# 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 Ashlyn Farley. Her topic, from Pre-Algebra: powers and exponents.

One class activity that will engage students while reviewing and/or teaching Exponent/Power concepts is “Marshmallow and Toothpicks.” This activity can be used for teaching the basic of exponents, as well as exponent laws. The idea is that the toothpicks are different colors, and the different colors represent different bases, thus the same color means it’s the same base. The marshmallows represent the exponent, i.e. the number of times the student needs to multiply the base. By following a worksheet of questions, the students should be able to solve exponent problems physically, visually, and abstractly. This activity, I believe, is best done with partners or groups so that the students can discuss how they think the exponents/exponent laws work. After the activity, the students are also able to eat their marshmallows, which encourages the students to participate and complete their work.

Exponents are used in functions, equations, and expressions throughout math, thus having a deep understanding of exponents and their laws is very important. By fully mastering exponents and exponent laws, the students will be able to more easily grasp more difficult material that uses these concepts. Some specific ideas that use exponents and/or exponent laws in future math courses are: multiplying polynomials, finding the volume and surface area of prisms and cylinders, as well as computing the composition of two functions. Exponents are also used in many other situations than just math, such as in science or even in careers. Some careers that consistently use exponents and/or exponent laws are: Bankers, Computer Programmers, Mechanics, Plumbers, and many more.

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 The website Legends of Learning focuses on creating educational games for students in kindergarten through 9th grade. One game that goes over exponents, as well as the exponent laws, is Expodyssey. This game has the students solve problems to “fix” a spaceship to get back to Earth. The problems are built upon each other, so it starts by having the student answer what an exponent is, then what multiplying two exponents same base is, and keeps building from there. Each concept has multiple problems to be solved before moving on so that the students can show their mastery of the content. I believe that this game also helps improve cognitive skills by having the students do various activities simultaneously, such as calculating, reading, maneuvering elements and/or filling answers as required.

References:
Blog: Number Dyslexia

# 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: Finding the equation of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

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

This student submission comes from my former student Noah Mena. His topic, from Precalculus: finding the equation of a circle.

The equation of a circle relies on knowing the definition of a circle, knowing the radius and deciding where the circle is centered at. All of these come into play when a student has to find the equation of a circle. It takes basic understanding of the cartesian grid and understanding the coordinate system.  The equation of a circle also builds on students being able to manipulate the equation to get it into standard form and identifying the equation of a circle when it is expanded out. The shape of a circle should also be known, which means with the equation of a circle, students should be able to construct the perfect circle according to the given specifications in the equation.

Learning to write the equation of a circle can be difficult. For one of my teaches last semester my mentor teacher suggested the use of a desmos paired with a worksheet to allow the students to explore what changes the standard equation of a circle. The worksheet had the students enter certain coordinates into the graphing calculator and write down what they thought was the equation of a circle. The next part of the assignment was student driven by having them share their conclusions on what the equation for a circle would be when it is centered at the origin vs. centered at (h,k). The worksheet shows that the students drove their own learning and came to their own conclusions which enhanced engagement through the lesson.

This topic can come up again in trigonometry, upper level calculus and in math modeling. In my TNTX math modeling course, we took a closer look at the derivation of this equation and the subtleties of the standard form. This topic may also be used in physics calculations or in general, science labs.  For a physics word problem, it may ask you to calculate the net force and acceleration of a moving object around a circle. In this instance, it would suffice to just know the definition and general shape of a circle to complete these calculations. The definition of a circle is also needed to calculate centripetal force.

# Engaging students: Solving exponential 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 comes from my former student Austin Stone. His topic, from Precalculus: solving exponential equations.

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

Exponential equations can be used in lots of different kinds of word problems. One that is pretty common but is very useful for students involves interest rate. “Megan has $20,000 to invest for 5 years and she found an interest rate of 5%. How much money will she have at the end of 5 years if the interest rate compounds monthly?” I would give them the formula A=P(1+r/n)rt. It is pretty easy to convince students that this is a real-world problem and would get the students engaged about exponential equations. You can also reword the problem to ask for how much Megan started with, what the rate is, or how much time the money was in there. That way students get used to solving equations when the variable is in the exponent and when it is not. This also can lead into or us prior knowledge of natural log to solve for the variable in the exponent. How could you as a teacher create an activity or project that involves your topic? Using the basis of the problem I mentioned above, a teacher could create a Project Based Instruction lesson using this idea. The teacher can set up a scenario where, over the course of a week or two, the students would have to decide which bank to make an investment in by calculating how much money they would profit at each bank. The students would have to research different banks and their interest rate. The teacher could also give each group different scenarios where some groups have more money to invest. Students would have to figure out how long they would like to invest. The teacher would give Do It Yourselves and Workshops that deal with solving exponential equations and also getting used to natural log. They would then make a presentation explaining what bank they have chosen and why. They would also have to explain the math that they would have used. How has this topic appeared in the news? To say that exponential equations have been in the news lately would be an understatement. It has virtually been the news this year. COVID-19 is a virus and viruses spread exponentially. This would get students engaged immediately because the topic would be relatable to their own lives. Doctors and scientists try to figure out different ways to “flatten the curve”, which essentially means to make the spread of the virus not exponential anymore. We have all heard people on the news telling the public how to stop the virus from spreading and how not make people around you at risk of contracting it (contributing the exponential spread). We all have most likely seen a doctor or scientist show a graph of the virus’s spread and their predictions on how it will look in the upcoming weeks. This would give students a chance to see that what they are learning can be applied to very crucial things going on in the world around them. References Exponential Functions # Engaging students: Compound interest 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 Mason Maynard. His topic, from Precalculus: compound interest. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.) • A deposit of$3000 earns 2% interest compounded semiannually. How much money is in the bank after for 4 years?
• A deposit of $2150 earns 6% interest compounded quarterly. How much money is in the bank after for 6 years? • A deposit of$495 earns 3% interest compounded annually. How much money is in the bank after for 3 years?

These word problems are some of the basic compound interest problems that your students learn how to do where you just plug in the correct values for their corresponding variables.

• If you invested $1,000 in an account paying an annual percentage rate (quoted rate) compounded daily (based on a bank year of 360 days) and you wanted to have$2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?
• If you invested $500 in an account paying an annual percentage rate compounded quarterly , and you wanted to have$2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?

These are the types of problems that get more difficult for the students. You want them to use compound interest to solve but then they must incorporate logs into their solutions because they are looking for time instead of interest.

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

With compound interest, students first learn about the simple interest formula. The only main difference is that you start to include exponents with compound interests. Then when you introduce your students to compound interest, you start to get into some more complicated problems. After they learn about compound interest and its basic problems, then you transition into logs with your students. This is used in compound interest and instead of just looking for the interest that will be accumulated after a specific amount of time, you then shift the variable around that you are looking for. The most coming type of problem that refers to this is they give you all of the information except for the amount time it takes to get a certain amount of interest. The last thing that leads up to compound interest in Calculus is when you transition from calculating the amount of interested over specific time intervals and a specific amount of times you compound it to calculating it with compounding it continuously over a specific time interval.

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

Interest is something you have to pay on a load. Depending on what side you are and how thinks go, you are either getting some more money back that what you invested or you are paying off a massive debt. Some think that the idea behind charging loans on interest came from the early days of neighbors loaning there cattle to one another. What is really unique about this is that the words in the Egyptian, ancient Greek and Sumerian languages is connected to cattle and their offspring. This leads some to believe that interest came about due to the natural increase of the herd that occurred when you loaned out your cattle.

The first evidence that comes of a compound interest problem dates back to 2000-1700 B.C. in Babylon. A clay tablet was found and the unique thing is that the interest rate use to solve it was not written. Some researchers assume that the rate was 20% due to that mainly all the other compound interest problems dating back closer to this used it. What is really crazy is that 20% worked to solve the problem. The only thing that was wrong was that the time was corresponding to the Babylon calendar of 360 days instead of our 365 days.

In 50 B.C. Cicero writes to a friend in Rome. The letter tells that he would not normally recognize more than 12 percent interest on a loan, even though a decree was passed which required money lenders to charge no more than 12 percent. Cicero would then write a few days later that they will pay back the loan in 6 years will 12 percent interest and more money will be added each year.

Resources:

Compound Interest History:

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/799CB1D40CDD46F3010767BFC60F24DB/S1357321719000254a.pdf/emergence_of_compound_interest.pdf

Word Problems:

https://www.basic-mathematics.com/compound-interest-word-problems.html

http://www.sosmath.com/algebra/logs/log5/log51/log51.html

# Engaging students: Infinite geometric series

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 Brendan Gunnoe. His topic, from Precalculus: infinite geometric series.

Curriculum:

Students can use the formula for an infinite geometric series to discover the formula for a finite geometric series. The teacher would start by posing the question “Can we use the infinite geometric series to come up with a formula for the finite version?” and writing out a series like so

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + ar^{n+1} + \dots$

Next, the instructor could ask questions like “If we’re looking for the sum up to the nth term, where do we need to chop off the terms to get what we want?,” “Does the ending part look familiar?”, and “How can we rewrite the chopped off part so that it looks like what we already know?”. The teacher guides the students into manipulating the formula to get this result

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + ar^{n+1} + \dots$

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + \sum_{j=n+1}^\infty ar^j$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle \sum_{i=0}^\infty ar^i - \sum_{j=n+1}^\infty ar^j$

The teacher notes that the last sum can be simplified to make it easier to see by doing a substitution of $k = j -n-1$. Adjusting the bounds and substituting in the new index, we get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle \sum_{i=0}^\infty ar^i - \sum_{k=0}^\infty ar^{n+1+k}$

$= \displaystyle \sum_{i=0}^\infty ar^i - \sum_{k=0}^\infty ar^{n+1}r^k$

$= \displaystyle \sum_{i=0}^\infty ar^i - r^{n+1} \sum_{k=0}^\infty ar^k$

Note that the two sums are identical, besides the index name, so we can factor and get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = a(1-r^{n+1}) \displaystyle \sum_{i=0}^\infty r^i$

Lastly, we utilize our formula for an infinite geometric series and get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = a(1-r^{n+1}) \displaystyle \frac{1}{1-r}$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle a\frac{1-r^{n+1}}{1-r}$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle a\frac{r^{n+1}-1}{r-1}$

Although the infinite series requires $|r|<1$ , the finite version works for all real $r$. Although the formal proof that this is the correct formula might be beyond the scope of the intended class, it can easily be done with induction.

Technology:

Sal Khan, one of recent history’s most well-known STEM educators, has a fantastic video that shows the relationship between a fractal known as the Koch snowflake and the geometric series. Khan works through the derivation of the formulas for the perimeter and area of an the nth­ iteration of the Koch snowflake. It turns out that both the area and perimeters for each iteration can be expressed using a geometric series, but the perimeter diverges to infinity while the area converges. Such a result makes sense intuitively since you can fit every iteration inside of a finite box that is slightly larger than the snowflake, and thus bounding the area, yet it would require an infinitely long wire to go around the perimeter of the limiting shape. Since fractals are not normally included in the math curriculum, showing how math can be used in interesting and different ways to solve problem can be very engaging for students.

Culture:

There is a strong connection between geometric series, fractals, and self-similarity, all with a relatively simple nature. Fractals have been used in architecture and art for a very long time. Examples of self-similarity seen in ancient cultures include Hindu temples, with their structure being composed of self-similar units, and Islamic geometric art found in the domes of mosques.

Since the invention of the computer in the mid-20th century, more detailed and intricate digital art has been made popular. Although not exactly a geometric series, the Mandelbrot set acts very much like a fractal and was among the first of the uses of a computer to investigate the properties of fractals. It has been used in many ways to make animations, photos and other digital arts.

Another link between fractals and art can be found in the Legend of Zelda games. One of the iconic symbols of the game is called the triforce, which is an equilateral triangle that’s been cut into 4 smaller triangles with the middle piece removed. Such a shape is the first iteration of a fractal known as the Sierpinski triangle. As you can see, fractals can be found in all kinds of art, coming in many different forms.

https://en.wikipedia.org/wiki/Fractal_art

# Engaging students: Computing trigonometric functions using a unit circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

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

This student submission comes from my former student Alizee Garcia. Her topic, from Precalculus: computing trigonometric functions using a unit circle.

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

Being able to compute trig functions using a unit circle will be the base of knowledge for all further calculus classes, as well as others. Being able to understand and use a unit circle will also allow students to start to memorize the trigonometric functions. One of the most important things from pre-calculus to all other calculus classes was being able to solve trig functions and having the unit circle memorized was very useful. Although there are trig functions and values outside of the unit circle, the unit circle almost is like the foundation for trigonometry. Most, if not all, calculus classes after pre-calculus will expect students to have the unit circle memorized. Although it can be solved using a calculator, this will allow equations and problems to be solves easier with less thought when a student knows the unit circle. Even outside of calculus classes, the unit circle is one of many important aspects in math classes.

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

Before students learn how to compute trigonometric functions using a unit circle, they learn about the trig functions by themselves. This usually starts in high school geometry where students learn sine, cosine, and tangent, yet they do not use them in the way a unit circle does. Most schools only teach the students how to use the calculator to compute the functions to solve sides or angles for triangles. As students enter pre-calculus, they use what they have learned about the trig functions in order to apply them to the unit circle. This will allow students to see that using trig functions can still be used to solve triangles, but it can also be used to solve many other things. Once they learn the unit circle, they will see more examples in which they will apply the functions and make connections to real-world scenarios that they can also be applied to.

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

There are probably many simulations and websites that can help students compute trig functions using the unit circle, but I think something that will engage the students is a Kahoot or Quizziz that will help the students memorize the unit circle. Giving students an opportunity to apply what they learned into a friendly competition not only gives them practice but will also let them be engaged. Other technology resources such as videos or a website that is teaching the lesson does not really allow the students to apply what they know rather than just being lectured. Although some websites and technology can be useful, I personally, enjoy giving students the opportunity to work out problems as well as being engaged. Also, using calculators could be helpful to check answers but if they have a unit circle it might not be necessary unless they do not have the unit circle in front of them.

# 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