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!

What do you know about Triangles? (Prezi multiple choice) October 31, 2012

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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.

Income and Debt with ZeroSum Ruler July 10, 2011

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One of the ZeroSum ruler’s main purposes is to calculate debt/income problems.   In the problem “I owe you $12 and pay you back just $7. How much do I still owe you?” how do you come to your answer?   Do you count backwards from $12 to $7?   Or do you count forwards from $7 to $12?   No really, how much do I owe you?   How did you figure this out?    The ZeroSum ruler allows the student to count forwards instead of backwards just like we do in real life!    So why do we make our kids count backwards in school?

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Multiplying Fractions with Pictures! June 15, 2011

Fraction Multiplication: Of what?

Fractions are probably the most troublesome topic in Math.  As soon as a problem involves a fraction, kids freeze up.  In Math, of tells us to multiply.  How many shrimp are in five pounds of shrimp?  We multiply the number of shrimp in a pound by five.  Once we know this, fraction multiplication becomes a bit easier to understand.

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The How of fraction multiplication is easy – multiply the numerators and multiply the denominators.  When we show fraction multiplication with pictures, we need to remember of.

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Now to the Why.  To start, we’ll look at a relatively easy problem so that we can develop a pattern to follow with more difficult fraction multiplication problems:

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(1/2)(1/2)

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Translated into English, this problem reads “one-half of one-half”.  Here’s a picture of  1/2:

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And the area below in red is “one-half of one-half”:

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It’s easy to see that one-half of one-half is (1/4).  And in fact:

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(1/2)(1/2)   =   1/4

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Before moving on, let’s look more closely at one aspect of the problem above: the denominator 4.  Where did this come from?  To get that denominator, we needed to keep the entire circle (whole) in mind.  In other words, we needed to say that the red piece was “1 out of something”.  (Confusingly, out of means to divide in Math!)  The denominator is 4 because the red pie piece is 1 out of 4 total pie pieces in the circle.  Always remembering the entire original area is key in fraction multiplication.  Later, we’ll see the same is true with fraction division.

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To make the denominator easier to see, we can divide the circle twice: first vertically for the first fraction, then horizontally for the second fraction:

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It’s then easy to see that the overlapped area (numerator) and the entire number of pie pieces in the circle (denominator) create our answer.  This will always be the case.  It wasn’t a coincidence that the denominator was naturally created as we divided the circle twice.  Let’s use this pattern to solve a more complicated fraction multiplication problem:

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(2/7)(3/5)

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Because these two fractions do not have a common denominator, it would be hard to divide a circle into 7 (and take 2), then into 5 (and take 3), and analyze the overlapped area.  So instead, we’ll use nice, easy rectangles.  First,  2/7:

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In the rectangle above, two of the rectangle’s 7 horizontal bars are colored green to represent 2/7.  Now, keeping the whole rectangle in mind, let’s take 3/5:

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In the rectangle above, three of the 5 vertical columns are colored blue to represent 3/5.  Where to the two colored areas overlap?

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In the above picture, we can see that the overlapped area consists of 6 purple boxes.  But 6 of what?  Remembering our easy example (1/2)(1/2), where our denominator was the total number of pie pieces in the circle after our two rounds of dividing, let’s count the total number of boxes in the above rectangle.  The total number of boxes is 35.

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And in fact:   (2/7)(3/5)   =   6/35

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OLD FASHIONED CHECK: We know that (2/7)(3/5) = 6/35 from the algorithm “multiplying across”.

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To see this more clearly, we can look at the below picture and see that the area the fractions share, or the overlapped region, is 6 boxes, and the area of the entire region is 35 boxes.  “6 out of 35 boxes are double shaded”.

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In this last picture we can see that the area of the double shaded region is 6, or (2×3), and the area of the entire region is 35, or (7×5), which is why we multiply the numerators (2×3) and the denominators (7×5) when we find the product of two fractions.

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Also see Dividing Fractions With Pictures!

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If this was helpful, here is a free poster download for your classroom:

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Also see Dividing Fractions With Pictures! and Differences of Squares with Pictures!

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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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contact blog author Shana Donohue: shanadonohue@gmail.com

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Math manipulatives lead to student failure May 19, 2011

During a 4th grade substituting assignment, the teacher left a set of word problems for the kids to do.  A bunch of these word problems involved division, and the students were directed to use their counting blocks.  As I walked around the room, I saw kids doing just about everything a kid will do with giant leggo-type blocks.  There were guns, there were swords, there were towers.  Some kids were using the blocks to work the word problems, but many of the students who wanted to use them for good were having trouble.  My role morphed from teaching math to teaching the kids how to use the counting blocks.  One word problem called for dividing 125 by a variety of numbers.  There is a large margin of error while counting 125 of anything, and with a string of problems that all rely on a 100% accurate count, it felt to me that the kids’ time could have been better spent.  When do manipulatives cross the line from helpful to hurtful?

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A great article titled Teacher Learning and Mathematical Manipulatives: A Collective Case Study About Teacher Use in Elementary and Middle School Mathematics Lessons  by Laurel Puchner, Ann Taylor, Barbara O’Donnell and Kathleen Fick, outlines one of the many problems that can arise while using manipulatives in math.  This article is a worthwhile read, especially for those teachers wondering why manipulatives don’t seem to work as well as advertised. 

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contact blog author Shana Donohue: shanadonohue@gmail.com

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AWESOME post on the “divisibility by 3″ trick April 3, 2011

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Author Josh Rappaport had written a blog post on the divisibility by 3 trick.  If you’re not familiar with this trick, it states that by finding if the sum of the digits in a number is divisible by 3 then the number itself is divisible by 3.  For instance, the number 12,345 is divisible by 3 because 1+2+3+4+5 = 15, which is divisible by 3.  Taking the trick even further, the digits in 15 – 1+5 – add to 6, which is also divisible by 3!  Neat stuff!

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But why does this work?  I guess if I had to, and I had a bunch of time!, I may have been able to figure this out (I’d disappoint my professors if I couldn’t), but I thought I’d ask Josh for the cheat.  He wrote the best blog post I’ve read in a while on WHY this trick works… How to See Why the Divisibility Trick for 3 Works.  Check it out!

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My Harvard Math for Teaching Thesis: Complete! And ready to share… March 20, 2011

After many many years of jumping through many many hoops, I am finally graduating with my MA in Mathematics for Teaching in May.  My thesis, Negative Number Misconceptions in High School: An Intervention Using the ZeroSum Ruler is right now at the printers being printed and bound.  I don’t know about you, but that instantaneous feeling of relief after taking a final exam or passing in a final paper stopped hitting me sometime in college.  So now, I’m just feeling a bit burnt out.  OK, completely burnt out.  But I’m sure it will hit me soon since it kind of needs to; I need to now get in a post-Bach program to get my Initial teaching license.  I like to do things backwards.

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So here it is for download!  For all to read!  Or maybe to just glance.  In my study, the ZeroSum ruler proved effective in reducing eleventh grade error on integer addition and subtraction problems (especially with negative integers).  If I wasn’t so burnt out, I’d want to test it with younger kids.  Imagine how our world would be if my eleventh graders actually mastered integers when they learned them in, and only in, 7th grade.  But that’s in my thesis.]