© 1996: The University of Newcastle: Faculty of Education
Sample Problem 1
The next three examples are drawn from the computer-based learning environment, Exploring Algebra (Arnold, 1993). They would be suitable for high school students and assume access to computer or graphics calculator facilities.
If x = 2 and x - 2 = 0 are identical mathematical statements, then why does squaring one produce an equation with TWO real solutions, while squaring the other produces only one? Investigate.
Now try to solve the equation
Graduated help was developed for this problem in the following format:
- * Level 1: General information
When we solve equations, we produce simpler equations by doing the same thing to both sides. We assume that this does not change the original equation, and that the solution remains the same.
But is this always true?
- * Level 2: Specific information
x = 2 and x - 2 = 0 are `equivalent' equations (with the same solution). We produce one from the other by subtracting 2 from both sides. But when we SQUARE both equations, we produce x2 = 4 and (x - 2)2 = 0.
- * Level 3: Suggest action
Begin by SOLVING both equations: x2 = 4 and (x - 2)2= 0. What do you notice?
Remember, these are `equivalent' equations, and yet after squaring both sides the solutions are different. Why?
- * Level 4: Suggest sequence
Viewing the GRAPHS of the various functions may help you to better understand what is happening here.
(How might we view the graph of x2= 4? By plotting both sides individually, this gives the parabola y = x2crossing the horizontal line y = 4 at two points).
- * Level 5: Demonstrate
Graphically, SQUARING `bends' a straight line into a parabola, often producing an additional place where the graph crosses the x-axis. So we must take care when we SQUARE both sides of an equation, as `spurious solutions' are likely to result.
In the case of the equation
squaring both sides does not remove all of the sqrt(radical) terms. Instead, we must add sqrt(6 - x) to both sides to produce x = sqrt(6 - x), which can then be squared - but beware of spurious solutions!
Constructing graduated help in this way is no simple task. It is important to avoid "giving the game away", even at the last level, while at the same time, providing increasingly clear directions which will assist in the solution process. Notice that level 1 provides broad, general mathematical information which (while relevant to the problem) does little to move the learner towards a solution, and level 2 is effectively a rewording of the problem. Both are important since students must be given every opportunity to provide a solution themselves, and often this simply involves re-reading the task. The increasing levels of support then move from a single action to a sequence, before finally giving fairly explicit instructions.
The restaurant problem.
Sample Problem 2.
Last updated: 1st May, 1996
Stephen Arnold
crsma@cc.newcastle.edu.au
© 1996 The University of Newcastle
© 1996: The University of Newcastle: Faculty of Education