A small square EFGH is constructed within a larger (unit) square, OABC, as shown.
The point W gives the locus of the defining point (x, 0) and the AREA of the smaller square.
Can you find an algebraic model which best describes this relationship?
Drag the point (x, 0) to explore this problem.
Before beginning the assessment task, you should activate the GeoGebra CAS engine - press the Activate CAS button which follows until you get a correct result. It may take a couple of attempts. Then commence the task.
Hint: When entering mathematical expressions in the math boxes below, use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.
Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". More?
1. Drag the point (x,0) to find the area of the small square when x = 0.5.
3. Study the triangle OCJ. The base is 1 but can you see that the height CJ = x units, making it easy to calculate ocj(x), the AREA of the triangle in terms of x? ocj(x) =
7. Focus now on triangle OCG. can you see the same congruent triangle repeated four times around the square? If we can find the area of this triangle, then it becomes easy to find the area of the enclosed square. To find the area of triangle OCG we know the height 'h' in terms of x, so we need the base OG. This is the length OJ - JG. First, give JG. jg(x) =
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This problem may also be modelled using CabriJr on the TI-83/84 series calculator. Click on the graphic below to download the file and try it yourself!
Try having students work in pairs: one generating data points using the CabriJr model and the partner entering these into lists to plot and then to help build their algebraic model!
