Category: Geometry

Engaging students: Perimeters of polygons

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 Geometry: perimeters of polygons.

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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.

 

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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.

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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

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

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

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

green lineWhat 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.

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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.

green lineHow 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: 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 Kelsi Kolbe. 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?

When students are learning how to complete the square they are usually told the algorithm take b divide it by two and square it, add that number to both sides. To the students this concept seems like a ‘random trick’ that works. This can lead to students forgetting the formula with no way to get it back. However, if we show students how to complete the square using algebra tiles they will be able to understand how the formula came to be (pictured to the left). This will allow the students to be able to have actual concrete knowledge to lean on if they forget the algorithm.

For an engage I would introduce them how to use the algebra tiles by representing different equations on the tiles. I would mix perfect squares and non-perfect squares. I would wait to do the actual completing the square as the explore activity. This way it’s something they can experiment with and really learn the material themselves.

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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?

Muhammad Al-Khwarizmi was a Persian mathematician in the early 9th century. He oversaw the translation of many mathematical works into Arabic. He even produced his own work which would influence future mathematics. In 830 he published a book called: “Al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala” Which translates to “The Compendious Book on Calculation by Completion and Balancing” This book is still considered a fundamental book of modern algebra. The word algebra actually came from the Latinization of the word “al-jabr” which was in the title of his book. The term ‘algorithm’ also came from the Latinization of Al-Kwarizmi. In his book he solved second degree polynomials. He used new methods of reduction, cancellation, and balancing. He developed a formula to solving quadratic equations. As you can see to the right this is how Al-Khwarizmi used the method of ‘completing the square’ in his book. It is very similar to how we use algebra tiles in modern day. You can really see the effect he had on modern algebra, especially in solving quadratic equations.

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

I found a fun YouTube video of the Fort Collins High School Math Department singing a parody of Taylor Swift’s song “blank space”. In the video they are teaching the steps for completing the square. It also addresses imaginary numbers for more complex problems. I think this could be a fun engage to get the students attention. The video incorporates pop culture into something educational. I have always liked watching mathematical parodies videos on YouTube. It not only engages the students, but if they already know the words to the song, they could also get the song stuck in their head, which will help them solve the problems in the future.

References:
Completing the Square. (n.d.). Retrieved September 14, 2017, from http://www.mathisradical.com/completing-the-square.html
Mastin, L. (2010). Islamic Mathmatics – Al-Khwarizmi. Retrived September 14, 2017, from http://www.storyofmathematics.com/islamic_alkhwarizmi.html

Solving a Math Competition Problem: Part 9

This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.

A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

Reflection
I didn’t really need the projection into the plane for the solution, but my problem-solving self needed it to be able to count points and regions in slow motion. So, I should present a cleaned-up solution:

 

Solution
Since there are 9 planes, each plane must intersect with every other in a line, creating two points on the surface of the sphere. Thus, there are (9∗8)/2 * 2 = 72 points of intersection, and for n planes, there are 𝑛(𝑛 − 1) points of intersection. With the first plane, there are zero points of intersection and two regions. Suppose we now have n planes and N regions. We add another plane, creating a circle on the sphere. For each segment that the circle intersects, it creates an additional intersection point as it enters, and it divides the region into two parts, adding one additional region. Hence, for each point added, a region is added as well. Since there are two
regions with zero points, there are thus 74 regions with 72 points of intersection.

Solving a Math Competition Problem: Part 8

This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.

A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.

The projection of four planes:

Conjecture: The max number of regions is the number of intersection points plus 2.
Proof (by induction)
If we have 1 plane, we have no intersection points and 2 regions. Suppose we have n planes with 𝑛(𝑛 − 1) intersection points and 𝑛(𝑛 − 1) + 2 regions. Now we add the next plane to our figure. The plane creates a circle on the sphere. To maximize the number of regions, we angle the plane so that our circle does not intersect any already-existing intersection points. So the circle goes through a number of segments. Each time it does, it cuts the region bounded by that segment into two. So for each new intersection point, we lose one region and gain two, for a net gain of one region. That is, however many intersection points are added, that will be the number of regions added as well. And since 𝑛 + 1 planes have (𝑛 + 1)(𝑛) intersection points, we will
have (𝑛 + 1)(𝑛) + 2 max regions. DONE.
For the original competition problem, we have 9 planes and hence 9*8 + 2 = 74 regions, answer e.