6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

The old army colonel and his oft-hiccuping son. The revered Native American Sohcahtoa. And Henry, who is unable to add tens or hundreds.

Just as the cast-off playthings on the Island of Misfit Toys want nothing more than to find a child to love, this lovable clan of well-meaning folk desires nothing more than to help your students remember the definitions of trigonometric ratios for acute angles.

Yep, they're all mnemonics used to help students remember that sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.

• The old army colonel and his son often hiccup.
• Silly old Henry can't add hundreds, tens, or anything.
• SOHCAHTOA

And of course, there are about a bajillion more.

Take your pick. Or, rather, let your students take their pick. It doesn't really matter to us which one they use, so long as they can remember which ratio is which.

Of course, we'd also like them to remember that, in similar triangles, corresponding sides are proportional and corresponding angles are congruent, which is where we get those trigonometric ratios in the first place.

Students would benefit from an investigation in which they find and compare these ratios. Perhaps give them a set of nested similar right triangles, each of which share one angle, and have them name the ratios of side lengths for each one.

The key here is that students work with the ratios in similar triangles so they can see that, even when the side lengths are different, the ratio of their lengths stays the same for a congruent angle. Growing or shrinking a triangle's side lengths will not affect the ratio. Only changing the size of the angle will cause the sine or cosine measurement of the angle to change.

After understanding the concepts of sine, cosine, and tangent, students should be able to use these as operators in order to find side lengths of right triangle given a side length and an angle. Then, they should be able to use the inverse trigonometric functions in order to find angle measures from side lengths.