Discovering tangent

This is an activity to help you learn more about the tangent function. The trigonometry function 'tangent' is often written simply as 'tan'.
‘Discovering’ the tangent ratio Starter: 1. Open the Geogebra file. Click on View … Construction protocol. Would you have created a right triangle in the same way? Why or why not? Return to this question after completing the assigned task. No. Name Construction Caption 1 Number n slider 2 Point A (0,0) 3 Angle α slider 4 Point C (3; α) 5 Ray a Ray through A, C 6 Point B (n, 0) 7 Segment b Segment [A, B] adjacent to α 8 Line c Line through B perpendicular to y = 0 9 Point D Intersection point of a, c 10 Segment d Segment [B, D] opposite α 2. What is your definition of a line? Compare your definition of a line with a ‘line segment’ and with a ‘ray.’ How are they similar? How are they different? 3. Both (0, 0) and (n, 0) are Cartesian coordinates. The point C is defined using a different coordinate system. The notation for point C does hint that it is different from points A and B. How? Main Assignment: 4. Play with the two sliders. What is the effect of each? 5. Give the definition of the trigonometry ‘tangent’ function. Calculate tangent for any set of values of n and α. My n = My α = tangent function for this triangle = 6. Scientists often keep one variable at a fixed value while adjusting the other variable in a systematic way. Try holding n at a particular fixed value while varying α. What happens to the tangent function? Now try holding α at a fixed value while varying n. What happens to the tangent function? Decide on a systematic vary to record your data. Analyse your data. What conclusion can you make? Extension tasks: 7. Do some research into the alternate coordinate system alluded to in question 3. 8. Modify or add to the Geogebra programme to make similar discoveries for the sine and cosine functions. 9. Metacognition, or thinking about how you think. Some teachers refer to this type of activity as ‘scaffolding’ while others call it ‘discovery based learning’. Does the name matter? Do you prefer to be told straight away what the result is, or do you prefer to find out for yourself? 10. Return to question 1 (perhaps weeks or months later). How would you write a Geogebra programme to teach the concept of tangent or another similar idea? Publish your Geogebra programme on the web (http://www.geogebratube.org/)