My Favorite One-Liners: Part 18

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This is a quip that I’ll use when a theoretical calculation can be easily confirmed with a calculator.

Sometimes I teach my students how people converted decimal expansions into fractions before there was a button on a calculator to do this for them. For example, to convert  $x = 0.\overline{432} = 0.432432432\dots$ into a fraction, the first step (from the Bag of Tricks) is to multiply by 1000: How do we change this into a decimal? Let’s call this number $x$.

$1000x = 432.432432\dots$

$x = 0.432432\dots$

Notice that the decimal parts of both $x$ and $1000x$ are the same. Subtracting, the decimal parts cancel, leaving

$999x = 432$

or

$x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}$

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time.

To make this more real and believable to them, I then tell them my one-liner: “I can see that no one believes me. OK, let’s try something that you will believe. Pop out your calculators. Then punch in 16 divided by 37.”

Indeed, my experience many students really do need this technological confirmation to be psychologically sure that it really did work. Then I’ll tease them that, by pulling out their calculators, I’m trying to speak my students’ language.

See also my fuller post on this topic as well as the index for the entire series.

My Favorite One-Liners: Part 14

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This quip is similar to the “bag of tricks” one-liner, and I’ll use this one if the “bag of tricks” line is starting to get a little dry.

Sometimes in math, there’s a step in a derivation that, to the novice, appears to make absolutely no sense. For example, to find the antiderivative of $\sec x$, the first step is far from obvious:

$\displaystyle \int \sec x \, dx = \displaystyle \int \sec x \frac{\sec x + \tan x}{\sec x + \tan x} \, dx$

While that’s certainly correct, it’s from from obvious to a student that this such a “simplification” is actually helpful.

To give a simpler example, to convert

$x = 0.\overline{432} = 0.432432432\dots$

into a decimal, the first step is to multiply $x$ by $1000$:

$1000x = 432.432432\dots$

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How did you know to do that?” To lighten the mood, I’ll explain with a big smile that I’m clairvoyant… when I got my Ph.D., I walked across the stage, got my diploma, someone waved a magic wand at me, and poof! I became clairvoyant.

Clairvoyance is wonderful; I highly recommend it.

The joke, of course, is that the only reason that I multiplied by 1000 is that someone figured out that multiplying by 1000 at this juncture would actually be helpful. Subtracting $x$ from $1000x$, the decimal parts cancel, leaving

$999x = 432$

or

$x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}$.

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. I learned this procedure when I was very young; however, in modern times, this procedure appears to be a dying art. I’m guessing that this algorithm is a dying art because of the ease and convenience of modern calculators. As always, I hold my students blameless for the things that they were simply not taught at a younger age, and part of my job is repairing these odd holes in their mathematical backgrounds so that they’ll have their best chance at becoming excellent high school math teachers.

For further reading, here’s my series on rational numbers and decimal expansions.

Engaging students: Fractions, percents, and decimals

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 Kim Hong. Her topic, from Pre-Algebra: fractions, percents, and decimals.

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

I think making the students create a foldable, a short and quick project, would be a good and concrete activity for teaching fractions, decimals, and percents. Each flap is a topic. There is a definition and example. On the back of the foldable the students could create a table going between fractions, decimals and percents with many “harder” values.

The foldable is portable and quick, and can be a helpful and quick resource.

The students can also draw pictures inside the flaps. E.g A pizza and its slices to show fractions.

#made4math- Converting Fractions, Decimals and Percents Foldable

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

This topic can be used in a students’ future course when they come across proportions and rates. They could see proportions when it appears in physics such a changes in time and speed. They could see rates of change when it appears in calculus involving derivatives. These values are factions that can be changed to decimals and percents because everything is a part of a whole.

Also, fractions, which are numbers over a whole, are the same as the term rational quantities. Rational quantities are numbers that can be written as a ratio that is a fraction. There is a subset of the Reals that are called the Rationals. In advanced logic and math courses, students will be able to work with this subset of the Reals.

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.

I found this really awesome website the students could play around with for the first minutes of class to get their juices flowing. Basically the objective of the game is to group the equal values in circles. There is a check answer option as well.

It starts off very simple with very easy mental math and then with each level, the difficulty increases.

http://www.mathplayground.com/Decention/Decention.html

Engaging students: Adding and subtracting decimals

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 Daniel Herfeldt. His topic, from Pre-Algebra: adding and subtracting decimals.

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

A great engage activity that I have thought about as a teacher would be to have the students add and subtract money. For this activity I would provide the students with play money (dollar bills, quarters, dimes, nickels, and pennies) needed to proceed. I would then ask the students to show me what 65 cents looks like. Most outcomes will probably look similar with two quarters, a dime, and a nickel while in fact there are many ways to show what 65 cents looks like. Some students might come up with a quarter and four dimes, or 13 nickels. After the students finish with their first example, I would ask them if they could find another way to add up the coins to get 65 cents. This is a very simple activity that refreshes the students’ knowledge on how to add decimals. The activity also shows the teacher which students have a harder time with the topic.

The concept of adding and subtracting decimals is all over the world. It is used for everyday things, such as sports. One of the most watched things on television is the summer or winter Olympics. People from all over the world compete in several events and get scores. For example, gymnasts compete for the highest score in the specific event they are doing and then add it to their total score to be declared the winner. After the first event, one gymnast may have the highest score of 16.543 while the person below her has a score of 15.785. Then in the second event, the person that previously had a higher score only scored 12.400, while the person that was behind her scored a 15.115. To declare the winner of the two, you would have to sum up both scores and see which of the two competitors had the higher score. You would get the total of 28.943 for the first gymnast, and 30.900 for the second. From here the winner would be the second gymnast.

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

This game would be a great tool to refresh a students’ memory on how to add decimals if you are planning to have a test. To start, you would hand out every student their own small whiteboard and marker. You would then put the game on the projector screen so that all of the students can see it. Start out with clicking the easy button so that you don’t start with a difficult problem. Ex: 31 + .4. This should be a problem that all students can answer. Have all the students write down on their own white board what they think the answer would be. After the students finish, ask them to put their white board face down. Once all of the students finish, have everyone raise up their answers. Afterwards plug in the answer that is most common amongst the students to see if the majority was correct. If most are correct, proceed to the next difficulty, which is medium, and repeat the steps that you did for the easy problem. If the majority of the class gets it right, then go to the final and hardest difficulty and repeat the steps one more time. If the majority of the students get the answer wrong for any difficulty, do the problem on the board to show the steps and try another problem of the same difficulty. The students will then remember the steps and have a higher chance of being correct. When the students get the hard problems correct, keep doing the hard problems until you feel the students have grasped the concept.

Lessons from teaching gifted elementary school students: Index (updated)

I’m doing 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 various lessons I’ve learned while trying to answer the questions posed by gifted elementary school students. (This is updated from my previous index.)

Part 1: A surprising pattern in some consecutive perfect squares.

Part 2: Calculating 2 to a very large exponent.

Part 3a: Calculating 2 to an even larger exponent.

Part 3b: An analysis of just how large this number actually is.

Part 4a: The chance of winning at BINGO in only four turns.

Part 4b: Pedagogical thoughts on one step of the calculation.

Part 4c: A complicated follow-up question.

Part 5a: Exponentiation is multiplication as multiplication is to addition. So, multiplication is to addition as addition is to what? (I offered the answer of incrementation, but it was rejected: addition requires two inputs, while incrementation only requires one.)

Part 5b: Why there is no binary operation that completes the above analogy.

Part 5c: Knuth’s up-arrow notation for writing very big numbers.

Part 5d: Graham’s number, reputed to be the largest number ever to appear in a mathematical proof.

Part 6a: Calculating $(255/256)^x$.

Part 6b: Solving $(255/256)^x = 1/2$ without a calculator.

Part 7a: Estimating the size of a 1000-pound hailstone.

Part 7b: Estimating the size a 1000-pound hailstone.

Part 8a: Statement of an usually triangle summing problem.

Part 8b: Solution using binomial coefficients.

Part 8c: Rearranging the series.

Part 8d: Reindexing to further rearrange the series.

Part 8e: Rewriting using binomial coefficients again.

Part 8f: Finally obtaining the numerical answer.

Part 8g: Extracting the square root of the answer by hand.

Lessons from teaching gifted elementary school students: Index (updated)

I’m doing 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 various lessons I’ve learned while trying to answer the questions posed by gifted elementary school students. (This is updated from my previous index.)

Part 1: A surprising pattern in some consecutive perfect squares.

Part 2: Calculating 2 to a very large exponent.

Part 3a: Calculating 2 to an even larger exponent.

Part 3b: An analysis of just how large this number actually is.

Part 4a: The chance of winning at BINGO in only four turns.

Part 4b: Pedagogical thoughts on one step of the calculation.

Part 4c: A complicated follow-up question.

Part 5a: Exponentiation is multiplication as multiplication is to addition. So, multiplication is to addition as addition is to what? (I offered the answer of incrementation, but it was rejected: addition requires two inputs, while incrementation only requires one.)

Part 5b: Why there is no binary operation that completes the above analogy.

Part 5c: Knuth’s up-arrow notation for writing very big numbers.

Part 5d: Graham’s number, reputed to be the largest number ever to appear in a mathematical proof.

Lessons from teaching gifted elementary school students: Index

I’m doing 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 various lessons I’ve learned while trying to answer the questions posed by gifted elementary school students.

Part 1: A surprising pattern in some consecutive perfect squares.

Part 2: Calculating 2 to a very large exponent.

Part 3a: Calculating 2 to an even larger exponent.

Part 3b: An analysis of just how large this number actually is.

Part 4a: The chance of winning at BINGO in only four turns.

Part 4b: Pedagogical thoughts on one step of the calculation.

Part 4c: A complicated follow-up question.

Thoughts on 1/7 and Other Rational Numbers: 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 the decimal expansions of rational numbers.

Part 1: A way to remember the decimal expansion of $\displaystyle \frac{1}{7}$.

Part 2: Long division and knowing for certain that digits will start repeating.

Part 3: Converting a repeating decimal into a fraction, using algebra.

Part 4: Converting a repeating decimal into a fraction, using infinite series.

Part 5: Quickly converting fractions of the form $\displaystyle \frac{M}{10^t}$, $\displaystyle \frac{M}{10^k-1}$, and $\displaystyle \frac{M}{10^t (10^k-1)}$ into decimals without using a calculator.

Part 6: Converting any rational number into one of the above three forms, and then converting into a decimal.

Part 7: Same as above, except using a binary (base-2) expansion instead of a decimal expansion.

Part 8: Why group theory relates to the length of the repeating block in a decimal expansion.

Part 9: A summary of the above ideas to find the full decimal expansion of $\displaystyle \frac{8}{17}$, which has a repeating block longer than the capacity of most calculators.

Part 10: More thoughts on $\displaystyle \frac{8}{17}$.

Calculator Errors: When Close Isn’t Close Enough (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: Propagation of small numerical errors.

Part 2: A tragedy during the 1991 Gulf War that was a direct result of calculator rounding.

Lessons from teaching gifted elementary school students (Part 4c)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

What is the chance of winning a game of BINGO after only four turns?

When my class posed this question, I was a little concerned that the getting the answer might be beyond the current abilities of a gifted elementary student. As discussed over the past couple of posts, for a non-standard BINGO game with 44 numbers, the answer is

$\displaystyle 4 \times \frac{4}{44} \times \frac{3}{43} \times \frac{2}{42} \times \frac{1}{41} = \displaystyle \frac{4}{135,751}$

What’s that in decimal?

With these gifted students, I encourage thinking as much as possible without a calculator… and they wanted me to provide the answer to this one in like fashion. For my class, this actually did serve a purpose by illustrating a really complicated long division problem so they could reminded about the number of leading zeroes in such a problem.

Gritting my teeth, I started on the answer:

At this point, I was asked the other question that I had anticipated but utterly dreaded… motivated by child-like curiosity mixed perhaps with a touch of sadism:

How long do we have before the digits start repeating?

My stomach immediately started churning.

I told the class that I’d have to figure this one later. But I told them that the answer would definitely be less than 135,751 times. My class was surprised that I could even provide this level of (extremely) modest upper limit on the answer. After some prompting, my class saw the reasoning for this answer: there are only 135,751 possible remainders after performing the subtraction step in the division algorithm, and so a remainder has to be repeated after 135,751 steps. Therefore, the digits will start repeating in 135,751 steps or less.

What I knew — but probably couldn’t explain to these elementary-school students, and so I had to work this out for myself and then get back to them with the answer — is that the length of the repeating block $n$ is the least integer so that

$135751 \mid 10^n - 1$

which is another way of saying that we’ve used the division algorithm enough times so that a remainder repeats. Written in the language of group theory, $n$ is the least integer that satisfies

$10^n \equiv 1 \mod 135751$

(A caveat:this rule works because neither 2 nor 5 is a factor of 135,751… otherwise, those factors would have to be taken out first.)

Some elementary group theory can now be used to guess the value of $n$. Let $G$ be the multiplicative group of integers modulo $135,751$ which are relatively prime which. The order of this group is denoted by $\phi(135751)$, called the Euler totient function. In general, if $m = p_1^{a_1} p_2^{a_2} \dots p_r^{a_r}$ is the prime factorization of $m$, then

$\phi(m) = n \left( \displaystyle 1 - \frac{1}{p_1} \right) \left( \displaystyle 1 - \frac{1}{p_2} \right) \dots \left( \displaystyle 1 - \frac{1}{p_r} \right)$

For the case at hand, the prime factorization of $135,751$ can be recovered by examining the product of the fractions near the top of this post:

$135751 = 7 \times 11 \times 41 \times 43$

Therefore,

$\phi(135,751) = 6 \times 10 \times 40 \times 42 = 100,800$

Next, there’s a theorem from group theory that says that the order $n$ of an element of a group must be a factor of the order of the group. In other words, the number $n$ that we’re seeking must be a factor of $100,800$. This is easy to factor:

$100,800 = 2^6 \times 3^2 \times 5^2 \times 7$

Therefore, the number $n$ has the form

$n = 2^a 3^b 5^c 7^d$,

where $0 \le a \le 6$, $0 \le b \le 2$, $0 \le c \le 2$, and $0 \le d \le 1$ are integers.

So, to summarize, we can say definitively that $n$ is at most $100,800$, and that were have narrowed down the possible values of $n$ to only $7 \times 3 \times 3 \times 2 = 126$ possibilities (the product of one more than all of the exponents). So that’s a definite improvement and reduction from my original answer of $135,751$ possibilities.

At this point, there’s nothing left to do except test all 126 possibilities. Unfortunately, there’s no shortcut to this; it has to be done by trial and error. Thankfully, this can be done with Mathematica:

The final line shows that the least such value of $n$ is 210. Therefore, the decimal will repeat after 210 digits. So here are the first 210 digits of $\displaystyle \frac{4}{135,751}$ (courtesy of Mathematica):

0.000029465712959757202525211600651192256410634175807176374391348866675015285338597874048809953517837805983013016478699972744215512224587664\
179269397647162820163387378361853688002298325610861061796966504850792996…