MathGrade 9ages 14–15
📐 tan
The tangent of an angle in a right triangle is always the opposite side divided by the adjacent side — and mixing up which side is which is the single most common slip in the topic.
Try this lesson freeNo card needed. The first lesson is free.
Download on the App StoreWhat your child learns
- The opposite side is the one that does not touch the angle.
- The side next to it touches the angle and is not the long side.
- The ratio is opposite over next to, and never the other way round.
- The triangle can grow and the ratio does not move, because both sides grow together.
The mistake most kids make here
Swapping the opposite side with the one next to the angle.
Try one question
This ratio divides the side opposite the angle by what?
Opposite over what
- the side next to the angle
- the long side
- the perimeter
Answer
the side next to the angle — That's it. Opposite on top, next to underneath.
How Maya teaches it
Your child tells Maya what they did in class. She teaches this topic at her desk, asks questions with tappable answers, and when the answer is wrong she explains why that answer was tempting instead of saying “try again”. She remembers what did not land and comes back to it.
Your child has it when they can: Say which two sides make up this ratio.
How to help at home
The opposite side is the one that doesn't touch the angle you're working with; the adjacent side does touch it (but isn't the longest side, the hypotenuse). Naming the two sides correctly before doing any calculation prevents most mistakes.
Draw a right triangle together and pick one of the non-right angles. Have your child physically point to the side across from that angle (opposite) and the side right next to it (adjacent, not the slanted long one).
When your child writes the ratio upside down, don't just correct the fraction — ask them to point at the two sides again and say out loud which is opposite and which is adjacent. The order (opposite over adjacent, never the reverse) becomes automatic once the pointing is automatic.
A sign they've got it: given any right triangle and any labeled angle, they can identify the opposite and adjacent sides correctly before writing a single number down.
Questions parents ask
- How do I remember which side is opposite and which is adjacent?
- Opposite is the side across the triangle from the angle — it doesn't touch the angle at all. Adjacent touches the angle but isn't the hypotenuse (the longest side, opposite the right angle).
- Does the tangent ratio change if the triangle gets bigger?
- No. If you scale the whole triangle up or down, both the opposite and adjacent sides grow by the same amount, so their ratio — the tangent — stays exactly the same.
- What's the difference between tan, sin, and cos?
- All three are ratios of two sides in a right triangle: tan is opposite over adjacent, sin is opposite over hypotenuse, and cos is adjacent over hypotenuse.
- How does Maya teach tan in a right triangle?
- Maya has the child identify and point to the opposite and adjacent sides before ever writing the ratio, so the labeling is solid before the formula is used.
No card needed. The first lesson is free.
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