Engaging students: Computing the composition of two 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 again comes from my former student Alexandria Johnson. Her topic, from Algebra II/Precalculus: computing the composition of two functions.

The following link is to a worksheet over composition of functions. The worksheet allows students to explore composition of functions without outright telling them what composition of functions is. Instead, the students are working on real world problems about shopping in a store that is having a 20% sale with mystery coupons. In the worksheet, students explore whether or not it matters which discount is applied first and the equations that go along with each scenario. This worksheet is interesting because it approaches composition of functions in an explorative way and it is using a real-world situation students in high school may find relatable, which can help hook students that are math-phobic.

https://betterlesson.com/community/document/1326462/going-shopping-student-materials-docx

Computing the composition of two functions requires prior knowledge of basic operations and combining like terms. This topic will expand upon their knowledge of basic operations by applying them to functions. Students will be able to add, subtract, multiply, and divide functions. Students should be able to use the distribution property; this is important when students are writing (fog)(x) and (gof)(x). During this topic, students should be able to expand upon their knowledge of creating functions from real world problems, which can be seen in the worksheet from the link above.

Musical composition is a way this topic can appear in high culture. Musical composition is the process of combining notes, chords, and melodies in a particular way. Arranging the notes, chords, or melodies in different ways can change the composition. Function composition is the combining of different functions f(x) and g(x) in different ways like addition, subtraction, multiplication, and division. Order usually matters in function composition just like in musical composition. If you have several band students, or musically inclined students, this would be a good hook to grab students interest.

Work cited
https://betterlesson.com/community/document/1326462/going-shopping-student-materials-docx

Engaging students: Using a recursively defined sequence

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

How could you as a teacher create an activity or project that involves your topic?
One activity that would be interesting to introduce recursion would be Fibonacci’s rabbit problem. In his book, Liber Abaci, Fibonacci introduced a problem where you start with one young pair of rabbits and try to find out how many rabbits you would have after a year. Every month, a grown pair of rabbits can give birth to a new pair, and it only takes one month for a young pair to grow up and be able to reproduce on their own, and the rabbits also never die. This is one of the most popular recursive sequences (the Fibonacci sequence), and, by itself, can be solved without a prior knowledge of recursion, but is a very good way to introduce the idea once the students begin to analyze the pattern of how many pairs of rabbits there are after each month. This problem is laid out in this video, https://youtu.be/sjQlW6cH3Ko but it is not necessary to show the video to introduce the problem.

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

One major place that a solid grasp of recursion can be used is in computer programming courses. Although not everyone takes these, they are becoming increasingly popular and the field is not likely to shrink any time soon. In programming, there are certain things that can either only be written recursively (as opposed to explicitly) or at least ones that are simpler to write and understand with recursion than with an explicit algorithm. There are also times, depending on the language and content, that a recursive function can be more efficient. Because of this, an understanding of recursion is becoming increasingly important for more people, and the ability to write and understand how it works is practically becoming necessary. So, even though not every student will go on to take computer science, many will, and the basic idea is still important to understand.

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

There is a series of Khan Academy videos on recursively defined sequences online. The first one is https://youtu.be/lBtb30SjU2Q and it shows how to read and understand what the basic frame for recursion is. Although Khan Academy videos are not always the most engaging for all students, they do work for many because of their consistent structure. This video in particular is about recursive formulas for arithmetic sequences. Without mentioning the vocabulary yet, the video does introduce the idea of a base case and the method for finding subsequent values. The video both shows how to look at a list of values and determine the recursive definition, as well as how to understand the recursive definition if that is what you are given. For a three minute video, it does a very good job of introducing important topics for recursive series and explaining the basic ideas so that students have a framework to build on later when more complex recursively defined sequences are introduced.

References:
1. https://youtu.be/sjQlW6cH3Ko
2. https://youtu.be/lBtb30SjU2Q
3. https://plato.stanford.edu/entries/recursive-functions/

Engaging students: Finding the focus and directrix of a parabola

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 Brittnee Lein. Her topic, from Precalculus: finding the focus and directrix of a parabola.

What are the contributions of various cultures to this topic?

Parabolas (as we know them) were first written about in Apollonius’s Conics. Apollonius stated that parabolas were the result of a plane cutting a double right circular cone at an angle parallel to the vertical angle (α). So, what does that actually mean?

Well, if we take a vertical line and intersect it with a straight line at a fixed point, and then rotate that straight line around the fixed point we form the shape below:

If the plane slices the cone at the angle β and β=α, a parabola is formed. This is still how we define parabolas today although you may not think about it that way. When you think of a parabola, you think of the equation $y = ax^2 +bx + c$ . This equation is derived using the focus and the directrix. This video shows how to do so:

Understanding how the focus and directrix affect the equation of a parabola is crucial to understanding what each word means. According to mathwords.com, “For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.” The directrix is a line perpendicular to the axis of symmetry and the focus falls on the line of the axis of symmetry.

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

This desmos activity can be used to show students how changing the focus, directrix, and vertex of the parabola affects the graph. https://www.desmos.com/calculator/y90ffrzmco

From this, students can shift values of the vertex and see that the directrix stays constant when the x-value is changed and that the focus remains constant when the y-value is shifted. If students change the value of the focus, they can see how it stretches and contracts the width of the parabola and how the directrix shifts. They can also see that when the focus is negative, the parabola opens downward and the directrix is positive. This website: https://www.intmath.com/plane-analytic-geometry/parabola-interactive.php Is also very helpful in showing the relationships between the focus, directrix and the graph of the parabolas because students can clearly see that the distance between a point on the parabola and the focus and the distance between that same point and the directrix are equal.

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

The website http://www.purplemath.com/modules/parabola4.htm has a lot of great real-world word problems involving finding the focus and the directrix of a parabola. For example, one of the questions is:

(This is a graph I made using desmos to model the situation at hand)

This problem requires a lot of prior knowledge of parabolas and really tests students’ ability to interpret information. From the question alone, the students can find the x-intercepts (-15,0) and (15,0) from the information “the base has a width of 30 feet”. They are also able to infer that the slope of the parabola will be negative because of the shape of an arch. The student must also know how to find the slope of the parabola using the x-intercepts, solving for the equation of the parabola using the x-intercepts and vertex and the equations for finding the focus and directrix from the given information. There are a few problems as involved as this one on the listed website above.

References
http://www.mathwords.com/d/directrix_parabola.htm

https://www.britannica.com/topic/conic-section
https://www.desmos.com/calculator/y90ffrzmco
https://www.intmath.com/plane-analytic-geometry/parabola-interactive.php
http://www.purplemath.com/modules/parabola4.htm

Engaging students: Finite geometric series

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 Caroline Wick. Her topic, from Precalculus: finite geometric series.

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

Finite geometric series was a concept that began over 4500 year ago in ancient Egypt. The Egyptians used this method of finite geometric series mainly to “solve problems dealing with areas of fields and volumes of granaries” but used it for many other uses too, including the pyramids and math problems similar to those one might find on a STAAR test today (see D1, and F1).

There are seven houses; in each house, there are 7 cats; each cat kills seven mice; each mouse has eaten 7 grains of barley; each grain would have produced 7 hekat. What is the sum of all the enumerated things?

Years passed and finite geometric series were not revisited until around 350 BC by the Greeks, namely Archimedes, who came up with a solution to the math problem V=1/3Ah by finite calculations instead of limits. In addition, the idea that a finite sum could be procured from an infinite series was created in what is called the “Achilles Paradox” (D2, F2).

Years after this came Mathematicians in the middle ages, like Richard Swineshead or Nicole Oresme, who aided the world by further refining these series. This eventually led to the renowned Physicist Isaac Newton to “discover the geometric series” after studying mathematician John Wallis’s method of “finding area under a hyperbola” (F1). We can attribute almost all of what we know about geometric series’ to these fine gentlemen above, and they can only attribute what they know from the ancient Egyptians and Greeks.

How has this topic appeared in pop culture?

In 2002, PBS came out with a kids’ TV show called CyberChase, which is an entertaining cartoon about a bunch of kids who get pulled into “Cyber Space” to fight the bad guy, named Hacker, all while discovering and using different mathematical concepts that they learned along the way. Eleven seasons have passed since the shows beginning and it is still going strong, but one episode that still sticks out to me was their version of explaining geometric series to kids. The episode was called “Double trouble” and was the 9th episode of the second season. The specific geometric series involved in the episode was doubling, but the “real world” clip at the end stood out more vividly to me. After losing a chess game, the main character has to decide between paying the winner $5.00 or paying one penny for the first space on a chess board, then two pennies on the second, then four on the third, and continuing to double the previous number for every space on the entire chess board. Since the main character thought pennies were less, he decided on the second option, only realize after that he would have to pay way more than $5.00 in the end. This helped me understand the most basic geometric series when I was a kid, and has stuck with me to this day, so I am certain that it has and can stick in other students’ brains as well.

Here is the clip from the show:

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

The idea of finite geometric series is typically lightly introduced around students’ sophomore year of high school when they take geometry, but it is not really expanded upon/explained until students reach Pre-Calculus. The specific TEKS related to this topic are located under Pre-Calculus in (5), (A)-(E) (Source B1). The concept is brought up again in Math Models with Applications and is used for understanding interest on a balance over a period of time, or “loan amortization.” The ideas can also be used to help understand difference equations that involve heat and cooling over a period of time, and how to predict what the temperature might be in the future, which is a concept that is important in the realm of science too.

When students get to college, finite geometric series are expanded upon even more when they take Calculus classes, and they will learn how to prove a series is finite using induction when they get to their Discrete Classes and Real Analysis classes. In the business realm, they will have to use it to predict monetary sums regarding interest and possible growth in a company, so likely no matter where a student ends up, s/he will have to use this important mathematical concept everywhere.

References:
D1, F1: http://mste.illinois.edu/courses/ci499sp01/students/ambucher/math306geo.pdf
D2, F2: https://www.britannica.com/topic/Achilles-paradox
B1: http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.html

Engaging students: Using radians to measure angles instead of degrees

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 Cody Jacobs. His topic, from Precalculus: using radians to measure angles instead of degrees

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

I believe the best way to get students to start using radians, or know how to convert from degrees to radians. Is giving students a lot of problems where they go back and forth between radians and degrees. It is a fairly simple task but you want all of your students to be able to easily recall how to do this. Another activity could be going over the unit circle with your students. Using the unit circle occurs frequently from pre calculus going forward, just be sure to express that the radian measure is the arc-length divided by the radius of the unit circle to the following point. I also know that my high school teacher always made us fill out the unit circle on every test giving us the degree measurement and making us fill out the radians. This really helped us concrete radian measurements in our head, just for those 5 points it would provide us.

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

I would say after radians and the unit circle is introduced, just about every math class after that has something to do with the two. Even if it is just looking for multiple solutions to a problem, where you just add two radians to the solution you already found. But why do we use radians instead of degrees in high math classes? This is because radians are a unitless measurement. Radians are defined as arc-length divided by radius, so the units cancel. This is helpful in higher level math classes because radians are easy to use in derivative and limit formulas. This is the main reason we use radians instead of degrees in higher level classes, the convenience.

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.com is yet again another great technological resource to use when introducing radians to the classroom. There is a great activity call “What is a Radian?” That introduces an activity student can do using a plate and folding it into different sections. I actually believe this is how I was introduced to radians in high school. The second part of the activity on desmos after you finish with the plate introduction, is asking students key questions. How many radians are in a certain degree measurements? How many degrees are in a certain radian measurement? As always desmos is still great at introducing radians and lets you easily monitor your students progress.

Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 3

Here’s another T-shirt that I found in my quest for the perfect ugly mathematical Christmas sweater: https://www.amazon.com/Fibonacci-Christmas-Tree-Holiday-Shirt/dp/B07KCF1F6D/

Unlike the shirt in my previous post, this one actually gets the first ten rows of Pascal’s triangle correct. So that’s a good thing.

There’s one small error: while the Fibonacci numbers can be found by adding along shallow diagonals of Pascal’s triangle, this really shouldn’t be called a “Fibonacci Christmas Tree.”

Oops.

Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 2

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

There’s only one small problem:

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

Oops.

Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 1

This year, I thought I’d surprise my family with matching ugly mathematical Christmas sweaters. Admittedly, I didn’t look very hard, but I couldn’t find a sweater that I liked both artistically and mathematically. However, I did stumble upon this T-shirt: https://www.amazon.com/Christmas-Math-Teacher-Tshirt-Lovers/dp/B077X14254/

I gave one to my wife and daughter, and it was a big hit.

However, I made the mistake of not inspecting the merchandise closely enough. About a minute after receiving her shirt, my daughter pointed at a digit in the sixth row of the decimal expansion and asked, “Shouldn’t this be a 5? Or maybe I’m mis-remembering.”

In that moment, I remembered that, a few years ago, she had memorized the first few dozen digits of $\pi$ for her elementary school’s talent show. Somehow, she had retained that bit of trivia all these years later. I didn’t miss the irony: I did not remember that she could remember the first few dozen digits of $\pi$ .

As I’ve learned not to daughter my daughter’s memory, I checked two different references (https://www.piday.org/million/ and https://www.wolframalpha.com/input/?i=N%5BPi,1000%5D), and, sure enough, she was right.

The shirt correctly wrote the first 47 digits of $\pi$ after the decimal point. But things went haywire after that. Not only did was the T-shirt’s 48th digit incorrect, but it skipped a few hundred digits in the decimal expansion of $\pi$ before picking it up again! Furthermore, after completing the “tree,” a few thousand more digits were skipped before constructing the base of the tree. And these latter digits were used twice!

The first 4,000 digits of $\pi$ are shown below (in blocks of 10 digits). The ones that appear on the T-shirt are marked in boldface and are underlined.

3.
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944
5923078164 0628620899 8628034825 3421170679 8214808651 3282306647
0938446095 5058223172 5359408128 4811174502 8410270193 8521105559
6446229489 5493038196 4428810975 6659334461 2847564823 3786783165
2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360
0113305305 4882046652 1384146951 9415116094 3305727036 5759591953
0921861173 8193261179 3105118548 0744623799 6274956735 1885752724
8912279381 8301194912 9833673362 4406566430 8602139494 6395224737
1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901
2249534301 4654958537 1050792279 6892589235 4201995611 2129021960
8640344181 5981362977 4771309960 5187072113 4999999837 2978049951
0597317328 1609631859 5024459455 3469083026 4252230825 3344685035
2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532
1712268066 1300192787 6611195909 2164201989 3809525720 1065485863
2788659361 5338182796 8230301952 0353018529 6899577362 2599413891
2497217752 8347913151 5574857242 4541506959 5082953311 6861727855
8890750983 8175463746 4939319255 0604009277 0167113900 9848824012
8583616035 6370766010 4710181942 9555961989 4676783744 9448255379
7747268471 0404753464 6208046684 2590694912 9331367702 8989152104
7521620569 6602405803 8150193511 2533824300 3558764024 7496473263
9141992726 0426992279 6782354781 6360093417 2164121992 4586315030
2861829745 5570674983 8505494588 5869269956 9092721079 7509302955
3211653449 8720275596 0236480665 4991198818 3479775356 6369807426
5425278625 5181841757 4672890977 7727938000 8164706001 6145249192
1732172147 7235014144 1973568548 1613611573 5255213347 5741849468
4385233239 0739414333 4547762416 8625189835 6948556209 9219222184
2725502542 5688767179 0494601653 4668049886 2723279178 6085784383
8279679766 8145410095 3883786360 9506800642 2512520511 7392984896
0841284886 2694560424 1965285022 2106611863 0674427862 2039194945
0471237137 8696095636 4371917287 4677646575 7396241389 0865832645
9958133904 7802759009 9465764078 9512694683 9835259570 9825822620
5224894077 2671947826 8482601476 9909026401 3639443745 5305068203
4962524517 4939965143 1429809190 6592509372 2169646151 5709858387
4105978859 5977297549 8930161753 9284681382 6868386894 2774155991
8559252459 5395943104 9972524680 8459872736 4469584865 3836736222
6260991246 0805124388 4390451244 1365497627 8079771569 1435997700
1296160894 4169486855 5848406353 4220722258 2848864815 8456028506
0168427394 5226746767 8895252138 5225499546 6672782398 6456596116
3548862305 7745649803 5593634568 1743241125 1507606947 9451096596
0940252288 7971089314 5669136867 2287489405 6010150330 8617928680
9208747609 1782493858 9009714909 6759852613 6554978189 3129784821
6829989487 2265880485 7564014270 4775551323 7964145152 3746234364
5428584447 9526586782 1051141354 7357395231 1342716610 2135969536
2314429524 8493718711 0145765403 5902799344 0374200731 0578539062
1983874478 0847848968 3321445713 8687519435 0643021845 3191048481
0053706146 8067491927 8191197939 9520614196 6342875444 0643745123
7181921799 9839101591 9561814675 1426912397 4894090718 6494231961
5679452080 9514655022 5231603881 9301420937 6213785595 6638937787
0830390697 9207734672 2182562599 6615014215 0306803844 7734549202
6054146659 2520149744 2850732518 6660021324 3408819071 0486331734
6496514539 0579626856 1005508106 6587969981 6357473638 4052571459
1028970641 4011097120 6280439039 7595156771 5770042033 7869936007
2305587631 7635942187 3125147120 5329281918 2618612586 7321579198
4148488291 6447060957 5270695722 0917567116 7229109816 9091528017
3506712748 5832228718 3520935396 5725121083 5791513698 8209144421
0067510334 6711031412 6711136990 8658516398 3150197016 5151168517
1437657618 3515565088 4909989859 9823873455 2833163550 7647918535
8932261854 8963213293 3089857064 2046752590 7091548141 6549859461
6371802709 8199430992 4488957571 2828905923 2332609729 9712084433
5732654893 8239119325 9746366730 5836041428 1388303203 8249037589
8524374417 0291327656 1809377344 4030707469 2112019130 2033038019
7621101100 4492932151 6084244485 9637669838 9522868478 3123552658
2131449576 8572624334 4189303968 6426243410 7732269780 2807318915
4411010446 8232527162 0105265227 2111660396…

I can understand getting a digit or two wrong on the T-shirt, but I have no idea how anybody could have possibly made a mistake like this.

Upon discovering this, my first reaction reflected my inner mathematician: “I want a refund.” After all, $\pi$ has been known to 47 decimal places since the 1700s, long before the advent of modern computers. However, upon further reflection, I decided that being able to tell this story of a Christmas $\pi$ T-shirt that incorrectly printed the digits of $\pi$ — and especially the story of how this error was brought to my attention — was by itself well worth the price of the shirt.

Christmas ornament

Last year, one of my former students gave me this Christmas ornament. I couldn’t wait to hang it on our family’s Christmas tree.