Engaging students: Finding the area of a square or rectangle

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 Trent Pope. His topic, from Geometry: finding the area of a square or rectangle.

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

On this website I saw that the Kelso High School GIC students went out and built a home as a class project. They were able to get a $13,000 grant from Lowe’s Home Improvement and blueprints from Fleetwood Homes to go out and build a physical house. I like the idea of having students use geometry in a real world application, as a teacher I would bring this idea to paper. Students would design and create blueprints for their dream house using squares and rectangles. I would start by giving them the total area their house will be. For example, I would tell them to make the blueprints for a 400 square foot house. They could have anywhere from 5 rooms to 20 rooms in their house. They will be responsible for showing the measurements for each room. After creating the layout of the house and calculating the areas of each room, students will be given a set amount of money to spend on flooring. They will then calculate the cost to put either carpet, wood, or tile in each room. This is to have students decide if they would have enough money to have a large room and if so what flooring would be best. There are other aspects you can add to this project to make it more personalized, but as teachers we just want to make sure we are having students find the area of square and rectangles.

http://tdn.com/news/local/geometry-in-construction-class-finishes-building-first-home/article_74143492-d7a9-11e2-995a-001a4bcf887a.html

 

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How has this topic appeared in the news?

There are many instances of where area made the news. I found multiple websites that talk about how schools are having building projects for geometry and construction classes. These students are building homes from 128 square feet to 400 square feet. Teachers are having students make these homes so that they can see that geometry is in the real world. By having a range of sizes, students have to adjust their calculations. When creating a house or mobile home you need to accommodate for walking space in each room. In order for students to know if there is enough space, they must find the area of each room. Teachers are using this project because blueprints for houses only use squares and rectangles, making it easier for students to practice solving for area of these shapes. This is just the start of teachers making the concept of area more applicable.

http://design.northwestern.edu/projects/profiles/tiny-house.html

 

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

The topic of finding area of squares and rectangles is used throughout many cultures. In the Native American culture along with todays, we see it in growing crops. A farmer must know how big their crop is so they can figure out how much food they will have at harvest. An instance used by many cultures is creating monuments in the shape of square pyramids. In order to build it the right way, you must know the area of the bottom base to build on top of that. A final use of it in our culture is in construction, when we decide how we want to build a building. The concept of area is something that many cultures use today because of how easy it is to calculate. This creates a great way for cultures that are less educated to become familiar with the same concepts as other cultures.

References

Geometry in Construction class finishes building first home. n.d. <http://tdn.com/news/local/geometry-in-construction-class-finishes-building-first-home/article_74143492-d7a9-11e2-995a-001a4bcf887a.html&gt;.
Tiny House Project. n.d. 6 10 2017. <http://design.northwestern.edu/projects/profiles/tiny-house.html&gt;.

 

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 Kelly Bui. 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 involves your topic?

As an activity, possibly the “exploration” part of the lesson, students will be paired in partners and the instructor will provide each pair with a different rectangle or square. The goal is to find the area of half of a rectangle.

The condition they must follow is that they cannot “draw” a straight line across the shape, they must “draw” a straight line starting from a corner. At some point, it should be evident that you can only draw a straight line from a corner to another corner. By drawing a diagonal line across the rectangle, they will now have two triangles (if that isn’t clear to them at this point, let them realize it on their own or go over it as a class at the end of the activity). Using rulers or meter sticks, they will have to discover on their own what the area of half of the rectangle is along with what the formula for that looks like. Most students will probably take the area of the entire rectangle and divide by 2. Once they come up with a formula for the area of half of a rectangle, it should look like A=1/2 bh, tell each student to raise half of the rectangle they cut, and announce: “congratulations, you have found the formula for the area of a triangle.”

 

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

Students begin to see the formula for the area of a triangle in 6th or 7th grade. They know the formula, but often times they don’t understand where it comes from. It can be useful for future homework/test problems that ask for the area of an irregular shape as well as in algebra with unknown lengths. These types of problems require students to think “outside of the box” in order to find the area of an irregular shape. It is not always evident that the irregular shape is simply made out of polygons.

Additionally, this topic will be useful when students are in algebra and they must solve for the area of a polygon that doesn’t have specific dimensions. For example, the trapezoid below has an unknown height as well as an unknown base. It is good for students to know how to apply the area formula of a triangle to solve for the dimensions as well as the area of the entire trapezoid itself. One important thing that should be stressed in the classroom is that formulas are extremely helpful on their own, but they’re even more helpful when they can be applied to different applications.

image1.jpg

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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?

Most students know the formula A=1/2 bh for the area of a triangle, but many students don’t remember the formula used to find the area when all three side lengths are known. Heron’s Formula: √[s(s-a)(s-b)(s-c)] is briefly mentioned in geometry and is often not used in other math courses in high school. Along with his derivation of Heron’s formula, he contributed greatly to ancient society.
Heron of Alexandria was a Greek engineer and mathematician who was known mostly for his work with geometry. He was also a lecturer at the Library/Museum of Alexandria where he would meet with other scholars and discuss work. Additionally, he wrote Metrica, a series of three books which included his work on area and volumes of different types of figures.
It is no secret that Heron had a brilliant mind, and with his engineering and mathematics background, he was actually ahead of the industrial revolution that would take place centuries later. He invented the “Hero Engine, also known as the “aeolipile,” which was powered by steam. Essentially, Heron was the first inventor of the steam engine.
Another one of Heron’s inventions was the “wind wheel,” which is very similar to the modern windmill.

Students will already know that there were many breakthroughs during the industrial revolution, but some of the machines and inventions implemented in the 1800s were actually ideas that were invented centuries before.

Irregular Shape Image: http://www.softschools.com/math/geometry/topics/the_area_of_irregular_figures/

Trapezoid Image: http://www.dummies.com/education/math/geometry/how-to-calculate-the-area-of-a-trapezoid/

Heron of Alexandria: https://www.britannica.com/biography/Heron-of-Alexandria

Hero Engine (aeolipile): http://www.ancient-origins.net/ancient-technology/ancient-invention-steam-engine-hero-alexandria-001467

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.

green lineHow 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.

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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

green lineWhat 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 \piand 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
https://www.youtube.com/watch?v=whYqhpc6S6g
(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

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

 

 

 

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: Finding 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 Daniel Herfeldt. His topic, from Geometry: finding the area of a circle.

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Learning area of a circle as well as the circumference in geometry is beneficial for anyone who is looking to pursue any career involving math. This includes anything from a math teacher to an architect. This helps with most future courses in mathematics. With this said, it will be very beneficial when going into pre-calculus. This is because in pre-calculus you will deal a lot with trigonometry. This includes such things as the unit circle, which is a great deal to pre-calculus itself. Being familiar with the equations of a circle helps to understand why things work in a unit circle. It can help with simple things such as why x2+y2=r2. Knowing the area of a circle will make the class easier to understand in all. This topic is also very important to future architects. The reason for this is because if an architect doesn’t know the area of a circle or any other shape, it would be very difficult to construct a building. If one cannot figure the dimensions of a pillar to help support the ceiling of a building, the building will have a possibility to collapse. This causes the structure of a building to rely highly on the dimensions, area, and volume of all shapes including the circle. This proves that the importance of the area of a circle to be very high. Most students will not know that everywhere they go, circles are needed. Informing them about these small details could have the students more eager to learn. Giving them great real world examples might also help the students understand and grasp the knowledge that you are trying to teach them because it relates to them.

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Circles will be anywhere you go. They are in your everyday TV show, video games, and movies. Although at first glance you might not actually see them, they really are there. When creating a character for an animated movie or a popular video game, artist first start to draw with simple circles and lines. They need to figure out a certain area of the character’s face to be able to fit the facial features. For example, they need to be able to fit eyes, a mouth, a nose and a few more features. From this, they will go on to the animation of the character. This also includes circles because in an animation, when you are wanting to move one object, you have to move it all. The same process applies when working on the landscape properties. It will mostly start with simple lines, circles, and boxes. From there, it will progress into more advance steps, putting more and more detail into it. When moving a character, it is also necessary to move the landscape and surroundings as well. This would be great to tell a class because students will be able to relate to the subject. Most kids in the high school level will play video games. Whether the game is on their phone or a gaming console, they still require the beginning steps. If the student doesn’t play video games, they can relate to it due to watching an animated movie. This will be a great way to engage the class in the first few minutes of class. Below is a picture of the progression of drawing a Pokémon.

 

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Many ancient civilizations have been fascinated with circles. Circles can be seen in many ancient structures and buildings from the Roman Coliseum to Stonehenge.

One ancient civilization fascinated with circles were the Greeks. The believed the circle to be a sacred divine shape mostly based on its multiple points of symmetry.  The Greeks also invented a puzzle called squaring the circle, in which the person had to construct a square with the exact area of a circle a compass and a straight edge. This puzzle has been proved mathematically impossible.

Other instances in which circles played an important role in history were the circles that appeared in the crops in different areas of the world. These crop circles have been argued to be a hoax while others indicate it is not possible for the crop circles to be the work of humans.  Regardless, of their origin, these crop circles continue to fascinate us.

Circles continue to have significance today. They are used in logos and other things usually to signify unity and harmony. Even the Olympic symbol is made up of five interlocking colorful rings. The circle is still found today enclosing the all seeing eye over the pyramid in the dollar bill on the US currency.

The significance of presenting this information to students, especially high school students, is to give them background information. I think many high schoolers will be interested to learn how circles have been significant in other cultures throughout history.  Students can be given a short introduction in the subject and asked to look for more instances in which circles played a role in an ancient civilization and then bring that trajectory to modern times.

 

Website: https://nrich.maths.org/2561

 

 

 

 

 

My Favorite One-Liners: Part 88

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In the first few weeks of my calculus class, after introducing the definition of a derivative,

\displaystyle \frac{dy}{dx} = y' = f'(x) = \lim_{h \to 0} \displaystyle \frac{f(x+h) - f(x)}{h},

I’ll use the following steps to guide my students to find the derivatives of polynomials.

  1. If f(x) = c, a constant, then \displaystyle \frac{d}{dx} (c) = 0.
  2. If f(x) and g(x) are both differentiable, then (f+g)'(x) = f'(x) + g'(x).
  3.  If f(x) is differentiable and c is a constant, then (cf)'(x) = c f'(x).
  4. If f(x) = x^n, where n is a nonnegative integer, then f'(x) = n x^{n-1}.
  5. If f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 is a polynomial, then f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + a_1.

After doing a few examples to help these concepts sink in, I’ll show the following two examples with about 3-4 minutes left in class.

Example 1. Let A(r) = \pi r^2. Notice I’ve changed the variable from x to r, but that’s OK. Does this remind you of anything? (Students answer: the area of a circle.)

What’s the derivative? Remember, \pi is just a constant. So A'(r) = \pi \cdot 2r = 2\pi r.

Does this remind you of anything? (Students answer: Whoa… the circumference of a circle.)

Generally, students start waking up even though it’s near the end of class. I continue:

Example 2. Now let’s try V(r) = \displaystyle \frac{4}{3} \pi r^3. Does this remind you of anything? (Students answer: the volume of a sphere.)

What’s the derivative? Again, \displaystyle \frac{4}{3} \pi is just a constant. So V'(r) = \displaystyle \frac{4}{3} \pi \cdot 3r^2 = 4\pi r^2.

Does this remind you of anything? (Students answer: Whoa… the surface area of a sphere.)

By now, I’ve really got my students’ attention with this unexpected connection between these formulas from high school geometry. If I’ve timed things right, I’ll say the following with about 30-60 seconds left in class:

Hmmm. That’s interesting. The derivative of the area of a circle is the circumference of the circle, and the derivative of the area of a sphere is the surface area of the sphere. I wonder why this works. Any ideas? (Students: stunned silence.)

This is what’s known as a cliff-hanger, and I’ll give you the answer at the start of class tomorrow. (Students groan, as they really want to know the answer immediately.) Class is dismissed.

If you’d like to see the answer, see my previous post on this topic.

Engaging students: Finding 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 Joe Wood. His topic, from Geometry: finding the area of a circle.

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The students would be greeted with Lion King’s “Circle of Life” song. While the song has nothing to do with area of a circle, it would create a different and exciting buzz in the classroom that wouldn’t always be offered in this form. (Plus, who doesn’t want to hear a little Lion King music?)

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

A great activity I found on the Mathbits website tackles both questions C1 and A1.
The two-page worksheet below is based off a scene from the movie Castaway. In the scene, Tom Hanks calculates the area of a circle to figure out the likelihood of his rescue. He then compares his calculated area to the area of Texas (which for young students who are all about Texas like I was, this is another attention getter on its own). I would show the clip (having sent a permission slip home since Tom Hanks is shirtless) which can be seen at https://www.youtube.com/watch?v=y89VE9_2Cig so that students can have a good laugh and also understand the scene described on the worksheet. While most, if not all students will never be stranded on a deserted island, this would be an interesting real world problem for the “survivalist” kid in the class.

https://www.youtube.com/watch?v=y89VE9_2Cig
The worksheet is great because it starts off asking if Tom’s calculations were even correct. It then has several example problems for area of a circle so they can practice, but it also brings in linear speed calculations, and a circumference problem which is great review (and a good warm up if you were maybe moving into angular speed later).

castaway1 castaway2

 

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Another interesting real world problem can be found at http://spacemath.gsfc.nasa.gov/geometry.html. The problem deals with solar energy on satellites (or solar panels in general). It talks about how much energy is needed to operate a satellite, tells the student how much energy is provided by solar cells per square centimeter, gives them different shaped solar panels, and ask is the solar panel can produce enough energy.

This specific worksheet only uses half of a circle on one problem, so it should be revised by the teacher to include more circles; however, once again, I think keeping all the different shapes is a great review for students. I also think having the semicircular shaped panel is a great idea to keep the students on their toes.

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A2. How could you as a teacher create an activity or project that involves your topic?

If one stereotype can be made about middle/high school students (especially the boys), it is that they love to eat! And, what do they like to eat? PIZZA! There are several ways this next idea could be carried out (pun intended), but for the purposes of this assignment I will call it a class project that ends in a pizza party.
The idea is that each pair of student will be assigned a pizza restaurant in the area, and they will do a presentation on why we should order pizza from this pizza place specifically. They will have find all the pizza sizes (small, medium, large, etc.) , their  prices, their diameters, the areas of  each pizza, the price per square inch of each of the pizzas, and the best buy. They can talk about anything else they want (such as quality vs price or customer service or whatever) so long as they are trying to sway the class on why the pizza should be purchased from this specific place. Finally, the students will need to provide some kind of proof of their work (menus, calculations, etc) in an organized fashion: PowerPoint, poster board, or some other method.

After the project is complete, the teacher can select the place to buy from, or hold it to a class vote, and have a pizza party during lunch hour or after school or in class.

 

 

 

 

Area of a Triangle and Volume of Common Shapes: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on different ways of finding the area of a triangle as well as finding the volumes of common shapes.

Part 1: Deriving the formula A = \displaystyle \frac{1}{2} bh.

Part 2: Cavalieri’s principle and finding areas using calculus.

Part 3: Cavalieri’s principle and finding the volume of a pyramid and then the volume of a sphere.

Part 4: Finding the area of a triangle using the Law of Sines.

Part 5: Finding the area of a triangle using the Law of Cosines.

Part 6: Finding the area of a triangle using the triangle’s incenter.

Part 7: Finding the area of a triangle using a determinant and the coordinates of the vertices.

Part 8: Finding the area of a triangle using Pick’s theorem.