# 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 Chris Brown. His topic, from Geometry: using a truth table.

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

Truth tables apply directly to the field in Computer Science, as in its essence, it runs on Boolean logic. Boolean logic simply means that everything has a result of True or False. This can be seen explicitly when dealing with logic gates, which are different paths that a computer program follows as it tests whether inputs are true or false based on given conditions. Based on the results, the program will continue to run, testing different cases, based on each result in a complex chain of tests. For example, for a simple program, let’s say you may input any integer, n, between 10 and 20 inclusive. If the number is divisible by 2, then it will compute n divided by 2. If the number is not divisible by 2, then it will return the original number. Then, if the resulting number is divisible by 2 as well, it will once again compute n divided by 2. If the resulting number is not divisible by 2, then it will return the resulting number. This sequence of tests follows the conditional statement, “If an integer between 10 and 20 inclusive is divisible by 2, and it’s resulting value is also divisible by 2, then the chosen integer has 22 within its prime factorization.” For the “and” truth table: if the integer chosen was 10, we see the True & False = False case; if the integer 16 was chosen, we see the True & True = True case; if the integer 19 was chosen, we see the False & False = False case. With variations and chains of logic gates, Computer Science has every single type of truth table embedded within the Boolean logic it uses.

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

Logical humor has often been used in the more intelligent based humor of popular culture, and truth tables and arguments are even more so apart of this. In the movie “Get Smart,” released in the year 2008, features a quirky, and humorous data analyst named Maxwell Smart who by an odd turn of events was promoted to field agent. On one of Smart’s missions to infiltrate the enemy base, he, Siegfried, and Shtarker wittingly enters into a logical argument that is a beautifully crafted logical argument. I have written the lines below.

Smart: I understand that you are the man to see if someone is interested in acquiring items of a nuclear nature

Siegfried: How do I know you are not Control

Smart: Since Neither of us are dead, so I guess I am not Control

Shtarker: That actually makes sense!

While this is not an example of a truth table per say, truth tables and propositional logic was the foundation of how this argument was created. What we see in lines 3-5 is the following propositional formula:

((p → q) ∧ (p → s))

Such that:

p = Smart being Control

By viewing the truth table, we see that when q and s are false, then p must be false; as stated in Line 5 of the movie.

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

The technology tool that I found was listed on the Stanford University website and is one that the students can easily use to check over their work. The website, attached below, allows students to enter in their propositional logic formulas for any complex length and has functionality for all necessary, binary logical operators. The site also allows for the usage of many logical expressions, not just 2. Inputting the formulas is very user friendly and allows for multiple representations of each logical operator. For instance, “or” can be represented by “\/” and also “or,” and can even both be used within the same formula chain. If a character or statement is used that the system does not recognize, the system will highlight the symbol in red and say, “illegal character,” which I personally find easily understandable for all ages. What I love most about this website is that as the formula is being entered, the student is able to see the table being created as it is being entered.

http://web.stanford.edu/class/cs103/tools/truth-table-tool/

# Engaging students: Finding the volume and surface area of spheres

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 Cameron Story. His topic, from Geometry: finding the volume and surface area of spheres..

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

As a geometry teacher, manipulatives and visuals are important for conceptual understanding. Rather than handing out a formula sheet, it is far more rewarding to have your students derive volume and surface area formulas for themselves using some kind of physical representation. Not only is this more engaging for students, but the concepts behind the formula are emphasized. Yes, the volume of a sphere is $V = \frac{4}{3} \pi r^3$, but why? Where does the fraction come from? These are important questions.

An example of an activity that could be useful when teaching the volume of a sphere is best shown by Megan Millan in the following YouTube Video:

Here, students fill up hollow solids with water and find ratios between the volumes of several different shapes.

Assuming students already know the formulas for cones and cylinders, it would make it much easier to visualize those volumes with water. Through pure experimentation, students conclude that the volume inside of a cone (whose height is twice the radius) plus the volume of a sphere is equal to the total volume of a cylinder equal height and radius.

From the student’s own experimentation (and some specifically sized manipulatives), the formula is found instead of given.

How has this topic appeared in the news?

An interesting news story by the Daily Galaxy reports that Saturn’s moon Titan has a methane cycle analogous to the water cycle on Earth; Titan has methane rain, methane clouds, and methane lakes. Ligeia Mare, Titan’s second largest methane lake, “occupies roughly the same surface area as Earth’s Lake Huron and Lake Michigan together,” (The Daily Galaxy, 2018). This news story is exciting as it hits on possible life outside earth, one that may even live in these liquid-methane lakes. As a math teacher, we can follow up this story with the following visual, illustrating the size of Earth compared the size of Titan. If these lakes are the same size, what fraction of the total surface area is the lake on Earth compared to the lake on Titan?

This can lead into how surface area changes as spheres grow or shrink. It also leads to some curiosity in the student. For example, what would Texas look like on Titan?

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

The Greek mathematician Archimedes discovered many things about solids and their properties long before calculus, and this is perfect for students in geometry; they can’t use calculus yet either. Archimedes is known for many mathematical discoveries, but in particular he is famous for finding that “…the volume of a sphere with radius r is two-thirds that of the cylinder in which it is inscribed,” (Toomer, 2018). This fact leads directly to the standard formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$. Supposedly, Archimedes was proud enough of this discovery to “leave instructions for his tomb to be marked with a sphere inscribed in a cylinder,” (Toomer, 2018).

What I like about this bit of history is that your students can discover this formula on their own with some support from the teacher. The great mathematician Archimedes found the same formula and found it so important that he had it be inscribed in his final resting place, so your students will have a sense of pride knowing that they overcame the same challenge that only the best mathematicians from 2,000 years ago could tackle.

References:

YouTube video by Megan Millan – “Cylinder, Cone, and Sphere Volume” https://www.youtube.com/watch?v=RZkhnIzBC_k

Toomer, Gerald J. “Archimedes.” Encyclopedia Britannica, Encyclopedia Britannica, Inc., 28 Mar. 2018, www.britannica.com/biography/Archimedes#ref=ref383380&tocpanel=sectionId~toc214869,tocId~toc214869.

“Cassini’s Final Encounter with Saturn’s Giant Moon Titan –‘Like the Early Earth.’” The Daily Galaxy, The Daily Galaxy, 14 Sept. 2018, dailygalaxy.com/2018/09/cassinis-final-encounter-with-saturns-giant-moon-titan-like-the-early-earth/.

# Engaging students: Defining the terms prism, cylinder, cone, pyramid, and sphere

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 Alejandro Rivas. His topic, from Geometry: defining the terms prism, cylinder, cone, pyramid, and sphere.

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

I as a teacher can create a research activity or project with prisms, cylinders, cones, pyramids, and spheres. The activity would entail having the students do some research over a particular building or structure of their choice. Once the students have decided on which building or structure I will ask them to identify all of the prisms, cylinders, cones, pyramids, and spheres the building or structure contain. The students will have to count the quantity of each, figure out a way that all of the 3-dimensional figures hold the building or structure together, have a picture, and  to present to the class. After the students have presented their projects, I will then explain how prisms, cylinders, cones, pyramids, and spheres are involved in our everyday lives. I will tie it in and explain that certain professions use these 3-dimensional figures such as Engineering, Architecture, Art, Graphic Design, etc.

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

This topic extends what my students should have learned in previous courses by them being able to identify simple shapes that form prisms, cylinders, cones, pyramids, and spheres. For examples, the most common and referred prism is the rectangular prism. The prior knowledge of the shapes the students need to have are rectangles and squares. To expand my student’s knowledge from previous courses I will have them build prisms, cylinders, cones, pyramids, and spheres out construction paper. Before they cut out and form the 3-dimensional figures the students will have to identify each shape. I will split the students up into different groups. Once the groups have been formed I will let the students choose between a prism, cylinder, cone, pyramid, and sphere. Once they choose the 3-dimensional figure they will create a poster that must contain the shapes that are being used in order to form the 3-dimensional shape, and the steps the students took to get the end result.

How can technology be used to effectively engage students with the topic?

A way that technology can be used to effectively engage students with defining the terms prism, cylinder, cone, pyramid, and sphere is by playing a game of Kahoot! I would begin the class with giving the students the definitions of the different 3-dimensional figures. Once they know the definitions I will break the students off into groups of 2 or 3 depending on the class size and have them come up with a team name. The Kahoot! will have different questions pertaining to the definition of prism, cylinder, cone, pyramid, and sphere. This should be able help me, the instructor, gauge how much the students know about prisms, cylinders, cones, pyramids, and spheres. This will also give me an opportunity to help the students understand major differences between the 3-dimensional figures. This will allow me to go into detail about the bases of certain 3-dimensional figures and how that ties into the reasoning behind their specific name.

# Engaging students: Completing the square

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 Victor Acevedo. His topic, from Algebra: completing the square.

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

Completing the square is an Algebra II topic that builds on students’ prior knowledge of areas and shapes. With a given quadratic equation, students can make a visual representation of what it looks like by using Alge-blocks or Algebra tiles.  The x-squared term becomes the starting point for the model. The x term gets split in half and placed on 2 adjacent sides of the x-squared term. The next step in the process requires the student fill in what is missing of the square. Students use their knowledge of squares and packing to complete the square and make the quadratic equation easily factorable.

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

Eddie Woo is an Australian High School Math teacher that also uploads videos to YouTube. He uploads his class lectures that he thinks will help others appreciate and understand math concepts better. He made this video where he makes a visual representation and informal proof for why the “Completing the Square” method works. By using the student’s knowledge of equations and shapes he can construct the square that appears when completing the square for a quadratic equation. The moment that he puts the blocks together you can hear the amazement by his students. Many of his videos have this some feeling to them in which he explores the beauty of math and makes logical connections between what students already know and what they need to know.

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

Completing the square was a method that was discovered in order to solve quadratic equations. This method was discovered by Muhammad ibn Musa al-Khwarizmi, a Persian mathematician, astronomer, and geographer. Al-Khwarizmi, also known as the father of Algebra, wrote “The Compendious Book on Calculation by Completion and Balancing” in which he presented systematic solutions to solving linear and quadratic equations. At the time Al-Khwarizmi’s goal was to simplify any quadratic equation to be expressed with squares, roots, and numbers (ax2, bx, and c constants respectively) to one of six standard forms. The method of completing the square is a simple one to follow, but it had not been put into words formally until Al-Khwarizmi laid out the steps. In his book he progressed through solving simple linear equations and then simple quadratic equations that only required roots. This method only came up once he got to quadratic equations of the form ax2+bx+c=0 that could not be solved simply with roots. The discovery of this method leads to a simpler way of visually representing quadratic equations and applying it to parabolic functions.

References

Mastin, Luke. “Al-Khwarizmi – Islamic Mathematics – The Story of Mathematics.” Egyptian Mathematics – The Story of Mathematics, 2010, www.storyofmathematics.com/islamic_alkhwarizmi.html.

# Engaging students: Midpoint

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 Tinashe Meki. His topic, from Geometry: deriving the term midpoint.

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

During political elections, we usually hear how candidates are projected to do as the election moves forward. An important marker that usually separates likely candidates to win is the midpoint. Different new channels and news castor tend to use the phrase “midpoint of the election…”, or “midway through the election…” as ways to signify a halfway marker in time or events. The use of midpoint in news is used to describe halfway mark of time, events, distance etc. It’s a flexible word which gives its viewers a marker of how they can predict future events, time or distance. The uses of midpoint is inherently powerful because it simplifies and organizes ideas for views. For example, during time election there are so many stories being reported, different polls and various interpretation of how candidates are doing. Once the midpoint of the elections is reached, news anchors and new outlets provide the viewers with a consensus on how the election is going. That information is better received by the viewers because they can organize all the information they have received and create the own opinions for the second half of the election.

How could you as a teacher create an activity or project that involves your topic?  What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

https://mathcs.clarku.edu/~djoyce/elements/bookIII/bookIII.html

This topic allows the teacher to simultaneously teach students about mathematical history and provide an engaging activity. I think the best way introduce students to the definition of a midpoint would be to have the students find the midpoint themselves, describe what they have found in their own words then provide them with a formal definition. A way to do that would be to show students how to bisect a line using Euclidian tool (ruler and compass) as the ruler, then have the students name the point where the line is bisected. Ask students to describe that point in their own words about the line. This activity would allow the instructor to introduce students to Euclidean geometry. The cool thing about using Euclidean geometry is that it allows students to visualize geometric concepts. It would provide them concrete understating of geometric topics.

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

https://www.learner.org/courses/learningmath/geometry/session1/part_c/index.html

https://www.ics.uci.edu/~eppstein/junkyard/origami.html

https://plus.maths.org/content/power-origami

An interesting approach to define midpoint would be to use origami geometry. Much like Euclid constructions, Origami offers similar constructions and definitions for geometry terms. Origami is Japanese art form that has been around since 200.AD. “Modern mathematicians Humiaki Huzita and Koshiro Hatori devised a complete set of axioms to describe origami geometry — the Huzita–Hatori axioms.” Among these axioms, one of them defines and constructs a proof for the midpoint. Having students construct the midpoint using Huzita and Hatori would be an interesting way to not only introduce the definition of midpoint, but also provide a different approach of explaining geometric concepts.

# Engaging students: Line Graphs

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 Tinashe Meki.  His topic, from Pre-Algebra: line graphs.

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

An engaging activity to introduce line graphs is to compare the height of boys v. girls in the classroom. I would pick 6 girls and 6 boys from the class and line each group up separately from shortest to tallest in front of the same board. Then, mark their heights by placing a point above their heads. Connect the points of the height of the boys in one color and the height of the girls in another. After the activity, I would ask students probing question to analyze and compare the data in the graph: Which group had the shortest height? Which group had the tallest height? Which group’s height increased the most? and Which two points has the greatest increase in height?  Then, create a x and y axis to provoke discussion on the naming of the axes.

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

Line graphs are the foundation for many other subjects within mathematics. For example, graphing linear equations in Algebra builds concepts of connecting two or more separate points to form a line. The ability to visualize relationship between points further enhances students’ understanding of linear equations. Understanding how to interpret a line graph based on data prepares students to be able to interpret linear equations. Topics such as slope would be easier to introduce to students who understand the concept of the “change in values”. Students would be able differentiate between increasing or decreasing slope. Although a line graph’s main purpose is to compare data, subtle lessons help students understand algebraic equations also.  Students could apply this line graph to slope by plotting different points on a coordinate plane. The students can randomly connect two points and compare the relationship of the lines they have created. They could differentiate how different lines are increasing and decreasing based on their direction. They could also compare the different rates of change between the lines.

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.

I open the weather app on my phone every morning after getting ready for class. Prior to this assignment, I never noticed how the predicted weather was displayed on my phone. The app uses a line graph to show the different temperature levels during the day and week. Weather apps and websites show students how line graphs can be used for scientific purposes. An engaging activity could be to observe how line graphs are utilized to predict change in different parts of the United States. To make things more interesting, students can be assigned different cities in the U.S to search on the weather website. Once the students have analyzed the graph, they can take turns sharing interesting trends about their cities temperature line graph.

# Engaging students: Circle Graphs

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 Phuong Trinh.  Her topic, from Pre-Algebra: circle graphs.

How has this topic appeared in pop culture?

Circle graphs, or pie charts, are regularly used to visualize data and information. As technology advances, pie charts do not appear only in statistic or scientific documents anymore. They have started to show up more regularly on social media as a mean for the younger generation to express themselves. One can easily type “funny circle graphs” into Google and get back plenty of results on various.

While the students might not be familiar with the formal documents, they can easily put themselves into the situation described in Figure 1. The students can discuss what the colors from the picture represent, as well as the meanings of their proportion. From there, the students can make connection to the data and information from more formal subjects such as statistic or science.  On other hands, showing them a funny example not only will get a chuckle out of them, it can also pique their interest in the topic.

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

Circle graphs can be used in many projects and activity.  An activity that can get the students to engage in the topic is having the students create circle graphs about themselves, more specifically, how they spent their time on an average day. The students will be given a circle graph that is divided into 24 equal sectors. Each sector represents one hour. The students will use different colors to record their activities for one day (24 hours), and provide a key to show which activity is presented by each color. The proportions of each activity will be different, depends on how much time they spent for each activity. Once the graphs are completed, the students will share and explain their circle graphs with their shoulder partner. With this activity, the students will learn how to create and interpret a circle graph while sharing who they are.

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

In this day and age, most students are familiar with technology. It is a great way to engage the students into the lesson.  The National Center for Education Statistics (NCES) provides a good website for getting students to understand the relationship between data and circle graphs (Reference A). The layout of the website is fairly simple and easy to understand with 2 tabs on the left side and 5 on the right. The left tabs include “Help” tab, which provides explanation for each element that appear on the right side of the page, and “Example” tab, which provides examples of how different types of graphs look like. The tabs on the right include “Design”, “Data”, “Labels”, “Preview”, and “Print/Save”. With the pie chart design, the site allows us to adjust the data amount, or “slices”, as well as input data as needed. On other notes, under the “Labels” tab, we can choose the type of value that will be shown (For example, value or % of total). As they explore the site, the students can compare their data with the graphs in order to make connection to how the arc length of each slice is proportional to the data it represents.

References:

# 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.