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

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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?

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.

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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:

 

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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: Geometric 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 Zachery Hasegawa. His topic, from Precalculus: geometric sequences.

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How has this topic appeared in pop culture? (movies, TV, current music, video games, etc.)

Geometric sequences appear frequently in pop culture.  One example that immediately comes to mind is the movie The Happening starring Mark Wahlberg and Zoe Deschanel.  In the movie, there is a scene where a gentleman is trying to distract another woman from the chaos happening outside the jeep they’re traveling in.  He says to her “If I start out with a penny on the first day of a 31 day month and kept doubling it each day, so I’d have .01 on day 1, .02 on day 2, etc.  How much money will I have at the end of the month?” The woman franticly spouts out a wrong answer and the gentleman responds “You’d have over ten million dollars by the end of the month”.  The car goes on to crash just after that scene but as a matter of fact, you’d have exactly $10,737,418.20 at the end of the 31-day month.  This is an example of a geometric sequence because you start out with 0.01 and to get to the next term (day), you would multiply by a common ratio of 2.

 

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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?

Geometric sequences are popularly found in Book IX of Elements by Euclid, dating back to 300 B.C.  Euclid of Alexandria, a famous Greek mathematician also considered the “Father of Geometry” was the main contributor of this theory.  Geometric sequences and series are one of the easiest examples of infinite series with finite sums.  Geometric sequences and series have played an important role in the early development of calculus, and have continued to be a main case of study in the convergence of series.  Geometric sequences and series are used a lot in mathematics, and they are very important in physics, engineering, biology, economics, computer science, queuing theory, and even finance.

 

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How can technology (specifically Khan Academy/YouTube) be used to effectively engage students with this topic:

 

I really like the video that Khan Academy does on YouTube about Geometric Sequences.  This particular video is a very good introduction to Geometric Sequences because he explains the difference between Geometric Sequences and Series, which I thought to be helpful because I always got the two confused with each other.  Mr. Khan starts out by explaining what exactly a Geometric Sequence is. He describes a Geometric sequence as “A progression of numbers where each successive number is a fixed multiple of the one before it.” He goes on to give numerical examples to specifically show you what he means.  He explains that a1 is typically our first term; a2 is the second term, etc.  He then explains that to get from a1 to a2, you will multiply a1 by the “common ratio” usually represented by “r. For example, “3, 12, 48, 192” is a finite geometric sequence where the common ratio, r, is 4 because to go from 3 to 12 or from 12 to 48, you multiply by 4. He goes on to explain that a Geometric Sequence is a list (sequence) of numbers (terms) that are being multiplied by a common ratio and that a Geometric Series is the sum of the terms (numbers) in the Geometric Sequence.  Using the same numbers as from the Geometric Sequence above, the geometric series is “3+12+48+192”.

 

 

References:

 

 

 

 

Arithmetic and Geometric Series: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how I remind students about Taylor series. I often use this series in a class like Differential Equations, when Taylor series are needed but my class has simply forgotten about what a Taylor series is and why it’s important.

Part 1: Deriving the formulas for the nth term of arithmetic and geometric sequences.

Part 2: Pedagogical thoughts on conceptual barriers that students often face when encountering sequences and series.

Part 3: The story of how young Carl Frederich Gauss, at age 10, figured out how to add the integers from 1 to 100 in his head.

Part 4: Deriving the formula for an arithmetic series.

Part 5: Deriving the formula for an arithmetic series, using mathematical induction. Also, extensions to other series.

Part 6: Deriving the formula for an arithmetic series, using telescoping series. Also, extensions to other series.

Part 7: Pedagogical thoughts on assessing students’ depth of understanding the formula for an arithmetic series.

Part 8: Deriving the formula for a finite geometric series.

Part 9: Infinite geometric series and Xeno’s paradox.

Part 10: Deriving the formula for an infinite geometric series.

Part 11: Applications of infinite geometric series in future mathematics courses.

Part 12: Other commonly-arising infinite series.

 

 

 

Advent calendar

Source: http://www.xkcd.com/994/

Engaging students: Geometric 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 Michelle McKay. Her topic, from Precalculus: geometric sequences.

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A.     What interesting word problems using this topic can your student do now?

In the movie Pay it Forward (2000), the young boy Trevor has the following idea: He can make the world a better place by encouraging people to help others.

payitforward

If Trevor helps three people and asks that they help three other people instead of repaying him, how can we represent this as a sequence? Write the first 5 terms.  (Hint: Let Trevor be represented by the number 1.)

What is a formula that can give us the amount of people affected after n terms?

When will 177,147 people be affected? 14,348,907 people?

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

For science classes, geometric sequences can be used to represent data collected for exponential growth or decay of a population or solution over time. Below are some examples of how geometric sequences can appear in a future science class.

Biology: A researcher is determining whether a certain species of mouse is thriving in its environment or becoming endangered. The total population of the mouse is calculated each year. What conclusions can you draw from the data below?

Year

Population

1

240

2

720

3

2,160

4

6,480

5

19,440

Chemistry: A student has been monitoring the amount of Na in a solution. Based off the data collected, when will the Na in the solution be negligible?

Day

Na %

1 95%
2 42.75%
3 19.24%
4 8.65%

Physics: Students in a physics class measure the following heights of a ball that has been dropped from 10 feet in the air. Each measured height is taken at the highest point in the ball’s trajectory.

10

8

6.4

5.12

4.096

Source: http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-256s.html

green lineA.     Application of geometric sequences.

The following prompt can be used as a short response or in-class debate:

A student is standing a distance of x meters away from the front of the classroom. If he decreases the distance between himself and the front of the classroom by half each time he moves, will he ever reach the front of the classroom? What if instead of a student, we use a point on a line? Justify your answer.

 

Formula for an infinite geometric series (Part 10)

I conclude this series of posts by considering the formula for an infinite geometric series. Somewhat surprisingly (to students), the formula for an infinite geometric series is actually easier to remember than the formula for a finite geometric series.

One way of deriving the formula parallels the derivation for a finite geometric series. If a_1, a_2, a_3, \dots are the first terms of an infinite geometric sequence, let

S = a_1 + a_2 + a_3 + \dots

Recalling the formula for an geometric sequence, we know that

a_2 = a_1 r

a_3 = a_1 r^2

\vdots

Substituting, we find

S = a_1 + a_1 r+ a_1 r^2 \dots

Once again, we multiply both sides by -r.

-rS = -a_1r - a_1 r^2- a_1 r^3 \dots

Next, we add the two equations. Notice that almost everything cancels on the right-hand side… except for the leading term a_1.  (Unlike yesterday’s post, there is no “last” term that remains since the series is infinite.) Therefore,

S - rS = a_1

S(1-r) = a_1

S = \displaystyle \frac{a_1}{1-r}

A quick pedagogical note: I find that this derivation “sells” best to students when I multiply by -r and add, as opposed to multiplying by r and subtracting.

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The above derivation is helpful for remembering the formula but glosses over an extremely important detail: not every infinite geometric series converges. For example, if a_1 = 1 and r = 2, then the infinite geometric series becomes

1 + 2 + 4 + 8 + 16 + \dots,

which clearly does not have a finite answer. We say that this series diverges. In other words, trying to evaluate this sum makes as much sense as trying to divide a number by zero: there is no answer.

That said, it can be shown that, as long as -1 < r < 1, then the above geometric series converges, so that

a_1 + a_1 r + a_1 r^2 + \dots = \displaystyle \frac{a_1}{1-r}

The formal proof requires the use of the formula for a finite geometric series:

a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1} = \displaystyle \frac{a_1(1-r^n)}{1-r}

We then take the limit as n \to \infty:

\displaystyle \lim_{n \to \infty} a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1} = \displaystyle \lim_{n \to \infty} \frac{a_1(1-r^n)}{1-r}

a_1 + a_1 r + a_1 r^2 + \dots = \displaystyle \lim_{n \to \infty} \frac{a_1(1-r^n)}{1-r}

On the right-hand side, the only piece that contains an n is the term r^n. If -1 < r < 1, then r^n \to 0 as n \to \infty. (This limit fails, however, if r \ge 1 or r \le -1.) Therefore,

a_1 + a_1 r + a_1 r^2 + \dots = \displaystyle \lim_{n \to \infty} \frac{a_1(1-0)}{1-r} = \displaystyle \frac{a_1}{1-r}

Pedagogical thoughts about sequences and series (Part 2)

After yesterday’s post about arithmetic and geometric sequences, I’d like to contribute some thoughts about teaching this topic, based on my own experience over the years.

1. Some students really resist the subscript notation a_n when encountering it for the first time. To allay these concerns, I usually ask my students, “Why can’t we just label the terms in the sequence as a, b, c, and so on?” They usually can answer: what if there are more than 26 terms? That’s the right answer, and so the a_n is used so that we’re not limited to just the letters of the English alphabet.

Another way of selling the a_n notation to students is by telling them that it’s completely analogous to the f(x) notation used more commonly in Algebra II and Precalculus. For a “regular” function f(x), the number x is chosen from the domain of real numbers. For a sequence a_n, the number n is chosen from the domain of positive (or nonnegative) integers.

2. The formulas in Part 1 of this series (pardon the pun) only apply to arithmetic and geometric sequences, respectively. In other words, if the sequence is neither arithmetic nor geometric, then the above formulas should not be used.

While this is easy to state, my observation is that some students panic a bit when working with sequences and tend to use these formulas on homework and test questions even when the sequence is specified to be something else besides these two types of sequences. For example, consider the following problem:

Find the 10th term of the sequence 1, 4, 9, 16, \dots

I’ve known pretty bright students who immediately saw that the first term was 1 and the difference between the first and second terms was 3, and so they answered that the tenth term is 1 + (10-1)\times 3 = 28… even though the sequence was never claimed to be arithmetic.

I’m guessing that these arithmetic and geometric sequences are emphasized so much in class that some students are conditioned to expect that every series is either arithmetic or geometric, forgetting (especially on tests) that there are sequences other than these two.

3. Regarding arithmetic sequences, sometimes it helps by giving students a visual picture by explicitly make the connection between the terms of an arithmetic sequence and the points of a line. For example, consider the arithmetic sequence which begins

13, 16, 19, 22, \dots

The first term is 13, the second term is 16, and so on. Now imagine plotting the points (1,13), (2,16), (3,19), and (4,22) on the coordinate plane. Clearly the points lie on a straight line. This is not surprising since there’s a common difference between terms. Moreover, the slope of the line is 3. This matches the common difference of the arithmetic sequence.

4. In ordinary English, the words sequence and series are virtually synonymous. For example, if someone says either, “a sequence of unusual events” or “a series of unusual events,” the speaker means pretty much the same thing

However, in mathematics, the words sequence and series have different meanings. In mathematics, an example of an arithmetic sequence are the terms

1, 3, 5, 7, 9, \dots, 99

However, an example of an arithmetic series would be

1 + 3 + 5 + 7 + 9 + \dots + 99

In other words, a sequence provides the individual terms, while a series is a sum of the terms.

When teaching this topic, I make sure to take a minute to emphasize that the words sequence and series will mean something different in my class, even though they basically mean the same thing in ordinary English.

Formulas for arithmetic and geometric sequences (Part 1)

I’m not particularly a fan of memorizing formulas. Apparently, most college students aren’t fans either, because they often don’t have immediate recall of certain formulas from high school when they’re needed in the collegiate curriculum.

While I’m not a fan of making students memorize formulas, I am a fan of teaching students how to derive formulas. Speaking for myself, if I ever need to use a formula that I know exists but have long since forgotten, the ability to derive the formula allows me to get it again.

Which leads me to today’s post: the derivation of the formulas for the nth term of an arithmetic sequence and of a geometric sequence. This topic is commonly taught in Precalculus but, in my experience, is often forgotten by students years later when needed in later classes.

green lineAn arithmetic sequence is specified by two numbers: the first term and the common difference between terms. For example, if the first term is 16 and the common difference is 3, then the sequence begins as

16, 19, 22, 25, 28, 31, 34, \dots

If the first term is 29 and the common difference is -4, then the sequence begins as

29, 25, 21, 17, 13, 9, 5, 1, -3, \dots

For those of us old enough to remember, our favorite arithmetic sequences came from Schoolhouse Rock:

Let’s discuss the first arithmetic sequence, whose first seven terms are:

16, 19, 22, 25, 28, 31, 34, \dots

How do we get the 8th term? That’s easy: we just add 3 to 34 to get 37.

How to we get the 100th term. That’s easy: we just add 3 to the 99th term.

Oops. We don’t know the 99th term. To get the 99th term, we need the 98th term, which in turn requires the 97th term. Et cetera, et cetera, et cetera.

The trouble (so far) is that an arithmetic sequence is recursively defined: to get one term, I add something to the previous term. Mathematically, the arithmetic sequence is defined by

a_n = a_{n-1} + d,

where d is the common difference. This can be very intimidating to students when seeing it for the first time. So, to make this formula less intimidating, I usually read this equation as “Each next term in the sequence is equal to the previous term in the sequence plus the common difference.”

It would be far better to have a closed-form formula, where I could just plug in 100 to get the 100th term, without first figuring out the previous 99 terms.

To this end, we notice the following pattern:

  • Second term: 19 = 16 + 3
  • Third term: 22 = 19 + 3 = 16 + 3 + 3 = 16 + 2 \times 3
  • Fourth term: 25 = 22+ 3 = 16 + (2 \times 3) + 3 = 16 + 3 \times 3
  • Fifth term: 28 = 25+ 3 = 16 + (3 \times 3) + 3 = 16 + 4 \times 3
  • Sixth term: 31 = 28+ 3 = 16+ (4 \times 3) + 3 = 16 + 5 \times 3
  • Seventh term: 34 = 31 + 3 = 16 + (5 \times 3) + 3 = 16 + 6 \times 3

It looks like we have a pattern, so we can guess that:

  • One hundredth term = 16 + (100-1) \times 3 = 313

In general, we have justified the closed-form formula

a_n = a_1 + (n-1)d,

where a_1 is the first term, and d is the common difference.  In words: to get the nth term of an arithmetic sequence, we add d to the first term n-1 times. (This may be formally proven using mathematical induction, though I won’t do so here.)

green lineA closed-form formula for a geometric sequence is similarly obtained. In a geometric sequence, each term is equal to the previous term multiplied by a common ratio. Mathematically, the geometric sequence is recursively defined by

a_n = a_{n-1}r,

where r is the common ratio. For example, if the first term is 3 and the common ratio is 2, then the first few terms of the sequence are

3, 6, 12, 24, 48, dots

By the same logic used above, to get the nth term of an geometric sequence, we multiply r to the first term n-1 times. Thus justifies the formula

a_n = a_1 r^{n-1},

which may be formally proven using mathematical induction.