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Blogging on math education and other related things

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.

On a Friday, my Geometry students couldn’t remember how to solve simple equations.  The next Monday, after giving them a hand-drawn flowchart, they were solving equations with variables on both sides.  Below is an improved flowchart, complete with “combining like terms” for those pesky equations with all the terms on one side.  By clicking the picture, you can download the PDF version that doubles as a handout and as a poster.  It has helped my students tremendously.

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New (free) ZeroSum ruler – for teaching addition with negative numbers

Below is a new version of the ZeroSum ruler.  This one needs no hardware to construct, just scissors and glue.  You can download, print and use this proven tool right now by clicking on the picture, which will bring you to the PDF file that contains 2 ZeroSum rulers.

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Simple “how to solve an equation” flowchartSeptember 14, 2012

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Since drawing this flowchart, I have created a printed one that includes “variables on both sides“.  You can see the post (and get the free poster download here: http://zerosumruler.wordpress.com/2012/09/30/solving-equations-flowchart-free-download-poster/

Is Common Core meeting its Goal?May 21, 2012

Is the original goal of Common Core being lost in the upper grades?

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One major difference between the U.S. and so-called ”A+ Countries” is, while we focus on breadth, they focus on depth.  While there is a natural progression throughout a student’s school years from one math topic to another in these high-achieving countries, in the U.S. we seem to have a “throw at the wall and see what sticks” mentality.  For example, in grade 8 we cover 32 unique mathematical topics.  In high-achieving countries this number is just 18.

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The new Common Core curriculum aims to bring our focus back to depth in the lower grades but seems to miss this mark once the abstract maths are reached.  While it is true that more topics have been cut out than added in most grade levels, topics traditionally covered in Algebra 2 (and some may say pre-Calculus and above) – piecewise functions, limits, logarithms, areas under curves, Algebraic proofs, and rational function graphing to name a few – are now part of Algebra 1.  Does adding so many advanced topics to the Gateway of Higher Math (ok, I’m biased) do what Common Core initially set out to do?

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Below is a comparison of the math topics taught each year in A+ countries (first chart) and those covered in the U.S. (second chart) each year (compiled by Professor W.H. Schmidt).  These comparison charts were created before, and as a support for, Math reform in the U.S.  Still, to meet the new upper-grade Common Core Standards, school districts are turning to hybrid-type courses: “Algebra/Geometry/Stats Year 1″, etc. (Yes, that’s ONE year’s course) to meet all of the new high school requirements.  While the Common Core sets out for mastery at the Elementary level, did it really mean to hybridize high school math?  If depth is more important that breadth, what are we doing?

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Grade 8 Math MCAS Review – PreziMay 2, 2012

Math MCAS are quickly approaching.  Below is a multiple-choice 8th Grade Math MCAS review Prezi (phew!) covering 25 question types that students often get wrong.  This Prezi is viewable (and downloadable) at http://prezi.com/8me0m014oo_u/mcas-prep/ (or by clicking the picture).   Please email me if you’d like editing rights to this Prezi.

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Thank you to Paolo Tosolini for his AWESOME YouTube tutorial on how to create a cool Prezi.