An example of effective interaction between teacher and students
T:
Here is a question “ sin   0.7 ”, so what is the meaning of sin  ?
S:
opposite side to hypotenuse
T:
That means “opposite side to the hypotenuse” is equal to 0.7, am I right?
S:
Yes
T:
So, what is the opposite side?
S:
The side which is facing the angle.
T:
The angle? Where is it? Can you draw a figure to show it to me?
S:
OK …… (Drawing a triangle)
The acute angle is the one I am talking about
T:
OK, do you know the value of this angle?
S:
I don’t …… is it the unknown which I need to figure out?
T:
Yes! So you don’t know the value of the angle, but you know “the opposite side
to the hypotenuse is 0.7”, right?
S:
Yes
T:
That means we need to find an angle, which can cause the length of the opposite
side to the hypotenuse to be 0.7, right?
S:
Yes, that’s the unknown we need to figure out.
T:
OK, do you remember that we made a table, which recorded different values of
angles and the ratio of the opposite side to the hypotenuse?
S:
Sure
T:
Do you think the table can help you to solve this problem?
S:
……
T:
What did we do to collect data for the table?
S:
We measured the length of the opposite side of a given acute angle and
hypotenuse, and finally get their ratio.
T:
Yes, you are right. And now, compare the different values of the angles. Can you
see anything special about the ratios?
S:
They are totally different.
T:
What is the value of the ratio of the opposite side to the hypotenuse which we
were talking about?
S:
0.7
T:
From the table, the ratio is totally different from different values of angles, and
you know that the ratio of the opposite side to the hypotenuse in the question is
0.7, what can you get from the table?
S:
From the table, the ratio is different from different angles …… I know the ratio is
0.7 ……
Can I check where 0.7 in the table is and figure out which angle it is?
T:
You can try
S:
(Searching……) I’ve got it from the table, when the angle is 45 , the ratio is
around 0.7.
T:
Only a 45 angle shows the ratio to be around 0.7?
S:
Yes, I can’t find another one…….
T:
(Drawing a figure) Refer to the figure, when the angle is 45 , the ratio of the
opposite side to the hypotenuse is around 0.7, am I right?
S:
Umm ……
T:
So, up till now, can you solve the equation sin   0.7 ?
S:
The answer is 45
T:
Why?
S:
It is because only a 45 right angled triangle can result in a ratio of 0.7.
T:
That’s right, you’ve got the answer.