How high
is it?
Space, measurement and trigonometric reasoning
Today: investigate first. Name the mathematics after you have found something.
Investigate → make sense of it → use it
Do not look for a formula yet.
Same angle.
Different triangles.
Your question
What can stay the same when the triangle gets bigger or smaller?
Build evidence before explaining it.
Make two triangles
Use the same acute angle, but make the triangles different sizes.
Label the sides
Choose the angle θ. Name the opposite, adjacent and longest sides relative to θ.
Measure and calculate
Find O/H, A/H and O/A for each triangle. Keep sensible precision.
Compare
Compare with another group. What pattern can you defend?
No sine, cosine or tangent names yet.
Make the pattern visible.
| Triangle | Angle θ | O | A | H | O/H | A/H | O/A |
|---|---|---|---|---|---|---|---|
| 1 | |||||||
| 2 | |||||||
| Another group |
What stays the same?
Fixed angle
What happens to the ratios when the size changes but θ does not?
Changed angle
What happens when θ changes?
Useful idea
How could a stable ratio help us find a length we cannot measure directly?
Make a claim. Show the evidence.
Be ready to change your claim if a new triangle challenges it.
Now make your own mind map.
No template.
Make your own map of the mathematics from the investigation. Decide what belongs in it and how the ideas connect.
Three ratios. Three names.
The reference angle decides which side is opposite and which is adjacent.
Check
Calculator in degree mode. Draw the angle you are using.
For the height task
If you know horizontal distance and angle of elevation, tangent is usually the useful ratio.
How could we find this height without measuring it vertically?
Work out the model before collecting data.
Measure an inaccessible height.
Choose an object
Tree, building, pole or another object with a clear top.
Collect data
Horizontal distance, angle of elevation and observer eye height.
Draw and calculate
Label the right triangle. Find the height above eye level, then add eye height.
Repeat
Take measurements from at least two more positions. Keep every estimate.
Important: round at the end, not during the calculation.
Which answer do you trust?
| Position | Distance | Angle | Estimated height | Notes |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 |
What did we assume?
Geometry
The ground is horizontal and the object is vertical.
Measurement
The distance and angle readings are close enough to use.
Technology
The calculator is in degree mode and rounding is delayed.
A model is useful because it simplifies reality.
It is also responsible for knowing what the simplification leaves out.
Show what you can now explain.
Finish with: “Trigonometry lets us…”