Blog: Alexi Makris, Secondary Development Lead

# Top tips for introducing trigonometry

Posted: 1/11/16

**As teachers, we know trigonometry is a tricky topic to teach. With a lot of new language and mathematical expressions to understand it can all seem quite overwhelming at first. No wonder many of us dread introducing it! **

Year 10 students following the Mathematics Mastery curriculum will be learning about trigonometry in right-angled triangles towards the end of the Autumn term. So how do we introduce trigonometry in a way that inspires mathematical thinking and allows students to proceed through Key Stage 4 with confidence in the subject?

### 1. Forget SOHCAHTOA and sine… for now

SOHCAHTOA is a great – if bizarre – mnemonic scaffolding for remembering the links between the trigonometric ratios. But it builds no real understanding of the underlying concepts of trigonometry. We recommend *only* mentioning this quick memory tool once students have developed a deep understanding of sine and its brethren.

Sine is a brilliant shorthand for the function that links the ratio of the sides in a right-angled triangle to a given angle, but it pays not to introduce it too quickly. It should be introduced as a *label* for concepts the students have already been taught to understand.

### 2. Resist the urge to rush through the topic

We can’t stress this one enough. Take your time to build the foundations of learning and unpick misconceptions as you go along. It pays off in the end.

As my colleague Laura said recently: teaching in a richer way is like investing more money on a builder. A rushed job will only mean spending more time and money fixing it and plugging the gaps down the line.

### 3. Establish labelling up front

Enable your students to become fluent in labelling the opposite, adjacent and hypotenuse sides of a right-angled triangle by challenging them with shapes in various orientations to ensure complete understanding.

### 4. Prime their ideas with investigations into 30° triangles.

Through measurement and investigation, students should find the opposite:hypotenuse ratios of 30° right-angled triangles are always 1:2. This understanding can be reinforced with a few missing value tasks.

### 5. Investigate all manner of similar right-angled triangles to solidify understanding

Once students are happy with this relationship, you can move students on to investigating right-angled triangles with angles of 10°, 20°, 40°, etc. Not only will this consolidate their understanding that all similar triangles have identical ratios of corresponding sides, this will allow students to start putting values on the opposite:hypotenuse ratios for right-angled triangles of various angle sizes.

### 6. Sine is just the cherry on top

Students should now understand that every right-angled triangle with a particular internal angle will have an associated side-length ratio associated with that angle. *Now* we can bring in sine as a shorthand for that relationship.

At this point it is understood that saying ‘sine 30° = 0.5’ is efficient shorthand for ‘in a right-angled triangle with an angle of 30°, the ratio of the side length opposite the 30° angle and the length of the hypotenuse is 0.5’. From here, cosine and tangent follow easily.

### 7. Get calculator fluent…but only once they’ve got it

Calculator fluency is crucial for topics such as trigonometry. But calculators should *only* be used once students genuinely understand the mathematical ideas behind the button-pressing.

Having this conceptual understanding will definitely help with students’ fluency on a calculator and their confidence in using them, both in the classroom and during exams.

**The Mathematics Mastery programme provides continuous development and resources to help introduce tricky topics like this. Find out how we can support mathematics teaching in your school by ****joining our information sessions****.**