Welcome to the Fly on the Math Teacher's Wall blog hop! In this recurring blog hop series, a great group of mathematics bloggers, covering all grade levels, band together to squash mathematical misconceptions. This time around, we're tackling fractions.
The misconception I am discussing is that a larger denominator means a larger fraction. Ask a handful of third graders (or 4th graders or 5th graders...) which fraction is greater, 1/8 or 1/4, and most are likely to quickly tell you 1/8. With big, proud smiles on their faces. You're nodding your heads out there--I see you! You've been there.
There's a reason this misconception is so widespread. Up to this point in their educational career, bigger numbers always meant bigger values. Eight is greater than four. When students begin to learn about fractions, they erroneously apply whole number reasoning to fractions. One-eighth must be greater than 1/4, because 8 is greater than 4.
So how do we address this misconception? First and foremost, students must have tons of experience with a variety of concrete and pictorial models of fractions. Use fraction tiles, fraction circles, Cuisenaire rods, number lines, and cut paper strips. It's pretty hard to look at models of 1/8 and 1/4 and not see that 1/4 is greater. The idea that fractions should be explored using manipulatives and models is very apparent from the wording of the 3rd grade Texas TEKS, but not so much in the CCSS. I cannot overemphasize, however, do NOT rush to abstract symbols.
The other way we can overcome this faulty reasoning is to help students truly understand the meaning of the denominator. The more parts an object is divided into, whether that object is a pizza or a number line, the smaller the parts.
Both the CCSS and new Texas TEKS address the issue of comparing fractions in a way that will help students deeply understand the denominator. The 3rd grade standards are really well written, once you get past understanding all the 1/b and a/b references, and will definitely result in better fraction number sense for our students.
Here are the standards for comparing fractions for both the CCSS and the TEKS:
- Students are only comparing two fractions, not ordering more than two
- The denominators are limited to 2, 3, 4, 6, and 8
- Students only compare fractions with either the same numerator or the same denominator
- The TEKS specifically mention words, objects, and pictorial models along with symbols
- Both include verbiage about reasoning about their size and justifying the conclusion
- The CCSS states that students must understand that the reasoning only works when referring to the same whole
Let's first consider comparing fractions with the same numerator. When two fractions have the same numerator, it emphasizes that a larger denominator means smaller parts. Look, for example, at the representations below of 1/8 and 1/4. When you look at one piece of each pizza, the idea that eighths are smaller than fourths is pretty clear. It works anytime the numerators are the same. For example 2/6 and 2/4, or 3/8 and 3/6.
Third grade students are also required to compare fractions with the same denominator. This emphasizes that the denominator describes how many parts the whole has been partitioned into (thereby influencing the size of the parts) and the numerator describes the number of those equal parts you have.
That's it! Those are the only two types of comparison 3rd grade students need to do. And if they truly understand and can explain these comparisons, the next generation of students we send up won't leave us scratching our head when they say 1/8 is greater than 1/4.
Do you like the cards pictured above? Well, guess what? They are my gift to you tonight. This freebie includes 16 cards--8 comparing equal numerators and 8 comparing equal denominators--along with a a recording sheet students can use to document their thinking in both words and symbols. Click here to grab your freebie!
Finally, who doesn't love a giveaway? In the spirit of a fraction blog hop, one lucky winner will receive my Fraction Bundle and five winners will select a fraction product of their choice! But hurry, the giveaway isn't around for long!
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Ready for the next stop along the hop? Head on over to Adventures in Guided Math!