# Engaging students: Half-life

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 Trenton Hicks. His topic: working with the half-life of a radioactive element.

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

The topic of half life is a direct intersection of math and chemistry. In addition to being a common precalculus problem, we see half life come up in radioactive decay in chemistry. Half life is a concept that extends all the way into upper college chemistry, physics, and even archaeology when it comes to carbon dating. If students use carbon dating to any extent, they can use half life to determine the age of organic material since carbon 14 is radioactive (Wood). Since half life has to due with nuclear chemistry, this can also tie into nuclear power, since half life is crucial in computations related to efficiency and nuclear engineering. Half life is a form of exponential decay. If students have a thorough understanding of half life, they can better understand other natural phenomena that exhibit properties that are consistent with exponential decay. These phenomena include RC circuits, atmospheric pressure, and toxicity.

In Chernobyl Ukraine, 1986, there was a disaster at a nuclear power plant that has had lasting effects on the environment, people, and culture. The initial explosion was harmful enough, as 2 people lost their lives. Furthermore, radiation leaked into the atmosphere, and it’s speculated that many individuals are suffering the health consequences. When this story first broke, it shook everyone, and scared people away from nuclear power. Lately, there was another documentary that came to light about the incident from HBO. Many people don’t know that the former power plant is still very dangerous to this day. Why? Because the highly radioactive byproducts of the meltdown have half lives that makes them stick around for quite a while. One particularly dangerous isotope, caesium 137, has a half life of about 30 years. This means that in 2016, about half of the caesium decayed. Half of the sheer amount of caesium that was leaked due to the meltdown is still an enormously dangerous amount. News and documentaries report that there’s still a massive constructive effort to contain the radiation. Showing these news stories to students will convey the importance of half life and give them a little bit of insight into how much care should be given to nuclear power.

Half life began as a model proposed by Rutherford in the late 1800’s and very early 1900’s. Rutherford discovered that radioactive decay would turn one element into another. This change happens at a rate that we recognize as exponential decay, hence the model we use is consistent with that idea. Rutherford’s work would soon earn him a Nobel Prize. Other disciplines have taken the idea of “half life,” and have created convincing arguments for how the universe behaves. For instance, toxicology uses half life to convey how potent a dose of toxin is versus long it takes for the body to metabolize the toxin. Another notable development is the blog post on the fs website (linked below) that discusses half lives in terms of how our brains retain information, as well as the information itself. Relaying that half life isn’t just a chemistry or math topic to students, and providing them with this history might just increase the half life on their retaining of the concept.

References:

Fs blog:

Half Life: The Decay of Knowledge and What to Do About It

Sources:
Author: Rachael Wood

Click to access ExponentialDecay.pdf

Click to access Section_4.5.pdf

# Engaging students: Half-life

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 Kerryana Medlin. Her topic: working with the half-life of a radioactive element.

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

Depending on when they take precalculus, this topic may appear earlier or later in chemistry. The following is the list of TEKS for this topic in chemistry.

112.35. Chemistry (12) Science concepts. The student understands the basic processes of nuclear chemistry. The student is expected to:
(A) describe the characteristics of alpha, beta, and gamma radiation;
(B) describe radioactive decay process in terms of balanced nuclear equations; and
(C) compare fission and fusion reactions.

This is likely the most immediate application the students will encounter, but this topic also appears in calculus and, later, in the topic of differential equations, since it involves exponential decay. This topic can also be brought up in environmental science to mention the lifetime of radioactive isotopes. When a student crunches the numbers on the lifetimes of these isotopes, they can see that sometimes a small action has a huge ripple effect, especially for isotopes that humans bring into the picture.

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

Ernest Rutherford received a Nobel Prize in Chemistry in 1908 for his discovery of the half-life of radioactive materials and his insistence that we apply this information to find the Earth’s age (Mastin, 2009). This later became more of a reality when Willard Libby started to develop carbon dating in 1946 (Radiocarbon Dating). Since then, carbon dating has been used to find the age of historical artifacts and bones, allowing historians to find more accurate time frames of events.

Carbon is not the only radioactive isotope. There are others which come to mind more readily when the word “radioactive” is used. These are typically the elements used for nuclear reactors. These are elements which readily undergo nuclear fission, which is the splitting of atoms, which releases energy. Uranium and Plutonium are the most common of these isotopes. Uranium-235 is the most commonly used for reactors and bombs (Brain and Lamb, 2000). This is probably the more interesting part of half-lives of elements and can extend the learning to an environmental issue such as nuclear waste, which takes an extremely long time to decay and which the U.S. Government has, in the past, not handled so well. (But I am not going into that, lest I go on a rant).

The last piece of history worth mentioning is fairly recent (and can be seen in real life and in the game mentioned later in this paper) which is that half-lives are not so clear cut. There is definitely a lot of estimating involved in the accepted half-life values. There is an article about this if you are interested (http://iopscience.iop.org/article/10.1088/0026-1394/52/3/S51/pdf), but I will leave it at this: much like most mathematical models, there is error in the half-life model, and the model formed may be a best fit, but there are always outliers for data and while carbon dating and half-lives of Uranium can give great estimates of what we are working with, they are not perfect.

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

For this topic, there is an interactive simulation posted on PHET. It lends itself to a guided worksheet which would allow students to use the simulations to create the functions for each half-life.
So the following would be an example of said worksheet without spaces for actual answers:

Radioactive Half-Life of Carbon-14 and Uranium-238

At the top of the game window are four different tabs: Half Life, Decay Rates, Measurement, and Dating Game. We will be going through each one in that order.

Some information about radioactive isotopes: An isotope is an element which has the same number of protons in its nucleus, but a differing number of neutrons, thus making it radioactive. These elements have lives which are defined by the time it takes to no longer be radioactive.

Part I: Half Life

Select the Carbon-14 atom and start placing the atoms in the white area. (The “add 10” tool is helpful here.) Then observe as each goes to Nitrogen-14 (This means the element is no longer radioactive and the radioactive isotope has run its course.)

What do you observe about the lives of the isotopes?

What time-frame do these lives fall into?

Do the same for Uranium-238 and record the time-frame.

Part II: Decay Rates

This part works by adjusting the slider and allowing the isotopes to run the course of their lives.

What does the graph on the bottom tell us?

How does one read the half-life of an isotope from this graph?

At what percent do we find the first half-life?

What is the half-life of Carbon-14 from this graph? Half-life of Uranium-238?

Part III: Measurement

On this one, you activate two separate events and then take readings of the amount of Carbon-14 and Uranium-238 in the objects.

Which item contains the Carbon-14? The Uranium-238?

Use the pause feature as you are taking the readings to find precise values of the half-lives.

At what percentages should we be reading the half-lives?

Use this data to create a function to model the half-life of both isotopes.

Part IV: Dating Game

Use your functions to estimate the date of two of the items (One C-14 and one U-238) in the dating game. Write down the name of the item and the estimated age of the item.

References:

Brain, Marshall and Lamb, Robert. (2000). How Nuclear Power Works. How Stuff Works. Retrieved from
https://science.howstuffworks.com/nuclear-power1.htm
Mastin, Luke. (2009). Important Scientists: Ernest Rutherford (1871-1937). The Physics of the Universe.

# 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 Jillian Greene. Her topic, from Precalculus: introducing the number e.

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

By this point in their mathematics career, the students have had plenty of experience with simple and compound interest formulas. Whether or not they discovered it them themselves through exploration in a class or their teacher just gave it to them, they’ve used it before. Now we can do an exploration activity that will connect that formula to the number e, and then to the limit. The activity will say: what if you invested $1 for 1 year at 100% compound interest? It’s a pretty good deal! But how much does the number of compounding periods affect the final value? Using the formula they have, A=P(1+r/n)^nt, they will calculate how much money they will make if it’s compounded: • Yearly • Biannually • Quarterly • Weekly • Daily • Hourly • Every minute • And every second The first time it’s compounded, the final value will be$2. However, the more compounding periods you add, the closer to e you’ll get. For instance, weekly would be A=1(1+1/52)^52=2.69259695. Every second will get you A=1(1+1/31536000)^31536000=2.71828162, which is pretty to 2.718. The last three calculations will actually begin with 2.718. We can have some discussion with this as a class, bringing in the concept of limits. Then we can assess and see if anyone has seen this number before. If not, they can pop out their calculators and you can have them type “e” and then hit enter, and blow their minds.

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

Though Euler does not receive credit for the first discovery of the number e, he does receive credit for naming it and first publishing it. Some say the e means exponential, some say he’d already published uses for a-d, and some say he named it after himself. He is quoted directly for saying “For the number whose logarithm is unity, let e be written, which is 2,7182817… [sic] whose logarithm according to Vlacq is 0,4342944… “ regarding the number e. He also has a couple of other choice quotes that illustrate his humor, ie “[upon losing the use of his right eye] ‘Now I will have less distraction.’” And “”Sir,

hence God exists; reply!” In response to the French philosophe Diderot, who was trying to convert the court of Catherine the Great of Russia to atheism. Diderot had no idea what Euler was talking about and left the court to a chorus of laughter.”  Back to e, however. If Euler did not first discover it, who did? A man name John Napier did the best he could to discover e. Napier was alive from 1550-1617, so he did not have access to a rich history of advanced algebra. Logarithm tables existed, some close to natural log, but none to identify this mystical number. Napier was merely trying to find an easier way to approach multiplication (and consequently exponentiation). His work, Construction of the Marvelous Rule of Logarithms, he states that X=Nap log y, where Nap log (107)=0. In today’s terms, with today’s math, we can translate that to Nap log y = 107 log1/e(y/107).

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

After some discussion on this topic, if my class is a pre-AP or particularly curious class, I will have them go around and read this poem about e out loud. Then from this poem, I can have the students split up into groups. Each group will be responsible for dissecting this poem for certain things and then presenting their most interesting/exciting/relatable findings. One group will tackle the names; what history lesson is given to us here? Another group will handle applications; what did the various figures say we can do with e? The final group will report back on different representations of e; what all is e equal to? My expectations here would be for the students to see the insanely vast history and application of this number and gain some appreciation. I would expect to see Napier, Euler, and Leibniz for sure from the first group. From the second group, I would expect continuous compound interest, 1/e in probability and statistics, and calculus. The third group would be expected to present the numerical value of e, the limit that e is equal to, its infinite sum representation, and Euler’s identity. A number worthy of a 500 word poem and a slew of historical mathematicians must be important.

The Enigmatic Number e

by Sarah Glaz

It ambushed Napier at Gartness,
like a swashbuckling pirate
leaping from the base.
He felt its power, but never realized its nature.
e‘s first appearance in disguise—a tabular array
of values of ln, was logged in an appendix
to Napier‘s posthumous publication.
Oughtred, inventor of the circular slide rule,
still ignorant of e‘s true role,
performed the calculations.

A hundred thirteen years the hit and run goes on.
There and not there—elusive e,
escape artist and trickster,
weaves in and out of minds and computations:
Saint-Vincent caught a glimpse of it under rectangular hyperbolas;
Huygens mistook its rising trace for logarithmic curve;
Nicolaus Mercator described its log as natural
without accounting for its base;
Jacob Bernoulli, compounding interest continuously,
came close, yet failed to recognize its face;
and Leibniz grasped it hiding in the maze of calculus,
natural basis for comprehending change—but
misidentified as b.

The name was first recorded in a letter
Euler sent Goldbach in November 1731:
“e denontat hic numerum, cujus logarithmus hyperbolicus est=1.”
Since a was taken, and Euler
was partial to vowels,
e rushed to make a claim—the next in line.

We sometimes call e Euler‘s Number: he knew
e in its infancy as 2.718281828459045235.

On Wednesday, 6th of May, 2009,
e revealed itself to Kondo and Pagliarulo,
digit by digit, to 200,000,000,000 decimal places.
It found a new digital game to play.

In retrospect, following Euler‘s naming,
e lifted its black mask and showed its limit:
e=limn→∞(1+1n)ne=limn→∞(1+1n)n
Bernoulli‘s compounded interest for an investment of one.

Its reciprocal gave Bernoulli many trials,
from gambling at the slot machines to deranged parties
where nameless gentlemen check hats with butlers at the door,
and when they leave, e‘s reciprocal hands each a stranger’s hat.

In gratitude to Eulere showed a serious side,
infinite sum representation:
e=∑n=0∞1n!=10!+11!+12!+13!+⋯e=∑n=0∞1n!=10!+11!+12!+13!+⋯

For Euler‘s eyes alone, e fanned the peacock tail of
e−12e−12’s continued fraction expansion,
displaying patterns that confirmed
its own irrationality.

A century passed till e—through Hermite‘s pen,
was proved to be a transcendental number.
But to this day it teases us with

e‘s abstract beauty casts a glow on Euler’s Identity:
eið + 1 = 0,
the elegant, mysterious equation,
where waltzing arm in arm with i and π,
e flirts with complex numbers and roots of unity.

We meet e nowadays in functional high places
of CalculusDifferential EquationsProbabilityNumber Theory,
and other ancient realms:
y = ex
e
is the base of the unique exponential function
whose derivative is equal to itself.
The more things change the more they stay the same.
e
gathers gravitas as solid under integration,
∫exdx=ex+c∫exdx=ex+c
a constant c is the mere difference;
and often e makes guest appearances in Taylor series expansions.
And now and then e stars in published poetry—
honors and administrative duties multiply with age.

References:

http://www.maa.org/press/periodicals/convergence/the-enigmatic-number-iei-a-history-in-verse-and-its-uses-in-the-mathematics-classroom-the-annotated

http://www.maa.org/publications/periodicals/convergence/napiers-e-napier

http://www-history.mcs.st-and.ac.uk/HistTopics/e.html

# Engaging students: Half-life

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 Brianna Horwedel. Her topic: working with the half-life of a radioactive element.

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

Half-Life of radioactive elements in Pre-calculus is generally used when introducing exponential decay. However, its main application is in the field of Chemistry and Archeology. If students go on to take any type of chemistry, they will definitely learn more about the half-life of radioactive elements and how long it takes to get rid of certain nuclear elements. The half-life of Carbon-14 is especially important in Archeology. Carbon-14 dating is a method used to determine the age of archeological artifacts of a biological origin using the half-life of Carbon-14. This process can date bone, wood, cloth, plant fibers, and more that are up to 50,000 years old. The way it works is as follows: as soon as a living organism dies, it stops taking in new carbon. The ratio of carbon-12 to carbon-14 is the same as every living thing. However, when an organism dies, the carbon-14 starts decaying with its half-life of 5,700 years. The carbon-12 does not decay. When an organism is found, they look at the ratio of carbon-12 to carbon-14 to determine the age based on the half-life of carbon-14.

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

I think this topic lends itself nicely to a project. Firstly, I would come up with several half-lives and place them in a bowl. Each student would pick a half-life and have to make up an element. Using poster-board, they would give a brief description of what their element is and then create a graph illustrating their particular half-life. They would then present it to the class explaining how they graphed their line and what equation they used. They could also include a table of input and output values. This would be a great refresher on graphing exponential decays along with allowing a little creativity. I think the students would have a lot of fun with this type of project.

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

I found this really great web-site (https://jeopardylabs.com/play/exponential-growth-decay) that has an exponential growth and decay form of Jeopardy. It allows you to pick how many teams there are and then it sets up a Jeopardy board. This would be a really fun way to review at the end of a unit over exponential growth and decay. To make the students more engaged, I would offer extra credit to the team with the highest score at the end. Because it is in a game form, students are more likely to pay attention to this type of review.

# 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 Loc Nguyen. His topic, from Precalculus: introducing the number $e$.

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

To be able to understand where the number e is produced in the first place, students need to understand how compound interest is calculated.  Before introducing the number e, I will definitely create an activity for the students to work on so that they can eventually find the formula for compounding interest based on the patterns they produce throughout the process.  The compound interest formula is F=P(1+r/n)nt.  From this formula, I will again provide students a worksheet to work on.  In this worksheet, I will let P=1, r=100%, t=1, then the compound interest formula will be F=(1+1/n)n. Now students will compute the final value from yearly to secondly.

When they do all the computation, they will see all the decimal places of the final value lining up as n gets big.  And finally, they will see that the final value gets to the fixed value as n goes to infinity.  That number is e=2.71828162….,

How has this topic appeared in the news?

To help the students realize how important number e is, I would engage them with the real life examples or applications. There were some news that incorporated exponential curves. First, I will show the students the news about how fast deadly disease Ebola will grow through this link http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary.  The students will eventually see how exponential curve comes into play. After that I will provide them this link, http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/, in this link, the article talked about the global population rate and it provided the scientific evidence that showed the data collected represent the exponential curve.  Up to this point, I will show the students that the population growth model is:

Those examples above was about the growth.  For the next example, I will ask the students that how the scientists figured out the age of the earth.  In this link, http://earthsky.org/earth/how-old-is-the-earth, the students will learn that the scientists used Modern radiometric dating methods to calculate the age of earth.  At this time, I will show them radioactive decay formula and explain to them that this formula is used to determine the lives of the substances such as rocks:

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

To introduce to the students what the number e is, I will engage them with two videos. In the first video, https://www.youtube.com/watch?v=UFgod5tmLYY, the math song “e a magic number” will engage the students why it is a magic number.  While watching this clip, the students will be able to learn the history of e.  Also the students will see many mathematical formulas and expressions that contain e.  This will give them a heads up that they will see these in future when they take higher level math.  It is also pretty humorous of how Dr. Chris Tisdell sang the song.

In the second video, https://www.youtube.com/watch?v=b-MZumdfbt8, it explained why e is everywhere.  The video used probability and exponential function to illustrate the usefulness of e, and showed how e is involving in everything.  It gave many examples of e such as population, finance…  Also the video illustrates the characteristics of the number e and the function that has e in it.  Watching these videos will enhance students’ perception and understanding on the number e, and help them to see how important this number is.

Reference

http://www.math.unt.edu/~baf0018/courses/handouts/exponentialnotes.pdf

http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/

http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary

http://earthsky.org/earth/how-old-is-the-earth

# 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 Nada Al-Ghussain. Her topic, from Precalculus: introducing the number e.

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

Not every student loves math, but almost all students use math in his or her advanced courses. Students in microbiology will use the number e, to calculate the number of bacteria that will grow on a plate during a specific time. Biology or pharmacology students hoping to go into the health field will be able to find the time it takes a drug to lose one-half of its pharmacologic activity. By knowing this they will be able to know when a drug expires. Students going into business and finance will take math classes that rely greatly on the number e. It will help them understand and be able to calculate continuous compound interest when needed. Students who do love the math will get to explore the relation of logarithms and exponentials and how they interrelate. As students move into calculus, they are introduced to derivatives and integrals. The number e is unique, since when the area of a region bounded by a hyperbola y= 1/x, the x-axis, and the vertical lines x=1 and x= e is 1. So a quick introduction to e in any level of studies, reminds the students that it is there to simplify our life!

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

In the late 16th century, a Scottish mathematician named John Napier was a great mind that introduced to the world decimal point and Napier’s bones, which simplified calculating large numbers. Napier by the early 17th century was finishing 20 years of developing logarithm theory and tables with base 1/e and constant 10^7. In doing this, multiplication computational time was cut tremendously in astronomy and navigation. Other mathematicians built on this to make lives easier (at least mathematically speaking!) and help develop the logarithmic system we use today.

Henry Briggs, an English mathematician saw the benefit of using base 10 instead of Napier’s base 1/e. Together Briggs and Napier revised the system to base 10, were Briggs published 30,000 natural numbers to 14 places [those from 1 to 20,000 and from 90,000 to 100,000]! Napier’s became known as the “natural logarithm” and Briggs as the “common logarithm”. This convinced Johann Kepler of the advantages of logarithms, which led him to discovery of the laws of planetary motions. Kepler’s reputation was instrumental in spreading the use of logarithms throughout Europe. Then no other than Isaac Newton used Kepler’s laws in discovering the law of gravity.

In the 18th century Swiss mathematician, Leonhard Euler, figured he would have less distraction after becoming blind. Euler’s interest in e stemmed from the need to calculate compounded interest on a sum of money. The limit for compounding interest is expressed by the constant e. So if you invest $1 at a rate of interest of 100% a year and in interest is compounded continually, then you will have$2.71828… at the end of the year. Euler helped show us many ways e can be used and in return published the constant e. It didn’t stop there but other mathematical symbols we use today like i, f(x), Σ, and the generalized acceptance of π are thanks to Euler.

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

Statistics and math used in the same sentence will make most students back hairs stand up! I would engage the students and ask them if they started a new job for one month only, would they rather get 1 million dollars or 1 penny doubled every day for a month? I would give the students a few minutes to contemplate the question, without using any calculators. Then I would take a toll of the number of the students’ choices for each one. I would show them a video regarding the question and idea of compound interest. Students will see how quickly a penny gets transformed into millions of dollars in a short time. Money and short time used in the same sentence will make students fully alert! I would then ask them another question, how many times do you need to fold a newspaper to get to the moon? As a class we would decide that the thickness is 0.001cm and the distance from the Earth to the moon would be given. I would give them some time to formulate a number and then take votes around the class, which should be correct. The video is then played which shows how high folding paper can go! This one helps them see the growth and compare it to the world around them. After the engaged, students are introduced to the number e and its roll in mathematics.

Money: watch until 2:35:

Paper:

References:

http://mathworld.wolfram.com/e.html

http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/

http://www.math.wichita.edu/history/men/euler.html

http://www.maa.org/publications/periodicals/convergence/john-napier-his-life-his-logs-and-his-bones-introduction

http://www.purplemath.com/modules/expofcns5.htm

http://ualr.edu/lasmoller/efacts.html

# 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 Kenna Kilbride. Her topic, from Precalculus: introducing the number e.

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

Students will add on to this constant from calculus up to differential equations and even further. In Calculus I students use the number e to solve exponential functions and logarithm function. Calculus II uses the number e when computing integrals. In Complex Numbers you see the number e written as the Taylor series

$latex e^x = \displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} Differential equations utilizes the number $e$ in $y(x) = Ce^x$. The number $e$ can be utilized in many other areas since it is considered to be a base of the natural logarithm. The number $e$ is also defined as: $e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$ Also the number $e$ can be seen in the infinite series$latex e = \sum_{k=0}^\infty \frac{1}{k!}

The number e can be seen in many different areas of mathematics and with many different series and equations. Stirling’s approximation, Pippenger product, and Euler formula are just a few more examples of where you can see the number e.

http://mathworld.wolfram.com/e.html

http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegralsCompTest.aspx

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.)

Introducing this constant can be a very hard thing for a teacher to do and using a word problem that involves a satellite that students can comprehend what they do in the sky will help.

A satellite has a radioisotope power supply. The power output in watts is given by the equation

P = 50e^(-t/250)

where t is the time in days and e is the base of natural logarithms.

Then when introducing, e, you can give them problems that they can easily solve without fully understanding what e is. Give them problems such as, how much power will be available in a year. The solution is:

P = 50e^(-365/250)

= 5Oe^(-1.46)

= 50 x 0.232

= 11 .6

Once e has been more formally introduced and the students can then become more familiar (this should only be added on when the students fully understand e) you can add onto this problem by giving them questions such as, what is the half-life of the power supply? Students must use natural log to solve this equation:

25 = 50e^(-t/250)

for t and obtain

– t/250 = ln O.5

= -0.693

t = 250 x 0.693

= 173 days

http://er.jsc.nasa.gov/seh/math49.html

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.)

John Napier was born in Scotland around 1550. Napier started attending St. Andrews University at the age of 13. After leaving St. Andrews without a degree he attended Cambridge University. Later he studied abroad, presumably in Paris. In 1614 Napier invented logarithms and later exponential expressions. Along with mathematics, Napier was interested in peace keeping and religion. Napier died on April 4, 1617 of gout.

Euler contributed to e, a mathematical constant. He was born 1707 in the town Basel of Switzerland. By the age of 16 he had earned a Master’s degree and in 1727 he applied for a position as a Physics professor at the University of Basel and was turned down. Due to extreme health problems by 1771 he had lost almost all of his vision. By the time of his death in 1783, the Academy of Sciences in Petersburg had received 500 of his works.

http://www.macs.hw.ac.uk/~greg/calculators/napier/great.html

http://www.pdmi.ras.ru/EIMI/EulerBio.html

# Exponential growth and decay (Part 11): Half-life

In this series of posts, I provide a deeper look at common applications of exponential functions that arise in an Algebra II or Precalculus class. In the previous posts in this series, I considered financial applications. In today’s post, I’ll discuss radioactive decay and the half-life formula.

One way of writing the formula for how a radioactive substance (carbon-14, uranium-235, brain cells) decays is

$A(t) = A_0 e^{-kt}$

In yesterday’s post, I showed how this formula is a natural consequence of a certain differential equation. Of course, students in Algebra II or Precalculus (or high school chemistry) are usually not ready to understand this derivation using calculus. Instead, they are typically given the final formula and are expected to use this formula to solve problems. Still, I think it’s important for the teacher of Algebra II or Precalculus to be aware of how the origins of this formula, as it only requires mathematics that’s only a year or two away in these students’ mathematical development.

There is another correct way to write this formula in terms of half-life. Sadly, my experience is that many students are familiar with these two different formulas but are not aware of how the two formulas are connected. As we’ll see, while yesterday’s post using differential equations is inaccessible to Algebra II and Precalculus students, the derivation below is entirely elementary and can be understood by good students in these courses.

Let $h$ be the half-life of the substance. By definition, this means that

$A(h) = \displaystyle \frac{1}{2} A_0$

Substituting into the above formula, we find

$\displaystyle \frac{1}{2} A_0 = A_0 e^{-kh}$

Let’s now solve for the constant $k$ in terms of $h$:

$\displaystyle \frac{1}{2} = e^{-kh}$

$\displaystyle \ln \left( \frac{1}{2} \right) = - k h$

$\displaystyle -\frac{1}{h} \ln \left( \frac{1}{2} \right) = k$

Let’s now substitute this back into the original formula:

$A = A_0 e^{-kt}$

$A = A_0 e^{ -\left[ -\frac{1}{h} \ln \left( \frac{1}{2} \right) \right] t}$

$A = A_0 e^{\ln \left( \frac{1}{2} \right) \cdot \frac{t}{h}}$

$A = A_0 \left[e^{\ln \left( \frac{1}{2} \right)} \right]^{t/h}$

$A = A_0 \displaystyle \left( \frac{1}{2} \right)^{t/h}$

This is the usual half-life formula, where the previous base of $e$ has been replaced by a new base of $\displaystyle \frac{1}{2}$ . For most applications, a base of $e$ is preferred. However, for historical reasons, the base of $\displaystyle \frac{1}{2}$ is preferred for problems involving radioactive decay. For example,

$A(2h) = A_0 \displaystyle \left( \frac{1}{2} \right)^{2h/h}$

$A(2h) = A_0 \displaystyle \left( \frac{1}{2} \right)^{2}$

$A(2h) = \displaystyle \frac{1}{4} A_0$

In other words, after two half-life periods, only one-fourth (half of a half) of the substance remains.

# Exponential growth and decay (Part 10): Half-life

In this series of posts, I provide a deeper look at common applications of exponential functions that arise in an Algebra II or Precalculus class. In the previous posts in this series, I considered financial applications. In today’s post, I’ll discuss radioactive decay and the half-life formula.

While these formulas are easy to state, not many high school teachers are aware of the physical principles from which they arise. The basic idea is that if amount $A$ of a radioactive substance (carbon-14, uranium-235, brain cells) is present, the rate at which the substance decays is proportional to the amount of the substance currently present. This can be rewritten as a differential equation, since the rate at which the substance decays is $dA/dt$. So we find that

$\displaystyle \frac{dA}{dt} = - k A$

The negative sign on the right-hand side isn’t strictly necessary, but it’s a reminder that amount present decreases as time increases.

This differential equation can be solved in several ways, including separation of variables (below, I’ll be sloppy with the constant of integration for the sake of simplicity):

$\displaystyle \frac{dA}{A} = -k$

$\displaystyle \int \frac{dA}{A} = - \displaystyle \int k \, dt$

$ln |A| = -k t + C$

$|A| = e^{-kt+C}$

$|A| = e^{-kt} e^C$

$A = \pm e^C e^{-kt}$

$A = C e^{-kt}$

To solve for the constant, we usually use the initial condition $A(0) = A_0$, a number that must be given in the problem:

$A(0) = C e^{-k \cdot 0}$

$A_0 = C \cdot 1$

$A_0 = C$

Plugging back in, we obtain the final answer

$A(t) = A_0 e^{-kt}$

Of course, students in Algebra II or Precalculus (or high school chemistry) are usually not ready to understand this derivation using calculus. Instead, they are typically given the final formula and are expected to use this formula to solve problems. Still, I think it’s important for the teacher of Algebra II or Precalculus to be aware of how the origins of this formula, as it only requires mathematics that’s only a year or two away in these students’ mathematical development.

# 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 Jesse Faltys. Her topic: solving exponential equations.

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

Once your students have learned how to solve exponential equations, they can solve many different kinds of applied problems like population growth, bacterial decay, and even investment earning interest rate.    (Examples Found: http://www.education.com/study-help/article/pre-calculus-help-log-expo-applications/)

Examples

1. How long will it take for $1000 to grow to$1500 if it earns 8% annual interest, compounded monthly?

$A = P \left( 1 + \displaystyle \frac{r}{n} \right)^{nt}$

• $A (t) = 1500$, $P = 1000$, $r = 0.08$, and $n = 12$.
• We do not know $t$.
• We will solve this equation for $t$ and will round up to the nearest month.
• In five years and one month, the investment will grow to about \$1500.

2. A school district estimates that its student population will grow about 5% per year for the next 15 years.  How long will it take the student population to grow from the current 8000 students to 12,000?

• We will solve for t in the equation $12,000 = 8000 e^{0.05t}$.

$12,000 = 8000 e^{0.05t}$

$1.5 = e^{0.05t}$

$0.05t = \ln 1.5$

$t = \displaystyle \frac{\ln 1.5}{0.05} \approx 8.1$

• The population is expected to reach 12,000 in about 8 years.

3. At 2:00 a culture contained 3000 bacteria.  They are growing at the rate of 150% per hour.  When will there be 5400 bacteria in the culture?

• A growth rate of 150% per hour means that $r = 1.5$ and that $t$ is measured in hours.

$5400 = 3000 e^{1.5t}$

$1.8 = e^{1.5t}$

$1.5t = \ln 1.8$

$t = \displaystyle \frac{\ln 1.8}{1.5} \approx 0.39$

• At about 2:24 ($0.39 \times 60 = 23.4$ minutes) there will be 5400 bacteria.

A note from me: this last example is used in doctor’s offices all over the country. If a patient complains of a sore throat, a swab is applied to the back of the throat to extract a few bacteria. Bacteria are of course very small and cannot be seen. The bacteria are then swabbed to a petri dish and then placed into an incubator, where the bacteria grow overnight. The next morning, there are so many bacteria on the petri dish that they can be plainly seen. Furthermore, the shapes and clusters that are formed are used to determine what type of bacteria are present.

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

The students need to have a good understanding of the properties of exponents and logarithms to be able to solve exponential equations.  By using properties of exponents, they should know that if both sides of the equations are powers of the same base then one could solve for x.  As we all know, not all exponential equations can be expressed in terms of a common base.  For these equations, properties of logarithms are used to derive a solution.  The students should have a good understanding of the relationship between logarithms and exponents.  Logs are the inverses of exponentials.  This understanding will allow the student to be able to solve real applications by converting back and forth between the exponent and log form.  That is why it is extremely important that a great review lesson is provided before jumping into solving exponential equations. The students will be in trouble if they have not successfully completed a lesson on these properties.

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

1. Khan Academy provides a video titled “Word Problem Solving – Exponential Growth and Decay” which shows an example of a radioactive substance decay rate. The instructor on the video goes through how to organize the information from the world problem, evaluate in a table, and then solve an exponential equation. For our listening learners, this reiterates to the student the steps in how to solve exponential equations.

2. Math warehouse is an amazing website that allows the students to interact by providing probing questions to make sure they are on the right train of thought.

For example, the question is $9^x = 27^2$ and the student must solve for $x$.  The first “hint” the website provides is “look at the bases.  Rewrite them as a common base” and then the website shows them the work.  The student will continue hitting the “next” button until all steps are complete. This is allowing the visual learners to see how to write out each step to successfully complete the problem.