Disclaimer: Not “solved”. Simplified!
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42 Folds to the Moon January 19, 2012
One of my students just can’t wrap his head around the power of exponents. Can you blame him? This week we learned that it would take just 27 folds of a piece of paper for the stack to reach the height of Mount Everest, and then just 15 more -a total of just 42 folds - to reach the moon. As we started the lesson, students guessed “one million” and “47 billion!” folds to reach the moon, so you can imagine the shock (and disbelief) in the actual number 42! Maybe the weirdest part is to think that it would take 41 of the folds to get just half-way to the moon and then just 1 more to make the second half of the journey.
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But how can this be? How is it possible that a thin sheet of paper easily ripped in half can reach the moon after a mere 42 folds? Well, let’s see….
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The simple Algebra 1 exponential growth formula is:
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As the thickness of a piece of paper is roughly 0.01 centimeter, we’d fill in our equation as:
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This gives us a very large number of centimeters: (43,980,465,111).
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Dividing this number by 100 will give us the equivalent number of meters: (439,804,651),
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and dividing by 1,000 will give us the equivalent number of kilometers: (439,804).
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For all us Americans stuck on the Imperial system, 439,804 kilometers is approximately 273,281 miles. The moon is, on average, 238,855 miles from Earth at any given time, so 42 folds of a piece of paper will actually get us PAST the moon!
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So how small would the surface area of the top paper on the stack be? How thin will be this paper tower to the moon?
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VERY thin!
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The exponential decay formula is almost exactly the same as the exponential growth formula except that there is a (1 – r) in place of the growth formula’s (1 + r). To write the equation for how thin this stack of paper to the moon will be, we have to think about a funny occurrence in the stock market…
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To go from 1 to 2 is a 100% increase: 100% of 1 is added to itself to get 2.
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But to get back to 1 is a different story: to go from 2 to 1 is a 50% decrease. Just 50% of 2 is removed to get to 1.
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So when your stock goes down 30% on Monday, it’s not back to where it was if it goes back up 30% on Tuesday. If your stock goes down 50% on Monday, it’s got to go up 100% on Tuesday to get back to where it was.
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Getting back on track (excuse me, not a fan of Wall Street), our decay equation would be written as:-
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This equation yields the incredibly small number: 6.37 x 10^-12 or .00000000000637 centimeters. P was set to 28 to because a 9.5×11 sheet of paper is about 28 centimeters long.
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We didn’t do this last part in class, which is a good thing because all of a sudden I’m having a hard time wrapping my head around exponents!
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Reducing fractions: One cookie = two cookies? July 11, 2011
Any kid will tell you that eating one of two cookies is not the same as eating two of four cookies. In the first case, you only get to eat one cookie and in the second case, you get to eat two! Yet in math, we are told that 1/2 is equal to 2/4. How can this be?
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First, we have to be able to read fractions to understand them. In other words, we have to remember that fractions are a sort of shorthand for longer phrases. For instance, let’s take 1/2.
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1/2 can mean:
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“one out of two”
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“one divided by two”
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“one out of every two”
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“one for every two“
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Writing “1/2″ is so much faster than writing any of the above phrases. And when we understand this, and that mathematicians often use abbreviations, we can begin to think about what “1/2” really is:
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Here’s two cookies:
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And here’s one out of two cookies:
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We took “one out of two cookies, or “1/2″ and showed the fraction “1/2″ with cookies! This seems obvious, but may be a little misleading. In our above example, it seems as though the numerator (1) represents the number of cookies we take and the denominator (2) represents the total number of cookies. And in a way this is true! But let’s look at one more example…
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Here we have four cookies…
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And here we take two of them…
We’ve taken two out of four cookies, or “2/4″. We’re told that 2/4 is the same as “1/2″, but how? Let’s remember our phrases. “1/2″ can also be read as “one out of every 2“, and in fact we have taken one cookie out of every two on the table. We can begin to see how 1/2 = 2/4.
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Dividing Fractions With Pictures! June 8, 2011
Of all my posts, this one gets the most hits. I think I know why. Fraction division seems like it should be simple. Afterall, ”flip the second fraction and multiply across” is a complete cake walk. But when we have to explain the process to a kid (or an overly-inflated interviewer), things can go very wrong. Why is it so hard? Recently, I met a new friend, Chris Fink, through my blog. Chris teaches Math in the California penal system. Through a series of emails back and forth, we both came to a better understanding of this tricky process. She was able to explain fraction division to her inmates (they all clapped and thanked her - yes, her - afterwards!) and I came to understand how to show the process through pictures a lot better thanks to her. I left my old post underneath the new stuff because, though wordy, it does give a bit more explanation. The following three screenshots (you can download the pdf here or by clicking on one of the three screenshots) are a decent start to How we show fraction division thorough pictures. Thanks Chris!
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Fraction Division: Not Just a How
Dividing fractions has got to be the algorithm we most often take at face value. The How – flip the second fraction and multiply across – is easy, while the Why can fill an entire chapter.
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Whenever we divide, we’re asking “How many groups of this will fit into that?” With, for example,
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10 ÷ 2, we’re asking “How many groups of 2 will fit into 10?” This is easy:
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Here’s 10:
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Here’s a group of 2:
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We can easily see that 5 groups of 2 will fit into 10:
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Unlike multiplication, division is not commutative. We cannot divide backward and forwards and expect to get the same result. For example:
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10 ÷ 2 ≠ 2 ÷ 10
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We always “flip the second fraction” in the fraction division algorithm as, contrary to logic, flipping the first fraction instead will not yield the same result. For example:
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The first number in a division problem is simply more important than the second number. The first number sets the stage while the second number asks, “How many groups of me will fit into your first number?” In 10 ÷ 2, we weren’t putting groups of 2 into any old number; we were putting groups of 2 into 10. We needed to keep the 10 in mind as we bundled our groups of 2. Division with fractions operates in the exact same way. Whenever a fraction is divided by another fraction, one of two possible outcomes occurs: a fraction less than 1 or a fraction greater than 1. Of the fractions greater than one, answers can be either whole numbers or mixed numbers. Whenever we deal with parts of wholes, things get interesting.
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We’ll start with a simple example where the result is a nice, easy whole number:
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If we ask “How many of the green pie piece will fit into the blue half of the circle?” we can see pretty easily that 3 will fit in perfectly. If we were to superimpose the green pie over the blue one, the centerlines on both pies would line up nicely, creating a common denominator of 6. Unfortunately, not all fractions superimpose over each other so nicely. To develop a pattern that we can use with more difficult fraction division problems, let’s look at 1/2 ÷ 1/6 in a slightly different way. First, we’ll set the stage with 1/2:
Here we have a circle and we colored half of it. This next part is where things can get weird. Remember how, in 10 ÷ 2, the 10 set the stage before we began bundling groups of 2? If we instead thought about 10 ÷ 2 as 10/1 ÷ 2/1 , we can begin to see why this problem was so easy: the 10 and the 2 already shared a common denominator. Just as we did there, we’ll create a common denominator in this problem. The easiest way to do this is to superimpose the 1/6′s denominator atop the 1/2 and see what shakes out:
When we divide the entire region into 6 equal pieces, essentially turning 1/2 into 3/6 , it will become very easy to then take 1/6 :
Just like in 10 ÷ 2, we now ask “How many 1/2 fit in 1/6 ?” In other words, how many green pie pieces fit into the original blue 1/2 ?
3 do. And in fact, 1/2 ÷ 1/6 = 1/2 • 6/1 = 3.
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Fraction division hasn’t earned its own post based on easy problems like 1/2 ÷ 1/6 . This problem was easy for a couple reasons: the answer was a whole number and it was very easy to create a common denominator. Next, let’s look at a slightly harder fraction division problem in a still slightly different way:
This problem is more difficult for a few reasons. First, the result will not be a whole number. Second, the result will be a fraction less than one. Third, the denominators 4 and 6 don’t overlap very easily, so we’ll need to create a common denominator that is larger than both 4 and 6. We’ll deal with these first two reasons as we work through the problem. To mitigate the third reason this problem is more difficult, we’ll create a larger common denominator. Fortunately, this larger common denominator will appear naturally as we begin to draw the problem.
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First, 3/4 :
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To show 3/4 , we divided an area into 4 columns and pink-boxed 3 of them. Keeping the entire area in mind as we have done before, we will now get ready to take 5/6 by first creating 6 rows:
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and then coloring in 5 of the rows:
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By using columns to show the first fraction, and rows to show the second fraction, we naturally created a common denominator of 24. This will happen every time we use the column and row method.
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Now let’s ask our question: “How many 5/6′s (orange boxes) fit into 3/4 (pink outline) ?” In other words, how many of the orange boxes from the group of 20 will fit into the pink-boxed 18 area? So it’s a bit easier to visualize, let’s move as many orange boxes inside as will fit:
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18 out of the group of 20 orange boxes will fit. And in fact, 3/4 ÷ 5/6 = 3/4 • 6/5 = 18/20
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So far we have seen a whole number answer and a fractional answer less than 1. In this third example, we’ll look at the last type of fraction division problem – one that yields a mixed number. This next problem was asked of me twice during two different interviews for middle school Math teaching positions in the Boston Public Schools:
Because we’ll need to create a larger common denominator here, as 2 and 3 don’t easily overlap, we will use the column and row method. Starting with 1/2 :
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we’ll get ready to take 1/3 by dividing the entire area into 3 equal rows:
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and coloring 1 of the 3 rows:
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We see a group of 2 blue boxes. As we’ve done before, let’s move the one on the outside into the inside:
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2 of the 2 blue boxes (2/2) will fit into the pink-boxed area. Additionally, another 1 out of the 2 blue boxes (1/2) will also fit:
And in fact, 1/2 ÷ 1/3 = 1/2 • 3/1 = 3/2
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With the new Common Core Standards, kids are being asked to divide fractions beginning in 5th grade. As with anything, once we develop a pattern for fraction division, showing the process with pictures becomes easy. Once a kid can see and feel what is happening in these problems, the process of dividing fractions will begin to make more sense.
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Please check the comments below for some good additional information on why the division algorithm works.
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Also see Multiplying Fractions with Pictures!
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Also see Multiplying Fractions with Pictures! and Differences of Squares with Pictures!
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You can download a PDF ebook that uses pictures to explain fraction division, multiplication and addition on CurrClick at Fractions: A Picture Book!
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contact blog author Shana Donohue: shanadonohue@gmail.com
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Adding and Subtracting Fractions… the crisscross method! June 6, 2011
Afterwards, you will invariably need to reduce your fraction. To do this quickly, and without thinking, bust out your TI-83 graphing calculator and press:
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MATH > 1:Frac > ENTER > ENTER
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Not knowing how to add and subtraction fractions with just a pencil and piece of paper is not good, but if you’re in a pinch and have a history term paper due, go for it. The above TI-83 button commands also work to convert rational decimals back into their fractions.
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Disclaimer: In no way will this method help do anything except get through your current homework assignment on adding and subtracting fractions. I know you don’t believe me because my 12-year old self wouldn’t believe me, but solving fraction problems without a calculator is an important piece of higher math.
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AWESOME post on the “divisibility by 3″ trick April 3, 2011
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Author Josh Rappaport had written a blog post on the divisibility by 3 trick. If you’re not familiar with this trick, it states that by finding if the sum of the digits in a number is divisible by 3 then the number itself is divisible by 3. For instance, the number 12,345 is divisible by 3 because 1+2+3+4+5 = 15, which is divisible by 3. Taking the trick even further, the digits in 15 – 1+5 – add to 6, which is also divisible by 3! Neat stuff!
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But why does this work? I guess if I had to, and I had a bunch of time!, I may have been able to figure this out (I’d disappoint my professors if I couldn’t), but I thought I’d ask Josh for the cheat. He wrote the best blog post I’ve read in a while on WHY this trick works… How to See Why the Divisibility Trick for 3 Works. Check it out!
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