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).

“History of mathematics – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/History_of_mathematics.

 

Engaging students: Solving proportions

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 Pre-Algebra: solving proportions.

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C. Culture: How has this topic appeared in the news?

Solving proportions, or the idea of a proportion being solved, appears in the news more often than not. One specific example that can be used is the effect of the economy on real estate companies. Say we are given 25% of 16 real estate companies that have closed their businesses due to poor economy. We can use proportions to determine the number of real estate companies that closed. We know that the percent is 25 and that the whole is 16. Therefore 25/100 = x/16 which gives us 4 real estate companies that closed (Review of Proportions). Proportions can also be used to determine how many miles we can drive on a certain amount of gas, and gas prices are constantly on the news. Also, this will be relevant to high school students who drive and need to find how much money they need to buy gas for the week, etc.

We can also use proportions to find the unit price of an item at a grocery store, or if an item costs a certain amount, you can find out how many of those items you can buy with a fixed amount of money you have. Buying items and saving money are also all over the news. If you find the unit price you can compare items therefore saving money by buying the item that you get the most out of your money.  Another way solving proportions can appear on the news is by the stock market. You can use proportions to find out how much the stock market will rise in a given amount of days given the current amount of points it has raised in a certain amount of days. Making a proportion problem for students to solve is relatively easy and can be related to anything that is on the news. We can use this to our advantage to get the students to be a little more interested in proportions (and mathematics) so they can see different ways it is related to real life.

 

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D. History: How was this topic adopted by the mathematical community?

The idea of proportions was adopted and used by many in the mathematical community. Proportions were used by Greek writers, including one named Nicomachus, who include proportions and ratios in arithmetic (Math Forum). Proportions were also adopted by Exodus who used them in geometry and by Theon of Smyrna who used proportions in music (Math Forum). In 2000 B.C., the Babylonians adopted proportions to represent place value notation (Pythagoras – Geometrical Algebra). Using proportions was accepted by mathematicians and was used to solve so many different equations used for so many different ideas, and is still used today. Early proportions were adopted by the Egyptians and were used to calculate fractions and measurement of farmland (Mathematics History). Later, proportions were adopted by so many more in the mathematical community like in Greece, China, India, and Babylonia in order to learn geometry. Greeks, like Plato, adopted proportions in order to study them with the Egyptians. I think that proportions were well liked by mathematicians and were adopted by many because you can use proportions to solve so many things.

 

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D. History: How did people’s conception of proportions change over time?

From the beginning, people have used proportions. Early humans used proportions to see if one tribe was twice as large as another or if one leather strap is only half as long as another (Math Forum). It is obvious that the idea solving proportions hasn’t really changed that much, but what we can use proportions to solve has changed. In 2000 B.C. Babylonians used proportions to evolve place value notation by allowing arbitrarily large numbers and fractions to be represented (An Overview of Egyptian Mathematics). Around 1600 B.C. in Egypt, proportions were used to calculate the fraction and superficial measure of farmland (Mathematics History). Egyptians then used proportions to find volumes of cylinders and areas of triangles.

Vitruvius thought of proportions in terms of unit fractions for their architecture calculations (Proportion (architecture)). Also, scribes used “unit fractions” for their calculations in Egypt and Mesopotamia. Egyptians based proportions on parts of their body and their symmetrical relation to each other; like fingers, palms, hands, etc. Multiples of body proportions would be found in the arrangement of fields and buildings people lived in (Proportion (architecture)) and from here, proportions evolved. In 600 B.C., the idea of using proportions evolved and was then used for geometry (Mathematics History). Proportions are still used in geometry, like in architecture and land, like it was 3000 years ago. When you think about it, proportions have evolved, but the use of proportions has evolved even greater. There are so many topics we can now solve using proportions!

Works Cited

“Math Forum – Ask Dr. Math.” The Math Forum @ Drexel University. 7 Mar. 2012. <http://www.mathforum.org/library/drmath/view/64539.html&gt;.

“Mathematics History.” ThinkQuest : Library. 7 Mar. 2012. <http://library.thinkquest.org/22584/&gt;.

“Proportion (architecture).” Wikipedia, the free encyclopedia. 7 Mar. 2012. <http://en.wikipedia.org/wiki/Proportion_%28architecture%29&gt;.

“Review of Proportions.” Self Instructional Mathematics Tutorials. 7 Mar. 2012. <http://www.cstl.syr.edu/fipse/decunit/ratios/revprop.htm&gt;.

“An Overview of Egyptian Mathematics.”  7 Mar. 2012. < http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_mathematics.html >

 

The Monty Hall Problem

In 1990 and 1991, columnist Marilyn vos Savant (who once held the Guinness World Record for “Highest IQ”) set off a small firestorm when a reader posed the famous Monty Hall Problem to her:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

She gave the correct answer: it’s in your advantage to switch. This launched an avalanche of mail (this was the early 90’s, when e-mail wasn’t as popular) complaining that she gave the incorrect answer. Perhaps not surprisingly, none of the complainers actually tried the experiment for themselves. She explained her reasoning — in two different columns — and then offered a challenge:

And as this problem is of such intense interest, I’m willing to put my thinking to the test with a nationwide experiment. This is a call to math classes all across the country. Set up a probability trial exactly as outlined below and send me a chart of all the games along with a cover letter repeating just how you did it so we can make sure the methods are consistent.

One student plays the contestant, and another, the host. Label three paper cups #1, #2, and #3. While the contestant looks away, the host randomly hides a penny under a cup by throwing a die until a 1, 2, or 3 comes up. Next, the contestant randomly points to a cup by throwing a die the same way. Then the host purposely lifts up a losing cup from the two unchosen. Lastly, the contestant “stays” and lifts up his original cup to see if it covers the penny. Play “not switching” two hundred times and keep track of how often the contestant wins.

Then test the other strategy. Play the game the same way until the last instruction, at which point the contestant instead “switches” and lifts up the cup not chosen by anyone to see if it covers the penny. Play “switching” two hundred times, also.

You can read the whole exchange here: http://marilynvossavant.com/game-show-problem/

For much more information — and plenty of ways (some good, some not-so-good) of explaining this very counterintuitive result, just search “Monty Hall Problem” on either Google or YouTube.

Source: http://www.xkcd.com/1282/

Seismic waves

xkcdseismic_waves

Source: http://www.xkcd.com/723/

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The above comic pretty much happened when a 5.8 earthquake hit Virgina in 2011, as people up and down the East Coast received tweets about an earthquake seconds before feeling the earthquake for themselves.

It also inspires the obvious word problem for Algebra I students.

When an earthquake hits, seismic waves travel at about 5 meters per second. Suppose that Alex tweets about the earthquake 30 seconds after feeling its seismic waves. The tweet travels at about 200,000,000 meters per second. How far away does someone have to be from Alex to receive the tweet before also feeling the seismic waves?

In-class demo: The binomial distribution and the bell curve

Many years ago, the only available in-class technology at my university was the Microsoft Office suite — probably Office 95 or 98. This placed severe restrictions on what I could demonstrate in my statistics class, especially when I wanted to have an interactive demonstration of how the binomial distribution gets closer and closer to the bell curve as the number of trials increases (as long as both np and n(1-p) are also decently large).

The spreadsheet in the link below is what I developed. It shows

  • The probability histogram of the binomial distribution for n \le 150
  • The bell curve with mean \mu = np and standard deviation \sigma = \sqrt{np(1-p)}
  • Also, the minimum and maximum values on the x-axis can be adjusted. For example, if n = 100 and p = 0.01, it doesn’t make much sense to show the full histogram; it suffices to have a maximum value around 5 or so.

In class, I take about 3-5 minutes to demonstrate the following ideas with the spreadsheet:

  • If n is large and both np and n(1-p) are greater than 10, then the normal curve provides a decent approximation to the binomial distribution.
  • The probability distribution provides exact answers to probability questions, while the normal curve provides approximate answers.
  • If n is small, then the normal approximation isn’t very good.
  • If n is large but p is small, then the normal approximation isn’t very good. I’ll say in words that there is a decent approximation under this limit, namely the Poisson distribution, but (for a class in statistics) I won’t say much more than that.

Doubtlessly, there are equally good pedagogical tools for this purpose. However, at the time I was limited to Microsoft products, and it took me untold hours to figure out how to get Excel to draw the probability histogram. So I continue to use this spreadsheet in my classes to demonstrate to students this application of the Central Limit Theorem.

Excel spreadhseet: binomial.xlsx

The Scale of the Universe

My former student Matt Wolodzko tipped me off about this excellent website that shows the scale of the universe, from the very large to the very small: http://htwins.net/scale2/. I recommend it highly for engaging students with the concept of scientific notation.

While I’m on the topic, here are two videos that describe the scale of the universe. The first was a childhood favorite of mine — I vividly remember watching it at the Smithsonian National Air and Space Museum when I was a boy — while the second is more modern.

Engaging students: Adding and subtracting decimals

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 Elizabeth (Markham) Atkins. Her topic, from Pre-Algebra: adding and subtracting decimals.

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Applications

Adding and subtracting decimals is a fun subject to learn about. Decimals are everywhere in the world! Sports use decimals when timing people. Let’s try this problem: “Billy Joe ran a lap in 61.7 seconds the first time and 59.3 seconds the second time. How long did both laps take Billy Joe?” We use decimals to measure rainfall. “On Monday it rained a total of 1.27 inches, measured in a rain gauge. By Tuesday .23 inches had evaporated. Tuesday night’s big storm gave us another 3.58 inches. How much rain was in the rain gauge after Tuesday’s big storm?” We also use decimals with money! “Let’s say you found a lost cat. You return it to its owner for a reward of $50.00.Then you receive your allowance of $50.00. You then get your pay check from work which states you earned $108.75 for a week after taxes were taken out. It’s been a good week! You decide to spend a little money. You put $10.03 of gas in your car. You then by three items: Shoes ($51.99), jeans ($71.27) and gun ($0.97). How much do you have left?”

 

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Technology

 Technology is an awesome tool that we have to use to engage your students. On YouTube there is a song called the decimal song about how to add, subtract, multiply, and divide decimals. There is also a website where you can buy mathematical songs like his YouTube hit the Rappin’ Mathematician Decimals. He has a catchy way to grab student’s attention and they still learn. Technology can be used to enhance a lesson, an anchor video for example. Many website provide games. Mathgamesfun.net is a good example. Calculators are not a good enhancement tool because students can simply have the calculator do all the work for them. Calculators are a good technology to use to check a student’s work! Math.harvard.edu provides examples of math in movies. This way a student can see how math is used in the world. Learnalberta.ca/content/mesg.html/math6web/index.html?page=lessons&lesson=m6lessonshell01.swf is a website devoted to fractions. Another good technology for the teacher’s advantage is kaganonline.com. It is a website of different tools to use when teaching mathematics!

 

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Curriculum

Decimals, along with fractions, numbers, and other basics, are a key foundational mathematical stepping stone to schooling and in life. Students will use math every day of their lives. In their science classes students will use decimals in measurement, weights, and time. Also when the student learns about scientific notation, they will use decimals. Students will use decimals to answer half-life questions. Decimals are used in economy. All of economy deals with money. Money deals with decimals. When learning about the stock market they use decimals. When looking at the mileage on their car, they use decimals. Students will have to learn decimals to help with percentages, sales, interest, sales tax, loans, and any sort of measurements in everyday life. Percentages are just decimals with a fancy symbol. If the students want to save money they need to know how to add and subtract decimals. Decimals are all around us we just have to teach the students how to see and use them!