Transcript 2 by 2 workshop

2 by 2....to infinity and beyond!!!
Primary Mathematics Conference
National STEM Centre,York
The pi Piper
Objectives
 How much mathematics can you teach or learn
with a 2 by 2 grid?
 How can we turn one simple task into higher level
learning?
 Reflection, questions, sharing, etc
Rich tasks in mathematics
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accessible
extendable
allow learners to make decisions
involve learners in making & testing hypotheses,
reflecting, interpreting, proving,
promote discussion and communication
encourage originality and invention;
encourage ‘what if’ ..........and ‘what if not’ questions;
are enjoyable and contain the opportunity for surprise.
“Better Mathematics”, WSIHE, (1988)
Primary learners
 DO… TALK… RECORD…
 Balance…..fluency, reasoning & problem solving
2 by 2.......by more!!!
Which different “themes” in school
mathematics can you teach / learn
with a 2 by 2 grid?
Place value:
•
•
•
Biggest add
Roll a dice & enter numbers in the
boxes.
Each player has own table
Write your numbers in any of your
boxes and then add your numbers
together
Place value:
•
T
U
•
•
Biggest add
Roll a dice & enter numbers in the
boxes.
Each player has own table
Write your numbers in any of your
boxes and then add your numbers
together
Variations
• Smallest add
• Biggest take-away
• HTU, TU.t
• what if you are allowed to put
numbers in another person’s boxes?
Addition squares
• Choose any 4 numbers ....2 at the
top and 2 on the side
• Add pairs of outside numbers
Addition squares
2
4
5
3
• Add these pairs of outside
numbers together
Addition squares
• Find all 4 numbers in this
way.
2
4
5
6
3
• Add the 4 numbers inside
the square
Addition squares
• Add pairs of outside
numbers
2
3
4
6
7
5
7
8
• Add the 4 numbers inside
the square...
• ..and add these 4 answers
to give a number in the
bottom square
Addition squares
• The number in the bottom square is
the sum of the 4 numbers.
2
3
4
6
7
5
7
8
• Is this number equal to double the
sum of the 4 outside numbers?
28
• Investigate other 2 by 2
squares
• What about 3 by 3 squares, 4 by 4,..?
• What about rectangles??
Addition squares...an afterthought
2
3
4
6
7
5
7
8
• Do you notice any patterns in
the numbers inside the
square?
28
8
11
14
17
• Can you find the outside
numbers if you just have the
inside numbers?
• Is this always possible?
Multiplication squares
• Multiply pairs of outside numbers
• Add these 4 new numbers
x
5
4
3
• What is the connection between
the 4 outside numbers and the
square total?
1
• Extend to bigger squares,
rectangles, etc
Grid multiplication
x
20
3
10
• Extend to HTU x TU
• Use with decimals
4
x
x
3
x
x²
3x
4 4x
12
• ...or with algebra
(x+3)(x+4) = x² + 7x + 12
Square frogs
• Move the red frog to the blank
square
• Only horizontal and vertical moves
are allowed.
• What is the fewest number of
moves?
• Use bigger squares, more frogs...
• Try rectangles.
• Record results & generalise
Four-ominoes
• These can be made with 4
squares.
• Are there any more?
Investigate
•
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Symmetries,
tessellations,
area, & perimeter.
3-D models (4 cubes)
• What about 5 squares, 6
squares, etc
Four-omino activities
1. Make 4-ominoes
2. Names
3. Symmetry
4. Tessellation
5. Area and perimeter
6. Joins and perimeter
7. Other “ominoes”
8. Using triangle
9. LOGO or Roamer
10. 3-D exploration
Use 5 squares joined edge to edge, how many different shapes
can you make?
Find names for all 4-ominoes? Which is a “snake” or the
“submarine”?
Which have line symmetry? Which have rotational symmetry?
Which 4-ominoes will tessellate? Will all 12 tessellate?
Which 4-omino has the biggest area?........longest
perimeter?
Investigate the number of joins and the perimeter.
Make some shapes using just 5 squares.....or 6 squares??
Use isometric paper to make shapes from 5 triangles
Write a LOGO programme to draw a 4-omino. .......or direct
a “pupil robot”
Use 5 multilink cubes to make a 3-D shape. How many can you find?
Braille
Your task is to design a new coding system for letters in the alphabet.
• The code is based on a 2 by 2 grid with up to 4 dots in the cells.
• Here are a few......
1 dot
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3 dots
How many different “Braille tiles” are there?
How many of these use 2 dots.......or just 3 dots, etc....?
Would you have enough for each letter of the alphabet?
Make some 3-dot, 5-dot, 6-dot........Braille tiles
2 dots
Braille 2
Brill shape
No dots
1 dot
2dots
3 dots
4 dots
1
4
6
4
1
5 dots
6 dots
Braille 3
Brill shape
No dots
1 dot
2dots
3 dots
4 dots
5 dots
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
6 dots
1
Sorting diagram
• Sort shapes by properties
3 sides
red
not red
4 sides
• Sort numbers [odd, prime,
multiples, etc]
• Make sets of criteria cards to
create a variety of problems.
• Use bigger diagrams [e.g. 3 by 3]
8
odd
factor of 30
multiple of 3
prime
factor of 12
2
square number
Always, sometimes, never...
Always true
Sometimes true
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Never true
Not sure
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Multiples of 3 are odd numbers
Squares have 4 right angles.
A 4-sided shape has a line of
symmetry
An even number cannot be a prime
number
A multiple of 3 cannot be a multiple
of 2.
You can draw a triangle with 2 right
angles
A shape with 4 sides is a square.
Graph & co-ordinate challenges
y=x–1
y=2
x=3
x+y=5
This graph passes
through (2,1)and (3,2)
Thank you
Check out The Pi Piper on the STEM Community resources
J