# Engaging students: Computing the determinant of a matrix

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 Brendan Gunnoe. His topic: computing the determinant of a matrix.

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

When students learn about the determinant of a matrix, they only learn about computing it and don’t learn about the applications of the determinant or what they signify. One interesting use of the determinant is finding the eigenvectors of a matrix. A visual understanding of what an eigenvector is can be done by showing what a matrix does to the any generic vector, and what it does to the eigenvectors. For a generic vector that is different from an eigenvector, the matrix knocks the vector off the span of the original vector. What makes an eigenvector special is the fact that the matrix transformation keeps the eigenvector on its span but rescales said eigenvector by its eigenvalue. For example, take the matrix

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right]$.

This matrix’s eigenvectors are $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$ with eigenvalues 8 and 2 respectively. That is,

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right] \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] = \left[ \begin{array}{c} 8 \\ 8 \end{array} \right] = 8 \left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$

and

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right] \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] = \left[ \begin{array}{c} 2 \\ -2 \end{array} \right] = 2 \left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$.

Eigenvectors have many useful applications in future math and science classes including electronics, linear algebra, differential equations and mechanical engineering.

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.

The YouTube channel 3Blue1Brown has a fantastic video on determinates and linear transformations. Grant, the channel owner, uses animations to visualize what a matrix transformation does to the plane . He starts by showing what a transformation does to a single square then shows why the change of change of that one area shows what happens to the area of any region. He also gives multiple explanations for what a negative determinate means in terms of orientation of the axes. Then he explains what happens when the determinate is 0. All of this is already extremely useful for understanding what a 2×2 matrix does, but Grant continues and extends all the same things for 3×3 transformations. Lastly, Grant gives a few explanations on why the formula for the determinate is what it is and poses a small puzzle for the viewer to contemplate. This video explains what and why we use determinates and how they can be useful all while showing pleasing visual examples and other explanations.

What interesting word problems using this topic can your students do now?

Using determinates to see if a set of vectors is a basis.

The determinant can tell you when a matrix squishes space into a lower dimensional space like a line or a plane. Thus, taking the determinate of a matrix composed of a set of vectors tells you if those vectors are a basis for the matrix’s dimension.

Part 1. A 3D printer’s axes are set up in such a way that the print head can only travel in the direction $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} -1 \\ 1 \end{array} \right]$. Assume that the place where the print head is right now is the origin $\left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$. Can you move the print head to the location $\left[ \begin{array}{c} x \\ y \end{array} \right]$ and $\left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$ by only moving in the directions of $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} -1 \\ 1 \end{array} \right]$?

Hint: Try to solve $\left[ \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \end{array} \right] = \left[ \begin{array}{c} x \\ y \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \end{array} \right]$?

Part 2. Next time you turn on your 3D printer, one of the motor’s broke and now the print head can only move in the direction of $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$. Assume that the place where the print head is right now is the origin $\left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$. Can you move the print head to the location  by only moving in the direction of $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$?

Hint: Try to solve $\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right] \left[ \begin{array}{c} a \\ b \end{array} \right] = \left[ \begin{array}{c} x \\ y \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \end{array} \right]$?

Part 3. You buy a new 3D printer that it can move in all three directions: up/down, left/right, forward/backwards. When you test out the movement of the print head, you see that each axis moves in the directions of $\left[ \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right]$, $\left[ \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right]$, and $\left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right]$. Can you use your new 3D printer to go to any location $\left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$, inside the printing space?

Hint: Think about solving $\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \\ c \end{array} \right]$? How do you know?

Part 4. Your little sibling messed around with your new 3D printer and now it moves in the directions $\left[ \begin{array}{c} 1 \\ 0 \\ 1 \end{array} \right]$, $\left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right]$, and $\left[ \begin{array}{c} 2 \\ 1 \\ 1 \end{array} \right]$. Is your 3D printer able to get to some point $\left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ inside the printing space as is, or do you need to fix your printer?

Hint: Think about solving $\left[ \begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$. Does this always have a solution $\left[ \begin{array}{c} a \\ b \\ c \end{array} \right]$? How do you know?

# Engaging students: Using a recursively defined sequence

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 Enrique Alegria. His topic, from Precalculus: using a recursively defined sequence.

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

Recursion is heavily emphasized within the branches of computer science. The technique can be used more than just in arithmetic and geometric sequences for finding the next term. Within computer science, recursion techniques can be utilized for sorting algorithms. The content will be able to transfer easily. Instead of finding the previous term to use to find the current term, within sorting algorithms, a set of numbers is chunked into smaller and smaller sets such that the original set of numbers becomes sorted.

We can take a deeper look at Merge Sort which is a recursive sorting algorithm. What occurs is the set of numbers repeatedly gets cut in half until there is only one element in the list. From there the elements are sorted in increasing order. Traversing back into the original size of the list with all of the elements contained except the final output is the list in increasing order.

Students can inspect the algorithm visually and need not to understand the implementation of code to comprehend the functionality of recursion. Guiding the students towards the smallest part of the process which is the single element and from there rearranging the elements of the list.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Recursively defined sequences influenced a renowned artist who is M.C. Escher. The concept of a sequence beginning at one point and continuing infinitely is how Escher exhibits recursion. Escher challenges the viewer of his work to determine the patterns from the artistic series.

For example, when observing the piece Drawing Hands, a student can predict what the ‘base case’ of the artwork would be followed by the next steps of the drawing. The spectator of this piece can break it apart into smaller and smaller partitions of the whole. And once they reach a starting point, they can put together the whole picture once again.

Similarly, students can view this piece titled Two Birds to follow the patterns. Without saying the name of the piece students can again predict the base case and determine how recursion techniques would be used for this sequence. Students can begin to learn how to think of how recursively defined sequences are applied through visual representations of M.C. Escher’s artwork.

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

Technology can be used to effectively engage students with recursion by showcasing the YouTube video “Recursion: The Music Videos of Michel Gondry” by Polyphonic. Through this video, students can compare recursively defined sequences to music they listen to. The video starts with singular notes and then repeating the notes to create a rhythm. Compiling the initial sounds into something familiar through loops of samples and sound bites. This video goes into the repetitive patterns of the small chunks of sound are shown through visual representations with the music videos by Michel Gondry. In the music video “Star Guitar” by The Chemical Brothers, the video starts off with the listener on a train ride going through a landscape. Slowly patterns emerge as buildings uniquely correspond to the notes and rhythms within the song. With this YouTube video students obtain a great introduction to recursion and hopefully continue to find patterns of recursion to music they listen to in the future.

References

Greenberg I., Xu D., Kumar D. (2013) Drawing with Recursion. In: Processing. Apress, Berkeley, CA. https://doi.org/10.1007/978-1-4302-4465-3_8

Miller, B., & Ranum, D. (2020). 6.11. The Merge Sort — Problem Solving with Algorithms and Data Structures. Runestone.academy. https://runestone.academy/runestone/books/published/pythonds/SortSearch/TheMergeSort.html.

# Engaging students: Powers and exponents

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 Austin Stone. His topic, from Pre-Algebra: powers and exponents.

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

“The number of people who are infected with COVID-19 can double each day. If it does double every day, and one person was infected on day 0, how many people would be infected after 20 days?” This problem can be a current real-life word problem that all students can relate to given the times we are in. This problem would be a good introductory for students to see how quickly numbers can get when using exponents. This would be an engaging introductory to exponents and will get the students interested because they can easily see that this can be used in current problems facing the world. This problem could also work later in Algebra if you ask how many days it would take to infect “blank” amount of people. This makes the question more of a challenge because they would have to solve for “x” (days) which is the exponent.

How has this topic appeared in the news?

This topic has been the news so far in 2020 if we are being honest. COVID-19 is a virus that has an exponential infection rate, just like any virus. When talking about COVID-19, news reporters and doctors usually use graphs to depict the infection rate. These graphs start off small but then grow exponentially until it slows down due to either people being more aware of their hygiene habits and/or the human immune system getting more familiar with the virus. Knowing how exponents work helps people better understand the seriousness of viruses such as COVID-19 and the everlasting impact it can have on the world. Doctors study what are the best ways to slow down the exponential growth so that a limited number of people contract and potentially die from the virus. To do this, they predict the exponential growth keeping in mind the regulations that may be enforced. Whatever regulation(s) slow down the virus the most are the ones that they try to enforce.

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

An easy way to introduce students who have never seen exponents or exponential growth before is to use a graphing calculator. By plugging in an exponential function into the calculator and viewing the graph and zooming out, students can easily see how quickly numbers start to get massively large. A teacher can set this up by giving the students a problem to think about such as, “how many people would be infected with the virus after “blank” amount of day?” Students then could guess what they believe it would be. After revealing the graph and the actual number, students will probably be surprised at how big the number is in just a short amount of time. After that, the teacher could show a video on YouTube about exponential growth and/or infection rates of viruses and how quickly a small virus can turn into a pandemic. This also has very current real-world applications.

# 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

# Engaging students: Using sequences

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 Gary Sin. His topic, from Precalculus: using sequences.

How has this topic appeared in pop culture?

Probably the most used sequence in pop culture or art is the Fibonacci sequence. I learned about the Fibonacci sequence myself from “The Da Vinci Code” by Dan Brown. The Fibonacci sequence has been explored by many mathematicians over the years and if we divided 2 successive numbers (larger divided by the smaller), the limit of the ratio is the golden ratio.

The golden ratio was heavily believed to be seen in nature itself. Naturally people were fascinated that such a number could be seen everywhere in nature. Many artists based their art on the golden ratio, believing that the ratio is aesthetically pleasing. A great example is the polyhedral seen in “’The Sacrament of the  Last Supper” by  Salvador Dali. Modern architects also utilize the golden ratio in their builds. It was also believed that the proportions of the different parts of the limbs of humans are in the golden ratio.

The Fibonacci Sequence is fascinating and is a great way to demonstrate to students the beauty in math and how even artists are influenced by it and is a beautiful link to how mathematics can also be seen in nature.

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

Sequences are fun to play around with as some sequences are infinite or finite and the series they form could converge to a number. Students could be given a starting sequence and are asked to find the nth term of a sequence. I could also point out how sequences can be seen in something as simple as the list of natural numbers, multiples of positive integers.

Students could also be given both arithmetic and geometric sequences and plot them on a graph accordingly to see if the sequence progresses linearly or exponentially. I could also introduce sequences that are neither and that are divergent.

One of the important usefulness of sequences is how it relates to limits of a sequence. I could provide a fun riddle for students to figure out the limit of a sequence using word problems like Zeno’s Paradox. Students can figure out the rule of a sequence and plot it on the graph to see how it converges toward a number.

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

The most amazing thing about sequences is that students use them from the moment they learn how to count as kids. Natural numbers are sequences that are obtained by adding 1 to the previous term. Naturally, the multiples of positive integers are also sequences. Students will also realize that the powers of a base are geometric sequences. When learning about plotting functions, linear, quadratic or cubic; the students are basically using sequences and basic pattern recognition to create tables of values and observing the rate of change.

Sequences are especially important in bridging a simple concept like a sequence to limits of functions, limits of infinity are an important abstract idea that provokes the students to think more about how a function would act if it  kept going forever.

When determining a recursive of exclusive formula for sequences, students will also have to apply basic algebra, order of operations, arithmetic, exponents in order to create or prove that a formula works for a sequence.

# 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 Jonathan Chen. His topic, from Precalculus: computing logarithms with base 10.

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

Computing logarithms with base 10 can appear in many scientific applications for word problems. To define the acidity or alkalinity of a substance, Chemists use the formula $pH = \log [H^+]$. “[H+] is the hydrogen ion concentration that is measured in moles per liter” (Stapel, n.d.). We know lemon juice is acidic because the pH value is less than 7. We know bleach is basic because the pH value is greater than 7. When a pH value is equal to 7, the solution is neutral. An example of something neutral would be pure water. Teacher can create word problems based on the information given about a liquid solution. Noise can be measured in decibels. The formula used to measure the strength of a sound is $dB = 10 \log(I \div I_0)$. “I0 is the intensity of ‘threshold sound,’ or sound that can be barely be perceived” (Stapel, n.d.). Teachers can create word problems based on the defined terms of how many times more intense a sound is than the threshold sound. Similar problems with the topic of computing logarithms can be made involving earthquake intensity.

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

As shown in the above answer, this topic can reappear in student’s future science course in the topic of pH levels, earthquake intensity, or “loudness” measured in decibels. In order to find the pH levels, [H+] concentration, or the [OH] concentration you may need to know how to calculate logarithms with base 10 when dealing with the equation $pH = \log [H^+]$. Similar things can be said about measuring “loudness” and earthquake intensity. Their formulas involve calculating logarithms with base 10. Other future topics students may encounter in mathematics are logarithmic functions, Euler’s number, natural log, and logarithm rules. While not all of these future topics are strongly related to the topic of calculating logarithms with base 10, they can be loosely connected to where the practice of calculating logarithms with base 10 makes it easier to understand and do things related to the future topics. With the topic of logarithmic rules, it can help better simply and calculate with logarithms with base 10.

Calculating logarithms with base 10 has been around since 1614. John Napier invented logarithms and ever since then small additions have been made. Additions such as a logarithmic table made it easier to solve logarithmic problems. The logarithmic tables are similar to the multiplication tables elementary schoolers memorize to calculate simple multiplication faster for their future problems. Many mathematicians made their contributions to add more to the logarithmic table to the point where the calculations reached up to 200,000. Aside from the logarithmic tables, there were other methods to calculate logarithms with base 10 such as the slide rule. It was also possible to memorize the values of the logs with base 10 of 1 through 10 and use the logarithmic rules to calculate bigger values. Because

$\log 400 = \log(100 \times 4) = \log 4 + \log 100$

by expansion and logarithmic rules, people can solve this problem my memorizing that $\log 4 = 0.602$ and knowing that $\log 100 = \log 10^2 = 2$. Knowing this makes the equation more clear to recognize and easier to solve by hand. Calculating logarithms with base 10 were used extensively until the creation of the calculator made it easier to calculate anything, including logarithms.

References

“The Log Log Duplex Trig” “Slide Rule”. (n.d.). Retrieved from Web Archive: https://web.archive.org/web/20090214020502/http://www.mccoys-kecatalogs.com/K%26EManuals/4081-3_1943/4081-3_1943.htm

Bourne, M. (n.d.). 4. Logarithms to Base 10. Retrieved from Interactive Mathematics: https://www.intmath.com/exponential-logarithmic-functions/4-logs-base-10.php

John Napier and the invention of logarithms, 1614; a lecture. (n.d.). Retrieved from Archive.org: https://archive.org/details/johnnapierinvent00hobsiala/page/18/mode/2up

# Engaging students: Finding the slope of a line

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 Austin Stone. His topic, from Algebra: finding the slope of a line.

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

Using “pull back” toy cars, you can create a fun little activity that students can compete in to see who wins. Students can be put into groups or do it individually depending on how many cars you have available. The idea of the activity would have students pull back the cars a small amount and record how far they took it back and how far the car went. After doing this from three or four different distances, the students would then graph their data with x=how far they took it back and y=how far the car went. Then the teacher would tell the students to find how far back they would need to pull for the car to go a specified distance by finding the slope of their line (or rate of change in this example). After students have done their calculations, they would then pull back their cars however far they calculated and the closest team to the distance gets a prize.

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

Students will continually use slope throughout their future math and science classes. In math courses, slope is used to graph data and predict what will happen if certain numbers are used. It is also used to notice observations about the graph such as steepness (how quickly it changes) and if the rate of change is increasing or decreasing. It is also used in science for very similar reasons. In physics, slope is used commonly to calculate velocity and force. In chemistry labs, slope is used to predict how much of a certain substance needs to be added to find observational differences. In calculus, when taking the first derivative of a function, if the slope is negative, then the function is decreasing during that interval and vice versa if it is positive. Slope is also widely used in Algebra II, so learning how to find the slope is very important for future math and science classes whether it be in high school or college.

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

Students should have already learned how to graph points on the coordinate plane. They can take this knowledge and now not only plot seemingly random points, but now see the relationship between these points. Plotting points is a skill usually learned around 6th grade and is used regularly after that. Also, finding the x and y axis can be used when finding the slope of a line. If you have a function with no points, finding the x and y axis can let you find the slope. Finding the x and y axis is learned in Algebra I so this would be fresh on students’ minds. Finding the slope of a line can be scaffolded with finding the x and y axis in lectures or in PBL experience. Also refreshing students on how to graph not only in the first quadrant, but in all four quadrants could be a quick little activity at the beginning of the PBL experience.

Reference:

http://www.andrewbusch.us/home/racing-day-algebra-2

# Engaging students: Multiplying binomials

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 Cire Jauregui. Her topic, from Algebra: multiplying binomials.

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

Khan Academy has a whole series of videos, practice problems, and models to help students learn about multiplying binomials. The first in this series is a video visualizing the problem (x+2)(x+3) as a rectangle and explains that multiplying the binomials would give the area taken up by the rectangle. This would help students connect multiplying binomials to multiplying numbers to find area. This can also help students who learn better with visual examples by giving them a way to show a picture demonstrating the problem they are multiplying. Khan Academy then moves from using a visual representation to a strictly alpha-numerical representation so students can smoothly transition from having the pictures drawn out to just working out the problem. The first video in the series of pages at Khan Academy can be found at this link: https://tinyurl.com/KhanAcademyBinomials

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

Multiplying binomials extends on two-digit times two-digit multiplication that students learn and practice in elementary and middle school courses. This video from the platform TikTok by a high school teacher Christine (@thesuburbanfarmhouse) shows the connection between vertical multiplication of two numbers and the multiplication of binomials together: https://tinyurl.com/TikTokFOIL By showing students that it works the same way as other forms of multiplication that they have already seen and hopefully mastered, it sets the students up to view the multiplication of binomials and other polynomials in a way that is familiar and more comfortable. This particular video is part of a miniature series that Christine recently did explaining why slang terms such as FOIL (standing for “first, outside, inside, last” as a way to remember how to multiply binomials) which many classrooms have used (including my own high school teachers), which are helpful when initially explaining multiplication of binomials, ultimately can be confusing to students when they move on to multiplying other polynomials. I personally will be staying away from using terms like FOIL because as students move on to trinomials and other larger polynomials, there are more terms to distribute than just the four mentioned in FOIL.

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

As I mentioned in the last question, learning to multiply binomials can lead students to success in multiplying polynomials. This skill can also help students factor polynomials in that it can help them check their answers when they are finished. It can also help them recognize familiar-looking polynomials as having possible binomials as factors. If a student were to see 12x2-29x-8 and couldn’t remember how to go about factoring it in other ways, a student could use a guess-and-check method to factor. They might try various combinations of (Ax+B)(Cx-D) until they find a satisfactory of A, B, C, and D that when the binomial is multiplied, creates the polynomial they were trying to factor. Without solid skills in multiplying binomials, a student would likely be frustrated in trying to find what A, B, C, and D as their multiplication could be wrong and seemingly no combination of numbers works.

# Engaging students: Solving absolute value 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 Conner Dunn. His topic, from Algebra: solving absolute value equations.

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

This topic is an excellent concept for algebra students wanting real life applications when learning math concepts. In creating an activity relevant to this, the “real life” concept I’d want to emphasize is distance, which conveniently is in the definition of absolute value. Distance can be expressed in words or in pictures, and specifically with absolute value, we model distance as a one-dimensional (one variable) function. To express a model like this, I’d want get students to know what the numbers and operations can mean for a distance problem. For example, a student should be able to know that |x-7| = 3 can be expressed as “the distance between x and 7 is 3.” The potential activity here is to get students to either express absolute-value equations in words or vice versus. The same concept of distance can be played out in pictural or graphical representations. Obviously, I can use absolute value graphs to model this, but I would specifically look at one-dimensional representation and maybe have students try and model a situation using absolute value equations. It’ll be in these activities that I could really nail down true meanings of 2-solution, 1 solution, or no solution problems and why, for example, they have to check for extraneous solutions when solving.

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

The concept of solving this type of equation is really relevant and similar to that of solving for quadratic equations as well as polynomial equations in general. When students are able to grasp the concept of having 0, 1, or 2 solutions in an absolute value equation and know why, they’ll be using this understanding when solving for polynomials of high degrees. I’d also like to imagine students might want to make the connection to midpoints in Geometry. Absolute value equations can tell the 1-dimensional distance from a point to another two points in either direction. When Geometry students see this modelled on a number line, they may be able to identify 3 points equidistant from one another forming 2 congruent segments.

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

The things I would teach about solving absolute value equations really build off students’ understanding of equivalence and the properties about it that they use when asked to “solve” for anything an algebra class. One of the big steps in solving a|bx+c| + d = e is described as “solving for the absolute value.” This step builds off students’ previous works of “solving for x.” The solution for connecting these is clear: just let the “x” or rather the variable to solve for be the absolute value, and then solve for it using those equivalence properties they know. The great thing about this is that it builds on the idea that when solving for unknown variables, it’s okay to not immediately know them. Equiveillance properties are tools that students can use to work towards solving for unknowns. The more accustomed students are to these tools, the better, so when throwing in absolute values into the mix, it makes for good practice in using “equivalence tools.”

# Engaging students: Adding and subtracting polynomials

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 Enrique Alegria. His topic, from Algebra: multiplying polynomials.

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

This topic can be used in students’ future courses in mathematics by simplifying expressions of increasing degree. In Algebra II students are expected to simplifying polynomials of varying degrees as they move on to multiplying and dividing polynomials. From there determining the factors of a polynomial of degree three and degree four. Real-world problems can be solved through the simplification of several like terms. Each term representing a specific part of the problem. We can even compare the addition and subtraction of polynomials to runtime analysis in Computer Science. Measuring the change in the degree and how that affects the output. In a way, this can translate to the runtime of a program. For example, a chain of commands with a constant time is run. A loop is nested in another loop that is placed after the first expressions. This has changed the overall runtime of the program from constant time to quadratic because of the degree of the nested loops. The overall time would be the addition of the expressions and their corresponding times.

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

This topic extends from the early concept, ‘Combining Like Terms.’ Starting with adding and subtracting items of similar groupings such as 8 apples and 4 apples altogether are 12 apples. Bringing students to place value such as adding 3 ones and 2 ones to adding multi-digit numbers. We then leap towards Algebra introducing expressions and equations. Learning about linear and quadratic equations and graphing them. Students should have learned about monomials in correspondence with coefficients and exponents. From there, students are familiar with algebraic terms. Those are the building blocks that we are going to be expanding upon. Once students familiarize themselves with several terms in an expression, they will focus on adding or subtracting like terms by focusing on both the coefficient, term, and exponents on the variables. Shortly after the students can continue to be challenged by using terms such as $6xy$ or $3a^2b^3+4a^2b^3c^2$ to focus on the terms and confirm if they are ‘like’ to be combined or just notice the fact that they have some common variables with the same exponents but with a slight difference other than the coefficient, the expression cannot be simplified as one may think.

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

Adding and subtracting polynomials can be engaging to students with the help of Brilliant. This site starts with helping students identifying polynomials and their degrees to help students understand how to describe them. Then moving to the arithmetic of polynomials performing addition and subtraction operations on the polynomial numbers. This source goes through polynomials through challenging and insightful exercises. For example, a quadrilateral of sides such as $5$, $3x+4$, $4x+1$, $17x-10$, and from there simplifying the expression. Students would be able to substitute values and determine if a specific quadrilateral has been made. I can have students go through a few exercises as a class or on their own and then they can come up with a problem on their own that would be posted to the ‘public’ (which would be only their class) so that the students will be able to have classroom interaction and grow as they challenge each other. Students can apply this concept by creating a large polynomial expression and then simplifying it and lastly graphing the equation.

References:

Polynomials. Brilliant.org., from https://brilliant.org/wiki/polynomials/

Simplifying Expressions. Brilliant.org., from https://brilliant.org/wiki/simplifying-expressions/