2 by 2.... to infinity and beyond!!! Primary Mathematics Conference National STEM Centre,York

1. 2 by 2....to infinity and beyond!!! Primary Mathematics Conference National STEM Centre,York The pi Piper

2. 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

3. 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

4. 2 by 2.......by more!!! Which different “themes” in school mathematics can you teach / learn with a 2 by 2 grid?

5. 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

6. 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?

7. Addition squares • Choose any 4 numbers ....2 at the top and 2 on the side • Add pairs of outside numbers

8. Addition squares • Add these pairs of outside numbers together

9. Addition squares • Find all 4 numbers in this way. • Add the 4 numbers inside the square

10. Addition squares • 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

11. Addition squares • 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??

12. Addition squares...an afterthought • 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?

13. Multiplication squares • 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

14. Grid multiplication • Extend to HTU x TU • Use with decimals • ...or with algebra (x+3)(x+4) = x² + 7x + 12

15. 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

16. 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

17. Four-omino activities • Make 4-ominoesUse 5 squares joined edge to edge, how many different shapes can you make? • NamesFind names for all 4-ominoes? Which is a “snake” or the “submarine”? • Symmetry Which have line symmetry? Which have rotational symmetry? • TessellationWhich 4-ominoes will tessellate? Will all 12 tessellate? • Area and perimeterWhich 4-omino has the biggest area?........longest perimeter? • Joins and perimeterInvestigate the number of joins and the perimeter. • Other “ominoes”Make some shapes using just 5 squares.....or 6 squares?? • Using triangle Use isometric paper to make shapes from 5 triangles • LOGO or RoamerWrite a LOGO programme to draw a 4-omino. .......or direct a “pupil robot” • 3-D explorationUse 5 multilink cubes to make a 3-D shape. How many can you find?

18. 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...... • 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

19. Braille2

20. Braille 3

21. Sorting diagram • 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]

22. 8 14 12 10 13 3 9 15 11 7 5 2 6 4 1

23. 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.

24. Graph & co-ordinate challenges This graph crosses the x-axis at (1,0) This graph passes through (4,2) This graph is parallel to the x axis. This graph passes through (2,1)and (3,2) This graph passes through (4,3) but not (3,4)

25. Thank you Check out The Pi Piper on the STEM Community resources J