Engaging students: Arithmetic 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 Danielle Pope. Her topic, from Precalculus: arithmetic sequences.

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

In the future, the topic of arithmetic sequences will be built upon by introducing another sequence, the geometric sequence. A geometric sequence is just a sequence of multiples instead of increasing by a constant. The next topic introduced will be finding the sum of a sequence of numbers. This will be introduced as a series. The summation symbol will also be introduced to kids and they will learn that new notation. Summations will bring along many formulas for finding the leading coefficient and will show up later in Calculus 2 classes when talking about convergence and divergence of series. Another one of the things that kids will always be doing with sequences and series is finding the general form of a given sequence or series. Through school, this idea will never change the sequence and series will just get harder to identify.
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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

An arithmetic sequence is a set of numbers that have a constant difference between each term. One of the main people that come up when researching these sequences is Carl Friedrich Gauss. Many math-loving people know him as the “Prince of mathematicians”. He is famous for coming up with the equations to solve the sum of an arithmetic sequence. This comes as no surprise that he came up with this formula. The surprising thing about this realization is that he made it at an age young enough to still be in grade school. Stories say that Gauss was asked to solve for the sum on the board in grade school and used the formula of M ( M + 1 ) / 2 to solve for the correct answer. This just goes to show that anyone can, in fact, contribute to the greater good of mathematics at any age.

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

One of the first civilizations that utilized sequences was the Egyptians. They used the sequence of multiples of 2 to do their multiplication. The basic sequence is 1, 2, 4, 8, 16, 32, … and we are trying to solve 24 x 13 with the process pictured below.

The process behind this is to write the multiple of 2 sequences down the left side of the paper until you reach the largest multiple of 2 without going over the second number being multiplied, in this case, 13. Once that is done set the first term on the right side equal to the first number being multiplied, in this case, 24. Next, multiply the right side by 4 until you get the same amount of terms on the left side. Lastly find the sum of numbers on the left that add to 13, which are 1, 4, and 8. Add the corresponding multiples from the side, 24 + 96 + 192 = 312. The right side sum of the corresponding numbers checked on the left gives the product of the original problem, i.e. 312. This trick is cool to show just on its own but it’s also cool because it uses something as simple as a specific list of numbers aka a sequence of numbers.

References

http://www.softschools.com/facts/scientists/carl_friedrich_gauss_facts/827/

https://rabungapalgebraiii.wikispaces.com/Arithmetic+Sequences+and+Series

http://www.math.wichita.edu/MathCircle/docs/egyptian_arithmetic.pdf

Engaging students: Introducing the number e

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 Deanna Cravens. Her topic, from Precalculus: introducing the number e.

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How was this topic adopted by the mathematical community?

The number e is a relatively newer irrational number if compared to pi. However, it first made its appearance very subtly in 1618. Napier was working on a table of natural logarithms, however it was not noted that the base was e. There were a few other appearances of e but mathematicians had not truly made a connection to it. Eventually in 1683, Jacob Bernoulli was looking at a business application dealing with continuously compounded interest and recognized that the log function and the exponential function were inverses. In 1690, a letter was written by Leibniz and e officially had a name, except it was called ‘b’ at the time. As it comes to no surprise, Euler had his hand in discovering e. He published Introductio in Analysin infinitorum in 1748 where he showed that e is the limit of (1 + 1/n)^n. Now Euler did not explicitly prove that e is irrational, however most people accepted it at that point, but it was indeed later proven.

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How could you as a teacher create an activity or project that involves your topic?
Where does the number e come from? Well, the answer is a business application dealing with continuously compounded interest. However, students in a pre-calculus class can easily discover the number e without having to use the calculus behind it. Simply give students this short activity at the beginning of class.

One of the good things about this activity is that it gives a brief snippet of the history of e before students begin to calculate it. Then, students can easily use a calculator and plug in the listed values in the table into the equation (1+1/n)^n. As the numbers get increasingly large, students will notice that they will all appear to be getting closer to 2.718… which is now known as the number e. As a teacher it is important to note that e is like pi, it is an irrational number that goes on forever and doesn’t have any sort of repeating pattern, yet it is extremely important in mathematics.

 

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

This video would be excellent to show students who are asking, “why is e so important or where does it come from?” The video starts out by stating what e is approximately equal to. Then it gives a brief history about e and talks about compounded interest. It does a great job at explaining compounded interest. It is executed in a way where pre-calculus students can easily understand the concept. It also uses good visual cues to show how it would work. Next it lists several applications of e. These applications include: statistics through the normal curve, biology by modeling population growth, and physics by the exponential decay of a radioactive material. Overall, it does a great job showing the importance of e in real world applications. Thus, showing the importance of e to a pre-calculus students.

References:
1. http://www.classzone.com/eservices/home/pdf/student/LA208CAD.pdf
2. http://www-history.mcs.st-and.ac.uk/HistTopics/e.html
3. https://www.youtube.com/watch?v=R0oUeLQIbIk

Engaging students: Synthetic Division

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 Deetria Bowser. Her topic, from Precalculus: synthetic division.

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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.

It is tough to find websites or technology that help with synthetic division, due to the fact that most websites consists of a long list of instruction, which is not engaging. One website that does seem helpful to a student learning synthetic division is: http://emathlab.com. Under the Algebra tab one can select the option for polynomial functions, then synthetic division. Once selected, a synthetic division activity will pop up. In this activity the student is given a polynomial (of third degree most of the time) divided by a degree one polynomial. The student is then expected to correctly fill the cells with the correct numbers for synthetic division. If they do not get the correct number, the cell turns red and they have to keep trying until they get the answer correct. This activity will be beneficial to students because they will be able to get a feel on the correct placement of numbers when using synthetic division. Additionally, this tool will help them realize what to do when they get polynomials, such as x^3-1. Finally this online tool will allow the students to evaluate themselves.

 

 

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The idea of synthetic division is used to find the zeros of a function. One may need to find zeros of a function in a variety of mathematics and science courses. For example, in physics, one may need to find the roots of a trajectory equation. To find said roots, one could use synthetic division. Also an example of finding roots could be used to help in computer programming. On math.stackexchange.com a programming student presents the following problem: “I am currently programming a simulation for a pinball game and want to calculate the time when the ball hits a circle (if they collide at some point). For the calculation part, I’m adding the radius of the ball to the radius of the circle, so that i only have to check if the midpoint of the ball collides with the circle. Of course, the circle is displayed with it’s original radius.

Now for the ball’s (midpoint) trajectory i’ve got these two equations who define the movement of the ball on the x- and y-axis (depending on the gravitational acceleration):

x(t)=s_x+v_x t, \quad y(t)=s_y+v_y t− \frac{1}{2} g t^2,

with (s_x,s_y)= starting point of the ball, (v_x,v_y)= initial velocity, g= gravitational acceleration and t= time.

To check for collision, I took these two equations and put them into the equation of a circle. Once multiplied out the student got something of the form: a t^4+bt^3+ct^2+dt+e=0. If the coefficients a,b,c,d,e are rational numbers, then he will be able to use synthetic division to find all of the roots, and successfully create his game.

 

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

In previous courses, students are taught to find zeros by either graphing, guessing and plugging in a number for x and hoping that the result is zero, or using long division. Synthetic division provides a more systematic way of finding zero’s than just guessing, and can prove to be quicker than graphing and using long division. Additionally, synthetic division can expand on the idea of showing something is not a factor. For instance, when one tries to synthetically divide the polynomial x^4-3x^2+5x-7 by x-2 one will get a remainder of 7. This is another way of proving that x-2 is not a factor of x^4-3x^2+5x-7. Also, one now knows what the polynomial x^4-3x2+5x-7 is when x = 2. Synthetic division, extends the idea of finding factors and non-factors of polynomials, as well as solutions to polynomials at a specific x.

References

https://math.stackexchange.com/questions/1462858/how-to-find-the-zeros-of-a-fourth-degree-polynomial-without-integer-coefficient
http://emathlab.com/Algebra/PolyFunctions/SyntheticDiv.php

 

 

 

 

 

Statistics for People in a Hurry

The following article was recommended to me by a former student: https://towardsdatascience.com/statistics-for-people-in-a-hurry-a9613c0ed0b. It’s synopsis is in the opening paragraph:

Ever wished someone would just tell you what the point of statistics is and what the jargon means in plain English? Let me try to grant that wish for you! I’ll zoom through all the biggest ideas in statistics in 8 minutes! Or just 1 minute, if you stick to the large font bits.