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.

TI1637

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: Determining the largest fraction

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 Perla Perez. Her topic, from Pre-Algebra: determining which of two fractions is largest if the denominators are unequal.

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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.)

 

Students are introduced to fractions in elementary school, but at a certain point this topic can become tedious. Trying to introduce new concepts to a topic they’ve seen and practiced for a while can be a challenge. A good idea can be to give them a problem at the start of class that they can answer after the day’s lesson is done. Students are given a word problem such as:

“James was arguing with John that he could eat more pizza than him, while John without a doubt believed the opposite. It got to the point where everyone in class had established their own opinions on it. So Nancy came up with a solution and ordered two large pizzas to see who could eat the most. Well, when the pizzas arrived they noticed that one pizza was cut into 10 equal pieces and the other into 16 equal pieces. After they devoured all that they could, John had eaten 7/10 and James had eaten 13/16. Now, who at the most pizza?”

After giving the students to time to think about the problem without any more information, get a show of hands to see who they think ate the most. Write up the number of students who voted for James and John somewhere visible. Then, at the end of the lesson, give and explain the answer.

 

 

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How has this topic appeared in the news?

 

The euro currently cost .890646 or 445323/500000 of a dollar. The British Pound .753423 or 753423/1000000 of a dollar. Now which currency is cheaper? If the fraction were only given to a student, some might be able to say the British pound because the 7 is greater that the 8 while others might say euro because of the 5 in denominator, and some that have no idea. There’s actually a formula to find out which is: AMOUNTto=(AMOUNTfrom X RATE from)/RATEto Although most times currency exchange is shown in decimal form, it gives a broader sense of how a simple concept relates to big-world topics. It is important for students to be able to determine if 3/7 is greater or less than 4/5, so that one day they can apply it to their daily lives. The exchange rate is just one example of different fractions being used in today’s society; in this case how the use of decimals and fractions translate to foreign relations. By relating the outside world to a classroom, educators can show students that there is more to numbers than just a grade in a class. These real world concepts can help students better understand the application of the material.

References:

http://www.mathinary.com/currency_conversion.jsp

http://www.x-rates.com/table/?from=USD&amount=1

 

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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.)

 

To students it may seem as if fractions have always been there. Some may have not thought much of its origin. A brief interesting part of history can be shared to spark some light in the matter. Well although there were contributions from the Babylonians, Arabs, and Ancient Rome, it was the Egyptians in 1800 BC seem to be the ones already using them. But interesting enough it isn’t like how it is seen today. Rather than seeing a fraction be an integer over another they used hieroglyphics and base ten.

For example, “The Egyptians wrote all their fractions using what we call unit 1 as its numerator (top number). They put a mouth picture (which meant part) above a number to make it into a unit fraction.”

It would be represented like,

Because of this method it was difficult to compute so they had to use numerous tables. Although our methods have changed one thing still remains the same; the way we use manipulatives in showing how fractions with different denominators compare. For instance, we have circle pictures that visually show fractions with different denominators can ease student into understanding them better.

Babylonians, though found a simpler way of representing fractious with symbols. All in all, it is interesting how visual description can be helpful still in today’s society.

 

References:

https://nrich.maths.org/2515

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.

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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

 

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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.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

 

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: Reducing fractions to lowest terms

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 Madison duPont. Her topic, from Pre-Algebra: reducing fractions to lowest terms.

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

Reducing fractions to lowest terms can be applied to future mathematics topics such as ratios and proportions, and scientific topics such as chemistry or physics. Ratios can be represented as fractions and are not typically reduced to lowest terms because they represent relationships of two subjects using numbers. Being able to reduce these ratios can help students better identify the underlying relationship and apply this relationship to other aspects of the math problem, such as problems using unit price or map scales. Proportions relate to the concept of reducing fractions to lowest terms when using cross-multiplication. Having both sides of the proportion reduced to lowest terms makes the cross-multiplication much easier to compute and derive a final reduced answer. Chemistry uses fractions reduced to lowest terms with topics, like stoichiometry, that use potentially small and large numbers in several ratios that are multiplied together to obtain a final converted and reduced answer. Physics often uses ratio-like formulas and problems that are applied to real-world scenarios, which typically require fractions reduced to lowest terms because answers like miles per one hour are the goal. All of these topics use concepts of reducing fractions to lowest terms to more easily accomplish problems using a series of fractional computations, or to get an answer that is in terms of a single unit or most reduced so that it makes sense to real-world application.

 

 

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

This topic extends previously learned topics such as concepts of unique prime factorizations, greatest common divisor, manipulating fractions, and multiplication facts. The concept of unique prime factorizations greatly aids students in finding the greatest common divisor, which is used to find the greatest factor of the value of both the numerator and denominator. Next, manipulation of fractions is used to properly divide the numerator and denominator by the greatest common divisor. This process of dividing both parts of the fraction utilizes multiplication facts as well to determine what the answer to the division problem on both the top and bottom of the fraction would be. These previously learned concepts are all subtle and important applications when reducing fractions to lowest terms.

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

 

 

This video reminded me of many students that I have tutored or encountered in classrooms that were determined that a calculator was all they needed when doing math. Applied to reducing fractions to lowest terms, this video is extremely relevant in displaying that technology cannot be the only source of intelligence when thinking mathematically. Reducing fractions with extremely large numbers or numbers that do not have well-known factors can seem exhausting or impossible. Punching several factors of the numerator and denominator into a calculator attempting to reduce numbers with each common factor, and then not being sure of whether the fraction appearing on their screen is truly in the most reduced form surely indicates the technology is not the only way of solving the problem. Many students hop on a procedural escalator when beginning varying types of problems (in addition to reducing fractions to lowest terms) using memorized steps, punching calculator buttons, feeling comfortable, until suddenly—there is a horribly unattractive fraction halting their progress. This is when using mathematical problem solving skills such as reducing the numerator and denominator by the greatest common divisor or checking to see that the numerator and denominator are relatively prime becomes pertinent. Using these conceptual skills can save someone that is stuck waiting for a calculator to do the work for them, or that has given up on finishing a problem because it seems impossible or difficult, from thinking they are incapable of working out a problem efficiently and successfully. This video highlights the importance of being capable of knowing when it is time to take the effort to climb the stairs to reach your destination.

 

References:

“Stuck on an Escalator” Video link:

https://www.youtube.com/watch?v=VrSUe_m19FY

found via Google video search

 

 

 

 

 

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.

Thoughts on the Accidental Fraction Brainbuster

I really enjoyed reading a recent article on Math With Bad Drawings centered on solving the following problem without a calculator:

I won’t repeat the whole post here, but it’s an excellent exercise in numeracy, or developing intuitive understanding of numbers without necessarily doing a ton of computations. It’s also a fun exercise to see how much we can figure out without resorting to plugging into a calculator. I highly recommend reading it.

When I saw this problem, my first reflex wasn’t the technique used in the post. Instead, I thought to try the logic that follows. I don’t claim that this is a better way of solving the problem than the original solution linked above. But I do think that this alternative solution, in its own way, also encourages numeracy as well as what we can quickly determine without using a calculator.

Let’s get a common denominator for the two fractions:

\displaystyle \frac{3997 \times 5001}{4001 \times 5001} \qquad and \displaystyle \qquad \frac{4001 \times 4996}{4001 \times 5001}.

Since the denominators are the same, there is no need to actually compute 4001 \times 5001. Instead, the larger fraction can be determined by figuring out which numerator is largest. At first glance, that looks like a lot of work without a calculator! However, the numerators can both be expanded by cleverly using the distributive law:

3997 \times 5001 = (4000-3)(5000+1) = 4000\times 5000 + 4000 - 3 \times 5000 - 3,

4001 \times 4996 = (4000+1)(5000-4) = 4000\times 5000 - 4 \times 4000 + 5000 - 4.

We can figure out which one is bigger without a calculator — or even directly figuring out each product.

  • Each contains 4000 \times 5000, so we can ignore this common term in both expressions.
  • Also, 4000 - 3\times 5000 and 5000 - 4 \times 4000 are both equal to -11,000, and so we can ignore the middle two terms of both expressions.
  • The only difference is that there’s a -3 on the top line and a -4 on the bottom line.

Therefore, the first numerator is the larger one, and so \displaystyle \frac{3997}{4001} is the larger fraction.

Once again, I really like the original question as a creative question that initially looks intractable that is nevertheless within the grasp of middle-school students. Also, I reiterate that I don’t claim that the above is a superior method, as I really like the method suggested in the original post. Instead, I humbly offer this alternate solution that encourages the development of numeracy.

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.

A subtle math joke

fifth1

 

In case you need a subtle hint, here’s another version:

fifth2