What do you see? (1)
Watch what happens.
Start again
What do you see? (1)
Describe what happens.
How many dots are added each time?
What do you see? (2)
Watch what happens; it starts with a square of
dots.
Start again
What do you see?(2)
Can you describe what happens?
Would the same thing happen if you
started with a larger square?
Or a smaller square?
Can you write this algebraically?
Start with any
square: a²
a
a
Take one
away: a²-1
a
a
Move the top row to
become a column…
a
a
Move the top row to
become a column…
a
a
Move the top row to
become a column…
a
a
Move the top row to
become a column…
a-1
a
Move the top row to
become a column…
a-1
a
Move the top row to
become a column…
a-1
a+1
Move the top row to
become a column…
a-1
a+1
(a-1)(a+1)
a-1
a+1
a²-1
=
(a-1)(a+1)
What do you see? (3)
Watch what happens; the initial shape is a
square.
The shape
removed is
a square.
Start again
What do you see? (3)
Can you describe what happens?
Would the same thing happen if you
with a larger square?
Or a smaller square?
Can you write this algebraically?
What do you see? (4)
Watch what happens.
Start again
What do you see? (4)
What would happen if it kept on going?
Describe what happens in words.
Can you describe what happens using
numbers?
1
2
1
2
1
8
1
32
1
16
What
should
these be?
1
4
Start again
What do you see? (4)
Does this help you to find the sum of the
following:
1 1 1 1 1
     ... 
2 4 8 16 32
What do you see? (5)
Watch what happens.
Start again
What do you see? (5)
Describe what happens in words.
What would happen if it kept on going?
Can you describe what happens using
numbers?
1
9
1
3
1
27
1
81
1
81
1
9
1
27
1
3
What do you see? (5)
Does this help you to find the sum of the
following:
1 1 1 1
 
  ... 
3 9 27 81
What do you see? (5)
Can you think of a similar geometrical proof to
find the sum of the infinite series:
1 1
1
1



 ... 
4 16 64 256
Teacher notes: What do you see?
This edition looks at a range of visual stimuli and asks ‘What do you
see?’ Many of the activities are visual representations of proof,
including sums of infinite series and algebraic equivalence.
The pedagogical focus for these activities is on generating discussion
amongst small groups of students in order that they arrive at an
understanding. At various times, the teacher might circulate and listen
in unobtrusively to discussions, ask probing questions, or draw out
conflicting or differing responses from groups to contribute to a whole
class plenary. The activities could be used as a series of starter
activities.
These are some of the approaches highlighted within the AfL session
run by Simon Clay and Debbie Barker at the MEI 2014 Conference.
Additionally, these are strategies which many girls find particularly
supportive – but that’s not to suggest that boys don’t find them
supportive too!
Teacher notes: What do you see? (1)
This activity shows the sum of consecutive odd numbers.
Show the sequence to students and ask
them to describe what they see.
Make it ‘girl friendly’:
Ask students to discuss
it with a friend before
giving an answer
At KS3 or 4, students could express in
words that adding odd numbers always gives a square number.
At a higher level students might use notation for the sum of a series.
Either: 1+3+5+…+(2n-1) =n2
Or:
𝑛
𝑖=1 2𝑛
− 1 = 𝑛2
Teacher notes: What do you see? (2)
This activity shows that a²-1 = (a-1)(a+1)
Show the sequence to students and ask
them to describe what they see.
A specific size of square is shown, ask
students to consider what would happen
with larger or smaller starting squares.
They might like to sketch them out to check,
so some squared paper might be useful.
Make it ‘girl friendly’:
Ask students to “think,
pair, share”: give students
silent thinking time before
a short discussion with a
partner and then others.
Teacher notes: What do you see? (3)
This activity shows that a²-b2 = (a-b)(a+b)
Show the sequence to students and ask
them to describe what they see.
A generic square is shown and a smaller
generic square is removed.
Make it ‘girl friendly’:
Ask students to work
with a partner; you
circulate, listening in and
asking questions quietly
to pairs.
Can students convince themselves that the rectangle ‘on top’ will always
fit ‘at the side’?
Encourage students to write this algebraically.
Teacher notes: What do you see? (4)
This activity shows the sum of the reciprocals of powers of 2.
Show the sequence to students and ask
them to describe what they see.
Transitions are timed – no mouse click
needed.
Make it ‘girl friendly’:
Show the sequence two
or three times through.
Accept and validate all
responses.
Students in KS3 and 4 may describe the
series in words.
Ask how much of the white square is used each time. The second part of
the sequence also shows the fractional values.
Teacher notes: What do you see? (5)
This activity shows the sum of the reciprocals of powers of 3.
Show the sequence to students and ask
them to describe what they see.
Transitions are timed – no mouse click
needed.
Make it ‘girl friendly’:
Ask students to
• Convince themselves
• Convince a friend
• Convince the class
Students in KS3 and 4 may describe the
series in words.
Ask how much of the white square is used each time. The second part of
the sequence also shows the fractional values.
The third part of the activity asks if students can think of something
similar for reciprocals of powers of 4. An example shown on the
following slides shows that the sum is 1/3.
Teacher notes: What do you see? (5)
In this case, divide the square into quarters,
colour one quarter dark and two light.
Teacher notes: What do you see? (5)
With the remaining space, divide it into quarters,
colour one quarter dark and two light.
Teacher notes: What do you see? (5)
With the remaining space, divide it into quarters,
colour one quarter dark and two light.
Teacher notes: What do you see? (5)
For every one dark section there will be two light
sections.