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.


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.

solar

 

 

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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.

 

 

Area of a Circle: Index

I’m using the Twelve Days of Christmas (with a week-long head start) 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 the formula for the area of a circle.

Part 1: Why the circumference function C(r) = 2 \pi r is the derivative of the area function A(r) = \pi r^2.

Part 2: Finding the area of a circle via integration by trigonometric substitution.

Part 3: Finding the area of a circle via a double integral.

Part 4: Justifying the formula A(r) = \pi r^2 to geometry students by slicing a circle into pieces and rearranging the pieces as a parallelogram (approximately).

 

 

 

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 again comes from my former student Juan Guerra. His topic, from Geometry: finding the area of a square or rectangle.

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E1.       How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The website below contains an activity that relates both perimeter and area. In particular, the activity stimulates the student’s mind by making them think of a way to get the amount of fencing that they would need in order to build the stable for animals. After the character in the game learns about perimeter, he is made to think about the area that would be created from the stable. Then the activity mentions the different possibilities of getting the same perimeter but at the same time, the area of each different possibility is also analyzed. The activity makes students realize that even though all stables have the same perimeter, the area was different most of the time. The activity also has the students practice taking measurements and finding the perimeter and area of rectangles. This activity targets multiple objectives and skills because students learn about perimeter, area, and go over measuring the sides with a virtual ruler. This website contains more interactive games that target multiple skills, which will be helpful to the teachers when planning a lesson. Aside from having interactive games, the website also contains videos on tutorials for some basic computations or definitions of terms in math.

http://www.mathplayground.com/area_perimeter.html

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F3.   How did people’s conception of this topic change over time?

Ancient civilizations have known how to compute the area of basic figures including the square and the rectangle. These civilizations include the Egyptians, Babylonians, and Hindus. The Babylonians actually had a different formula for the area of a square or rectangle. The formula we know today is a*b, where a and b are the lengths of the figure. The Babylonian formula for multiplying two numbers, which was essentially the same as finding the area was [(a + b)2 – (ab)2]/4. Looking at the formula, it is clear that they had a different perception of what it was to find the product of two numbers and also the area of a square or rectangle. It turns out that the Babylonians were the only ones who used a different formula for the area of a rectangle or square, which means that they saw area differently than the other two civilizations. Another person that represented area was Euclid. In his book named Euclid’s Elements, he showed how multiplying two numbers would look geometrically, which was by taking a segment with length a and another segment with length b and putting them together so that they form a right angle at the ends and completing the rectangle by adding the other two missing sides. This method was used for visualizing the multiplication of numbers but it was also the representation of what area looked like geometrically although Euclid did not mention in his book that this was called area.

http://en.wikipedia.org/wiki/Egyptian_geometry

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html

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B1.   How can this topic be used in your students’ future courses in mathematics or science?

In the future, students will need to know the concept of area in general in order to solve other types of problems in courses like calculus. To illustrate a better example, suppose you have the equation y = x. What if you wanted a student to find the area of the triangle formed on the interval from 0 to 5? It would seem obvious that when the student graphs it and creates the triangle from that interval, he or she would use the formula for the area of a triangle once they are able to find the base and the height of the triangle. Another example where they would have to find the area of a rectangle would be when they have an equation like y = 5. Let’s say that you wanted to find the area of the rectangle formed from 0 to 4. The student would naturally use the formula that has been known to them for a long time and plug in the numbers. So what if we asked them to find the area of the function y = x^2 from 0 to 10? Would the student be able to use the formulas for area that he or she knows? This is where the concept of integration can be introduced to the student. The student might develop the curiosity of wanting to find out how it would be possible to find the area under a curve since the formulas for area that he or she has known all along do not apply. This is only one example where area can be seen in future courses but it seems like an activity like this would naturally lead into integration in a calculus class.

 

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 Erick Cordero. His topic, from Geometry: finding the area of a triangle.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

Students in high school usually take geometry during the first or second year, and after that they might not see it again until college. Three years might be the wait until a student sees geometry again, nevertheless, geometry does come back in the form of trigonometry. Trigonometry is a class taken right before pre-calculus and it is here where students truly see geometry again. The importance of the triangle in geometry is enormous and in fact, there would not be any trigonometry if it were not because of triangles. Students learn in this class different ways of getting the area of a triangle because they are no longer given the height and the length of the base, now students are given angles or other information and they have to somehow find the area. The topic of area is also used throughout college in math classes, although we are not always finding the area of a triangle, we are nonetheless finding the area of something. To make everything even better, those students who decide to become teachers have to take a course called foundations of geometry. Now it is here were the student really understands the triangles and the axiomatic method of doing proofs.

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

http://www.britannica.com/EBchecked/topic/194880/Euclid

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

In ancient Greece, mathematicians did not deal with the concept of area as we do today. In fact, numbers were not used in geometry and mathematicians had other creative ways of expressing algebraic expression. The great mathematician, Euclid, whom was born in 300 BC, would be the person who would unify all the geometry that was around at the time. Euclid’s greatest contributions and perhaps the most famous book in the history of mathematics, The Elements, is a book that for hundreds of years was the standard way of doing geometry. Euclid’s approach is what is referred to as axiomatic geometry in which one proves geometric expression on the basis on a few assumptions that are assumed to be obvious. In many of his proofs, Euclid compares different triangles in order to learn more about the situation or scenario he is trying to prove. Euclid has a nice way of defining the area of a triangle. He first proves that one can construct a parallelogram and then he proves that two triangles fit into this parallelogram, and thus the area of a triangle is half a parallelogram.

Thus, Euclid defines the area of a triangle in terms of parallelograms. He proves this by using the basic properties of a parallelogram, such as the fact the opposite angles and sides are congruent, to prove that in fact two congruent triangles can fit into a parallelogram.

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E. How can technology be used to effectively engage students with this topic?

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

The website above is a great website for high school students to look at, but because of the language (sounds ancient) I would prefer to go and explore this website with the students. This website contains Euclid’s elements and although the students would not be expect to know how to do all the proofs, I would expect them to know how to prove the formula for the area of a triangle using Euclidian methods. I think the history that this website contains is amazing and it also has diagrams of the way Euclid did his proofs and students like pictures, especially with math, so this would be good. The wording on the website could cause students some problems but for the immense knowledge they can learn from visiting this website, I believe its worth it. Students will get introduce to this beautiful way of proving geometric theorems, methods that were developed hundreds of years ago and are still being used in universities today. I believe this is something incredibly amazing and every student in geometry should at least be familiar with this method of proving things. I believe students will enjoy this way of doing proofs because it is new (it is new to them) and it is not so rigid and mechanical as algebra might have seemed to them. Also, I believe it is only right that they get to know, from reading some of the proofs, who this great mathematician that we know as Euclid was and the immense influence he had in the history of mathematics.

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 Andrew Wignall. His topic, from Geometry: finding the area of a right triangle.

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.)

To introduce the topic of the area of a right triangle early in a lesson, we can first examine the area of a rectangle, which students should already know how to do.

Say you have a large rectangular garden, 60 feet wide and 10 feet long. Home Depot sells sod (which is a pre-grown grass on a net that can be spread on the ground) at a rate of $3/square foot. What is the area of the garden, in square feet? How much sod should you order? How much would it cost to cover the entire garden with sod?

Instead of having the entire garden covered with sod, suppose you wanted to cover part of the garden with sod and leave the rest as soil for planting flowers. To make it more visually interesting, you decide to set the sod as a triangle? The sod triangle will have a base of 60 feet and a height of 10 feet. What is the area of this triangle in relation to the area of the entire garden? What is the area of this triangle? How much sod should you order? How much would it cost to cover the triangular area with sod?

Through this activity, we can investigate a relationship between right triangles and rectangles, and also the relation of the area of a triangle compared to the angle of a rectangle.

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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 tool to show the area of a right triangle quickly and easily is the Area Tool on Illuminations (http://illuminations.nctm.org/Activity.aspx?id=3567). With trapezoids, parallelograms, and triangles available, you can click and drag the three vertices of a triangle and instantly see how the area is affected. You can create a quick table and keep a running tally of the base, height, and area, so you can recalculate in front of the class.

Illuminations has a sample lesson plan available online for discovering the area of triangles, and integrates this tool into the plan. If not using this tool as part of a similar plan, we must understand that this tool will not be great for introducing the lesson, as there is no button to lock onto a right triangle. However, there is a button to lock the height, so when you move the vertex opposite the base, you can see how the area does not change, see how the height can be outside the triangle, and extend the formula for the area of a right triangle to the area of any triangle. This tool can then be used in further lessons when discussing the area of parallelograms and trapezoids.

 

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

Since triangles are one of the most basic shapes, the area of triangles comes up time and time again. Triangles will also be used to find the area of more complex polygons, such as hexagons and irregular polygons, by breaking down complex shapes into simple triangles and quadrelaterals. Trigonometry uses right (and non-right) triangles extensively; in Precalculus, we will revisit the area of triangles, and learn how to find the area of triangles without explicitly being given the base and height.

Outside the classroom, the area of a triangle is used extensively in architecture, as triangles are strong, and triangular trusses and frames are used in many steel structures. As the inside empty area of the triangle increases, then the stress on the triangle increases, and architects must take this into consideration.

Triangles are also used in 3d computer graphics, as the 3d shapes they design actually consist of lots of little triangles, and they have to fit textures of a certain size (say 512 pixels x 512 pixels) onto a few triangles, so it is important that they know how and where for these textures to lie.

 

References

Math is Fun, “Activity: Garden Area”. http://www.mathsisfun.com/activity/garden-area.html

Illuminations: Resources for Teaching Math, “Discovering the Area Formula for Triangles”. http://illuminations.nctm.org/Lesson.aspx?id=1874

Illuminations: Resources for Teaching Math, “Area Tool”. http://illuminations.nctm.org/Activity.aspx?id=3567

Home Depot, http://www.homedepot.com/p/StarPro-Greens-Centipede-Southwest-Synthetic-Lawn-Grass-Turf-Sold-by-15-ft-W-rolls-x-Your-L-2-97-Sq-Ft-Equivalent-RGB7/202025213

Math is Fun, “Heron’s Formula”. http://www.mathsisfun.com/geometry/herons-formula.html

Maths in the City, “Most stable shape – triangle”. http://www.mathsinthecity.com/sites/most-stable-shape-triangle

Andre LaMothe, “Texture Mapping Mania”. http://archive.gamedev.net/archive/reference/articles/article852.html

 

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 Dale Montgomery. His topic, from Geometry: the area of a circle.

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History

Archimedes was the mathematician who we attribute with finding the area of a circle to be  Where r is the radius and π is the ratio of circumference to diameter of a circle. (Note that Archimedes was not the first to find the area of a circle, but was the first to find π). I would really like to start the class with something along the lines of introducing Archimedes supposed final words “Do not disturb my circles.” And then go into the death of Archimedes and the mystery surrounding his tomb, such as the account of Cicero and the fact that no one knows where the tomb is now. Cicero said that his tomb had a sphere inscribed in a cylinder, which Archimedes considered to be his greatest mathematical proof. From there, the class should have great interest in what is going on. And we can talk about the fact that the area of a circle is the same as the area a triangle with the same base as the circumference and the same height as the radius. ArchimedesCircle

Rorres, Chris. “Tomb of Archimedes – Illustrations”. Courant Institute of Mathematical Sciences. Retrieved 2011-03-15.

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Culture

http://newsfeed.time.com/2013/02/02/are-crop-circles-more-than-just-modern-pranks/

I would show this article in class, most likely passing it out to read. I would ask if they thought it was a prank, and then give them a similar picture as presented in the article but mapped out with radiuses. Then I would say that the average person could do so many square feet of crop’s per hour. If it gets dark at 9 pm and the sun comes up at 6 am, could a person pull a prank like this?

After we discussed how to find the area of a circle I would have found one that it was impossible for one person to do. Then I would display this youtube video.

Seeing that there were 2 people working on it could display that it is possible for it to be a hoax. I like this because it gives the students a way to analyze information that they are given. Does it make sense for these things to be aliens? Not really, so let’s find other explanations. It both introduces the concept and teaches some critical thinking skills.

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You could apply the area of a circle to the diameter of a pizza. When you order pizza you order things like an 8 or a 12 inch. These are diameters and do not give the best idea of how much pizza you are actually getting. You can even include this lesson with a pizza party or something similar. This would easily get kids excited since it is something that most kids like, and they would have the possibility of getting pizza afterwards.

pizza

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 Kayla (Koenig) Lambert. Her topic, from Geometry: finding the area of a square or rectangle.

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B) Curriculum: How can this topic be used in student’s future courses in math or science?

 Finding the area of a square or rectangle can be applied in many other subjects throughout a student’s school career. This topic is learned around 4th or 5th grade, and around this time students will just be using the formulas to find the areas. In middle school, they might be finding the areas by way of more difficult problems, like word problems. The real fun for this subject, in my opinion, doesn’t start until high school. In high school you can use the area of squares and rectangles to find the solutions to many problems. In high school geometry, the Pythagorean Theorem is taught. The area of squares is related to this depending on how the teacher presents this to the student. The Pythagorean Theorem states that “in any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs of the right triangle” (Square-geometry).

In college, possibly high school calculus, students will learn to approximate the total area under a curve (or integral) using the Riemann Sum. To approximate the integral, you find the area of each rectangle, and all of the rectangles areas added together give you the approximated integral. The area of rectangles is also used in Statistics. When creating a histogram, you multiply the height (density) and width of the bars (rectangles).  Then adding the areas (relative frequencies) of all of the bars should be equal to one. Students will also need to use the area of squares and rectangles on college placement exams and standardized testing.

 

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

 In my opinion, anything and everything is a form of art, so the area of squares and rectangles can appear in an infinite amount of high culture. M.C. Escher has used squares and rectangles to create tessellations and “portrayed mathematical relationships among shapes, figures and space” (MC Escher). The area of a rectangle was used to Polykeitos the Elder who was a Greek sculptor. He used the area of a rectangle to create the perfect ratio for the human body. Painters also needed to figure out how to depict 3D scenes onto 2D canvas during the Renaissance (Mathematics and Art).

However, one of the more well-known applications of mathematics in art is the Golden Rectangle, which just so happens to involve the area of squares and rectangles. The Golden Rectangle is the area of the original rectangle to the area of the square, which is also the Golden Ratio. In other words, the Golden Rectangle is a rectangle wherein the ratio of its length to its width is the Golden Ratio (Golden Rectangle). Many ancient art and architecture have incorporated the Golden Rectangle into designs. The Golden Rectangle was used in the floor plans and design of the exterior of The Parthenon, which was a Greek temple dedicated to goddess Athena in 5th century BC (Mathematics and Art). Leonardo DaVinci also used the Golden Rectangle in his work. When painting the Mona Lisa, he used this to “draw attention to the face of the woman in the portrait” (Mathematics and Art). DaVinci also used the Golden Rectangle in the Last Supper using it to create a “perfect harmonic balance between placement of characters in the background” and also used it to arrange the characters around the table (Mathematics and Art).

 

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D) History: Who were some of the people who contributed to the development of this topic?

 Finding the area of squares and rectangles didn’t just come out of the blue; we can thank geometry and ancient mathematics for the development of this topic. One person in particular who contributed to the development of this topic was Euclid, or Euclid of Alexandria, who was a Greek mathematician and known as the “Father of Geometry” (Euclid). He was said to revolutionize geometry and his book The Elements is considered the most influential textbook of all time (History of Mathematics). The collection of his books, all thirteen of them, contain all traditional school geometry (Solomon).

However, Euler wasn’t the only one to contribute to this topic. Pythagoras and his students discovered most of what high school students learn in geometry today (History of Mathematics). In the classical period, Aryabhata wrote a treatise including the computation of areas. From the kingdom of Cao Wei, Liu Hui edited and commented on The Nine Chapters of Mathematics Art in 179 AD (History of Mathematics). There are so many people who contributed to this topic, and people are still contributing and developing to the area of squares and rectangles today!

 

Works Cited

“Euclid – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia.  20 Feb. 2012. http://en.wikipedia.org/wiki/Euclid.

“Golden Rectangle.” Logicville : Puzzles and Brainteasers.  20 Feb. 2012. http://www.logicville.com/sel26.htm.

“M. C. Escher – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/M._C._Escher.

“Mathematics and art – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia.  20 Feb. 2012. http://www.en.wikipedia.org/wiki/Mathematics_and_art.

Solomon, Robert. The Little Book Of Mathematical Principals, Theories and Things. New York: Metro Books, 2008.

“Square (geometry) – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/Square_(geometry).

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Area of a circle (Part 4)

Math majors are completely comfortable with the formula A = \pi r^2 for the area of a circle. However, they often tell me that they don’t remember a proof or justification for why this formula is true. And they certainly don’t remember a justification that would be appropriate for showing geometry students.

In this series of posts, I’ll discuss several ways that the area of a circle can be found using calculus. I’ll also discuss a straightforward classroom activity by which students can discover for themselves why A = \pi r^2.green line

In the previous three posts, I discussed various ways that calculus can be used to show that A = \pi r^2. Still, most future high school teachers would like to know a justification for why A = \pi r^2. After all, the definition of \pi is

\pi = \displaystyle \frac{ \hbox{Circumference} }{ \hbox{ Diameter}}, or C = 2\pi r

So there ought to be a reasonable explanation for why \pi reappears in the formula for the area of a circle. Furthermore, this explanation should within the grasp of geometry students — so that the explanation should not explicitly use calculus. Even better, they’d prefer a hands-on classroom activity so that students could discover the formula for themselves.

The video below shows a completely geometric justification for why A = \pi r^2 that meets the above criteria. I have a couple of small quibbles with the narrated text — I’d prefer to say that the each rearrangement of pieces is approximately a parallelogram (as opposed to a rectangle), and that figures get closer and closer to a real parallelogram with area A = \pi r^2.

In other words, I would avoid saying that we ultimately divide the circle into infinitely many wedges of infinitesimal width to get a perfect rectangle, as this promotes a misconception concerning the definition of a limit that they shouldn’t carry into a future calculus course.

However, the graphics are excellent in this video. In my mind, that more than counterbalances the preferred way that I would describe the process of taking a limit to students.

Pedagogically, I would recommend a hands-on activity along these lines. Let the students use a protractor to draw a 5- or 6-inch circle on a piece of paper. Then have them mark 18 points on the circumference of the circle at every 20^o, and then draw the lines connecting these points and the center of the circle. Then have the students cut out these wedges and physically rearrange them as in the video. They should discover for themselves that the wedges approximately form a parallelogram, and they know how to find the area of a parallelogram.

After they do this activity, then I would show the above video to geometry students.

If anyone knows a video that (1) is as visually appealing as the one above and (2) correctly states the principle of limit for geometry students, please let me know.