In today’s era, the use of technology simplifies our lives to such an extent that we sometimes forget how to cope without these luxuries. The use of appropriate software can for example draw an angle of 90˚ with the minimum of effort and without the use of special tools or instruments.
Most of our learners do not have access to computers outside the school environment. How can we assist these learners with thinking outside of the box, or in this case, outside of the CPU? Can your learners complete the task of rotating a point with coordinates (2; 5), 90˚anticlockwise around the origin without the use of technology or instruments?
Your contribution and ideas will assist learners not only to complete their homework during the electricity load-shedding period but also contribute towards the conceptualization when applying technology to the same situation.
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This posting addresses a crucial issue - building the bridge between technology and subject matter. Technology is only an enabler; it is essential that learners must be helped to understand basic principles. I would be most interested to see the solution to the problem that has been posed!
I feel like McGyver now, looking for different household things to solve this problem. There is a very simple paper solution, but I am going to suggest others which are a bit more complicated and requires other pieces of home-based technology.
Solution A : Extend the line to point (-2,-5). Use a piece of string, pencil and drawing pin to draw an arc using point (2,5)- in the second quadrant.Use the same tools to draw an arc in the second quadrant using point (-2,-5). Where these to arcs meet, draw a line down to the centre of the line joining (2,5) and (-2-5).
Solution B
If you have a piece of clear plastic, draw a circle on it. Divide the circle into quarters. Place the circle on the set of axes so that it fits perfectly. Now Mark off point (2,5). Now rotate the plastic circle through 90 degrees and you should land up at the correct answer.
These solutions takes much more effort, but with many ideas innovation is stimulated.
Thinking outside the box is not included in curriculum delivery by most educators. When the requirements for the learner portfolio are completed, the job for the year is done.
Listening to a radio interview on SAFM on Friday 22 February 2008, the participants felt that South Africans in general are not critical thinkers. Our poor showing in international mathematics competitions confirms this. Names like Nelson Mandela, Chris Barnard, Mark Shuttleworth and Jacob Zuma popped up as great and critical thinkers.
Coming back to the question posted. Educators still will have to teach learners to find the solution using graph paper and pen or pencil. The educator’s task is to allow learners to explore the challenge, find solutions, write it down and to critically discuss the it. This will help the learners to get clarity on the thinking process and will help them in their thinking/solution strategy. Learners will not be able to use the CPU in an examination or test. The purpose of software in this specific case will be to show the solution instantaneously.
How can we develop critical thinkers? Some solutions are the use of brain teasers, card games and mathematics competitions once a week or fortnight. It will motivate learners to find solutions to a challenge. It will also stimulate creative thinking and can start a debate on solutions submitted. Non-routine or unseen questions must be included in the curriculum cross all learning outcomes from grade R to 9 in mathematics. (It is already included in Grade 10, 11 and 12).
A possible technique is to fold the graph paper.
Step 1 – fold the paper on the x-axis so that point A(2;5) is on the outside. An angle of 180 degrees is formed.
Step 2- fold the paper on the y-axis so that the point A(2;5) is still on the outside. An angle of 90 degrees is formed.
Step 3 – fold the paper a final time so that the x-axis and y-axis are on top of one another. An angle of 45 degrees is formed.
Use a needle to make a small hole through the paper at point A(2;5). Take the needle out. Fold the paper back to its original form. Eight holes will be visible, two per quadrant. One hole in the second quadrant is the reflection of A in the y-axis. The coordinate is (-2;5)
The other point in the second quadrant is the rotation of A through 90 degrees anti-clockwise. The coordinate is (-5;2)
As an average Grade 11 learner an algebraiec approach would be to think of the line from the origin to that point as the hypotenuse; therefore there can only be two possible solutions; either (-2;5) or (-5;2) since that point can only be in the second quadrant; and the hypotenuses should be equadistant; a wild guest by the average learner should be (-5;2). By using the rules for perpendicular lines I can now proof that those two lines are perpendicular; therefor the angle at the origin is equal to 90 degrees.
The technology use mathematical rules/principles/facts to provide only the answer. It is therefor essential that the educator or the learner must still apply the basic mathematical principles to get to the rule/answer.
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