# The Pythagorean theorem to five decimal places

Piers Morgan, mathematician extraordinaire:

I don’t know how to begin describing how his attempt at insulting the intelligence of one of the Love Island evictees went horribly wrong.

# Trigonometry for the heavens

I enjoyed this article from the magazine Physics Today about the historical background behind three-dimensional spherical trigonometry: https://physicstoday.scitation.org/doi/10.1063/PT.3.3798

# Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 1

This year, I thought I’d surprise my family with matching ugly mathematical Christmas sweaters. Admittedly, I didn’t look very hard, but I couldn’t find a sweater that I liked both artistically and mathematically. However, I did stumble upon this T-shirt: https://www.amazon.com/Christmas-Math-Teacher-Tshirt-Lovers/dp/B077X14254/

I gave one to my wife and daughter, and it was a big hit.

However, I made the mistake of not inspecting the merchandise closely enough. About a minute after receiving her shirt, my daughter pointed at a digit in the sixth row of the decimal expansion and asked, “Shouldn’t this be a 5? Or maybe I’m mis-remembering.”

In that moment, I remembered that, a few years ago, she had memorized the first few dozen digits of $\pi$ for her elementary school’s talent show. Somehow, she had retained that bit of trivia all these years later. I didn’t miss the irony: I did not remember that she could remember the first few dozen digits of $\pi$.

As I’ve learned not to daughter my daughter’s memory, I checked two different references (https://www.piday.org/million/ and https://www.wolframalpha.com/input/?i=N%5BPi,1000%5D), and, sure enough, she was right.

The shirt correctly wrote the first 47 digits of $\pi$ after the decimal point. But things went haywire after that. Not only did was the T-shirt’s 48th digit incorrect, but it skipped a few hundred digits in the decimal expansion of $\pi$ before picking it up again! Furthermore, after completing the “tree,” a few thousand more digits were skipped before constructing the base of the tree. And these latter digits were used twice!

The first 4,000 digits of $\pi$ are shown below (in blocks of 10 digits). The ones that appear on the T-shirt are marked in boldface and are underlined.

3.
1415926535 8979323846 2643383279 5028841971 6939937
510 5820974944
5923078164 0628620899 8628034825 3421170679 8214808651 3282306647
0938446095 5058223172 5359408128 4811174502 8410270193 8521105559
6446229489 5493038196 4428810975 6659334461 2847564823 3786783165
2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360
0113305305 4882046652 1384146951 9415116094 3305727036 5759591953
0921861173 8193261179 3105118548 0744623799 6274956735 1885752724
8912279381 8301194912 9833673362 4406566430 8602139494 6395224737
1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901
2249534301 4654958537 1050792279 6892589235 4201995611 2129021960
8640344181 5981362977 4771309960 5187072113 4999999837 2978049951
0597317328 1609631859 5024459455 3469083026 4252230825 3344685035
2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532
1712268066 1300192787 6611195909 2164201989 3809525720 1065485863
2788659361 5338182796 8230301952 0353018529 6899577362 2599413891
2497217752 8347913151 5574857242 4541506959 5082953311 6861727855
8890750983 8175463746 4939319255 0604009277 0167113900 9848824012
8583616035 6370766010 4710181942 9555961989 4676783744 9448255379
7747268471 0404753464 6208046684 2590694912 9331367702 8989152104
7521620569 6602405803 8150193511 2533824300 3558764024 7496473263
9141992726 0426992279 6782354781 6360093417 2164121992 4586315030
2861829745 5570674983 8505494588 5869269956 9092721079 7509302955
3211653449 8720275596 0236480665 4991198818 3479775356 6369807426
5425278625 5181841757 4672890977 7727938000 8164706001 6145249192
1732172147 7235014144 1973568548 1613611573 5255213347 5741849468
4385233239 0739414333 4547762416 8625189835 6948556209 9219222184
2725502542 5688767179 0494601653 4668049886 2723279178 6085784383
8279679766 8145410095 3883786360 9506800642 2512520511 7392984896
0841284886 2694560424 1965285022 2106611863 0674427862 2039194945
0471237137 8696095636 4371917287 4677646575 7396241389 0865832645
9958133904 7802759009 9465764078 9512694683 9835259570 9825822620
5224894077 2671947826 8482601476 9909026401 3639443745 5305068203
4962524517 4939965143 1429809190 6592509372 2169646151 5709858387
4105978859 5977297549 8930161753 9284681382 6868386894 2774155991
8559252459 5395943104 9972524680 8459872736 4469584865 3836736222
6260991246 0805124388 4390451244 1365497627 8079771569 1435997700
1296160894 4169486855 5848406353 4220722258 2848864815 8456028506
0168427394 5226746767 8895252138 5225499546 6672782398 6456596116
3548862305 7745649803 5593634568 1743241125 1507606947 9451096596
0940252288 7971089314 5669136867 2287489405 6010150330 8617928680
9208747609 1782493858 9009714909 6759852613 6554978189 3129784821
6829989487 2265880485 7564014270 4775551323 7964145152 3746234364
5428584447 9526586782 1051141354 7357395231 1342716610 2135969536
2314429524 8493718711 0145765403 5902799344 0374200731 0578539062
1983874478 0847848968 3321445713 8687519435 0643021845 3191048481
0053706146 8067491927 8191197939 9520614196 6342875444 0643745123
7181921799 9839101591 9561814675 1426912397 4894090718 6494231961
5679452080 9514655022 5231603881 9301420937 6213785595 6638937787
0830390697 9207734672 2182562599 6615014215 0306803844 7734549202
6054146659 2520149744 2850732518 6660021324 3408819071 0486331734
6496514539 0579626856 1005508106 6587969981 6357473638 4052571459
1028970641 4011097120 6280439039 7595156771 5770042033 7869936007
2305587631 7635942187 3125147120 5329281918 2618612586 7321579198
4148488291 6447060957 5270695722 0917567116 7229109816 9091528017
3506712748 5832228718 3520935396 5725121083 5791513698 8209144421
0067510334 6711031412 6711136990 8658516398 3150197016 5151168517
1437657618 3515565088 4909989859 9823873455 2833163550 7647918535
8932261854 8963213293 3089857064 2046752590 7091548141 6549859461
6371802709 8199430992 4488957571 2828905923 2332609729 9712084433
5732654893 8239119325 9746366730 5836041428 1388303203 8249037589
8524374417 0291327656 1809377344 4030707469 2112019130 2033038019
7621101100 4492932151 60
84244485 9637669838 9522868478 3123552658
2131449576 8572624334 4189303968 6426243410 7732269780 2807318915
4411010446 8232527162 0105265227 2111660396…

I can understand getting a digit or two wrong on the T-shirt, but I have no idea how anybody could have possibly made a mistake like this.

Upon discovering this, my first reaction reflected my inner mathematician: “I want a refund.” After all, $\pi$ has been known to 47 decimal places since the 1700s, long before the advent of modern computers. However, upon further reflection, I decided that being able to tell this story of a Christmas $\pi$ T-shirt that incorrectly printed the digits of $\pi$ — and especially the story of how this error was brought to my attention — was by itself well worth the price of the shirt.

# Engaging students: Defining sine, cosine and tangent in 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 comes from my former student Jessica Williams. Her topic, from Precalculus: defining sine, cosine and tangent in a right triangle.

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

I know of a good project/activity for the students to do that will be extremely engaging. You could either do this for an elaborate activity for your students or maybe an opening activity for day 2 of a lesson. For my class, I would get a square cookie cake, and have the slices cut into right triangles. I would allow each student to have a piece (but not eat it just yet). The students will be provided with rulers and a protractor. The students will each measure the hypotenuse of their cookie cake and the degree of whichever angle you would like them to measure, however each student should be measuring the same parts so do this unanimously). As a class, decide on an average for the measurements for everyone to use so that the data is not off. Then take the supplies away from the students and ask the students to find the rest of the missing sides and angles of their piece of cookie cake. They will also be provided with a worksheet to go along with this activity. This is a good review activity or al elaborate activity to allow further practice of real world application of right triangle trigonometry. Then go over as a class step by step how they solved for their missing angles and side lengths and make each group be accountable for sharing one of them. This allows the students to all be actively participating. Through out the lesson, make sure to tell the kids as long as they are all participating they will get to eat their slice when the lesson is done. Lastly, allow the students to eat their slice of cookie cake.

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

Prior to learning about right triangle trigonometry the students will know how to use the Phythagorean Theorem to find how long the missing side length is of a right triangle. The students know basic triangle information such as, the sum of the angles in a triangle is 180 degrees. The students already know the difference between the hypotenuse and the other two legs. The students know that hypotenuse will be the longest leg and the leg across from the 90 degree angle. The students will also know the meaning of a fraction or ratio. The students may need some refreshing of memory on some parts of prior knowledge, but as teachers we know this is an extremely important part of a lesson plan. Even as teacher we tend to forget things and require a jog of memory. A simple activity such as headbands or a kahoot with vocabulary would be an excellent idea for accessing the students prior knowledge. This allows the students to formally assess themselves and where they stand with the knowledge. Also, it allows the teacher to formally assess the students and see what they remember or parts they are struggling on. This allows the teacher to know what things to spend more time on.

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

Technology is always an amazing aspect of the classroom. Like stated above, a vocab review using headbands or kahoot would be a good idea for this type of lesson that DEFINITELY needs prior knowledge to be applied in order to succeed. Also, showing the students how to plug in sine, cosine, and tangent is crucial. They have seen these buttons on the calculator but they do not know what they mean or how to use them. Using an online TI on display for the class is great. I had to do this with my 10th grade students to make sure they understood how to use the 3 buttons. Also, when using arcsin, arccos, and arctan it can be confusing. Using technology to show the class as a whole is the best route to go. Also, technology can used as review for a homework assignment or even extra credit for the students. It benefits them by getting extra review and extra credit points. I found a website called http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/object_interactives/trigonometry/use_it.html , which is a golf game that requires review of triangles and trigonometry. It allows the students to practice the ratios of SOH-CAH-TOA using a given triangle.

# 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.

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

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

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: Using a truth table

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 Saundra Francis. Her topic, from Geometry: using a truth table.

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

An activity that can be done with truth tables is notice and wonder. You could give students an example of a truth table, say finding the results of p and q, and have them write down what they notice and what they wonder about the table. This can lead to a great discussion about what they notice about the truth table and how they think it works. Some students might be able to figure out what the table represents or it could help display misconceptions before the lesson starts. Students can then present their wonders, which you can make sure to answers by the end of the lesson. This activity is a good way to discover if students have prior knowledge of the concept and you can see what they hope to have learned by the end of the lesson.

How has this topic appeared in pop culture?

In this Alice in Wonderland clip where Alice attends the Mad Hatter’s tea party we notice the importance of truth tables and logic to distinguish what the Mad Hatter, March Hare, and the Dormouse are discussing. Alice is using logic to discover what the Mad Hatter means and argue with him about his logic. Once you have students watch this video you can ask students to share examples of where logic is displayed. You can choose one example to transfer into a truth table later in the lesson to show students that we can use the logic displayed outside of the classroom. This will engage students and display the importance of using logic to distinguish what others mean.

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

Bertrand Russell, one of the contributors that made truth tables as we use them today, worked within the philosophical branch that combined mathematics and logic. Russell was born in Trelleck, United Kingdom in 1872. He attended Trinity College in Cambridge achieved a First Class distinction in philosophy. In 1903 Russell wrote The Principles of Mathematics with Dr. Allan Whitehead. This book extended the work of Peano and Frege on mathematical logic. This was the book where he contributed to truth tables. In 1918 Russell was imprisoned for six months due to a pacifist article he wrote. While incarcerated he wrote Introduction to Mathematical Philosophy. Russell received the Nobel Prize in literature in 1950 in recognition of his thoughts on freedom of thought and humanitarian ideals. He is known for his humorous and controversial writing.

# Engaging students: Writing if-then statements in conditional form

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 Sarah McCall. Her topic, from Geometry: writing if-then statements in conditional form.

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

For this exercise, I would like to kill two bird with one stone (engage students as well as deepen their understanding) by choosing word problems that are funny and have to do with topics that students are familiar with. For this activity, students could work individually or in groups to construct if then statements out of two given clauses, and then construct two if then statements out of their own clauses: one true and one false. This activity would encourage students to be as funny and outrageous as possible. For example:

1. Construct a logical if then statement using the following:
I do not take a nap today.
I will cry about my homework.
2. Construct a logical if then statement using the following:
Miss. McCall’s dog is the cutest dog on the planet. Pigs can fly.
3. Construct a false if then statement using any two clauses you can think of.
4. Construct a true if then statement using any two clauses you can think of.

Questions 3 and 4 are important because they engage students by requiring them to use their creativity and humor, as well as their logical skills. Students should ask themselves, if I want my entire statement to be false, should my “if” and “then” clauses be true or false? This could also be used as a class project where students present their creations to the class afterwards, which would further facilitate student understanding and involvement.

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

To get students engaged in this material, I would like to switch it up from the usual mathematics lesson by introducing students to a whole new side of mathematics: logic! For this activity, students would work in groups to solve the following riddle (citations listed below):

Somewhere on an island far far away, there are two large families quite different from you and I. The people in one of these families, the Truthtellers, tell the truth all the time, even when they’d rather lie. They can’t say something false even by mistake! And the other family, the Liars, always lies, even when they wish they could tell truth. Everything they say is a lie. Now suppose you meet two people on this island: Ashley and Amanda, and Ashley tells you “We are both liars”. What family is Ashley from? What about Amanda?

The goal of this exercise is to get students to use their logical intuition and eventually conclude that Ashley must be a Liar, because if she is a Truthteller, then she cannot say she is a Liar. Hopefully students will be introduced to how to communicate their ideas in a simple if then format and realize that logic is fairly intuitive and natural.

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

Many students often worry (or grumble) that they may never use the topics they are learning in geometry/math in the real world. However, as more and more students are considering jobs in STEM fields, I think it is important to acknowledge that these if then statements are not going away! Not only are if then statements used to develop hypotheses for science experiments, they will also be used in upper level math courses like calculus. Learning how to apply if then logic to everyday situations as well as theorems is a skill that will be helpful to all students who plan to go on to any STEM, medical, scientific or research fields. I believe starting off a lesson on if then logic with these reminders would be helpful in getting students to be engaged in their own learning, because they will see that their learning now will affect them later on.

# Engaging students: Defining sine, cosine and tangent in 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 comes from my former student Jessica Williams. Her topic, from Geometry: defining sine, cosine and tangent in a right triangle.

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

I’ve actually had the opportunity to teach this lesson to my 10th graders last semester. It is a difficult concept for the students to understand, however if you teach it in a way the students are actively engaged, it helps extremely. Prior to this lesson, the students knew about the hypotenuse and knew the other sides lengths as “legs.” We started by calling 3 students up to the front to hold up our three triangle posters. (triangle cut outs with the 90 degree angle showing and then there was an agle missing). We asked the students how we could find a missing angle given only one side length. For starters, I demonstrated on one triangle by placing a spray water bottle at the missing angle given, and spray the water across.

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

Trigonometry was originally developed for the use of sailing as a navigation method. The origins can be traced back to ancient Egypt, the Indus Valley, and Mesopotamia. This was over 4000 years ago. Measuring angles in degrees, minutes, and seconds comes from the Babylonian’s base 60 system of numbers. In 150 B.C.E, Hipparchus made a trigonometric table using sine to solve triangles. Later on, Ptolemy extended the trig calculations in 100 C.E. Also, in interesting fact is the ancient Sinhalese used trig to calculate for water flow. Persian mathematician Abul Wafa introduced the angle addition identities. As you can see, there are MANY different mathematicians who distributed to the topic of trigonometry. A lot of them built upon previous work and discovered new formulas, identities, etc. It’s amazing to see how even trigonometry is used to every day life. You always hear people say, “When will I ever use this is life?” and it bugs me to hear this. However, I always have examples of how math is used in our everyday world and from a past long ago that advanced us to where we are today.

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

I would show this video that I found on youtube. I would exclude the first movie example that involves shooting, however the rest are great examples.

Showing movie clips to students is always a great way to grab their attention. Visually showing them that math is a part of movies, and every day life shows them that it is important. This video would also be great to use as practice problem, but blur out one of the side lengths or angles missing. You could play the movie scene then pause it on the part with the triangle and have the students solve for missing angle or side length. It would be a fun activity for the students and involve great practice. You could even make this a homework assignment. It’s engaging to watch and keeps the student’s attention while doing homework. The video shows that math is involved in dancing, buildings, etc. This activity also can excite students to try to find math in the movies or tv shows that the watch. You could assign the students to pay attention to to the next couple of shows or movies the watch and to bring back to class an example or two of how math was incorporated in it. Mathematics goes unnoticed because it is honestly part of our everyday norm/lifestyle.

http://www.newworldencyclopedia.org/entry/Trigonometry

# Engaging students: Deriving the distance formula

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 Peter Buhler. His topic, from Geometry: deriving the distance formula.

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

Although the distance formula may be introduced as part of the Geometry curriculum, it also has applications in Algebra and even Pre-calculus. This allows for many possible applications, as it can be used in various ways. One project that students could be assigned to is by modeling something in real life on a coordinate grid, and using the distance formula to calculate various distances within that real life object or place. An example of this could be to take a baseball diamond and use the fact that the bases are 90 feet apart, and calculate the distance between the corners on opposite sides. Another example could be to overlay a map of their town onto a coordinate grid and measure the distance between places that they usually visit. These students can fact check the distances by plugging them in to Google Maps. One aspect of this project to be careful of is to make sure that students are using the distance formula, and not the Pythagorean Theorem. Allowing the students to present their findings could spark curiosity into how mathematics is used in everyday life by city planners, architects, engineers, and in other careers.

How has this topic appeared in high culture?

The following piece of artwork was created by Mel Bochner and titled, Meditation on the Theorem of Pythagoras.

While immediately this picture appears to be related to the proof of the Pythagorean Theorem, There are also applications to the distance formula. This artwork could be a great engaging activity for students as they come into class, simply by reflecting on what can be seen. A challenging question would be to ask students to guess how many hazelnuts they think the artist used to create this artwork (without counting each piece). It should be noted that each corner of the triangle consists of two corners of the squares, so the answer is not simply 9+16+25, but you must subtract off how many are shared.
We can apply this to the distance formula by asking students how to relate the Pythagorean Theorem with the distance formula. Having students compare and contrast these two mathematical equations could provide excellent discussion. As an instructor, you can also overlay this artwork onto a coordinate grid and have students use the distance formula to calculate the various side lengths and confirm that it works.

The three mathematics who are primarily responsible for what we know as the distance formula are: Euclid, Pythagoras, and Descartes. Euclid stated in his third Axiom that “it is possible to construct a circle with any point as its center and with a radius of any length”. This matters because the distance formula is a corollary of the circle formula. Pythagoras then took this idea, and proceeded to invent the Pythagorean Theorem, which can be easily converted to the distance formula. Later on, Descartes applied this to the coordinate system, in an event consisting of the union of algebra and geometry.
While this material may seem fairly dry to middle school or high school students who are first learning the Pythagorean Theorem, there are certainly some applications that can make the history more appealing. One such application is to ask the students to connect the formula of a circle with the distance formula, and discuss how they are related. This would provide excellent discussion about how Euclid and Pythagoras may have begun their study of the distance formula. Another application could include assigning students to study one of these three mathematicians, and having them provide several interesting facts about the person they chose to study. Consequently, when introducing the distance formula, students will be familiar with those who had a huge impact on the development of the distance formula.

# Engaging students: Radius, Diameter, and Circumference 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 Michelle Contreras. Her topic, from Geometry: radius, diameter, and circumference of a circle.

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

A way to relate circumference, radius, and diameter to my student’s real life would be by incorporating an end of unit project where they find 5 circular things that are cool to them in their home. I will ask the students to measure those 5 circular objects and find the radius, diameter, and circumference: rounding to the tens place. The students will also be asked to divided the found circumference by the diameter for every object and estimate the number they find in the hundredths place. The students should keep getting the same estimated number and realize how the estimated number for π was discovered. The students are to label the 5 objects with their radius, diameter, circumference, and to present all their finding to the class including the special number they found when dividing the circumference by the diameter. I would also participate with my students in this project by finding 5 objects around my house and present it to the class as well. There is so much the students can gain by this project, not just mathematically. Students will get an opportunity to show their classmates a little bit about themselves as well as gaining confidence in their perception about their knowledge of this topic.

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

Way before William Jones, observations were made regarding the circumference and diameter of a circle. The human race became curious about the circle and made some discoveries; the people saw a relationship between π (pi), the circumference and the radius. The people observed that every time you tried to see how many times the diameter goes into the circumference a similar number was computed. There was talk about the special number being around 22/7 or 355/113 making it seem that the special number was a rational but Jones believed it was an irrational number. Not only Jones but many others before him saw that this special number approached but never quite reached a specific number because it kept going. William Jones introduced the symbol known today for this special number: π in 1706. Though there is a belief by many that Leonhard Euler was the first to introduce and talk about the symbol π, Jones however, published his second book Synopsis Palmanorum Matheseos in 1706 using the symbol π. William Jones was a self-taught mathematician that was born in 1675 that only had a “local charity school education”. Interesting enough Jones was served for the navy before becoming a math teacher. William Jones would charge a fee to those who come to a coffee shop and listen to his lectures in London. Based on the website historytoday.com William Oughtred “used π to represent the circumference of a given circle, so that his π varied according to the circle’s diameter, rather than representing the constant we know today.” The symbol pi is an important irrational number that connects the circumference to the radius and diameter of a circle. There has been many mathematicians who have contributed in some way to this symbol pi regarding circumference.

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

YouTube is a great tool to use as a classroom teacher, there are many educational videos that can be beneficial to the student’s education. There are videos that include examples and visuals that your students may or may not relate to. Particularly for the topic regarding circumference and its different features radius and diameter there were many intriguing videos that I came across with. There was one video that I felt I would totally use in one of my lesson about circumference. The video is called “Math Antics: Circles, What is Pi?” I would only show about 3 minutes of the video as an engage at the beginning of the lesson, I really liked however how they explained the definition of a circle with visuals. I believe this video will be very beneficial for my students before starting the unit over the circumference, it also does a very good job at capturing the attention of the audience and explaining pi. I have always believed YouTube to be a great tool for educational purposes but there is a website called mathisfun.com which is my go to for a better explanation or summary of certain concepts. This website gives you really good real world examples that anyone can relate to and great ideas for short engaging activities. The definitions are simplified so any middle school student can understand a concept. The website not only have great examples that you can talk about in your classroom as an engage but also have easy to follow explanations. I would definitely use this website when having trouble explaining, in a simpler form, a certain topic to my students.

References:
“William Jones and his Circle: The Man who invented Pi”. July 2009 http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi

“Math Antics: Circles, What is Pi?” https://www.youtube.com/watch?v=cC0fZ_lkFpQ

https://www.mathsisfun.com/geometry/circle.html