Engaging students: Defining the terms acute triangle, right triangle, and obtuse triangle

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 Taylor Vaughn. Her topic, from Geometry: defining the terms acute triangle, right triangle, and obtuse triangle.

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

As soon as you think triangles are gone, they are not. In pre-calculus you will address these triangles again, but in a different outlook. In pre-calculus you will notice patterns associated with sin, cos, tan and the different triangles, acute, obtuse, and right. Also there is a cool theorem called Pythagorean Theorem, a²+ b² =c², where a and b are the legs and c is the hypotenuse. This theorem you will forever use, no matter how up in math you get. In calculus right triangles are used for trig substitutions. Trig substitution is instead of using the number, you use sin, cos, tan, sec, to solving different equations. So triangles you want to always remember because in math everything is linked together amd almost everything is a pattern.

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

This semester I have the pleasure of working at the Rec. Being a supervisor for intramurals causes me to a lot of the behind the scenes work that I didn’t know happened. One is turning a patch of grass into a football field. I know you probably thinking what does this have to do with anything, but I actually used 3-4-5 triangles, right triangles, to draw the field. So when laying down the basics of the field we had to mark of 15 yards from a fence so that participants would hurt themselves. Then I placed the stake at that spot. Then we tied twine around the stake and walked down 100 yards and placed a stake. Then wrapped a new piece of twine to the new stake and measured of 40 yards for the width (measurements comes from NIRSA handbook, which are the rules we go by for flag football). Then did the same for the other side to get a rectangle of a length of 100 yards and with of 40 yards. When I saw this paint can, it then hit me that we had to actually paint this. SO my question was “How am I supposed to get straight line?” Well to my shock, my boss pulls up the measuring tape and said “a 3-4-5 triangle!” Who knew! So for the first corner we measured down the twine 3 yards and then 4 yards going into the field and placed a stake. Then we had to twine the two together measuring to see if it was 5 yards. If it wasn’t we had to keep moving the stakes till they were. Once it was it was for sure that the twine was straight and you could use the paint machine and just push along the line. You do this process and until all the lines are done, even for the yard marking lines , like the 20 yard line, and 40 yard line, that you see on the field. Just as shocked as I was, I bet students will be too. Here is a video to show what I am saying so if it is a little confusing the students will have a visual. Or definitely and visual you could do to show this.

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.

One cool activity I found was an online game called Triangle Shoot, where you had to classify the triangles. The game has a lot of floating triangles and on the bottom of the cursor it says what triangle you need to click. Before you start the game, it gives definitions and pictures of the triangles before starting. I played it myself and actually found it fun. For me, the timed mode was more fun due to the fact as time got closer to 0 the more pressure I felt trying to beat my previous score. And since the shapes are floating you try to click them before they float away. I also liked that the shapes are not always facing the same way, some are rotated on its side or flipped, which made it a little more difficult. It also calculates a percentage and tells you how many you got wrong and right. The only thing I wish it did was break down the hits and miss according to the triangle that way students know what triangle that understand ad don’t. I really thought this was a fun activity after introducing the vocabulary. The website is actually a good tool for students to practice what each triangle is and how they differ. Even if a school doesn’t have computers that students could actually try this in class, it is something that students could use as a practice. Also the game has a mode where you can do equilateral, isosceles, and scalene triangles. http://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/triangles_shoot.htm

References

Ricalde, Paul. “3-4-5 Method, How to Get a Perfect Right Angle When Building Structures.” YouTube. N.p., 28 Mar. 2013. Web. 7 Oct. 2015.

“Triangle Shoot.” Sheppard Software. N.p., n.d. Web. 7 Oct. 2015.

Engaging students: Deriving the Pythagorean Theorem

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 Emma Sivado. Her topic, from Geometry: deriving the Pythagorean Theorem.

How has this topic appeared in pop culture?

What if I told you that knowing the Pythagorean Theorem could help you become a millionaire? We’re all familiar with the popular game show “Who Wants to be a Millionaire” so let me take you back to 2007 when Ryan was playing for $16,000. The question asks “which of these square numbers is the sum of two smaller square numbers.” We see the sweat immediately begin to accumulate on his brow as he struggles to find the right answer. He quickly goes to his life lines and asks the audience. The majority say the answer is 16. Ryan contemplates for a minute before going with the audience and selecting 16. Disappointment follows as we discover this is the wrong answer and Meredith explains that the answer is 25 or 4²+3²=5².

https://www.youtube.com/watch?v=BbX44YSsQ2I

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

The Pythagorean Theorem is first taught in Geometry, according to the TEKS, and is expected to be defined, proved, and executed by these students. However, many people say that the Pythagorean Theorem is the basis of trigonometry, which is studied in depth in the student’s pre-calculus course. Beyond pre-calculus applications, the Pythagorean Theorem is used in physics to calculate kinetic energy, in computer science to compute processing time, and in social media to prove Metcalfe’s Law. Beyond math and science, the theorem is used in architecture and construction to determine distances, heights, and angles, in video games to draw in 3-D, and in triangulation to locate cell phone signals.

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

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.

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.

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 19^th century that non-Euclidean geometry was considered.

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

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. Here’s my series on the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

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

Part 3a, 3b, and 3c: A geometric magic trick (see also here).

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

Part 5a, 5b, 5c, 5d: Predicting a digit that’s been erased from a number.

Part 6: Finale.

Part 7: The Fitch-Cheney 5-card trick.

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

Langley’s Adventitious Angles: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my short series on a couple of easily stated but remarkably difficult geometry problems.

Part 1: The world’s second hardest easy geometry problem.

Part 2: The world’s hardest easy geometry problem.

Different Ways of Computing a Limit: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on different ways of computing the limit

$\displaystyle \lim_{x \to \infty} \frac{\sqrt{x^2+1}}{x}$

Part 1: Algebra

Part 2: L’Hopital’s Rule

Part 3: Trigonometric substitution

Part 4: Geometry

Part 5: Geometry again

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).

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.

Hippopotenuse

Source: https://www.facebook.com/sandraboynton/photos/a.206184542749790.56269.114554295246149/1075341509167418/?type=3&theater

My Mathematical Magic Show: Part 3c

Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else.

In the last couple of posts, I discussed a trick for predicting the number of triangles that appear when a convex $x-$ gon with $y$ points in the middle is tesselated. Though I probably wouldn’t do the following in a magic show (for the sake of time), this is a natural inquiry-based activity to do with pre-algebra students in a classroom setting (as opposed to an entertainment setting) to develop algebraic thinking. I’d begin by giving the students a sheet of paper like this:

trianglechart

Then I’ll ask them to start on the left box. I’ll tell them to draw a triangle in the box and place one point inside, and then subdivide into smaller triangles. Naturally, they all get 3 triangles.

Then I ask them to repeat if there are two points inside. Everyone will get 5 triangles.

Then I ask them to repeat until they can figure out a pattern. When they figure out the pattern, then they can make a prediction about what the rest of the chart will be.

Then I’ll ask them what the answer would be if there were 100 points inside of the triangle. This usually requires some thought. Eventually, the students will get the pattern $T = 2P+1$ for the number of triangles if the initial figure is a triangle.

Then I’ll repeat for a quadrilateral (with four sides instead of three). After some drawing and guessing, the students can usually guess the pattern $T=2P+2$ .

Then I’ll repeat for a pentagon. After some drawing and guessing, the students can usually guess the pattern $T=2P+3$ .

Then I’ll have them guess the pattern for the hexagon without drawing anything. They’ll usually predict the correct answer, $T = 2P+4$ .

What about if the outside figure has 100 sides? They’ll usually predict the correct answer, $T = 2P+98$ .

What if the outside figure has $N$ sides? By now, they should get the correct answer, $T = 2P + N - 2$ .

This activity fosters algebraic thinking, developing intuition from simple cases to get a pretty complicated general expression. However, this activity is completely tractable since it only involves drawing a bunch of figures on a piece of paper.

My Mathematical Magic Show: Part 3b

This is a magic trick that my math teacher taught me when I was about 13 or 14. I’ve found that it’s a big hit when performed for grade-school children. Here’s the patter:

Magician: Tell me a number between 5 and 10.

Child: (gives a number, call it $x$ )

Magician: On a piece of paper, draw a shape with $x$ corners.

Child: (draws a figure; an example for $x=6$ is shown)

Important Note: For this trick to work, the original shape has to be convex… something shaped like an L or M won’t work. Also, I chose a maximum of 10 mostly for ease of drawing and counting (and, for later, calculating).

Magician: Tell me another number between 5 and 10.

Child: (gives a number, call it $y$ )

Magician: Now draw that many dots inside of your shape.

Child: (starts drawing $y$ dots inside the figure; an example for $y = 7$ ) While the child does this, the Magician calculates $2y + x - 2$ , writes the answer on a piece of paper, and turns the answer face down.

Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other.

Child: (connects the dots until the shape is divided into triangles; an example is shown)

Magician: Now count the number of triangles.

Child: (counts the triangles)

Magician: Was your answer… (and turns the answer over)?

The reason this magic trick works so well is that it’s so counter-intuitive. No matter what convex $x-$ gon is drawn, no matter where the $y$ points are located, and no matter how lines are drawn to create triangles, there will always be $2y + x - 2$ triangles. For the example above, $2y+x-2 = 2\times 7 + 6 - 2 = 18$ , and there are indeed $18$ triangles in the figure.

This trick works by counting the measures of all the angles in two different ways.

Method #1: If there are $T$ triangles created, then the sum of the measures of the angles in each triangle is $180$ degrees. So the sum of the measures of all of the angles must be $180 T$ degrees.

Method #2: The sum of the measures of the angles around each interior point is $360$ degrees. Since there are $y$ interior points, the sum of these angles is $360y$ degrees.

The measures of the remaining angles add up to the sum of the measures of the interior angles of a convex polygon with $x$ sides. So the sum of these measures is $180(x-2)$ degrees.

These two different ways of adding the angles must be the same. In other words, it must be the case that

$180T = 360y + 180(x-2)$ ,

$T = 2y + x - 2$ .

I’m often asked why it was important to choose a number between 5 and 10. The answer is, it’s not important. The trick will work for any numbers as long as there are at least three sides of the polygon. However, in a practical sense, it’s a good idea to make sure that the number of sides and the number of points aren’t too large so that the number of triangles can be counted reasonably quickly.

After explaining how the trick works, I’ll again ask a child to stand up and play the magician, repeating the trick that I just did, before I move on to the next trick.