## Teaching Tip #2: Tackling those Dreaded Fractions

03/28/2013

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A week into teaching fractions and your students are dividing M&Ms evenly into 4 piles and cutting pies into 3, 4, even 6 equal parts. You think they’re ready to move on so you start teaching equivalent fractions or adding fractions or comparing fractions...Only to be met with blank stares. So it’s back to basics. Here are some strategies for teaching fractions that worked in my classroom:
1. Start with what kids already know: sharing fairly.
Students come to school with a sense of “fairness.” Build
on their informal understanding with sharing exercises: if
you have 12 jelly beans, how do you divide them equally
among 2 people? How about 3 people? Now 4? When
students can divide discrete objects, present them with
whole objects, such as apples or oranges: How would you
divide this orange so that you and your friend get the
same amount? Don’t introduce the language of fractions
(one-half, thirds, quarters) or begin labeling the fractions until students demonstrate that they can correctly divide objects. If you don't have fraction manipulatives in your classroom, try Donna Young's printable paper manipulatives.  Students can explore fractions (and other math topics) virtually using this fun tool from McGraw Hill.

2. Use the sharing examples above to introduce comparing and ordering fractions. As the
number of people sharing something increases, the amount they each receive
decreases. Would you rather share the jelly beans with three people (receiving 1/3) or
five people (receiving 1/5)? 1/3 of the jelly beans is greater than 1/5. When comparing
and ordering fractions, I also present my students with a chart, similar to this one, that
helps students easily visualize how fractions are related to one another and identify
equivalent fractions.
3. Connect fractions to students’ everyday lives. For
example:
• How much time is left until lunch? 1/4 of an hour.
• Have students take a survey of the class: How many people like chocolate ice- cream?  1/3 of the class.
• How much glue is left in the bottle? About 5/8.
• Would you rather strike out 2 out of 5 times or 3 out of 4 times? 2/5 is less than 3/4.
See how many representations of fractions students can come up with. Check out this
lesson plan on Thirteen.org explaining how fractions are used in everyday life. Using Cuisinaire rods as fraction manipulatives
4. Make fractions visual by using physical
manipulatives. One study demonstrated
that students average test scores were
higher when fraction concepts were
presented using physical manipulatives
instead of drawings of fractions.
Manipulatives include fraction sticks,
Cuisinaire rods, and folded strips of paper.
A fun alternative to the usual classroom
fraction manipulatives: Play-doh!  When students are able to demonstrate their understanding of fractions using physical manipulatives, you can present concepts using virtual manipulatives. One of my favorite resources is the National Library of Virtual Manipulatives (scroll down for the “fraction” resources). Students can also create fraction models and convert them into decimals and percents with Illuminations.

5. Use an “anchor” in all of your fraction lessons. For example, keep an image of a
fractions labeled on a number line posted throughout your lessons. This reminds
students that the numerator and denominator aren’t separate whole numbers but that
the fraction as a whole is a rational number. If your lesson involves examples where
students are creating fractions out of apples, keep an image or actual apple visible: this
helps students understand that each piece of apple is part of the whole apple.
6. I can’t stress the use of NUMBER LINES
enough. The core of so many problems
with fractions stem from a lack of
understanding this simple fact: fractions
are rational numbers that lie between 0
and 1 on the number line (excluding
improper fractions). Demonstrate how
to plot fractions on a number line.
Create giant number lines on the floor
and have students tape fractions on the
number line. Teach comparing and
ordering fractions using a number line.
Use virtual number lines.

7. Does this sound familiar: “The num..ater.. of the fraction 1/3 is 3.” Even after my tenth
lesson on fractions, even after my students could demonstrate how to add fractions
with unlike denominators, they just couldn’t remember the difference between the
numerator and denominator or correctly pronounce the terms. This is where you step
in.  On each lesson or worksheet, have
prominently displayed. Color code them. Use the terms constantly in your fraction
lessons along with the definitions (“The numerator, top number, of the fraction…”) and
have students use them when explaining how they solved a problem. Keep the words
displayed until every student can use them correctly.
Fractions are important. If we can help our students build a better foundation with fractions today, they will be better equipped to calculate ratios, proportions, probability and rates tomorrow. What are some resources or techniques you’ve used to help your kids understand fractions?

## The Trouble with Fractions

03/18/2013

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Fractions were one of the trickiest topics for my students to learn. Prior to fractions, numbers made sense: they were whole, they increased in value as they got bigger, and they usually became larger when multiplied and smaller when divided.

With fractions, numbers were suddenly stacked on top of one another!  Confusing words like “simplify” and “common factor” had to be memorized!  When the numbers grew larger (1/4, 1/5, 1/6) their value decreased!  In fact, they weren’t even called “numbers” anymore, but “numerators” and “denominators!”

It’s no wonder that fractions left my students scratching their heads.

But fractions aren’t only difficult for students with learning disabilities. On the NAEP (National Assessment of Educational Progress), half of all 8th graders couldn’t correctly order 3 fractions from least to greatest. So why are fractions so tricky for our students? Here are some common mistakes my students have made and why they made them:

 Students view the numerator and denominator as separate whole numbers instead of a single amount. They know that 7 is larger than 4, so naturally 2/7 must be larger than 2/4.

 Students treat the numerator and denominator as whole numbers. Here they apply the properties of whole numbers to fractions: just as you would add 3+1=4 and 4+4=8, students apply their knowledge of addition to each separate part of the fraction.

 Students struggle with multiplying and dividing fractions because the operations don’t result in the answer they expect. With whole numbers, students learned that division results in smaller numbers, and multiplication results in larger numbers. However, the opposite is typically true for fractions. Division yields a larger quotient while multiplication yields a smaller product. While students are often taught the procedural steps for multiplying and dividing fractions, they do so without understanding the underlying concept, leading to confusion in the later grades.

 Students may have difficulty simplifying fractions because of a shaky understanding of division.

 Students often order fractions incorrectly and cannot place them on a number line. They cannot easily count fractions the way they count whole numbers (1,2,3,4…). Instead, there are infinite number of fractions between each fraction-- a fact that even college students have difficulty grasping.

 Students overgeneralize rules for computing fractions. Once they learn that to keep the denominator the same when adding and subtracting fractions, they do the same when multiplying and dividing fractions.

Now that we know common mistakes students make with fractions and the underlying misunderstandings, what can we do about it? Our next blog post will explore simple instructional techniques teachers and parents can use to build their kids’ confidence with fractions.