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This year, after teaching regular education Math for 8 years in Massachusetts, I took a position teaching Special Education Math.  I’ve learned how to write IEPs, manage a caseload, write progress reports, process IEPs, make accommodations, write accommodations, and a bunch of other things that will come to me as soon as I click post.  I love teaching Special Education.  I’d love to stay here for the rest of my Life.  Teaching kids who are afraid of Math to be less afraid is where I feel my strength lies.  The only problem is… certification.

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To be more precise, certification isn’t exactly my issue.  I get that we all need to be certified to teach, which is why I am certified in both Math and Biology and why I went to graduate school for Math for Teaching.  I want to be qualified; every teacher does.  I studied for and passed the Foundations of Reading MTEL exam this year to move forward on Special Education certification.  Unlike subject-matter licenses, to obtain a Special Education license a certain number of courses/seminars/letters of endorsement need to be collected.  I get all this.  I understand the importance of being both knowledgeable in Math and Special Education if students in front of me need that.  This is fine.  My issue isn’t with any of this but rather with the complete lack of available resources, or rather of ONE QUALIFIED PERSON WITH A PHONE NUMBER, that will answer whether this grad class from this college or this seminar offered during this PD session will cover this requirement for the Special Education certificate.

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There is no one. No one will answer the simple question, “Will this $400 course satisfy this Special Education requirement?”  As it stands now, I take the $400 course and then just pray to God that it counts.  It shouldn’t be this way.

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After waiting on hold for 29 minutes, I finally got through to someone at the MA DESE to ask how I can guarantee that a course will count.  The woman told me to write DESE a letter, addressed to “TIM”.  Apparently there is only one Tim who works at the DESE.  It is June.  The DESE is currently reviewing materials sent to them in October.  My class starts August 1 and wasn’t even created until late May.  She then gave me the option to take the course and send in the paperwork afterwards to see if it will be approved.  Seriously?  The reason I called was to avoid this, was to avoid paying for countless courses that won’t count!  This is total bullshit.

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I’m under no illusions that all majors are created equal.  I do believe that we should warn kids about the dangers of taking it too easy in college, but steering young people away from majors that will lead to fulfilling careers is where I draw the line.  Have you ever noticed that doing nice things for others makes you feel good?  That pulling in a big paycheck still doesn’t put a smile on that rich lady’s wrinkled face?  That in a society we need to help each other and that getting paid for it is a bonus?  It’s been shown that once basic needs are met, additional income makes little difference to a person’s happiness (Forbes’s Tim Hartford reports on this here.  The film I Am is another wonderful example).  I don’t think Alex Planes had Life in mind when penning his nauseatingly narrow minded article The Five Worst College Degrees for Your Career.  It was less an analysis of bad degree choices and more an attack on the selfless act of helping others.  Every one of Alex’s targeted career choices is one that makes society run a little smoother.  It left me wondering if he had recently been jilted by a teacher.

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Alex’s 5 worst college degrees are, along with my responses to each:

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5: Special Education – I’m a special education teacher.  On top of being in the most rewarding teaching position that I have ever held, Special Education teachers are in very high demand.  A dual-certified Special Education teacher will never want for work.  Alex’s median mid-career salary is also way off, unless he only surveyed teachers in the most rural parts of the US.

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4: Human Development – Admittedly I have no idea what Human Developers do, but seeing as this career choice is sandwiched between two very important careers, I can only assume that Alex has no idea what he is talking about.

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3: Elementary Education – It was an elementary school teacher who gave Alex the ability to later string together nonsense and call it fact.  An elementary school teacher taught him to identify letters, sounds, words, the parts of a book, how to punctuate a sentence.  Ditto on the median salary here too, Alex.

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2: Social Work – Yeah, not important.  Of the billions of photos one can easily pull from a Google Search, Alex chooses to punctuate this bullet point with one of a nurse helping an elderly man.  Screw old people, right Alex?  Gotta get dat money, son!  He even adds that “the field is projected to add to its numbers at a faster clip than the national average (161,200 new social workers will be needed by 2020).”  So if job security was not a consideration here, what was?  Is Alex’s definition of a good career choice one that helps as few people as possible?

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1: Child and Family Studies – We all want our kids to take school seriously and graduate.  It’s been shown that being a part of a Head Start program as a small child is directly linked to the student’s high school attendance.  This paper by the Baltimore Education Research Consortium is a compilation of the data.  Head Start programs don’t run themselves, Head Start teachers do.  They inspire our kids to be the best they can be.

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At the very end of Alex’s article is link for a “free”, “shockingly true” retirement report.  Sell out much?  There’s nothing worse than a 25 year old working every day for his future retirement.  But since he baited me to go there, I will.  While it’s true that a teacher will never get the opportunity to make $150k per year, it all averages out in the end.  Every two weeks, 11% of my paycheck goes into a little closed-system thing called a pension fund.  Once I hit 60 years old, I can walk away from teaching and continue to collect 80% of the average of my best three years for the rest of my Life.

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Write about that one, Alex!

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!

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“We show that, when anticipating an upcoming math-task, the higher one’s math anxiety, the more one increases activity in regions associated with visceral threat detection, and often the experience of pain itself (bilateral dorso-posterior insula). Interestingly, this relation was not seen during math performance, suggesting that it is not that math itself hurts; rather, the anticipation of math is painful. Our data suggest that pain network activation underlies the intuition that simply anticipating a dreaded event can feel painful.”

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You can read the article When Math Hurts: Math Anxiety Predicts Pain Network Activation in Anticipation of Doing Math by Ian M. Lyons and Sian L. Beilock  here.

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We already knew that Math anxiety causes decreased brain function, explained here in Huffington Post’s article Math Anxiety Linked With Differences In Brain Functioning, Study Finds.  Even without an article stating so, this fact is obvious to any teacher, parent or even student who can solve for x during a warm up (with one eye closed while catching up with a friend) but then chokes on the subsequent quiz.  Now there’s proof that just thinking about our Math classes may be causing our kids physical pain.  I sort of feel bad.

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Well, not that bad.  Alleviate your pain by doing your homework!

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.

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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Clicking the picture below will bring you to the What do you know about Triangles? Prezi.  This Prezi consists of just 11 questions and filled up about 60 minutes of time.  I had my kids put up cards (red:A, orange:B, yellow:C, green:D) to answer each of the multiple choice questions.  I also had them record their answers on an easy answer sheet.  They had a blast and it became a more valuable “fun” activity than I had anticipated.

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Click the flags to go to the Prezi.