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.

Adding Fractions With Pictures! (The Crisscross Method) December 3, 2012

Fraction Addition (And Subtraction): We’re not in kindergarten anymore

Addition and subtraction are only easy in elementary school.  Once middle school starts, continuing throughout any Math class taken that point forward, addition and subtraction are much harder than multiplication and division.  Why?  The Common Denominator.  To a kid who is not fluent in his multiplication facts, finding The Common Denominator is an exercise in torture.

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What is a common denominator?  A common denominator is a multiple of both denominators in a fraction addition (or subtraction) problem.  For example:

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In the above example, 6 is a common denominator of 2 and 3.  But is it the only one?  No.  How many common denominators are there between two fractions?  Infinite.  For example:

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Why would we want to use 7830 as a common denominator?  Why not?  The point is that any number that both denominators divide into evenly can act as a common denominator.  We are far less restricted than we thought.

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So if we’re virtually unrestricted in choosing a common denominator, why not pick the one that is the product (multiply) of the two denominators?  For example:

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Just multiply the denominators to find a common denominator.  This is easy.

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At this point in the traditional method of adding fractions, we’d begin to ask our questions: “How many 8’s go into 16?”  Ok, 2.  “2 times 3 is …?”  Ok 6.  So 3/8  =  6/16 .  Though this process is easy to a person who is fluent in their multiplication and division, it will give reason for a non-fluent Math student to seize up.

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A great alternative way of adding fractions is the Crisscross Method of adding (and subtracting) fractions.  In this method, we use the common denominator just once (this method will not create two equivalent fractions to the original two) and multiply “crisscross” to find two new numerators.

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In  3/8  +  5/2, we’ll first multiply the denominators to find our new, common denominator:

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Next, we’ll multiply 3 • 2 (always starting our crisscross in the top left corner) to find the first missing numerator:

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And then 8 • 5 to find the second missing numerator:

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But why are we allowed to do this?  Let’s back up to see what really happened.-

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First, we found the common denominator 16 by multiplying the denominators (8 and 2) of both fractions.  We’re guaranteed that our denominator is common if we created it by multiplying the two original denominators to get it.  To get the first numerator 6, we multiplied the numerator of the first fraction (3) by the denominator of the second fraction (2).

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In the process, we multiplied both numerator and denominator by 2.  In other words, we multiplied  3/8 by  2/2.  Any number divided by itself is just a fancy 1, and multiplying any number by 1 does not change the number’s value.  As a check to see if this process worked,  3/8  =  6/16 .  The old and new fractions are equivalent.

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The same is true to get the second numerator 40:

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Both numerator and denominator were multiplied by 8.  In other words, we multiplied  5/2  by 8/8, which is just a fancy 1.  Multiplying by 1 does not change a number’s value.  As a check,  5/2   =  40/16.  The old and new fractions are equivalent.

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Now we simply add the numerators:

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The Crisscross method also works for fraction subtraction – we’d have a subtraction in the numerator.  Why was this method not taught in school?

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Hurray for Fraction Addition (and Subtraction)!

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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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Top 10 Myths about Math Education December 29, 2011

The following article and its 10 myths about Math education were posted on the Gideon Learning Blog.  From inquiry being the catch-all to memorization being a dirty word, this article hits all the kinks in the way we teach, or at least how we’re told to teach (see Myth #10), math today.  You can see the full article by clicking on the picture or by following the link underneath.  p.s. It’s not just the reps of the Boston Teacher’s Union who roll their eyes at the mere mention of TERC (see Myth #2)….

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http://gideonlearning.wordpress.com/2011/12/28/ten-myths-about-math-education-and-why-you-shouldnt-believe-them/

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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!

 

 

 

 

 

Surface Area to Volume Animation September 26, 2011

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The following [totally awesome] video comes courtesy of Dr. Brian Biswell.

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The Language of Math Poster August 19, 2011

Below is a poster I hang in my classroom every fall.  Each year it grows longer as more and more terms come up for the different operations of math.  When I was a kid, no one told me to look out for these words, or that math was even a language at all, which made word problems pretty tough.  By clicking on the poster you will be sent to the original Excel file on Google Docs.  Do you have any words to add?

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Link to Google doc:

https://spreadsheets.google.com/spreadsheet/ccc?key=0Asra4GjkRBNidGhoZlZYcjk4dmhISDlSNHJDbjBPTXc&hl=en_US

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…-

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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