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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 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: 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?
T U
Addition squares
• Choose any 4 numbers ....2 at the top and 2 on the side
• Add pairs of outside numbers
Addition squares
2 3
4
5
• Add these pairs of outside numbers together
Addition squares
2 3
4 6
5
• Find all 4 numbers in this way.
• Add the 4 numbers inside the square
Addition squares
2 3
4 6 7
5 7 8
• Add pairs of outside numbers
• Add the 4 numbers inside the square...
• ..and add these 4 answers to give a number in the bottom square
Addition squares
2 3
4 6 7
5 7 8
28
• The number in the bottom square is the sum of the 4 numbers.
• Is this number equal to double the sum of the 4 outside numbers?
• 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
28
• Do you notice any patterns in the numbers inside the square?
• Can you find the outside numbers if you just have the inside numbers?
• Is this always possible?
8 11
14 17
Multiplication squares
x 4 3
5
1
• Multiply pairs of outside numbers• Add these 4 new numbers
• What is the connection between the 4 outside numbers and the square total?
• Extend to bigger squares, rectangles, etc
Grid multiplicationx 20 3
10
4 • Extend to HTU x TU• Use with decimals
• ...or with algebra(x+3)(x+4) = x² + 7x + 12
x x 3x x² 3x4 4x 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• Symmetries, • tessellations, • area, & perimeter.• 3-D models (4 cubes)
• What about 5 squares, 6 squares, etc
Four-omino activities1. Make 4-ominoes Use 5 squares joined edge to edge, how many different shapes
can you make?2. Names Find names for all 4-ominoes? Which is a “snake” or the
“submarine”? 3. Symmetry Which have line symmetry? Which have rotational symmetry?4. Tessellation Which 4-ominoes will tessellate? Will all 12 tessellate?5. Area and perimeter Which 4-omino has the biggest area?........longest
perimeter?6. Joins and perimeter Investigate the number of joins and the perimeter.7. Other “ominoes” Make some shapes using just 5 squares.....or 6 squares??
8. Using triangle Use isometric paper to make shapes from 5 triangles9. LOGO or Roamer Write a LOGO programme to draw a 4-omino. .......or direct
a “pupil robot”10. 3-D exploration Use 5 multilink cubes to make a 3-D shape. How many can you
find?
BrailleYour 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......
• 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
1 dot 2 dots3 dots
Braille 2
Brill shape No dots 1 dot 2dots 3 dots 4 dots 5 dots 6 dots
1 4 6 4 1
Braille 3Brill shape No dots 1 dot 2dots 3 dots 4 dots 5 dots 6 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 1
Sorting diagram
3 sides 4 sides
red
not red
• Sort shapes by properties
• 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]
odd factor of 30 square number
multiple of 3
prime
factor of 12
8
46
5
1
7
9
12
2
11
153
1310
14
Always, sometimes, never...
• 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.
Always true Sometimes true
Never true Not sure
Graph & co-ordinate challenges
y = x – 1 x = 3
y = 2 x + y = 5
This graph crosses the x-axis at (1,0)
This graph passes
through (4,2)
This graph passes through (2,1)and (3,2)
This graph passes through (4,3) but not (3,4)
This graph is parallel
to the x axis.
Thank you
Check out The Pi Piper on the STEM Community resources
J