Fractions are numbers, too – Part 2

We have a lot to say about CCSSI’s treatment of fractions, which starts tentatively with 1.G.3, but we’ll initially hone in on Grade 3, which is where Common Core begins its big push.  We’ll discuss Common Core’s sequence, and compare or contrast it to our own preferences for how fraction concepts should be introduced, and if we differ, provide a (hopefully justified) rationale for our choices.

3.NF.1 states, ``Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.’’

Certainly when formally introducing fractions and fraction notation, understanding the meaning of a fraction comes first.  In 3.NF.1, though, CCSSI is already on the wrong track, for it assumes fractions only refer to parts of whole objects.  This limited definition excludes a more basic (and easier to understand) use of fractions to refer to numbers in a group, such as the NAEP umbrella problem we first cited in Part 1:
 
Students in Grade 3 already understand cardinality, so using counting as an entry point into fractions makes sense, but Common Core ignores this obvious segue.  Students do not need to recognize the entire group of umbrellas as ``a whole’’ in the same way as one pizza is a whole in order to get the meaning of fraction notation and its myriad uses.  Introducing fractions in multiple ways—as numbers in a group (umbrellas), then as parts of a whole (slices of regular pizza, or rectangular Sicilian pizzas), as liquids in measuring cups, as points on a number line (activity: measuring fractional lengths with a ruler)—abstracts the concept more effectively.

We’ve said many times in this blog that multiple applications of a concept is the way to go.

3.NF.1 and 3.NF.2 are unnecessarily wordy and use confusing algebraic notation to describe what should be a simple, single concept: what does a fraction represent?

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Following the concept of a fraction, Common Core ventures into the topic of equivalent fractions. 3.NF.3a&b:

3. Explain equivalence of fractions in special cases...

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

This is a huge leap in concept and a terrible choice.

Although the concept of equality (learned before Grade 1) followed logically after understanding the cardinality of whole numbers, equivalence of fractions is not the analogous next step from understanding the meaning of a fraction, because equivalence and equality are not the same.  (Think of 3 red umbrellas in a group of five umbrellas compared to 6 red umbrellas in a group of ten.)

What might an open-ended assessment question look like under 3.NF.3?
One already exists.  This NAEP question was given to 4th graders in 2003, but only 19% could finish the task completely.

In Grade 3, students under Common Core have yet to memorize the times tables, yet embedded in the concept of equivalent fractions are not only multiplication, but other, more complex skills, such as division, factoring, LCDs, and auxiliary lines—requiring multiple steps—all of which are also involved in the addition and subtraction of fractions with unlike denominators, a topic 3.NF.3 does not broach.

Common Core wants to introduce equivalent fractions early on while avoiding its more complex aspects, but developing a concept incompletely is worse than ignoring it.

Of course equivalent fractions belong in the standards, but not yet.  This standard as written exposes a common flaw in Common Core: failing to consider the age-appropriateness of a task and how it fits into the overall sequence.


A more logical progression from the concept of fractions is to compare fractions (first with visual models, then with numbers only) that have common denominators such as ⅖ and ⅗.
Students will recognize the more basic concept that fractions can be compared, just like whole numbers.

After comparing fractions, students can add ⅕ + ⅖ (again, first with visual models, then with numbers only) and recognize that fractions can be added and subtracted, just like whole numbers.

Instilling in students the notion that fractions behave like regular numbers (but have special rules for the mechanics of arithmetic) is essential, and should come well before more complex concepts like equivalence of fractions.


One caveat to educators: visual fraction problems can be poorly conceived:
Although used commonly, this type of example will confuse students because it doesn’t account for how two separate partially shaded wholes mysteriously combine into one.  More effective is to use a visual that partially shades one whole and then poses a question: ``The picture shows ⅕ of the circle shaded.  If you shade another ⅖ of the circle, how much of the circle is now shaded?’’

Or pour the liquid from one partially filled measuring cup into another partially filled one.  Or add fractions along a number line.  Typical word problem: Fran ate ⅛ of pizza in the morning and ⅜ of a pizza in the afternoon.  Fred ate ⅝ of a pizza at lunch.  Who ate more pizza?


Our take: third graders should be working extensively first with visual fraction models and then with problems that have fractions with common denominators (of course, different problems should use different denominators), which will lead to a clear understanding that (1) fractions only add or subtract when the denominators are the same, (2) denominators behave differently than numerators, and (3) fractions are never defined by the numerator or denominator alone—the entire expression is the number.  These critically important properties of fractions alone should be given an entire year to sink in.

Why do we advocate this limited, focused plan for Grade 3?  Because extensive practice with the Big Three of Grade Three—basic meaning, comparing, and adding and subtracting fractions with common denominators—will cement these essential skills, and will help avoid nightmares like...
 
…where 29% of 4th graders taking the 2005 NAEP thought the answer was either A or C.

The foundations of fractions must be laid correctly, or the negative repercussions will extend until high school and beyond:
 
(28% correct on Grade 12 2009 NAEP—but a greater number, 31%, added both the numerators and denominators and got answer A.)

These unacceptably poor results in the most elementary of fraction skills lay bare a long term mathematics education issue—a cognizance lacking in Common Core—that our alternative sequence addresses squarely and resolutely.


It's lunchtime.  Let's get a couple of Sicilian slices; that's 2/9, right?


Continue to Fractions are numbers, too – Part 3