2. Lesson plan to accompany the article Breaking the equation “empirical argument = proof” (published in Mathematics Teaching, issue 213, pp. 9-14) Andreas J. Stylianides University of Cambridge, UK

3. Acknowledgements This lesson plan is derived from the dataset of the research project “Enhancing Students’ Proof Competencies in Secondary Mathematics Classrooms,” funded by the UK’s Economic and Social Research Council (Grant Number: RES- 000-22-2536). This is an adapted version of another lesson plan developed in a joint research project with Gabriel Stylianides, funded by the Spencer Foundation (Grant Number: 200700100). The opinions reflected in the lesson plan as it appears in these PowerPoint slides are those of the author. The author is grateful to the two Year 10 teachers and their students who participated in the research.

4. Disclaimer This lesson plan should be considered together with the accompanying article. The lesson plan is intended to describe one possible way to help students begin to realise the limitations of empirical arguments as methods for validating patterns. The lesson plan is presented neither as an exemplar nor as a “recipe” for how the aforementioned learning objective can be achieved. Teachers who wish to promote this learning objective in their classes are advised to consider carefully their particular contexts (the ability level and prior knowledge of their students, their teaching styles, etc.) and modify the lesson plan accordingly.

5. Request If you end up using this lesson plan or a modified version of it, I would be very interested to hear how you used it and how it played out in your class. Andreas J. Stylianides University of Cambridge Faculty of Education 184 Hills Road Cambridge CB2 8PQ, UK E-mail: as899@cam.ac.uk Tel.: +44 (0) 1223 767550

6. The “Squares” problem 1.How many different 3-by-3 squares are there in the 4-by-4 square above? 2.How many different 3-by-3 squares are there in a 5-by-5 square? 3.How many different 3-by-3 squares are there in a 60-by-60 square? Are you sure that your answer is correct? Why? Do you think we have responded appropriately to this question? Why?

7. Place different numbers of spots around a circle and join each pair of spots by straight lines. Explore a possible relation between the number of spots and the maximum number of non-overlapping regions into which the circle can be divided. When there are 15 spots around the circle, is there an easy way to tell for sure what is the maximum number of non-overlapping regions into which the circle can be divided? The “Circle and Spots” problem

8. 1 5 432 6 7 8 9 10 29 25 22 16 21 30 26 23 19 20 27 17 15 12 11 28 24 18 13 14 31 When there are 6 spots around a circle, ____ is the maximum possible number of non-overlapping regions: 31 What does the “Circle and Spots” problem teach us?

9. 1 5 432 6 7 8 9 10 29 25 22 16 21 30 26 23 19 20 27 17 15 12 11 28 24 18 13 14 31 When there are 6 spots around a circle, ____ is the maximum possible number of non-overlapping regions: 31 What does the “Circle and Spots” problem teach us? The “Circle and Spots” problem teaches me that checking 5 cases is not enough to trust a pattern in a problem. Next time I work with a pattern problem, I’ll check more cases to be sure. What do you think about this student’s comment?

10. Consider the following statement: The expression 1+1141n 2, where n is a natural number never gives a square number. People used computers to check this expression and found out that it does not give a square number for any natural number from 1 to 30 693 385 322 765 657 197 397 207. BUT It gives a square number for the next natural number!!!

11. The “Circle and Spots” problem teaches me that checking 5 cases is not enough to trust a pattern in a problem. Next time I work with a pattern problem, I’ll check more cases to be sure. Do you have any further thoughts about the student’s comment on the “Circle and Spots” problem?