accidentally used NCAA men’s basketballs (which have a circumference of 29.5-30 inches) instead of the standard women’s basketballs (circumference of 28.5-29 inches). Because the balls were not regulation size, the victories did not count. Students could use the given circumferences to find the surface area and volumes of each ball and see how significant the difference is, then discuss with their peers what the significance of different sized basketballs is. Although this is not an advanced practice idea, it is still a way for students to compute volume and surface area, as well as discover the significance of each of those properties in a way that could interest them, as many students are interested in sports and do not often think of math as playing a significant role in them. Computing the volume and surface area of the basketballs would also help them recognize the relationship between those and circumference.

Engaging students: Equations of two variables

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 Trent Pope. His topic, from Algebra: equations of two variables.

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.)

I found a website that has many word problems where students can solve for two variables. An example of one of these problems is “If a student were to buy a certain number of $5 scarfs and $2 hats for a total amount of $100, how many scarfs and hats did they buy?”. This example would give students a real world application of how we use two variable equations. It would show students that there are multi variable problems when we go to the store to shop for things, like food or clothing. An instance for food would be when a concession stand sells small and large drinks at a sporting event and want to know how many drinks they have sold at the end of the night. After using a two variable linear equation and knowing the price of the cups, total amount earned, and total cups sold, students would be able to solve for the number of small cups as well as large cups sold.
https://sites.google.com/site/harlandclub/Home/math/algebra/word2var

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

This topic extends on the students’ ability to graph and solve a linear equation, which should have been taught in their previous classes. The only difference is that the variable, y, that you solved for in Pre-Algebra is now on the same side as the other variable. For instance, the equation y =(-1/4) x + 4 is the same as x + 4y = 16. We see that we solve for the same variables, but they are both on the same side. This is because you are solving the same linear equation. A linear equation can be written in multiple forms, as long as the forms have matching solutions. This is something that students could prove to you by graphing and solving the equations. They would solve the equations to see that they have the same variables. This makes students more aware that they need to be able to compute for other variables besides x if the question asks for it.

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? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The most effective way to engage a student about this topic is by using a graphing calculator. This is to help students make the visual connection with the topic and check to see if they have graphed the equations the correct way. Students learn more effectively through visual demonstration. Because students are the ones to solve for the equation and plug it into the calculator to check their work, they are going to be able to make that connection, and we will be able to verify that they understand the material. As teachers, we need to incorporate more technology into the ways of learning because we are surrounded by it daily. Using graphing calculators would be a great way to show and check the work of a two variable equation. This gives students a chance to see what mistakes they have made and what lose ends need to be tied up.

References

Solving Word Problems using a system with 2 variables. n.d. <https://sites.google.com/site/harlandclub/Home/math/algebra/word2var>.

Predicate Logic and Popular Culture (Part 172): Clement Clarke Moore

Let $C$ be the set of all creatures, let $H(x)$ be the proposition “ $x$ is in the house,” and let $S(x)$ be the proposition “ $x$ is stirring.” Translate the logical statement

$\forall x \in C (H(x) \Rightarrow \lnot S(x))$ .

Of course, this matches the first two lines of one of the most popular poems in the English language.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 171): Hunter Hayes

Let $P$ be the set of all people, and let $H(x,y)$ be the proposition “ $x$ has $y$ .” Translate the logical statement

$\forall x \in P (x \ne I \Rightarrow \exists y \in P(H(x,y)) \land \forall y \in P (\lnot H(I,y))$ .