Comparing and Ordering Fractions

Here's a simple one which gets the technicalities of fraction comparison out of the way for a spell so that students can focus on how the numerators and denominators affect magnitudes. This is now in the Order and Absolute Value lesson app for Grade 6.

You can enter numerators and denominators on the right to model the corresponding fractions. Those greater than 1 are not shown—they extend past the viewing area—but the bar does turn black when the fraction is greater than 1.

One thing I emphasize a bit in the lesson app that doesn't get enough attention, I think, in mainstream materials is that rational numbers are those numbers that CAN be written as the ratio of two integers \(\mathtt{\frac{a}{b}}\) such that \(\mathtt{b \neq 0}\). But they don't HAVE to be written that way.

Making this definition correct makes the whole topic a little more interesting, to me at least. We can ask whether something like \(\mathtt{\frac{0.5}{3}}\) is a rational number, for example, or perhaps work up to whether any ratio of terminating decimals is rational.

At any rate, what we use the above to do, mostly, is to (a) systematically change the denominators to show that the fraction increases when the denominator shrinks and decreases when the denominator grows, (b) systematically change the numerators to show how they affect the magnitude of the fraction, (c) check equivalences and improper fractions, and (d) then of course compare and order rational numbers.