Trick or Strategy?

I remember being in the classroom and prior to beginning a new unit, I would typically have a "pow wow" with my students about what we were about to learn and how it would be useful in their practical lives. As we entered the fraction unit and discussed how to compare fractions, I would always have a students shoot their hand up and yell out "I know! We can "Cross Multiply!"

 

 

 

And each time, my skin would crawl as my Cruella DeVille would rise up and say "No, we will NOT!" I would be infuriated that the kid wanted to use some lame trick to compare fractions." But I soon realized I couldn't be upset at the student. It was the 4th grade teacher I needed to spit venom at. They were making us 5th grade teachers work harder (we had to break the old, bad habit then teach a new habit) when really we should have been teaching the concept from a clean slate (or at least building on a foundational fraction concept).


Does "Cross Multiplication" work? Absolutely! But what does it really represent?
If students could not find a common denominator to turn both fractions into equivalencies, they could easily multiply the two denominators together (In the example above, they would get 75).
Then the numerator of both fractions, respectively, would be the product results of cross-multiplication. But this is a bit advanced for 3rd and 4th graders.

So what happens when teachers teach "Tricks" like "Cross Multiplication" is students lose conceptual understanding essential to their development in mathematical learning. They learn a trick but have no understanding of the "why" behind its' process. In actuality, they learn to be lazy and settle for a drive-thru junk food meals for a quick fix rather than wait for and feast on a home-cooked meal that won't only satisfy their craving, but provide nutritional value.

So what's the remedy? Learn the foundation for concepts yourself (even if it seems new to you).

First let's look at foundational progression of fractions.

Watch the progression:
3rd Grade- 3.3H "Compare two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models.

4th Grade- 4.3D "Compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, <.

Prior to the NEW TEKS this TEK progression applied to 4th and 5th grade, respectively.

 Students can understand fractions on a number line by relating Fraction tiles to number lines. Why? Because in previous grades (even in 3rd grade), the standard encourages open play with fraction tiles! LET.YOUR.STUDENTS.EXPLORE! Take the manipulatives off the shelves and let them build and create and make connections. They're capable!

Once students have grasped that (for example) three-thirds is equivalent to one whole they should be able to see and understand that one-third is close to zero, while two-thirds is closer to one whole. This leads to the understanding of "benchmark fractions"

Benchmark fractions can be seen and taught fairly quickly. Once students know 0, 1/2 (and its equivalencies) and 1 on a number line, they can justify which fractions are closer to zero, which are closer to half and those which are closer to 1 whole. This is more than half way to the learning expectation!!!

*Coach's note: What I would do is have my students build a blank number line. Then take a fraction tile (such as a fourth) off the tray and place it on the number line close to zero. Have them make a tick mark where the 1/4th tile ends. Continue this pattern until you reach 4/4ths = 1 whole.

Using the Number line as a basis, comparing then becomes feasible because students can plot two fractions on a number line or compare them both to benchmarks (five-sixths) would be closer to one whole (six-sixths) which is larger than, say, five-eighths because 5/8 is closer to four-eighths or one half.

Coincidentally, once students understand that the larger the denominator, the small the fractional piece (through exploration)...and vice versa, they will begin to compare fractions easily. They can justify that three-eighths is 3 small pieces, while one-half is 1 large very piece so one half MUST BE larger than three-eighths.

Now THAT'S strategy! And though it takes some doing (or developing) to get students to this point, their understanding goes a long way.

Ever wonder how students will use "cross multiplication" when they must compare (or "order" is the mathematical SE verb) three or more fractions? That's how you know the "Trick" you taught them, was unreliable.