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