How to teach SOH CAH TOA

Why this matters

Plenty of students can recite SOH CAH TOA without knowing why sin 37° is the same number on a skate ramp and on a mountain. When a problem gives the opposite side, they grab whichever ratio comes to mind and write 8 ft where the answer is 6. On a worksheet one wrong number looks like any other, and the teacher has to wait for quiz day to find out.

SOH CAH TOA and why the ratios work

SOH CAH TOA stands for the three trig ratios in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The hypotenuse is the side across from the right angle, and students name the opposite and adjacent sides from the acute angle they're working with, not from the right angle. Right triangles with the same acute angle are similar, so their side ratios match however big the triangle is (Common Core G-SRT.C.6). That is why sin 37° is about 0.60 on every 37° triangle, and why inverse sine can turn a ratio back into an angle.

In the Flint activity, each student drags a right triangle bigger and smaller and watches the three ratio boxes stay the same until the angle changes. Then they use the ratios on problems set in their own interest. On a worksheet, a student who picks the wrong ratio finds out when it's graded. Here, a student who uses cosine for the height gets sent back to the triangle by Sparky, Flint's AI tutor, to find the side across from the angle, and a right answer gets an angle to find with inverse sine.

Maya's chat: a skate ramp at 37°

Maya is a 10th grader who skateboards. She knows the Pythagorean theorem and can use a trig ratio once someone tells her which one. She hasn't worked out which side is opposite and which is adjacent, so she guesses.

The whole class gets the same activity: the same greeting, the same resizable triangle and the same first ramp problem. Each student's chat is their own. Sparky sets the problems in what the student is into, and its next message depends on what the student just wrote.

Sparky uses the student's interestSparky sends the student back to the triangle she already triedA right answer gets a harder question

M

Maya

Interest: SkateboardingStarting point: Knows SOH CAH TOA, mixes up the sides

Sparky

Hi! I'm Sparky. Before we start: what's something you're into, like a sport, game or hobby? And how do sine, cosine and tangent feel right now: new, shaky or solid?

Maya

I skateboard, and trig feels shaky

Sparky

Skateboarding works for trig. Drag the glowing corner to make the triangle bigger or smaller and watch the three ratio boxes. Then move the Angle slider. What do you notice when you resize, and what changes when you move the angle?

Maya

when i make it bigger the sides change but the ratios stay 0.60, 0.80, 0.75. when i move the angle they change

Sparky

That's it: only the angle changes the ratios. Here's a short video about two skate ramps at 37°. While it loads, put it in your own words: why does resizing leave the ratios alone?

Maya

because a bigger ramp with the same angle is just the small one scaled up, so all the sides grow by the same amount

Understanding check

Exactly. Now use it. A skate ramp's surface is 10 ft long and makes a 37° angle with the ground. How tall is the ramp? Which ratio connects the height and the surface?

Maya

cos(37) x 10 = 7.99 so about 8 ft

Sparky

Your calculation is right, but check which side you're finding. Go back to the triangle and find the 37° angle. Which side is across from it, and which side touches it? The ramp's height is the vertical side. Which of those is it?

Maya

the height is across from the 37 so its opposite. sin(37) x 10 = 6.02, about 6 ft

Sparky

Right: sin 37° × 10 ≈ 6.02 ft. You found your own mistake by going back to the triangle. Here's a harder one. A ramp is 8 ft long along its surface and rises 3 ft. What angle does it make with the ground?

Maya

3 is opposite and 8 is the hypotenuse so sin x = 3/8. inverse sin of 0.375 = 22 degrees

Sparky

sin⁻¹(0.375) ≈ 22°. Right setup and right answer: you named the sides, picked sine, then undid it with inverse sine.

Fixed her own ratio, then found a ramp angle with inverse sine

Maya used cosine because she knew a ratio was needed but not which side was which. Sparky didn't give her the 6 ft or name sine. It sent her back to the triangle she had just resized, and she labeled the sides from the 37° angle herself. Her harder question needed the same labeling plus inverse sine.

Maya is an example student, and Sparky's replies are shortened.

What Flint did for Maya

• Sparky built a triangle Maya could resize, so she saw the ratios hold at 37° before anyone told her why.
• When Maya said what she noticed, Sparky made a 71-second narrated video about two skate ramps at the same angle.
• When Maya used cosine for the height, Sparky asked her to find the side across from 37° on the triangle instead of giving the answer.
• Sparky checked 6.02 ft and 22° with its calculator before replying.
• After Maya got the height right, Sparky gave her a ramp angle to find with inverse sine.

A teacher can't do this by hand for 30 students, and Sparky does it in every student's chat.

Who mixed up opposite and adjacent

When students finish, Sparky reads every chat and sums up the class on the activity page. The teacher sees who is stuck right now, which mistake came up most, and what to practice next.

Set up the activity for free

The right triangle activity is in Flint's public library, with Sparky's guidelines, greeting and greeting suggestions already written. Open it, select Use activity to copy it into a free Flint account, and move it to a class. In the activity's Preview tab, select Try session myself to test it as a student before you assign it.

For more math activities, see Flint for math.