Engaging students: Using Pascal’s 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 Jaeda Ransom. Her topic, from Precalculus: using Pascal’s triangle.

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

A great activity that involves Pascal’s Triangle would be the sticky note triangle activity. For this activity students will be recreating an enlarged version of Pascal’s Triangle. To complete this activity students will need a poster of Pascal’s Triangle, poster board, markers, sticky notes, classroom wall (optional), and tape (optional). The teacher’s role is to show students Pascal’s Triangle, along with an explanation of how it was made. Students will be working in pairs and grabbing the necessary materials needed to complete this activity.On the poster board the students will recreate Pascal’s Triangle. Students will write a number 1 on a sticky note and place it at the top of the posterboard, they will then write 2 number 1’s on a sticky note and place it directly under. The students will continue recreating the triangle on their poster board until they run out of space. You can also consider having students use smaller sticky notes so that students are engaged with creating more rows.

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

Pascal’s Triangle was named after French mathematician Blaise Pascal. At just the age of 16 years old Pascal wrote a significant treatise on the subject of projective geometry marking him as a child prodigy. Amongst that, Pascal also corresponded with other mathematicians on probability theory, which vastly encouraged the development of modern economics and social science. Pascal was also one of the first two inventors of the mechanical calculator when he started pioneering work on calculating machines, these were called Pascal’s calculators and later Pascalines. Pascal impressively created and invented all of this as a teenager. Though the Pascal Triangle was named after Blaise Pascal, this theory was established well before Pascal in India, Persia, China, Germany, and Italy. As a matter of fact, in China they still call it the Yang Hui’s triangle, named after Chinese mathematician Yang Hui who presented the triangle in the 13th century, though the triangle was known in China since the early 11th century.

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

This topic can be used in my students future mathematics course to introduce binomial expansions, where it is known that Pascal’s Triangle determines the coefficients that arise in binomial expansion. The coefficients aᵢ in a binomial expansion represents the number of row n in the Pascal’s Triangle. Thus, $a_i = \displaystyle {n \choose i}$ .

Another useful application of this topic is in the calculations of combinations. The equation to find the combination is also the formula to find a cell for Pascal’s Triangle. So, instead of performing the calculations using the equation a student can simply use Pascal’s Triangle. In doing this you can continue a lesson over probability or even do an activity using Pascal’s Triangle while implicating probability questions.

Resources:

https://en.wikipedia.org/wiki/Pascal%27s_triangle#Formula

https://study.com/academy/lesson/pascals-triangle-activities-games.html

Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on slightly incorrect ugly mathematical Christmas T-shirts.

Part 1: Missing digits in the expansion of $\pi$ .

Part 2: Incorrect computation of Pascal’s triangle.

Part 3: Incorrect name of Pascal’s triangle.

Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 2

This was another T-shirt that I found in my search for the perfect ugly mathematical Christmas sweater: https://www.amazon.com/Pascals-Triangle-Math-Christmas-shirt/dp/B07KJS5SM2/I love the artistry of this shirt; the “ornaments” at the corners of the hexagons and the presents under the tree are nice touches.

There’s only one small problem:

$\displaystyle {8 \choose 3} = \displaystyle {8 \choose 5} = \displaystyle \frac{8!}{3! \times 5!} = 56$ .

Oops.

Engaging students: Using Pascal’s triangle

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 Rachel Delflache. Her topic, from Precalculus: using Pascal’s triangle.

How does this topic expand what your students would have learned in previous courses?

In previous courses students have learned how to expand binomials, however after $(x+y)^3$ the process of expanding the binomial by hand can become tedious. Pascal’s triangle allows for a simpler way to expand binomials. When counting the rows, the top row is row 0, and is equal to one. This correlates to $(x+y)^0 =1$ . Similarly, row 2 is 1 2 1, correlating to $(x+y)^2 = 1x^2 + 2xy + 1y^2$ . The pattern can be used to find any binomial expansion, as long as the correct row is found. The powers in each term also follow a pattern, for example look at $(x+y)^4$ :

$1x^4y^0 + 4x^3y^1 + 6x^2y^2 + 4x^1y^3 + 1x^0y^4$

In this expansion it can be seen that in the first term of the expansion the first monomial is raised to the original power, and in each term the power of the first monomial decreases by one. Conversely, the second monomial is raised to the power of 0 in the first term of the expansion, and increases by a power of 1 for each subsequent term in the expansion until it is equal to the original power of the binomial.

Sierpinski’s Triangle is triangle that was characterized by Wacław Sieriński in 1915. Sierpinski’s triangle is a fractal of an equilateral triangle which is subdivided recursively. A fractal is a design that is geometrically constructed so that it is similar to itself at different angles. In this particular construction, the original shape is an equilateral triangle which is subdivided into four smaller triangles. Then the middle triangle is whited out. Each black triangle is then subdivided again, and the patter continues as illustrated below.

Sierpinski’s triangle can be created using Pascal’s triangle by shading in the odd numbers and leaving the even numbers white. The following video shows this creation in practice.

What are the contributions of various cultures to this topic?

The pattern of Pascal’s triangle can be seen as far back as the 11th century. In the 11th century Pascal’s triangle was studied in both Persia and China by Oman Khayyam and Jia Xian, respectively. While Xian did not study Pascal’s triangle exactly, he did study a triangular representation of coefficients. Xian’s triangle was further studied in 13th century China by Yang Hui, who made it more widely known, which is why Pascal’s triangle is commonly called the Yanghui triangle in China. Pascal’s triangle was later studies in the 17th century by Blaise Pascal, for whom it was named for. While Pascal did not discover the number patter, he did discover many new uses for the pattern which were published in his book Traité du Triangle Arithméthique. It is due to the discovery of these uses that the triangle was named for Pascal.

Reference:
• https://en.wikipedia.org/wiki/Pascal%27s_triangle
• http://mathforum.org/workshops/usi/pascal/images/fill.comb.gif
• https://www.britannica.com/biography/Blaise-Pascal#toc445406main
• https://en.wikipedia.org/wiki/Sierpinski_triangle

Pascal’s Triangle and a British game show

So this happened on the popular British game show “University Challenge” on Monday, April 2. This game show pits teams of four from various British universities and is a severe test of the breadth and depth of their knowledge of many fields, including mathematics. A contestant’s response to one math question, asking for the seventh row of Pascal’s triangle, took the UK by storm this week (start at the 26:42 mark of the video below).

Twitter immediately went ablaze. Amazingly, a write-up of this encounter made it into the Times of London, one of the world’s most venerated newspapers (as opposed to the tawdry English tabloids). The above link requires a subscription; here’s a photo of page 13 from the April 4 edition:

I must admit that I’m a little amused by the amount of press that this little encounter received. When I was a kid, I memorized the first few rows of Pascal’s triangle simply from working with it so often, so when a family member told me about this story earlier this week, I knew the answer to the question instantly. I suspect that’s exactly what the contestant did here. (Whether I could have gotten the answer right under the pressure of a quiz show and a national TV audience, on the other hand, is another matter entirely.)

I have a theory as to why this appeared to be a mighty feat of mental arithmetic. The audience may have thought that he was adding the numbers quickly, but I’m guessing that the real purpose of the introductory clause “If 1,1 is the second row of Pascal’s triangle…” is to label that row as the second row instead of the first row (following the usual convention of starting the row and column counts with 0.)

Engaging students: Using Pascal’s triangle

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 Lisa Sun. Her topic, from Precalculus: using Pascal’s triangle.

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

To introduce Pascal’s Triangle, I would create an activity where it involves coin tossing. I want to introduce them with coin tossing first before bringing in binomial expansions (or any other uses) because coin tossing are much more familiar to majority, if not all, students. Pascal’s Triangle can show you the probability of any combination of coin tossing (aka binomial distribution). Below are a few of the results and how they compare with Pascal’s Triangle:

Afterwards, I would ask the students guiding questions if they see anything interesting about the numbers that we gathered. I want them to notice that each number is the numbers directly above it added together (Ex: 1 + 2 = 3) and how those three numbers form a triangle hence, Pascal’s Triangle.

B2: How does this topic extend what your students should have learned in previous courses?

In previous courses, students should have already learned about binomial expansions. (Ex: (a+b)²= a²+ 2ab + b²). This topic extends their prior knowledge even further because Pascal’s Triangle displays the coefficients in binomial expansions. Below are a few examples in comparison with Pascal’s Triangle:

If any of the students are having difficulties expanding any of the binomials or remembering the formula, they can remember Pascal’s Triangle. Using the Pascal’s Triangle for solving binomial expansions can aid the students when it comes to being in a stressful environment (ex: taking a test). Making a connection between their prior knowledge on binomial expansion and Pascal’s Triangle, I believe it would give the students a deeper understanding as to how Pascal’s Triangle was formed.

C2: How has this topic appeared in high culture?

There’s a computer scientist, John Biles, at Rochester University in New York State who used the series of Fibonacci numbers to make a piece of music. How do the Fibonacci numbers relate to Pascal’s Triangle you ask? Well, observe the following:

As you can see, the sum of the numbers diagonally gives you the Fibonacci numbers (a series of numbers in which each number is the sum of the two preceding numbers).

John Biles composed a piece called PGA -1 which is based on a Fibonacci sequence. Note that on a piano, from middle C to a one octave C, there are a total of eight white keys (a Fibonacci number). Also, when you do a chromatic C scale which includes all the black keys, there are a total of five black keys (another Fibonacci number) which are also separated in a group of two and three black keys (see the pattern?). When you’re creating chords, let’s take the C chord for example, it consists of the notes C, E, and G. Notice that harmonizing notes are coming from the third note and the fifth note of the whole C scale. So following similar ideas on the use of these numbers/sequences, John Biles was able to compose music.

Here is John Biles full article: http://igm.rit.edu/~jabics//Fibo98/

Here is his composed song: http://igm.rit.edu/~jabics//Fibo98/PGA-1.mp3

The following may be a bit extra, but I also want to include this youtube link of this blogger who was very precise and compared the sequences to current pop music:

[I found this to be super interesting!]

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

Hundreds of years before Blaise Pascal (mathematician whom Pascal’s Triangle was named after), many mathematicians in different societies applied their knowledge of the Triangle.

Indian mathematicians used the array of numbers to represent short and long sounds in poetic meters in their chants and conversations. A Chinese mathematician, Chu Shih Chieh, used the triangle for binomial expansions. Music composers, like Mozart and Debussy, used the sequence to compose their music to guide them what notes to play that would be pleasing to the audience. In the past, arithmetic composing was frowned upon however contemporary music to this day is now filled with them. When Pascal’s work on the triangle was published, society began to apply the knowledge of the Triangle towards gambling with dice. In the end, all cultures began to use Pascal’s Triangle similarly in their daily lives.

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

The Youtube video above is a great tool for students who are visual learners. This video is to the point and clear with the message as to what Pascal’s Triangle is, the uses of it, and who aided in the discovery of it. I also believe the characters that were being used in this video would be appealing to students. This video was filled with facts that I want my students to know therefore, I would like them to follow along and write down important facts about Pascal’s Triangle. I would like to conclude that technology can be a “force multiplier” for all teachers in their classroom. Instead of having the teacher being the only source of help in a classroom, students can access web site, online tutorials, and more to assist them. What’s great is that students can access this at any time. Therefore, they can re-watch this video again once they’re home when they need a refresher or didn’t understand something the first time.

References:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#othermusic

http://www.mathsisfun.com/pascals-triangle.html

http://ualr.edu/lasmoller/pascalstriangle.html

Engaging students: Using Pascal’s triangle

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Daniel Herfeldt. His topic, from Precalculus: using Pascal’s triangle.

A great activity for Pascal’s triangle would be to first have the students find a pattern of odds and evens. The first thing that you would do is to print out blank Pascal’s triangle. You would give each student a paper for them to fill out. They would have to first fill out the triangle themselves. This would give them practice on which numbers to add as well as further see a pattern of what the next one would potentially look like. After they finish, they would have to color in all of the odd numbers a certain color, and followed by coloring all of the even ones a different color. From here, they will see that once you color it is, the even numbers will make an upside down triangle. Next to the biggest triangles, you will see smaller triangles. An example is shown below. When the students have finished, you will show them why it is like that. Then explain what the name of the colored triangle is, which is called the Sierpinski Triangle.

Pascal’s Triangle is used all over mathematics. It is mainly recognized as how to find the coefficients of binomials, as well as a lot of other uses for binomials. What students and many other people do not know, is that this triangle can be used for much more. For example, you are able to use Pascal’s triangle to find the Fibonacci sequence. Although it may be a little harder to find than the coefficients of binomials, it is still possible. If you add up the numbers in a diagonal pattern from right to left, you will be able to find the Fibonacci sequence. Below will be a picture of how this is implemented. Another way that this will help in future courses is that it allows you to find squares of a number easily. If you look at the 3^rd diagonal row, adding two consecutive numbers from left to right will give the square of a number. A picture of this will also be posted below. Another way that this is implemented in future courses is statistics and probability. This triangle can be used to find the probability of many different things. This is only a few ways that the triangle can be used in future courses, considering that there are plenty of other ways it can be used. In all, this is a very important topic for someone that is pursuing mathematics.

Fibonacci sequence:

Squares of a number:

This video would be a great way to either start a lesson on Pascal’s Triangle or to review the lesson before a test. The video shows different ways that you can implement the triangle to solve different things in mathematics. If this was the video to start the lesson, I would have each student take out a notebook and writing utensil while watching the video. Throughout the video the students would have to find at least three different ways a person may use Pascal’s triangle that they found particularly interesting. This should lead to most of the ways to be picked by at least one student. After they share their answers, explain further why these work. This could make students more intrigued with the subject. If the video was for a review of the topic, I would also have the students have out a writing utensil and a notebook. For this instance, I would have each individual write down what they had forgotten about Pascal’s triangle. From here the teacher will review the points that were most forgotten, serving as a review.

My Favorite One-Liners: Part 91

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

Everyone once in a while, a student might make a careless mistake — or just choose an incorrect course of action — that changes what was supposed to be a simple problem into an incredibly difficult problem. For example, here’s a problem that might arise in Calculus I:

Find $f'(x)$ if $f(x) = \displaystyle \int_0^x (1+t^2)^{10} \, dt$

The easy way to do this problem, requiring about 15 seconds to complete, is to use the Fundamental Theorem of Calculus. The hard way is by multiplying out $(1+t^2)^{10}$ — preferably using Pascal’s triangle — taking the integral term-by-term, and then taking the derivative of the result. Naturally, a student who doesn’t see the easy way of doing the problem might get incredibly frustrated by the laborious calculations.

So here’s the advice that I give my students to trying to discourage them from following such rabbit trails:

If you find yourself stuck on what seems to be an incredibly difficult problem, you should ask yourself, “Just how evil do I think my professor is?”

My Favorite One-Liners: Part 32

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s story is a continuation of yesterday’s post. I call today’s one-liner “Method #1… Method #2.”

Every once in a while, I want my students to figure out that there’s a clever way to do a problem that will save them a lot of time, and they need to think of it.

For example, in Algebra II, Precalculus, or Probability, I might introduce the binomial coefficients to my students, show them the formula for computing them and how they’re related to combinatorics and to Pascal’s triangle, and then ask them to compute $\displaystyle {100 \choose 3}$ . We write down

$\displaystyle {100 \choose 3} = \displaystyle \frac{100!}{3!(100-3)!} = \displaystyle \frac{100!}{3! \times 97!}$

So this fraction needs to be simplified. So I’ll dramatically announce:

Method #1: Multiply out the top and the bottom.

This produces the desired groans from my students. If possible, then I list other available but undesirable ways of solving the problem.

Method #2: Figure out the 100th row of Pascal’s triangle.

Method #3: List out all of the ways of getting 3 successes in 100 trials.

All of this gets the point across: there’s got to be an easier way to do this. So, finally, I’ll get to what I really want my students to do:

Method #4: Write $100! = 100 \times 99 \times 98 \times 97!$ , and cancel.

The point of this bit of showman’s patter is to get my students to think about what they should do next as opposed to blindly embarking in a laborious calculation.

As another example, consider the following problem from Algebra II/Precalculus: “Show that $x-1$ is a factor of $f(x)=x^{78} - 4 x^{37} + 2 x^{15} + 1$ .”

As I’m writing down the problem on the board, someone will usually call out nervously, “Are you sure you mean $x^{78}$ ?” Yes, I’m sure.

“So,” I announce, “how are we going to solve the problem?”

Method #1: Use synthetic division.

Then I’ll make a point of what it would take to write down the procedure of synthetic division for this polynomial of degree 78.

Method #2: (As my students anticipate the real way of doing the problem) Use long division.

Understanding laughter ensures. Eventually, I tell my students — or, sometimes, my students will tell me:

Method #3: Calculate $f(1)$ .

Lessons from teaching gifted elementary school students: Index (updated)

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on various lessons I’ve learned while trying to answer the questions posed by gifted elementary school students. (This is updated from my previous index.)

Part 1: A surprising pattern in some consecutive perfect squares.

Part 2: Calculating 2 to a very large exponent.

Part 3a: Calculating 2 to an even larger exponent.

Part 3b: An analysis of just how large this number actually is.

Part 4a: The chance of winning at BINGO in only four turns.

Part 4b: Pedagogical thoughts on one step of the calculation.

Part 4c: A complicated follow-up question.

Part 5a: Exponentiation is multiplication as multiplication is to addition. So, multiplication is to addition as addition is to what? (I offered the answer of incrementation, but it was rejected: addition requires two inputs, while incrementation only requires one.)

Part 5b: Why there is no binary operation that completes the above analogy.

Part 5c: Knuth’s up-arrow notation for writing very big numbers.

Part 5d: Graham’s number, reputed to be the largest number ever to appear in a mathematical proof.

Part 6a: Calculating $(255/256)^x$.

Part 6b: Solving $(255/256)^x = 1/2$ without a calculator.

Part 7a: Estimating the size of a 1000-pound hailstone.

Part 7b: Estimating the size a 1000-pound hailstone.

Part 8a: Statement of an usually triangle summing problem.

Part 8b: Solution using binomial coefficients.

Part 8c: Rearranging the series.

Part 8d: Reindexing to further rearrange the series.

Part 8e: Rewriting using binomial coefficients again.

Part 8f: Finally obtaining the numerical answer.

Part 8g: Extracting the square root of the answer by hand.