We started the lesson with a few quick questions on Pythagoras' theorem (I used these ones). I taught them Pythagoras' theorem way back in November, but most of them remembered how to solve the problems with a bit of prompting. I was aiming to refresh identification and use of the word "hypotenuse" and also provide a link from something they'd previously learned about right-angled triangles. After we'd done this, it was on to the first bit of the trigonometry introduction!
1. The relationship between lengths and angles
After trying a few different approaches in my first few years of teaching, I've settled on introducing trigonometry by drawing and measuring triangles. When I taught this initially, I used three nested similar triangles, but I found that this caused confusion for some pupils, as they couldn't see the three separate triangles clearly. I adapted this to look at the triangles separately last year (see this video on YouTube if you're interested in a tutorial for pupils to follow), but I decided to try two similar and one different this year.
We discussed what we saw from the results; why were the results for triangles A and B identical, but the results for C were different? We got to the idea that there was a link between the side measurements and the angle we'd measured, and that if the triangle was enlarged, both the angle and the ratio of the measurements would stay the same.
2. Using trig tables
When I first tried this way of introducing trigonometry, I got pupils to then work out sine, cosine and tangent of the angles on their calculators, and spot that these came out with the same results, moving quite quickly on to missing angle problems the same lesson. However, in the last couple of years, I've avoided calculators altogether for the first lesson or two, and got pupils using trig tables instead.
So after the activity above, I handed out a set of trig tables (download a copy here) and got them to look at the rows corresponding to the angles in their triangles. We spotted that our results matched fairly well with the values in the tables, and discussed how inaccuracy in measurements could have contributed to our results being slightly out.
By this point we were reaching the end of the lesson, so to finish off, I taught them the names of the three ratios, emphasising that these are just relationships between two sides on a right-angled triangle (for now!).
After a few more Pythagoras problems to kick the lesson off, I set the pupils off on a task focused on labeling sides and writing trigonometric ratios correctly. I gave them a set of triangles with all three side lengths shown and asked them to write down the ratios for sine, cosine and tangent for each triangle. Once they'd done this, they worked out the decimal values of each one on their calculators, then it was back to the trig tables to work out an approximate value for the angle.
By this time, they were getting a bit fed up with the tables; a couple of them were complaining that their eyes were going funny from squinting at the values, and one pupil said "but surely we don't get this in the exam?". So at this point I showed them three examples of some more traditional exam-type problems (with two sides and a missing angle), and also demonstrated how they could use the inverse sine, cosine and tangent buttons on their calculators rather than struggling with the tables, explaining that the inverse buttons were just working backwards from the trigonometric ratio to the angle. The rest of the lesson was spent practising these new skills using questions from a textbook.
(I almost felt the need to write an apology there for referring to textbook use, then stopped myself. I think it's absolutely vital that time is built into maths lessons for skills practice, and sometimes the best way to do that is to get them to work through some carefully selected problems independently).
I'll blog next week about where we go from here. I'm in two minds about continuing to finding missing sides (which is what I'd usually do), or alternatively doing more in-depth problem solving using angles first - but I've got a few days to think about it!
http://TrianCal.esy.es - Open in Google Chrome.
(Triangles online calculator developed by Jesus S.)
I propose this free online calculator triangles without advertising to help students with geometry, not exercising, because their calculations formulas are not displayed. It is designed in a didactic way to check and view the
TrianCal is online calculator triangles that works with any combination of values including sides, heights, angles, the area or perimeter of any triangle, calculating it with the minimum possible value (typically three).
- Draw the triangle (s) with GeoGebra.
- Set the range of values allowed into each element.
- The type of angle.
- The type of triangle by its sides and angles.
- Selection of language (English or Spanish).
- Select and angles [degrees (°), Radians, Degrees, minutes and seconds (°
'") or degrees and minutes (°')] is.
- Number of decimal places to show in the results (0-15).
- You can use the arrow keys and the Tab key to navigate through the
- Drop-down menu to select values comfortably.
- Create a link (URL) to the current triangle.
- An icon mail to communicate with the author.
NOTE: You must use the Google Chrome browser to display correctly TrianCal.
Examples of possible combinations:
- The area, perimeter and other data (side, height or angle), if the outside
equilateral triangle would not need the third data.
- 2 angles and other data (if the value of the other data is not put aside
the value of "a" at the time of drawing the triangle is 10).
- One hand, one high and one angle.
- 3 heights.
- 3 sides.
- 2 heights and perimeter.
- Any other combination of values.