Category: Geometry

Engaging students: Proving that the measures of a triangle’s angles add to 180 degrees

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 A’Lyssa Rodriguez. Her topic, from Geometry: proving that the measures of a triangle’s angles add to 180 degrees.

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

People generally do not believe something until they can see it for themselves. So this activity can help do just that. Each student will receive a sheet of paper. They are then asked to draw a triangle on that sheet of paper and cut it out. Having each student draw their own triangle allows for many types of triangles and further proving the point later. Once the triangles are cut, each student will rip off each angle from the triangle. Next, they will arrange those pieces so that each vertex is touching the other. Once all the vertices are touching, they will notice that a straight line is formed and therefore proving that the sum of a triangles angles all add up to 180 degrees.

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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?

Euclid proves that the measures of a triangle’s angles add up to two right angles (I. 32) in the compilation geometrical proofs better known as Euclid’s Elements. This compilation was actually all the known mathematics at the time.  So not all of the theorems were written or discovered by Euclid, rather by several individuals such as Pythagoras, Hippocrates, Theudius, Theaetetus and Eudoxus. Euclid’s Elements actually consist of 465 theorems, all of which are proven with only a ruler (straight edge) and compass. This book was so important to the mathematical community that it remained the main book of geometry for over 2,000 years. It wasn’t until the early 19th century that non-Euclidean geometry was considered.

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

Students can be given a variety of images such as the Louvre, the pyramids in Egypt, certain types of sports plays, and the Epcot center in Disney World and then be asked what they all have in common. It may or may not be hard for them to notice but they all have triangles. Then, hand the students the same images but with the triangles outlined and with the measurement of all the angles. They can then compute the sum of the angles for each triangle. Each triangle obviously looks different and all the angles are different but the sum will always be 180 degrees.

 

Resources
http://www.storyofmathematics.com/hellenistic_euclid.html

 

 

 

 

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 comes from my former student Avery Fortenberry. His topic, from Algebra: finding the volume and surface area of spheres.

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

The topic of volume and surface area of spheres is building upon the students’ knowledge of area and circumference of a circle.  A sphere is similar to a circle in that a circle is a closed shape with all points equidistant from the centerpoint (the distance is the radius) and a sphere is a closed object with all points at an equal distance from its centerpoint (the distance is also r).  Students will be familiar with the area of a circle formula, which is A=πr2 and will be able to easily use and understand the formula for volume of a sphere V=(4/3)πr3.  The same is true for circumference in relation with surface area.

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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?

Archimedes was the first mathematician to discover the most important ratio in all of mathematics, π.  He did this by finding the area of a circle using shapes that were incrementally closer and closer to the same size as that circle.  In other words, he would start with a circle and enclose it within a square, then a pentagon, then a hexagon, and so on until he came extremely close to the same shape.  He used this same method to find the volume of a sphere by enclosing it within a cylinder of a known volume and cutting out piece by piece and measuring until he found the parabolic segment is 4/3 that of an inscribed triangle.

Source: http://www.storyofmathematics.com/hellenistic_archimedes.html

 

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What are the contributions of various cultures to this topic?

This topic had many cultures contribute to the understanding of it.  These contributions came from Greek, Chinese, and Arabic mathematicians.  The Greek contribution came mainly from Archimedes, which I discussed in D1.  The Arithmetic Art in Nine Chapters is a Chinese book written in the 1st century that gave a formula that was close but not exact to finding the volume of a sphere.  The author of the book calculated pi as being equal to 25/8 or even as just 3 at times.  Ancient Arabic mathematicians submitted very similar ideas to the Chinese in terms of the volume of a sphere.  While it is known the Chinese derived some ideas from the Greeks, it is still unclear today how the ideas were spread to the Arabic mathematicians.

Source: http://www.muslimheritage.com/article/volume-sphere-arabic-mathematics-historical-and-analytical-survey#sec2.2

 

 

 

 

Folding a New Tomorrow: Origami Meets Math and Science

From the YouTube description:

Origami, the art of paper folding, has been practiced in Japan and all over the world for centuries. The past decade, however, has witnessed a surge of interest in using origami for science. Applications in robotics, airbag design, deployment of space structures, and even medicine and bioengineering are appearing in the popular science press. Videos of origami robots folding themselves up and walking away or performing tasks have gone viral in recent years. But if the art of paper folding is so old, why has there been an increase in origami applications now? One answer is because of mathematics. Advances in our understanding of how folding processes work has arisen due to success in modeling origami mathematically. In this presentation we will explore why origami lends itself to mathematical study and see some of the math that has allowed applications to become so fruitful.

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a recent picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

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Another poorly written word problem (Part 5)

Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy.

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

 

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This one really annoys me. The area is less than 55 square inches, and so the appropriate inequality is

\frac{1}{2} (5)(2x+3) < 55

5(2x+3) < 110

2x + 3 < 22

2x < 19

x < 9.5

However, part (c) asks for the maximum height of the triangle. But there isn’t a maximum possible height. If the height was actually equal to 9.5 inches, then the area would be equal to 55 square inches, which is too big! Also, if any height less than 9.5 is chosen (for the sake of argument, say 9.499), then there is another acceptable height that’s larger (say 9.4995).

Technically, the problem should ask for the greatest upper bound (or supremum) of the height of the triangle, but that’s too much to expect of middle school or high school students learning algebra.

This problem could have been salvaged if it had stated that the area is less than or equal to 55 square inches. However, in its present form, part (c) of this problem is unforgivably awful.