The Diminishing Square - GeoGebra Dynamic Worksheet

©2019 Compass Learning Technologies ← Live Mathematics on the Web ← GeoGebra Assessment Showcase →The Case of the Diminishing Square
The Case of the Diminishing Square

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.

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Assessment

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?

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1. Drag the point (x,0) to find the area of the small square when x = 0.5.

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2. Now find the value of x that gives a small square half the area of the large square.

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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) =

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4. We can also let OJ be the base, and GC the perpendicular height (call it 'h'). Give an expression for the length of OJ in terms of x: oj(x) =

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5. Now calculate the area of OCJ another way using the height h and base oj(x): ocj2(x) =

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6. Use these two expressions for the same area to find height(x), height in terms of x: height(x) =

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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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8. Now give an expression for the base (OG) in terms of x. base(x) =

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9. The area of triangle OCG is area(x) =

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10. The area of square EFGH is dimsquare(x) =

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How difficult did you find this activity? Please assign a rating value from 1 (very easy) to 5 (extremely difficult). Comments? Suggestions? What did you learn from this activity?

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Steve Arnold, 2018, Created with GeoGebra

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!

Just wait until you try this with TI-nspire CAS!

And you really should have a look at GXWeb and Geometry Expressions.

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This dynamic figure was created using the latest version of the free GXWeb from Saltire Software, as shown below.

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Solve this problem using GXWeb.

See a complete solution to this problem.

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©2019 Compass Learning Technologies ← Live Mathematics on the Web ← GeoGebra Showcase