2 by 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 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? TU

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

Addition squares Add these pairs of outside numbers together

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

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

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??

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?

Multiplication squares x 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 multiplication x Extend to HTU x TU Use with decimals...or with algebra ( x +3)( x+ 4) = x ² + 7 x + 12 x x 3 x x²x²3x3x 4 4x4x 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 activities 1.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 triangles 9.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?

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 1 dot 2 dots 3 dots

Braille 2 Brill shape No dots1 dot2dots3 dots4 dots5 dots6 dots 14641

Braille 3 Brill shape No dots1 dot2dots3 dots4 dots5 dots6 dots

Sorting diagram 3 sides4 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]

oddfactor of 30square number multiple of 3 prime factor of

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 trueSometimes true Never trueNot sure

Graph & co-ordinate challenges y = x – 1x = 3 y = 2x + 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.

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