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

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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.

 

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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.

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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

 

 

 

Field Guide to Math on the National Mall

For anyone visiting my old stomping grounds of Washington, D.C., this summer, the Mathematical Association of America has compiled its Field Guide to Math on the National Mall. For example:

Washington, D.C., was planned around a large right triangle, with the White House at the triangle’s northern vertex and the U.S. Capitol at its eastern vertex, linked by Pennsylvania Avenue (as the hypotenuse). A 1793 survey established the location of the triangle’s 90° vertex, and Thomas Jefferson, when he was Secretary of State, had a wooden post installed to mark the spot. This post was replaced in 1804 by a more substantial marker, which came to be known as the Jefferson Pier.

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

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

As a teacher I think that a fun activity that is not too difficult but will need the students to be up and around the room is kind of like a mix and match game. I will give a bunch a students, a multiple of three, different angles. And then I will give the rest of the students cards with acute, obtuse, and right triangle listed on them. The students with the angles will then have to get in groups of three to form one of the three triangles. Once the students are in groups of three, they will then find another student with the type of triangle and pair with them. They will then present and explain to rest of the class why they paired up the way that they did. I think that it would be a good way for the students to be up and around and decide for themselves what angles for what triangles and then to show their knowledge by explaining it to the class.

 

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

The topic of defining acute, right, and obtuse triangles extend what my students should already know about the different types, acute, right, and obtuse, angles. The students should already know the different types of angles and their properties. We can use their previous knowledge to build towards defining the different types of triangles. I will explain to the students that defining the triangles is like defining the angles. If they can tell me what angles are in the triangle and then tell me the properties of the triangles then they can reason with it and discover which triangle it is by looking at the angles.

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

I found an article that I like that was written about a soccer club, FC Harlem. FC Harlem was getting a new soccer field as part of an initiative known as Operation Community Cup, which revitalizes soccer fields in Columbus and Los Angeles. This particular field, when it was opened, had different triangles and angles spray painted on the field in order to show the kids how soccer players use them in games. Time Warner Cable was the big corporation in on this project.

 

References:

http://www.twcableuntangled.com/2010/10/great-day-for-soccer-in-harlem/

Engaging students: Area of a 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 again comes from my former student Lucy Grimmett. Her topic, from Geometry: finding the area of a triangle.

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

This topic is perfect for creating a mini 5E lesson plan or a discovery activity. Students can easily discover the area of a triangle after they know what the area of a square is. I would give my students a piece of paper (can also use patty paper) that is cut into a square. The students would be asked to write down the area of a square and from there would derive the formula for a triangle by folding the paper into a triangle. They will see that a triangle can be half of a square. The students will be able to test their formula by finding the area of the square and dividing it by 2 and then using the formula they derived. If the two answers match then the student’s formulas should be correct. The teacher would be floating around the room observing, and asking probing questions to lead students down the correct path.

 

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

Finding the area of a triangle is important for many different aspects of mathematics and physics. Students will discuss finding the surface area of a figure, finding volume, or learning further about triangles. When discussing surface area and volume students will have to find the area of a base. In many examples a base can be a triangle. For examples, if a figure is a triangular pyramid and students are finding the surface area they will have to find the area of 4 triangles (3 of which will be the same area if the base is equilateral.) The Pythagorean theorem is also a huge aid when finding areas of non-right triangles. Mathematics consistently builds on itself. In physics triangles are very often used to find the magnitude at which force are being applied to an object. They use vectors to show this relationship and then use trigonometry functions (derived from the area) to find the magnitude of the force.

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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?

Finding the area of a triangle can be performed using different methods. Heron (or Hero) found Heron’s formula for finding the area of a triangle using its side lengths. Heron was considered the greatest experimenter of antiquity. Heron is known for creating the first vending machine. Not the type of vending machines we have today, but a holy water vending machine. A coin would be dropped into the slot and would dispense a set amount of holy water.  The Chinese mathematicians also discovered a formula equivalent to Heron’s. This was independent from his discovery and was published much later. The next mathematician-astronomer who was involved in the area of a triangle was Aryabhata. Aryabhata discovered that the area of a triangle can be expressed at one-half the base times the height. Aryabhata worked on the approximation of pi, it is thought that he may have come to the conclusion that pi is irrational.

 

Information found from: https://en.wikipedia.org/wiki/Hero_of_Alexandria

https://en.wikipedia.org/wiki/Area#Triangle_area

 

Engaging students: Finding the area of a right 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 again comes from my former student Jessica Bonney. Her topic, from Geometry: finding the area of a right triangle.

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

 

Since students have learned the area of a rectangle, we can use this previously learned knowledge to help students better understand the area of a right triangle. To start off the class you could say that a farmer needs our help developing his pasture into two hay meadows, one for warm-season grass and the other for cool-season grass. The large, rectangular pasture measures 250 yards wide and 600 yards long. Hancock Seed Company sells bahia grass(warm-season grass) seed for $140 per 50-lb bag per acre and ryegrass (cool-season grass) seed for $25 per 50-lb bag per acre. Have the students initially calculate the area of the pasture, then the area of the area of each triangle. From there the students can calculate how many acres are in each triangular section of pasture to determine how many pounds of seed the farmer will need. This activity allows the students to investigate and see the relationship between the area of a triangle compared to the area of a rectangle in a real world setting.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

 

Khan Academy has a great tool for showing students the area of a right triangle (https://www.khanacademy.org/math/geometry-home/geometry-area-perimeter/geometry-area-triangle/a/area-of-triangle). This tool allows students to see how the area of a triangle correlates to the area of a rectangle. By clicking on the dot and dragging it, the user can see why the formula for the area of a triangle works. Students should have previously learned that the area of a rectangle is the base multiplied by the height (A=bh). This interactive tool shows students that the area of a triangle is one half the area of a rectangle (A= ½ bh). Through further interactions on the website the students then can transform the triangles to rectangles and solve to find the area of the triangle. For further explanation of the formula, Khan Academy has a video demonstrating and proving the area of a triangle using methods from Euclid’s Elements, but in a much simpler form so that students will be able to follow along.

References:

 

Khan Academy: https://www.khanacademy.org/math/geometry-home/geometry-area-perimeter/geometry-area-triangle/a/area-of-triangle

 

Trigonometry for Physics: http://www.lshsstem.com/uploads/3/9/1/4/39145399/phy_1_trig_for_physics.pdf

 

Khan Academy: https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles

 

Radford University: http://www.radford.edu/rumath-smpdc/Units/src/Poles_Sports.pdf

 

Hancock Seed Company (Bahia Grass Seed): https://hancockseed.com/hancocks-pensacola-bahia-grass-seed-50-lb-bag-4.html

 

Hancock Seed Company (Ryegrass Seed): https://hancockseed.com/hancocks-ryegrass-seed-50-lb-bag-14.html

 

 

Engaging students: Deriving the proportions of a 45-45-90 right 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 Amber Northcott. Her topic, from Geometry: deriving the proportions of a 45-45-90 right triangle.

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

There are ways to make the 45-45-90 right triangle not only interesting, but make it fun. A project or activity that I made up involves architecture using the special right triangle 45-45-90. In the project the students become architects. Their job is to create their own architecture, whether it is a bridge or house, etc. by using 45-45-90 right triangles. They must use a three to ten 45-45-90 right triangles. Once the students figured out how many they will use, they are going to draw their architecture. Then the students will label the sides and angles of what they drew. At the end of the activity or project they will solve the 45-45-90 triangles they used. An option for a long project is to actually build the architecture using measurable materials. The project will allow them to be creative and connect real life to the 45-45-90 right triangle. The students will also present their projects.

Another way to do the activity or project is make it a group activity and give the students some word problems dealing with architecture and have them choose one of those word problems. The students will then take the word problem and create the architecture in the word problem. They can draw it or create it, but it has to be measured and labeled along with finding the missing piece. Then they can present their findings, which includes how they came up with their measurements of sides and angles.

All the ways to do the activity or project will still need the student to be able to answer any questions that their peers or myself may ask. Also, at the end their will be a reflection on the project and their interpretation of how to solve the 45-45-90 right triangle.

 

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

Triangles can be seen everywhere. For example, they can be seen on bridges and buildings. The website geometrinarchitecture.weebly.com has a section talking about the special right triangles, which includes the 45-45-90 right triangle. On the bottom of the page the website shares pictures of windows, roofs, and even a front door is seen within a triangle. The webpage also gives examples of how the special triangles can be used in architecture.

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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 dynamicgeometry.com website talks about the Geometers Sketchpad. After checking it out, I find that the program can be useful. The students can create their own 45-45-90 right triangles and explore the idea of 45-45-90 right triangles on their own after instructions on how to use the program. This engages them because the student will be able to think, how can I create a 45-45-90 right triangle? What is a 45-45-90 right triangle?  The students will have these questions and more, but those questions will soon be answered throughout the lesson itself.

References

http://geometrinarchitecture.weebly.com/special-triangles.html

http://www.dynamicgeometry.com/index.html

 

 

 

 

 

 

The birth of a right triangle

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Source: https://www.facebook.com/BrainyMiscellany/photos/a.376986692399535.77951.353887201376151/942186395879559/?type=3&theater

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.

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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.

 

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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.

 

 

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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

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

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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.

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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

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