FAEEZA
5.5 The simplification of the real world problem
It was evident that the learners had no idea as to how they could proceed further with finding an accurate solution to the problem. It was fine that they knew that the point could be dragged around to find an approximate solution, but determining points in this way is
time consuming as well, and if one looks at find solutions correct to a few decimal places then finding accurate solutions is essential.
RESEARCHER
FAEEZA
RESEARCHER
FAEEZA
Now, besides just measuring and dragging the point around like we were doing earlier. There might be other methods in doing this. Do you know or can you think of any other way in which you could find the point?
No.
Are you sure?
Yes ... I am positive that I don't know of any other way.
Pravanie expressed a similar response but she acknowledged that there might be a method for finding an accurate solution.
PRAVANIE RESEARCHER PRAVANIE
I don't know ... that is the only method I can think of.
Do you think that there might be a mathematical method of finding the most suitable position?
Maybe, but I don't know it.
This was the general response from all the learners. In order to simplify the problem, the researcher now asked the learners to consider the same situation with only two villages.
RESEARCHER Let's simplify this problem by taking two villages - let's suppose we had two villages represented in this diagram by I and J, okay?
How would I find a point that is exactly the same distance away from I as it is from J?
All the learners responded that the most suitable point would be the midpoint. Christina's responded as follows:
CHRISTINA RESEARCHER CHRISTINA RESEARCHER CHRISTINA RESEARCHER CHRISTINA
You find the centre.
Centre of what?
Centre of I and J.
What would that centre be called?
The midpoint.
So you would find the midpoint of! and 17 Let's find the midpoint of
I and J ... there it is , the midpoint is Q. What do you observe?
Distance QI is 2, 2 the distance of QJ is 2,2 as well, so that means Q is equidistant from I and 1.
She showed a lot of confidence and did not hesitate with her responses to the researcher's questions. The responses ofthe others were similar.
NIGEL
RESEARCHER NIGEL
At the centre of the line.
What do we call that point?
Midpoint.
When Roxanne was asked to find the most suitable point between two villages her immediate response was:
Yes, that will be easy.
Where will that be?
ROXANNE RESEARCHER
ROXANNE In the middle, the midpoint of! & J.
Clearly the simplified version of the problem was easy to solve for the learners. More importantly for the learners, they were getting immediate feedback by using Sketchpad. By constructing the midpoint of the segment IJ and measuring the distances to I and J, they could see that it was equidistant.
Again using the idea that the midpoint may not be suitable as a real-life solution, the learners were asked if they could determine another point which was also equidistant from I and J. The purpose was to investigate whether learners could find other points and realize that there were infinitely many and that all lay in a line, that is, the perpendicular bisector.
It should be pointed out to the reader that traditionally in South Africa, the perpendicular bisector is usually introduced in Grade 8 or 9 (or sometimes earlier), but only as a construction. In other words simply as a line that passes perpendicularly through the midpoint of a line segment, but its equidistance properties are traditionally never investigated (and perhaps only alluded to in passing).
Most learners gave the same response as Faeeza did.
RESEARCHER
FAEEZA RESEARCHER
FAEEZA
RESEARCHER FAEEZA
RESEARCHER FAEEZA
As we said earlier, in real life there might be a problem, in the sense that the best position that we chose may not be suitable, or cannot be used for construction, for various reasons. Do you think that I
can find another point? .
You can locate it around there (pointing), somewhere where it would be equal distance apart.
Do that construction .... (after a point and the segments were constructed) Okay how can we check whether it is equidistant apart?
You measure the distance from the village to the point.
But they are not, what must you do now?
We move it... I'll drag this point around.
How long should we move it around?
Till we get the.. till the distances are equal.
IQ =3.2 em JQ =3.2 em
Q
J
• D
• E
Figure 29
ID = 3.4 em JD =3.4 em
IE =4.1 em JE =4.1 em
But Christina's response was different. She could see that the solution to the question lay on a line, but she could not determine exactly which line it should be, though she stated that it should be a line through Q.
RESEARCHER
CHRISTINA
RESEARCHER
CHRISTINA
RESEARCHER
CHRISTINA
Earlier on we said that it might be possible that that point will not work for various reasons. Do you think there might be another point, which might be equidistant from J and I?
If you move further away.
If you move further away, and do what?
Make a straight line on Q Would it be just any line?
I think that it is a line but I'm not sure what line ... .
Initially the researcher was unsure what Nigel meant, but it became clear that he was basically saying the same thing as the others. But it did indicate that different learners
conceive these ideas differently and they develop their understanding of a situation in their own unique ways.
RESEARCHER
NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
Click on the button for that construction. (after constructing) There you have the midpoint
Q
it is 2,2 away from I and 2,2 away from J.So it confirms that it is at the centre. Now as we said earlier there might be a problem at that point. What will you do?
1'11 draw a triangle.
Draw a triangle?
Yes, so that you can measure the distance from I to the apex and from J to the apex.
Oh, you want to construct a point here and draw a triangle so that you could measure the two sides of the triangle?
Yes to measure distances from I and J ...
All you want to do is measure the distances?
Yes.
The learners were quite clear about the fact that if one point was not suitable then another could be used - another point that satisfied the condition of being equidistant from two points. The researcher was initially unsure whether the learners would be able to make the deduction that all points that are equidistant from two points will lie on the perpendicular bisector. But the learners made this connection to the perpendicular bisector quite easily.
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER
How many points do you think we can find?
Many .... hundreds.
What do you notice about all ofthese points we have drawn?
All the points seem to be on a straight line
Construct a line and see if it is on a straight line. (after a while) Do they lie on the same line?
Yes
What else do you notice about this line?
It divides ... it cuts the other line in two ... it is equal What else do you notice?
It looks like a right angle here ... .
Do you think it is a right angle? Do you want to check?
Yes
Measure it ... (after a while) it is 90 degrees Yes .... I thought so.
So what does this mean? If I asked you for more points where would we find them?
On that line.
You are telling me that you can find many points on that straight line?
NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
RESEARCHER NIGEL
Yes.
Describe this line.
It is a straight line. It bisects there, it forms a right angle
So what is a line which forms a right angle with another line called?
A perpendicular line.
Is this only perpendicular?
It bisects too ... a perpendicular bisector.
Faeeza's response was similar.
RESEARCHER FAEEZA
RESEARCHER FAEEZA RESEARCHER
FAEEZA RESEARCHER FAEEZA RESEARCHER FAEEZA RESEARCHER
FAEEZA
RESEARCHER FAEEZA
RESEARCHER
FAEEZA
RESEARCHER
FAEEZA RESEARCHER FAEEZA RESEARCHER FAEEZA
RESEARCHER FAEEZA
RESEARCHER FAEEZA
RESEARCHER FAEEZA
Faeeza how many points do you think you can get like this?
Quite a few, it depends on which ... millions ... if we are working with two villages ....
So if we work with two villages we will have millions of points?
Ya.
What do you notice about the points? Look at the points and what can you see there?
Going in the same direction.
What does that mean?
(hesitant) They look like they are on a line ... . How will I know that for sure?
Draw a line ... ?
Then draw a line. Okay, there you are - you were right - they lie on a straight line. What else can you observe about this straight line?
Is it special in any way?
Hmmm.
Okay, it might not necessarily be obvious but look at it carefully ....
They look perpendicular.
How can you be sure that it is perpendicular? It looks perpendicular, how can we know for sure that it is perpendicular?
Hmmm ... Measure the angle?
Ok, try that. (after the angle is measured) There's the angle there, what is the angle?
90 degrees.
What does that imply?
It is perpendicular!
So if you had 2 villages where should you build a water purification plant that will be equidistant from both the villages?
Hmmm ... We can go on a straight direction and move further away, it will still be equal distant that you can find ... move on the straight line ...
How can you describe this line?
It is perpendicular.
Is it only perpendicular?
It cuts in the middle.
What do we call a line that is perpendicular and cuts a line into equal parts?
Roxanne took a little longer to arrive at her conclusion but she knew what transpired and found it only slightly more difficult to articulate her response.
RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE
How many points do you think?
I don't know, just a lot.
Looking at these points does it show any pattern?
Yes .... they are forming a straight line.
How can we know that for certain?
I can draw a line through these points.
(after a while) What do you observe?
It passes through all the points.
What else can you observe about this line?
I don't know.
Look at the line that we constructed. Is there something special about it?
Well .. it looks like it is forming a 90 degrees angle here.
Do you think it might be?
Yes .. .it looks like that.. . How can we be certain?
Measure the angle. (after a while) It is 90 degrees.
So what do we call this line?
A straight line.
Yes it is straight line ... but it is a special line.
A bisector.
Yes it is a bisector. .. but it also has another property.
Mmm ...
What is this other property?
It is perpendicular.
Yes ... so what can we say about the line ...
It is a perpendicular bisector.
The researcher tried to determine whether the learners really understood the concept of the equidistant property of the perpendicular bisector. The following is Roxanne's response.
RESEARCHER ROXANNE RESEARCHER ROXANNE RESEARCHER ROXANNE
So if I asked you to find points that are equidistant from two points how will you do it?
Draw the perpendicular bisector and then any point on the line will be correct.
So if we had two villages, where should we build the reservoir?
Just determine the perpendicular bisector and any point on it will work.
Let us say that I drew a perpendicular line here (pointing to a point that is not the midpoint) ... will it work?
No, it is not the same distance away.
At the end of this section of the interview, learners had discovered that in order to find any point equidistant from two points, the perpendicular bisector could be drawn. It was also evident that this conclusion was relatively easy for them to arrive at.