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!

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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Simple “how to solve an equation” flowchart September 14, 2012

(click on the picture for a downloadable pdf file)

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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/

Factoring Trinomials: A video ‘how to’ when A > 1 February 12, 2012

42 Folds to the Moon January 19, 2012

One of my students just can’t wrap his head around the power of exponents.  Can you blame him?  This week we learned that it would take just 27 folds of a piece of paper for the stack to reach the height of Mount Everest, and then just 15 more -a total of just 42 folds - to reach the moon.  As we started the lesson, students guessed “one million” and “47 billion!” folds to reach the moon, so you can imagine the shock (and disbelief) in the actual number 42!  Maybe the weirdest part is to think that it would take 41 of the folds to get just half-way to the moon and then just 1 more to make the second half of the journey. 

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But how can this be?  How is it possible that a thin sheet of paper easily ripped in half can reach the moon after a mere 42 folds?  Well, let’s see….

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The simple Algebra 1 exponential growth formula is:

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As the thickness of a piece of paper is roughly 0.01 centimeter, we’d fill in our equation as:

 

 

 

 

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This gives us a very large number of centimeters: (43,980,465,111).

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Dividing this number by 100 will give us the equivalent number of meters: (439,804,651),

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and dividing by 1,000 will give us the equivalent number of kilometers:  (439,804).

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For all us Americans stuck on the Imperial system, 439,804 kilometers is approximately 273,281 miles.  The moon is, on average, 238,855 miles from Earth at any given time, so 42 folds of a piece of paper will actually get us PAST the moon!

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So how small would the surface area of the top paper on the stack be?  How thin will be this paper tower to the moon?

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VERY thin!

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The exponential decay formula is almost exactly the same as the exponential growth formula except that there is a (1 – r) in place of the growth formula’s (1 + r).  To write the equation for how thin this stack of paper to the moon will be, we have to think about a funny occurrence in the stock market…

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To go from 1 to 2 is a 100% increase:  100% of 1 is added to itself to get 2.

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But to get back to 1 is a different story:  to go from 2 to 1 is a 50% decrease.  Just 50% of 2 is removed to get to 1.

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So when your stock goes down 30% on Monday, it’s not back to where it was if it goes back up 30% on Tuesday.  If your stock goes down 50% on Monday, it’s got to go up 100% on Tuesday to get back to where it was.

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Getting back on track (excuse me, not a fan of Wall Street), our decay equation would be written as:-

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This equation yields the incredibly small number: 6.37 x 10^-12 or .00000000000637 centimeters.  P was set to 28 to because a 9.5×11 sheet of paper is about 28 centimeters long.

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We didn’t do this last part in class, which is a good thing because all of a sudden I’m having a hard time wrapping my head around exponents!

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If Pigeons can do it, so can you December 24, 2011

Here is a fascinating article on the mathematical intelligence of…. pigeons. 

(to read full article, click on the pigeons)

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http://news.discovery.com/animals/pigeons-math-animals-111222.html

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How Observant Are You? November 28, 2011

How many times do the people in white pass the ball? 

The answer is somewhere between 13 and 19.