Grade 10 Math MCAS Review – A Prezi! April 27, 2013

You can get to the Grade 10 Math MCAS review Prezi here or by clicking the above screenshot.

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It’s about that time again. No wait, it is that time again: Grade 10 Math MCAS. Forget about grades 11 and 12 as these grades are meaningless. The new Grade 12 is Grade 10. The new SAT is MCAS.

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I write this with both a migraine and with my tongue in cheek. As a Special Education 10th grade Geometry teacher, I have very mixed feelings about this dreaded test. As a teacher who never had to take MCAS, I think that my students will come out perfectly fine without proving their 10th grade knowledge on some expensive test. I did. All [most] of my friends did. As a person who has taken a boat load of tests and who has become very aware of the unique sense of accomplishment that comes from passing the seemingly-impossible, I want to give my students every tool to show this test who is boss. There is no better feeling than whipping a test’s ass. I want my students to experience this feeling.

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I learned an important lesson from an unlikely source at UMass: Kids in Mr. Chandler’s inorganic Chemistry class who had internet and could access Chandler’s old exams would do better than me. Why? Because his tests were partially-recycled. Above is a screenshot of a Prezi I made from 2012′s Grade 10 Math MCAS multiple choice questions.  Every kid gets 4 colored index cards with either “A”, “B”, “C”, “D” written on it. My “A” is red, but that part doesn’t matter so much. As we click through the slides, kids do their work on scrap paper then hold up the colored card that corresponds to their answer choice. This does two things: makes the kids feel that they’re playing a game and lets me see the class-wide weaknesses to focus on during explicit cramming. In addition to practicing the concepts exemplified in these multiple-choice questions, we’ve been doing the open response questions in class, being sure to review Statistics. MCAS creators love mean, median, mode, range, box-and-whiskers, stem-and-leaf, line plots. “When will I ever use a box-and-whiskers thing in real Life?” Never kids, just possibly on May 13th.

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The best of luck to your students!

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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How Romney’s Bain Gutted My Business October 22, 2012

My husband wrote this article.  He is quite fired up about what went on at Bain Capital, and he wants you to know how Romney and Bain Capital bankrupt US businesses.

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“I have done a decent job of staying away from the Facebook politics, up to now. I struggle every time someone posts that Romney is a job creator. Please don’t take this as an endorsement of Obama. I am not particularly fond of the job Obama has done, and his push for the Race to the Top and similar initiatives have little research basis, and the economy is a constant worry.

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That said, Romney’s record as a job creator is a joke. Romney made his money off the backs of hardworking Americans. People are overlooking his time at Bain Capital. Don’t take my word on it either – really go investigate it – because it is terrible to read about. Romney and his leadership are responsible for ripping the heart out of many companies.

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What Bain Does: Bain uses a tiny bit of money to secure loans to purchase/take over already-viable companies. No big deal, smart business. Keep in mind, these are viable companies – not struggling ones – that are in the black enough to be approved for loans. But it is the next steps that are not only highly questionable, but some consider incredibly immoral. Once the company in question is acquired by Bain Capital, the loans are shifted from Bain to the company. Bain then charges the company millions of dollars in fees for their services. Bain then sells the company and walks away, leaving the company fundamentally bankrupt.

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Don’t believe me? Look up the following companies and see what happened to them:

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Armco Steel
Georgetown Steel
American Pad and Paper (They fired all 250 workers and invited them to reapply for their jobs at lower pay and a 50% cut in healthcare coverage)
Bealls Brothers
Palais Royal
Dade International
Holson Burnes
The Damon Corporation (while under Bain’s “leadership” was charged with and fined over 100 million in Medicare fraud).

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Certainly, Bain found great success with Staples. Interestingly, Romney did not want to invest in Staples. Let’s also forget about the 1.5 Billion (yes billion) he took from the federal government to finance the 2002 Winter Olympics or the 10 million he took in 1993 to prevent Bain from hitting bankruptcy.  There are other successes –Wesley Jessen VisionCare and Calument Coach and the Gartner Group and Sports Authority (Bain was a minority investor). However, his ability to acquire a successful company, saddle it with debt and move on overshadows these serious successes. The Wall Street Journal indicates that 1 out of 5 companies Bain acquires end up bankrupt. 1 out of 5.  “Bankrupt” is not the same as “out of business”, though we sometimes accidentally use these terms interchangeably.  A bankrupt company is one whose debt just disappears, much like a black hole makes light disappear.  But unlike black holes, the banks that held the loans – and thereby the government - are left with big deficits they need to fill in some other way, if they can at all. The Associated Press estimates that Bain cost over 4,000 workers their jobs in Romney’s first 10 years of leadership. Can we take that chance with all of America?

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Bain Capital literally ripped the profits out of companies in order to secure profits for investors.  This was a company, led by Romney, that didn’t care about the American worker or the middle class or helping put food on the table. Bain and Romney cared about only themselves – how to make money – how to profit – at the expense of the middle class.

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Add this to Romney’s constant flip-flopping and I am totally turned off:
- Bailouts – He criticizes the bailout of the auto industry and credits/defends the bailout of Goldman Sachs (a Bain partner).
- Abortion – In 1994 and 2002 he supported women’s rights to abortion; in 2007 and 2011 the opposite.
- Don’t Ask Don’t Tell – In 1994 he supported Gays and Lesbians serving openly – 2007 the opposite.
- Vietnam – In 1994 he “had no plans of serving in the military” – 2007 he “longed” to serve
- Climate Change – June 2011 “I believe the world’s getting warmer. I can’t prove that, but I believe based on what I read that the world is getting warmer. I believe humans contribute to that – October 2011 “We don’t know what is causing climate change on this planet.”
- Bush Taxes Cuts
- Health Care Reform
- Gun Control
- Hunting (yes – he even changed his mind about hunting)

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I really want to support the Republican Party this election. I really do. I don’t like how the Obama administration has handled the debt or job growth or health care. I tend to be a middle of the road type of guy, but I can’t in good conscience vote for Mitt Romney. He has set an example of corporate and personal greed gone amuck. He would be terrible for America. When you are looking at the better of two bad choices, I will take Obama. His record for supporting America is just better.”

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Selling Math To Students – number tricks, cell phones and fractals

Do you have a number trick you like?  I’d love if you posted it in the comments below!  Also see Super Cool Math Tricks.

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To begin each new school year, I like to sell Math a little.  So each fall I like start the year with my favorite number trick:

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1: Have a student choose an item: pen(3), book (4), board (5), or eraser (6).  You can also do this with fruit of different word lengths or anything else, really.  The important thing is that the words are all different lengths.

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2: Have the student:

a: Turn the word into its number

b: Multiply by 5

c: Add 3

d: Double the number

e: Add in favorite single digit

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3: Now ask for the number. 

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To “undo” the trick, simply subtract 6 (in your head is most impressive).  The first digit in the number you now have corresponds to the item the student chose.  The second digit is the student’s favorite number.

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The debunk:

Let’s say we chose “eraser”.  We’ll use E for “eraser”:

5E

Let’s say my favorite number is 3:

5E + 3

Now I double it:

2(5E + 3) = 10E + 6

Add in my favorite digit “n”:

10E + 6 + n

The 10 shifts the “eraser” (E) number to the tens placement and the favorite digit is in the ones spot.

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If this doesn’t work (it usually does), then I move to more drastic measures.  This Prezi has embedded clips from Nova: Hunting the Hidden Dimension because it is the most awesomest Nova special (or any type of special) ever.  The kids enjoyed this intro to their school year:

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Dirty Word of the Day: Memorization October 21, 2012

As a high school Math teacher, I hear all the time, “I suck at Math!”, especially, considering that everything else in the world is found at the push of a button, when my students are faced with problems they can’t immediately solve.  I hear “I hate Math” when we’re solving equations, when we’re factoring, when we’re plugging x values back in to find angle sizes.   I hear it all the time.  but it’s when I hear it that got me thinking.

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Since 2003, I’ve been keeping a mental log of all the times I have heard “I hate Math” or “I suck at Math”, mainly because each one has left its own little crater on my Math soul.  I want – need, really - to figure out why kids feel this way.

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It took nearly 9 years, but yesterday I finally figured out why some kids hate Math with all of their being.  It’s because they can’t multiply.  This had been my suspicion for a few years, but yesterday it became clear that multiplication makes the difference. 

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“But multiplying is easy, it’s not that.  Math is WAY harder than just multiplying!” you may say.  And I agree with you.  However, Math is 90% confidence, and when a kid loses this confidence because “multiplying is easy” and he can’t multiply, then he feels like a loser and closes off to the rest of his years of problem solving. 

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The Conclusive Evidence

I had never seen a little kid do Math until yesterday when a former student of my husband came over with her Mother for lunch.  She’s in 4th grade now and has been having trouble with Math, so we sat down with her current homework: multi-digit multiplication problems.  The algorithm “multiply then carry, then multiply again and add what you carried” is a little weird, but she got that part.  Then all of a sudden out of nowhere, with fists slamming on homework…

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“I stink at Math!”

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“I hate Maaaaath!”

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Within the giant problem my husband gave her to do, she needed to multiply 8 x 5.  When it didn’t come immediately, she exploded.  And up went the walls.  Single-digit “multiplication is easy”, right?  Not if you don’t know it.  If you don’t know 8×5, then Math is the shittiest subject there ever was, ever is, or ever will be.  It totally blows.

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So what would help her?  Memorizing her multiplication tables.  Sounds simple and ridiculous, right?  Hold on a second.  Below are a few excerpts from an article, “Chess Experts Use Brains Differently Than Amateurs”:

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Experts use different parts of their brains than amateurs, maximizing intuition, goal-seeking and pattern-recognition, says a new study that examined players of shogi, or Japanese chess.

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Researchers believe that experts who train for years in shogi are actually perfecting a circuit between the two regions that helps them quickly recognize the state of the game and choose the next step.

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“Being ‘intuitive’ indicates that the idea for a move is generated quickly and automatically without conscious search, and the process is mostly implicit,” said the study.

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Math is a lot like chess: strategy, visualization of next moves, attack!  When a kid is a multiplication amateur, strategy can never develop, patterns will never be recognized, Math will always be counterintuitive.  Multiplication facts take a lot less time to master than chess.

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Memorization: Mathematicians’ Dirty Word

Memorizing sight words doesn’t make reading Harry Potter easy, but it does make it easier.  This is why we do it.  So why not memorize multiplication facts to make Math easier?  At some point between being a Math student and being a Math teacher, ”memorization” became a dirty word.  I agree that we shouldn’t force kids to memorize every Mathematical formula or the digits of pi, but I remember a deep sense of pride in having my multiplication facts memorized.  Maybe the way it was done – calling us up one by one to recite the facts to our 3rd grade teacher – was not the best method and probably contributed to my high-strung demeanor.  But when I got to pre-Algebra, I could cross-multiply; in Algebra I could quickly find factors; and in Geometry I could “plug it in” without a calculator.  All of these seemingly-unrelated abilities contributed to my feeling that Math wasn’t impossible.  I had confidence because I could multiply quickly.  I was fluent, solving came easy.  I could do more advanced problems because I had confidence.  I had confidence because I didn’t need to stop and think through every instance of multiplication.

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A recent brain study done by Dr. Carolyn McGettigan in the UK yielded unexpected results.  Contrary to hypothesis, the expert beatboxer uses less of his brain than the novice:

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The novice used many more brain areas, suggesting a need to plan each sound and a lack of automatic processing.

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Dr Carolyn McGettigan, a neuroscientist at University College London, compared magnetic resonance imaging (MRI) scans during two tasks – counting and beatboxing.

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Dr McGettigan says: “When you think about an expert you might think they activate extra bits of the brain – not just the bits you use to make sounds, but something exciting and different that you might not expect.”

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“What we have at the moment is a demonstration that being an expert doesn’t mean you activate more of your brain. The phrase ‘less is more’ is sort of appropriate here.”

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Thinking is Overrated

Thinking is important, but not that important.  When it comes to the building blocks of any language, a level of fluency is essential.  Without it, reading is exhausting, and things that are exhausting are avoided.  I doubt that my husband – a true bookworm - would have read the entire Harry Potter series [more than once] if he had to individually sound out each word.  It just wouldn’t have happened.  J.K. Rowling would have never earned the necessary funds to  go on to write The Casual Vacancy (is it any good?) if everyone struggled through her Harry Potter books.  Is it any mystery that kids hate solving equations or finding missing side lengths if they have to “sound out” each instance of multiplication?

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So what to do?

We have to stop making our kids think so much!  It’s exhausting them.  When I was a kid, my parents gave me this Math toy that tricked me into learning my multiplication facts.  Flashcards for facts up to 12×12 are also great.  It’s got to be fun, not forceful, of course, but it’s got to happen.  It will make all the difference later and is really that simple.

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Free dyslexic fonts from OpenDyslexic October 3, 2012

“OpenDyslexic is a new open sourced font created to increase readability for readers with dyslexia. The typefaces include regular, bold, italic and bold-italic styles. It is being updated continually and improved based on input from dyslexic users. There are no restrictions on using OpenDyslexic outside of attribution.  You can get the free font HERE.

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Your brain can sometimes do funny things to letters. OpenDyslexic tries to help prevent some of these things from happening. Letters have heavy weighted bottoms to add a kind of “gravity” to each letter, helping to keep your brain from rotating them around in ways that can make them look like other letters. Consistently weighted bottoms can also help reinforce the line of text. The unique shapes of each letter can help prevent flipping and swapping.

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OpenDyslexic also takes a different approach to italic styles. It is generally recommended that italics be avoided in reading material for dyslexia. However, instead of taking the normal approach of “slant x% for italic,” OpenDyslexic’s italic style has been crafted to allow for its use for emphasis while maintaining readability.”  (http://dyslexicfonts.com/)