Reducing fractions: One cookie = two cookies? July 11, 2011

Any kid will tell you that eating one of two cookies is not the same as eating two of four cookies.  In the first case, you only get to eat one cookie and in the second case, you get to eat two!  Yet in math, we are told that 1/2 is equal to 2/4.  How can this be?

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First, we have to be able to read fractions to understand them.  In other words, we have to remember that fractions are a sort of shorthand for longer phrases.  For instance, let’s take 1/2.

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1/2 can mean:

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“one out of two”

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“one divided by two”

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“one out of every two”

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“one for every two“

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Writing “1/2″ is so much faster than writing any of the above phrases.  And when we understand this, and that mathematicians often use abbreviations, we can begin to think about what “1/2” really is:

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Here’s two cookies:

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And here’s one out of two cookies:

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We took “one out of two cookies, or “1/2″ and showed the fraction “1/2″ with cookies!  This seems obvious, but may be a little misleading.  In our above example, it seems as though the numerator (1) represents the number of cookies we take and the denominator (2) represents the total number of cookies.  And in a way this is true!  But let’s look at one more example…

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Here we have four cookies…-

And here we take two of them…

We’ve taken two out of four cookies, or “2/4″.  We’re told that 2/4 is the same as “1/2″, but how?  Let’s remember our phrases.  “1/2″ can also be read as “one out of every 2“, and in fact we have taken one cookie out of every two on the table.  We can begin to see how 1/2 = 2/4. 

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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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“Unschooling”: A Movement? May 23, 2011

I recently came across what I thought was a rogue blog about “unschooling”. Out of curiosity, I Googled “unschooling” and was aghast to find that it’s an actual movement. This to me feels as legal as polygamy and kidnapping. Maybe you think my analogies are a bit extreme, but you should know I looked into “unschooling” further to see what kinds of parents “unschool” their kids. I found that parents who have strong educational backgrounds, often with advanced degrees, “unschool” their kids.

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On the surface, this sounds reasonable. These parents are well-educated and therefore can give their kids a strong education. There is no doubt kids with educated parents grow up to be smarter and more educated themselves. In fact, there is research that links a kid’s mother’s education level with how well the kid does in school. Why? Research points to the number of words the kid hears as a youngster even before kindergarten.

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But thinking more about “unschooling” and extrapolating out to when a kid is, say, 18 and deciding whether or not to go to college, a few kinks surface. The kid will have to take the SATs. Unschooler’s philosophy on math is to teach just the basics one needs to cook a meal (measuring) and a few other things. With a poor math SAT score, the kid doesn’t stand a chance of getting a score any higher than what they for bubbling in your name (will the kid even know how to bubble? Frightening.). Thinking further into the future when the unschooled kid has a kid of his own, how will this now unschooled parent teach his children? He can unschool them, but he definitely wouldn’t be able to homeschool or answer homework questions brought home from a traditional school. The grandkids then of the original parents who decided to “unschool” are left with the extreme disadvantage of having no useless school knowledge, which is, consequently, what gets you further in life.

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It seems to me, and correct me if I am wrong, that “unschooling” is a selfish, thoughtless decision made by people whose parents gave them an excellent education. “Unschoolers” are setting their kids up for limited career choices and failure. The unschooled kids may be well-spoken and seemingly intelligent, but there’s a big difference between well-spoken and well-educated, and as a teacher I know that a kid who is well-spoken is often hiding a learning deficiency. It is sad to me that these “unschooled” kids will be hiding a learning deficiency imposed on them by their parents, who no doubt had good intentions- ”let’s keep our kids with us and teach them what they really need to know about the world”- but who fail to realize that someday those kids may want to make decisions for themselves, like going to college, and will be left with no tools to do so. They will do poorly on the SATs thereby having limited college choices. They will then never be able to get the strong college educations their parents had. And this says nothing of LSATs, GREs, or any other exam that goes along with deciding to further one’s career.

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“Unschooling” to me is a one-generation dead-end decision that is rooted in a selfishness to keep kids home, like one would keep an indoor cat. “Unschooling” parents pride themselves on letting their kids make their own education decisions, but the decision to [not] go to college is made for these “unschooled” kids way before they have a voice.

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Already ill from the thought of rich kids all over the US “unschooling” their kids, I get an email from the tutoring company WyZant from a parent in the South End of Boston who’s looking for help for her 11-year old son:

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Our son is 11 years old and attends the Sudbury Valley School. We would like tutoring for him on the weekend, starting in September. He has not had any academics at all, as this is the philosophy of his school. He taught himself to read and his word comprehension has been tested at a 5th grade level (by an outside evaluator). His reading comprehension is not good, however. The main point of what he has read often eludes him! He has never had formal math and only knows the plus and minus signs, but is quite intelligent and does the math in his head for daily life easily and effortlessly. He has matured and thrived at Sudbury Valley, but we want some formal academics taught so he has choices for attending traditional schools, should he so choose. His weakest point is writing, and is at kindergarten level or less. He has a mild case of NonVerbal Learning Disorder, according to the evaluator, and it is clear that writing is agonizing for him. We live in the South End.

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For those of you who don’t live in Boston, the South End should be called the “High End” in that it’s price-prohibitive for 99% of Bostonians. What struck me about this email is 1) the kid’s age. He’s 11. He’s on the verge of being a teen and his mom is freaking out that 2) he writes like a kindergartener even though he 3) GOES TO SCHOOL in 4) Framingham! That really sucks. This parent drives her kid 30 miles every day to a school that costs $7000 per year (I looked it up) only to have him be 11 years old with a kindergarten education.

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But the “GOES TO SCHOOL” thing is what really struck me. Is the “unschooling” movement becoming an institution?

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Oooo, that shiver just cooled me off a bit.

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(To be sure, I know that unschooling is much different from homeschooling. Homeschooling involves a curriculum and parents who care to teach their kids subject matter that will prepare them for work in the real world. I applaud any parent who has decided to take on this huge endeavour! I was in a Calculus class at Harvard with kids who had been homeschooled; they were half my age and scored twice as many points. I wanted their parents to homeschool me.)

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ZeroSum ruler’s 62% success rate! March 9, 2011

 

The ZeroSum ruler improved my student’s understanding

of integers

by 62%

in a very short 2 weeks. 

Surpassed even my high expectations!

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HaPpY Calculating!

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US vs A+ Countries: Breadth vs Depth in Math. Which is better? December 6, 2010

(Click chart to enlarge)

Schmidt, William H., Wang, Hsing Chi., McKnight, Curtis C., J Curriculum Studies, 2005, volume 37, number 5, pages 525–559

 

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When does -22 + 5 = -27?

My graduate thesis is a study of the long-term effects the ZeroSum ruler has on eleventh grade student understanding of negative integers.  By eleventh grade, students should easily be able to answer “-22 + 5 =”, but on a diagnostic test given to 57 students, 40.35% of the students answered this problem incorrectly.  Why does this matter?  It matters because it shows that students did not learn the relationship between negative and positive numbers in elementary or middle school.  By the time they get to me in eleventh grade and need to be fluent in equation manipulation, answering “-22 + 5 = -27″ is a real problem. 

 

My thesis was set up the following way:

1: Diagnostic test: eight simple sums and differences of integers  (ie: ’22 + 5=”) without a ZeroSum ruler or calculator

2: Introduction to the ZeroSum ruler with examples

3: Three activities, spaced out over 2 weeks,  using the ZeroSum ruler

4: A post test within days of the last activity (no ZeroSum ruler or calculator)

5: A delayed retention test one month after the last activity (no ZeroSum ruler or calculator)

 

Because the attendance rates of students in Boston Public Schools is not the best, especially by the 11th and 12th grades,  a subgroup of 31 students was identified who took the diagnostic test, participated in at least 2 of the 3 activities with the ZeroSum ruler, took the post test, and took the delayed retention test.  The data shows a 62% decrease in student error from the diagnostic test to the delayed retention test because of the ZeroSum Ruler!  These results indicate that the ZeroSum ruler works to improve student comprehension long-term even without the ruler.

 

Pretty exciting stuff.

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Negative Numbers. OH NO! October 6, 2010

In our BPS high school, there’s a big focus on the “broken window theory”, made famous recently in The Tipping Point.  One broken window we’ve identified in the school as far as discipline goes is hats and ipods.  So, there’s been a big push to get rid of them.

 

I’d like to mention to you a “broken window” that has somehow gotten lost in the mess of school closings, going charter, union fighting, pension plans, longer days, MCAS scores.  As a high school math teacher, the biggest broken window I face – in fact, it’s a gaping hole not even bothered to be temporarily covered with plastic- is… negative numbers.

 

What do I mean by negative numbers?  I’ve done my research as they’re the topic of my Harvard thesis.  Students using the TERC Investigations curriculum in Boston elementary schools do not do problems like “-22 + 5″.  One TERC representative told me they “leave that topic to middle school”.  So, I looked at the middle school Connected mathematics Project 2 (CMP2) curriculum, and negative integer problems, like “-22 + 7″ are taught for 20 days total in the 7th grade.  20 days.  From then on, students are assumed to know how positives and negatives interact and to be able to evaluate “-22 + 5″.

 

Then students get to me, their 11th grade Algebra 2 teacher, and they can’t solve for y in “y + 22x = 5x – 7″ because they don’t know what “5 – 22″ is.  The kids think -22 + 5 = -27.  Why?  Maybe the rules of multiplication get mixed in.  I don’t know.  Or maybe it’s because these problems were taught to them for a total of 20 days four years earlier and were never touched n again except in the context of other problems.  Understanding why and how kids think is beyond the scope of my thesis and my means for data collection.  What I can tell you is that because my students don’t know what “5 – 22″ is, they can’t solve y + 22x = 5x – 7 for y.  Because they can’t solve the equation for y, they can’t graph the equation.  I assume you know where I’m going with this.

 

Please, as someone on the front lines of math education in Boston, I’m telling you that the biggest difficulty our students have in math is adding and subtracting positive and negative integers.  It seems ridiculous and that there are bigger fish to fry, some of which I have listed, but if you want more competency in math, please, heighten the focus on negative numbers.  It will lead to better test scores, more understanding, but most of all, to students who feel good about themselves when they’re not still making silly 7th grade mistakes in high school.