Area Ratios

The dynamic geometry activities below are from my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Areas Worksheet & Teacher Notes.

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Brief background
Since young inexperienced students at school tend to be naive empiricists, they are usually easily convinced of the truth of a statement with only a few examples. In addition, the dragging & checking feature of dynamic geometry is quite convincing to students. Hence, it is a real challenge within dynamic geometry to create suitable pedagogical examples to illustrate the verification function of proof, which is what this activity specifically tries to focus on. More examples of this kind can be found in some of the Related Links below, in the Verification section of the Rethinking Proof Activities as well as in De Villiers (2007) and De Villiers & Heideman (2014).

Some Area Ratios

Notes
1) Click the 'Show Ratio of Areas' button. What do you notice?
2)You are likely to conjecture that the ratio of the area of ABCD to that of IJKL is exactly equal to 5. But how sure are you that this conjecture is correct?
3) Ensure that you check your conjecture in 1) by dragging.
4) Further check your conjecture in 1)-3), by clicking on the 'Link to Areas Further Check' button to navigate to a new sketch, which has more accurate measurements.
5) Do you still agree with your conjecture in 1)? Check more by dragging.
6) Navigate to a new sketch where the same construction has been carried out on a parallelogram by clicking on the 'Link to Areas 2' button.
7) What do you notice? Can you make a conjecture? How sure are you that your conjecture is correct?
(Here we could argue that even though the result appears to be true, measured to 5 decimal accuracy, we cannot know with 100% certainty - as we've seen with the 1st example - that the calculated/measured ratio isn't changing in the 50th or 100th decimal position & that the displayed invariance we're observing is merely due to round-off error).
8) Challenge: Can you prove that (verify that) your conjecture in 7) is true?
(Hint: Click on the 'Half turn triangles' button. What do you notice? Use this observation to prove the area ratio.)
9) Regarding a correct, modified conjecture made by a student related to 1)-3), see Sylvie's Theorem.
10) Despite the increased sophistication of automated theorem provers in mathematics, even in dynamic geometry, there appears to be some theoretical & practical constraints regarding validation using this type of software (e.g. Botana & Recio, 2016).
11) Lastly, as pointed out in Hanna, Reid & De Villiers (2019, p. 7), computer proofs only provide verification — they generally do not, for example address mathematicians’ needs for understanding, explanation, clarification and systematization.

References
Botana, F & Recio, T. (2016). On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving. Math. Comput. Sci., No. 10, pp. 5–25.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2007). Some pitfalls of dynamic geometry software. Learning & Teaching Mathematics, No. 4, pp. 46-52.
De Villiers, M. & Heideman, N. (2014). Conjecturing, refuting and proving within the context of dynamic geometry. Learning & Teaching Mathematics, No. 17, pp. 20-26.
Hanna, G., Reid, D. & De Villiers, M. (2019). Proof Technology in Mathematics Research and Teaching. Springer Nature, Switzerland AG.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 11 July 2025; updated 13 July 2025.