# 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 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 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: Computing logarithms with base 10

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 Katelyn Kutch. Her topic, from Precalculus: computing logarithms with base 10.

How has this topic appeared in the news?

http://www.seeitmarket.com/the-log-blog-trading-with-music-and-logarithmic-scale-investing-14879/ . This website gives an insight into logarithms that many students would not know and I think that what is has to say is quite interesting. While this may not be a news article, it includes many methods in which logarithms can and are being used in the world. It also gives some insight into the history of logarithms. I feel like showing the students this website would get them interested in logarithms because they can see what logarithms can do, like tell us the magnitude of an earthquake on the Richter Scale. Students may not find logarithms interesting, but I feel like most would find this interesting.

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

http://mathequalslove.blogspot.com/2014/01/introducing-logarithms-with-foldables.html . This website gives multiples games that teachers can do with logarithms, not just base 10, but for all logarithms. The teacher had foldables that the students put their notes in for logarithms and personally, as a kinesthetic learner, that is something that I loved when teachers did it. It helped me visually put down the notes and it was something that I could keep to refer to. The teacher also had Log War, Log Bingo, and Log Speed Dating. Students always respond better when a sense of fun is involved in the lesson and this teacher proved that when one of her students asked about another game involving the subject. The games are ones that students interact with the teacher, with each other, and it enhances their own thinking because they are having to do calculations, correctly, in order to win the game. This seems like a wonderful website to pull from when wanting to do something fun with a lesson.

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

In 1614 a Scottish mathematician by the name of John Napier published his discovery for logarithms. Napier worked with an English mathematician by the name of Henry Briggs. The two of them adjusted Napier’s original logarithm to the form that we use today. After Napier passed away, Briggs continued their work alone and published, in 1624, a table of logarithms that calculated 14 decimal places for numbers between 1 and 20,000, and numbers between 90,000 and 100,000. In 1628 Adriaan Vlacq, a Dutch publisher, published a 10 decimal place table for values between 1 and 100,000, which included the values for 70,000 that were not previously published. Both men worked on setting up log trigonometric tables. Later, the notation Log(y) was adopted in 1675, by Leibniz, and soon after he connected Log(y) to the integral of dy/y.

# Engaging students: Solving Equations with Rational Functions

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 Heidee Nicoll. Her topic, from Precalculus: solving equations with rational functions.

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

To jog the students’ memory of what rational functions look like and what some of their properties are, I would do a relay race with them.  The class would be divided into two groups, and each group would have a different rational function, not anything too difficult, but something for which they could easily compute values, something like f(x)=-2/x and g(x)=3/x.  On the board would be two large papers, each with a table of values and a blank graph.  The x-values would be filled in, but the y-values would be blank.  The students would line up, and the first student in each line has to compute the y-value for the first given x-value, then grab the one marker for his/her team, go up to the board and write that value in the table.  The next student will compute the next value, and so on.  The students would be able to use the calculators on their phones if necessary, but they would not be able to use graphing calculators since they would be able to just plug the function in and look at the table.  Once the teams had all the y-values written down, the next student would have to come up to the board and plot the first point on the graph, and so on, until all the points were plotted.  The very last student would connect the dots to make a curve.  Then we could have a class discussion about vertical asymptotes, and how they show up in the table as an error or undefined value.  We could talk about what they remember of end behavior, horizontal asymptotes, x- and y-intercepts, and that could lead into the rest of the lesson.

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

Desmos online graphing calculator is quite nifty.  The functions show up in different colors, and you can graph points as well as lines and curves.  I found a sort of online worksheet on Desmos talking about rational functions, and modified it.  This is the link to the modified version: https://www.desmos.com/calculator/zi62lrxnim It leads the student step by step, as they click on each function to see it on the graph, through looking at the vertical asymptotes, x- and y-intercepts, any holes or slant asymptotes, and at the very end gets them thinking about intersections and solving equations.  The purpose would be to remind the students of all the properties of rational functions that we should think about when solving, and how graphing the functions to get a solution is a viable option.  In the activity, the students are also asked to move a few slides to graph the correct asymptotes.  In this way they are not just taking in information, but are required to provide some answers of their own.  All of this information should be already learned, so it would just be a review for the students as they take what they already know and learn how to apply it to solving equations with rational functions.

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

This is a paragraph from Encyclopedia Britannica about Apollonius of Perga and his contributions to geometry.

Greek geometry entered its golden age in the 3rd century. This was a period rich with geometric discoveries, particularly in the solution of problems by analysis and other methods, and was dominated by the achievements of two figures: Archimedes of Syracuse(early 3rd century bc) and Apollonius of Perga (late 3rd century bc). Apollonius is best known for his Conics, a treatise in eight books (Books I–IV survive in Greek, V–VII in a medieval Arabic translation; Book VIII is lost). Apollonius presented a comprehensive survey of the properties of these [parabolas, hyperbolas, and ellipses]. A sample of the topics he covered includes the following: the relations satisfied by the diameters and tangents of conics (Book I); how hyperbolas are related to their “asymptotes,” the lines they approach without ever meeting (Book II); how to draw tangents to given conics (Book II); relations of chords intersecting in conics (Book III); the determination of the number of ways in which conics may intersect (Book IV); how to draw “normal” lines to conics (that is, lines meeting them at right angles; Book V); and the congruence and similarity of conics (Book VI).  (Knorr).

We would read it as a class and I would point out that a hyperbola is the parent function for rational functions, y=1/x, and that when we are talking about asymptotes, we are using information that Apollonius worked on and studied.

# Works Cited

Biographical Dictionary. n.d. Image. 18 November 2016.

Knorr, Wilbur R. Encyclopedia Britannica: Greek Mathematics. n.d. Website. 18 November 2016.

Original Desmos Activity: https://www.desmos.com/calculator/3azkdx4llk

Modified Desmos Activity: https://www.desmos.com/calculator/zi62lrxnim

# 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 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 3rd 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.

# Engaging students: Finding the domain and range of a function

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 Brittany Tripp. Her topic, from Precalculus: finding the domain and range of a function.

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

One of my favorite games growing up was Memory. For those who haven’t played, the objective of the game is to find matching cards, but the cards are face down so you take turns flipping over two cards and have to remember where the cards are so when you find the match you can flip both of the matching cards. To win the game you have to have the most matches. I think creating an activity like this, that involves finding domain and range, would be a really fun way to get students’ engaged and excited about the topic. You could place the students in pairs or small groups and give each student a worksheet that has a mixture of functions and graphs of functions. Then the cards that are laying face down would contain various different domains and ranges. In order to get a match you have to find the card that has the correct domain and the card that has the correct range for whatever function or graph you are looking at. You could increase the level of difficulty by having functions, graphs, domains, and ranges on both the worksheet and the cards. This would require the students to not only be able to look at a graph of a function or a function and find the domain and range, but also look at a domain and range and be able to identify the function or graph that fits for that domain and range.

These pictures provide an example of something similar that you could do. I would probably adjust this a little bit so that the domain and ranges aren’t always together and provide actual equations of functions that the students’ must work with as well.

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

Finding the domain and range of a function is used and expanded on in a variety of ways after precalculus. For instance, one way the domain and range is used in calculus is when evaluating limits. An example is the limit of x-1 as x goes to 1 is equal to zero, because when looking at the graph when the domain, x, is equal to 1 the range, y, is equal to zero. Finding domain and range is something that is applied to a variety of different type of functions in later courses, like when looking at trigonometric functions and the graphs of trigonometric functions. You look at what happens to the domain of a function when you take the derivative in calculus and later courses. You work with the domain and range of different equations and graphs in Multivariable calculus when you are switching to different types of coordinates such as polar, rectangular, and spherical. There are also multiple different science courses that use this topic in some way, one of those being physics. Physics involves a lot of math topics discussed above.

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

I found a website called Larson Precalculus that technically is targeted toward specific Precalculus books, but exploring this website a little bit I found that is would be a super beneficial tool to use in a classroom. This website has a variety of different tools and resources that students could use. It has book solutions which if you weren’t actually using that specific textbook could be a really helpful tool for students. This would provide them with problems and solutions that are not exactly the same to what they are doing, but similar enough that they could use them as examples to learn from. This website also includes instructional videos that explain in depth how to tackle different Precalculus topics including finding domain and range. There are interactive exercises which would give the students ample opportunities to practice finding the domain and range of graphs and functions. There are data downloads that give the students to ability to download real data in a spreadsheet that they can use to solve problems. These are only a few of the different resources this website provides to students. There are also chapter projects, pre and post tests, math graphs, and additional lessons. All of these things could be used to engage students and help advance and deepen their understanding of finding domain and range. The only downfall is that it is not a free resource. It is something that would have to be purchased if you chose to use it for your classes.

References:

http://esbailey.cuipblogs.net/files/2015/09/Domain-Range-Matching.pdf

http://17calculus.com/precalculus/domain-range/

http://www.larsonprecalculus.com/pcwl3e/

# Engaging students: Solving logarithmic equations

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 Anna Park. Her topic: how to engage Algebra II or Precalculus students when solving logarithmic equations.

Application:

The students will each be given a card with a) a logarithmic equation solution and b) a new logarithmic equation. The student that has a number one on the back of their card will begin the game. The student will stand up and tell the rest of the class what they have for b) the Log equation they have, then the student with the corresponding card will read their solution a) to the first students problem. If that student is correct they will read part b) the new log equation. Then another student that has the logarithmic solution will stand up and say their solution a) and then read their new log equation b). This will continue until the last student stands with their new equation and it loops back to student number one’s solution. This will end the game. This game requires students to solve logarithmic equations and recognize how to rewrite a logarithmic equation. There will be an appropriate amount of time before the game begins so the students can work backwards to find their logarithmic equation that matches their solution.

History:

John Napier was the mathematician that introduced logarithms. The way he came up with logarithms is very fascinating, especially how long it took him to develop the logarithm table. He first published his work on logarithms in 1614. He published the findings under “A Description of the Wonderful Table of Logarithms.” He named them logarithms after two Greek words; logos, meaning proportion, and arithmos, meaning number. His discovery was based off of his imagination of two particles traveling along two parallel lines. One line had infinite length and the other had a finite length. He imagined both particles starting at the same horizontal positions with the same velocity. The first line’s velocity was proportional to the distance, which meant that the particle was covering equal distance in equal time. Whereas the second particle’s velocity was proportional with the distance remaining. His findings were that the distance not covered by the second line was the sine and the distance of the first line was the logarithm of the sine. This showed that the sines decreased and the logarithms increased. This also resulted in the sines decreasing in geometric proportion and the logarithms increasing in arithmetic proportion. He made his logarithm tables by taking increments of arc (theta) every minute, listing the sine of each minute by arc, and the corresponding logarithm. Completing his tables, Napier computed roughly ten million entries, and he selected the appropriate values. Napier said that his findings and completing this table took him about 20 years, which means he probably started his work in 1594.

Technology:

I have found that when it comes to remembering rules, sometime the cheesiest of songs help student’s to remember the rules. It is also a very good engage before the students start with the lesson. The chorus is typically the most important content for the student’s to remember. Here are two videos that would help the student’s to remember how to compute logarithms.

The first video is a song from Youtube set to the song Thriller by Michael Jackson. The song is produced very well and is very engaging throughout the whole song.

The Second video is of a student’s project  on Youtube of how to remember how to compute logarithms to the song Under the sea by the little mermaid. Though the production isn’t as good as the first video, the young girls do a good job at explaining how to solve logarithms.

# Applying Science to Speed Training

I enjoyed this surprising (well, surprising to me) application of exponential functions: training sprinters and other runners.

# Clowns and Graphing Rational Functions

I thought I had heard every silly mnemonic device for remembering mathematical formulas, but I recently heard a new one: the clowns BOBO, BOTU, and BETC for remembering how to graph rational functions.

• BOB0: bigger (exponent) on bottom, $x = 0$
• BOTU: bigger on top, undefined
• BETC: bottom equals top eponent, coefficients (i.e., the ratio of coefficients)

Which naturally leads to this pearl of wisdom: