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
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: RES000-22-2536). 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.
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.
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
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?
The “Circle and Spots” problem
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?
31 is the maximum possible
When there are 6 spots around a circle, ____
number of non-overlapping regions:
1
2
7
18
3
6
5
4
10
11
12 13
20
14
17
19
15
16
21
22
23
25
24
30
28 29
31
26
8
27
9
What does the “Circle and Spots”
problem teach us?
31 is the maximum possible
When there are 6 spots around a circle, ____
number of non-overlapping regions:
1
2
7
18
3
6
5
4
10
11
12 13
20
14
17
19
15
16
21
22
23
25
24
30
28 29
31
26
8
9
What does the “Circle and Spots”
problem teach us?
27
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?
Consider the following statement:
The expression
1+1141n2, 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!!!
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?