# ZeroSum Ruler (home)

## Blogging on math education and other related things

### Difference of Squares (and binomial multiplication) With Pictures!January 12, 2013

We’re starting to see a difference of squares emerge…

Multiplying binomials.  FOILing.  Whatever you call it, and however bad we want it, there’s no real shortcut.  So why does (x + 5)2   ≠   x2 + 25?  Let’s take a look:

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Above is a representation of (x + 5)2.  We can see along the top edge “x 1 1 1 1 1”, representing x + 5.  Whenever we square something, we multiply it by itself, so we see the same x + 5 along the left edge.  Since (x + 5)2 = (x + 5) times (x + 5), let’s multiply to find the area of each colored region:

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If we put all the pieces together, we get:

(x + 5)2   =   x2 + 10x + 25

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When we say that (x + 5)2   =  x2 + 25, we miss out on all of those little blue 1x’s.  Multiplying two expressions together will always give us an area.  For example, a rectangle with length 5 and width 3 will have an area of 15.  Multiplying two binomials together, like we did above with (x + 5)(x + 5), usually yields a trinomial.  I say usually because there is one case when this is not true…

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Let’s multiply (x + 5)(x – 5).  A great way to do this is with the Box Method:

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Above, we see (x + 5) along the top of the Box and (x – 5) along the left.  If we multiply these two binomials together:

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and then combine like terms, we get:  x2 – 25.  Since both x2 and 25 are square numbers, and they are being subtracted, we literally have a difference of squares.  There is no middle term because the +5x and the -5x cancel each other out.

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To see how this problem translates into areas like our first example (x + 5)(x + 5), let’s start at the end and work our way back to the beginning….

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Here we see two squares: one is green and one is white.  The white one is being subtracted (difference) from the green one.

Since “difference” means subtract in the language of Math, we quite literally have a difference of squares.  Above, we see 52 being subtracted from x2.  To make things more interesting, let’s overlap the regions:

Because the green shape is pretty lopsided now, let’s draw some dotted lines to think about the green shape in terms of three nice, regular shapes:

And now let’s multiply to find the areas of each of the nice, regular shapes:

If we simplify each of the white expressions, we get:

5(x – 5)  =  5x – 25

5(x – 5)  =  5x – 25

(x – 5)(x – 5)  =  x2 – 5x – 5x + 25   =   x2 – 10x + 25

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And then if we add them up:

(5x – 25)   +   (5x – 25)   +   (x2 – 10x + 25)   =   x2 – 25   It’s a difference of squares!

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But can we express this x2 – 25 as the product of two expressions, like we did with x2 + 10x + 25  –>(x + 5)(x + 5)?  When we ask this question, we’re asking if we can go backwards; we’re asking if we can factor the expression to find out where it originally came from.

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In the first example, x2 + 10x + 25 factored to (x + 5)(x + 5).  Can we do the same with x2 – 25?

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Let’s go back to our overlapped picture to find out:

Maybe if we break up the green region:

And begin to rearrange the pieces, first sliding one rectangle up:

and then chopping that bottom part, rotating it 90° and putting it on the left:

We made a rectangle!  And what are its dimensions?

(x + 5)(x – 5)!

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So x2 – 25 came from (x + 5)(x – 5).  In this situation we didn’t get a middle x term when we multiplied the two binomial expressions together.  Instead, we got a difference of squares, which makes sense since that’s where we started!

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Here’s a video that shows why (a + b)2 ≠ a2 + b:

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Contact this blog’s author at shanadonohue@gmail.com.

### Archimedes says “cone + sphere = cylinder”?April 14, 2012

Filed under: algebra,math education,number theory,proof — ZeroSum Ruler @ 5:19 pm
Tags: ,

Archimedes of Syracuse (87 BC – 212 BC), before the eternal time sucks of Facebook and DrawSomething, sat and imagined a cone and a sphere balancing one side of a scale, and on the other side balancing against these two objects was a cylinder.  The sphere he imagined has the same diameter as height of the cylinder and the cone and he reasoned that the scale would remain perfectly balanced.

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But would it?  Using just their volume formulas, which Archimedes did not yet have, can we see that this works?

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Does sphere + cone = cylinder?

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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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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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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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### Super Cool Math Tricks: Be a Human Computer!June 29, 2011

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Also see Selling Math to Students – number tricks, cell phones and fractals

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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:

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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You can download a PDF ebook that uses pictures to explain fraction division, multiplication and addition on CurrClick at Fractions: A Picture Book!

contact blog author Shana Donohue: shanadonohue@gmail.com

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### I can read your mind! (well, Algebra can :)January 9, 2011

If you have a good handle on your Algebra, you can read anyone’s mind!  This video (click the red triangle to go to the YouTube video) is just one case where I can read your mind straight through the computer! It’s true!   Can you develop an algebra trick that reads your friend’s mind? I bet you can!

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### can anyone prove this algebraically?June 27, 2010

Filed under: algebra,math,proof — ZeroSum Ruler @ 10:13 pm
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1^3 + 2^3 + 3^3 + . . . + n^3 = (1 + 2 + 3+ … + n)^2