Code breaking is so cool. Above is a short Prezi on Kryptos in Washington, DC. Below is an activity that allows students to encrypt, and decode, their own secret messages using Algebra.
The Distributive Property (“FOIL”) Through Pictures December 15, 2011
The transitive property was always my favorite as it could be applied to so many situations. I like chocolate, there is chocolate in those cookies, so I like those cookies. Totally useful.
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But a close runner up to this cookie property has got to be the Distributive Property. With strange rules of “first, outer, inner, last”, I liked its mystery. I could multiply two things together with no mention of a multiplication sign and somehow it meant something. Something big. I was doing real Algebra now.
It wasn’t until I became a teacher that I really had to think about what was being done. My students would make mistakes when “F.O.I.L.ing” (I do not like this acronym. What if one piece is a trinomial?) and I would attempt to explain what was happening. It’s difficult to explain something that has been taken for granted for 15 years. But as I made my way through my graduate program where being able to explain math was seen as the most important, I began to rethink this important property.
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The Example:
I always like to start with a concrete example. Let’s take the problem “14 x 7”
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“14 x 7” is no easy problem for most of us as neither of these numbers is easy to work with. To begin, let’s look at “14 x 7” as a geometric area in a picture:
We can easily count up the small rectangles to find how many there are, though that would take time and leaves a lot of room for error. Or, we could break the picture down into smaller pictures to make it easier to work with:
Here, we’ve broken “14 x 7” down into (10 + 4) x (5 + 2), or simply (10+4)(5+2). Is this form familiar?
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Now we can see that “14 x 7” = (10 + 4)(5 + 2). And now we can simply use multiplication to find the areas of the different colored pieces and add them up:
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10 x 5 = 50
10 x 2 = 20
4 x 5 = 20
4 x 2 = 8
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50 + 20 + 20 + 8 = 98! And in fact, 14 x 7 = 98.
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The Generalization:
Now let’s make a generalization that we can apply to other similar problems:
Here, we’ve replaced all of the numbers with letters and we can rewrite the problem as:
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(a + b)(c + d)
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Using the method we used before, we multiply each colored piece to find its area and then add up all the areas to find the total:
(a) x (c) = ac
(a) x (d) = ad
(b) x (c) = bc
(b) x (d) = bd
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The area is: ac + ad + bc + bd !
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Not the prettiest of answers, but done correctly. Using this model, can you multiply (3x + 4)(5x + 2)?
We’ll use the same picture because “x” can stand for any number at all.
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We have:
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(3x + 4)(5x + 2)
(3x)(5x) = 15x2
(3x)(2) = 6x
(4)(5x) = 20x
(4)(2) = 8
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Putting the pieces together, we have the trinomial:
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15x2 + 26x + 8 !
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The Error:
The biggest error I have seen with the Distribute Property is forgetting to multiply a piece or two. Students sometimes will answer:
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(3x + 4)(5x + 2) = 15x2 + 20x + 8
Can you see what they forgot? Can you imagine what other mistakes could be made?
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If you always remember the area of each piece, you will be The Best Distributor and Master of the Distributive Property!
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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