Counting Squares Working in pairs Discuss how many

Counting Squares Working in pairs: • Discuss how many squares are in this image

Royal Institution Primary Maths Masterclasses Off the shelf Masterclass: Being Systematic rigb. org @Ri_Science Image credits: Anon. Moos via Wikimedia Commons

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Royal Institution videos • CHRISTMAS LECTURES – on the Ri website

Royal Institution videos • CHRISTMAS LECTURES – on the Ri website • Ri on You. Tube – experiments, videos & talks for all ages

Royal Institution videos • CHRISTMAS LECTURES – on the Ri website • Ri on You. Tube – experiments, videos & talks for all ages • Expe. Rimental – science experiments at home

Counting Squares Working in pairs: • Discuss how many squares are in this image

Counting Squares Can you use a systematic approach to ensure you have counted all of them? How can you record your data?

Counting Squares Square size How many along/up? How many on the whole board

Counting Squares Square size How many along/up? 1 x 1 2 x 2 8 7 3 x 3 6 4 x 4 5 5 x 5 4 6 x 6 3 7 x 7 2 1 8 x 8 How many on the whole board 64 49 Can you fill in the missing boxes?

Counting Squares Square size How many along/up? How many on the whole board 1 x 1 8 64 2 x 2 7 49 3 x 3 6 36 4 x 4 5 25 5 x 5 4 16 6 x 6 3 9 7 x 7 2 4 8 x 8 1 1 Number of squares on 8 by 8 chessboard = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204

Counting Squares: can you generalise? Square size Upwards/along On whole board 1 x 1 n n 2 2 x 2 (n-1)2 3 x 3 (n-2)2 nxn 1 1 So the number of squares in a n x n chessboard = n 2+ (n-1)2+ (n-2)2 +…. . + 22 +12

Counting Squares: can you generalise? So the number of squares in a n x n chessboard = n 2+ (n-1)2+ (n-2)2 +…. . + 22 +12 Square size Upwards/ On whole along board 1 x 1 n n 2 2 x 2 (n-1)2 3 x 3 (n-2)2 How many squares in a 1 x 1 chessboard? 1 How many squares in a 5 x 5 chessboard? 25+16+9+4+1= 65 What would the calculation be to find the number of squares in a 20 x 20 chessboard? 202+192+182+…+22+12 =2870 nxn 1 1

How many squares? 1 x 1 squares: 24 2 x 2 squares: 13 3 x 3 squares: 4 4 x 4 squares: 1 Total: 24+13+4+1 = 42 squares Can you work out how many squares are in this shape? Can you be sure you haven’t missed any? Can you be sure you haven’t counted any twice? Problem taken from: www. transum. org

Counting cubes! How many cubes? Image credits: Acdx via Wikimedia Common

Counting cubes! Inner cube Upwards/ size along/ backwards In whole cube 1 x 1 x 1 3 27 2 x 2 x 2 2 8 3 x 3 x 3 1 1 Number of cubes in 3 by 3 cube = 33 + 23 +13 = 27 + 8 + 1 = 36 Image credit: modified from Thomas 003 on Wikimedia Commons

Counting cubes! Inner cube Upwards/ size along/ backwards In whole cube 1 x 1 x 1 4 64 2 x 2 x 2 3 27 3 x 3 x 3 2 8 4 x 4 x 4 1 1 Number of cubes in 4 by 4 cube = 43 + 33 + 23 + 13 = 64 + 27 + 8 + 1 = 100 Image credits: Mike Gonzalez via Wikimedia Common

Counting Cubes: can you generalise? Cube size Upwards/along/ backwards In whole cube 1 x 1 x 1 n n 3 2 x 2 x 2 (n-1)3 3 x 3 x 3 (n-2)3 nxnxn 1 1 So the number of cubes in a nxnxn cube = n 3+ (n-1)3+ (n+2)3 +…. . + 23 +13

Being systematic Today we have… • Developed strategies for counting squares in larger 2 D shapes • Extended this to 3 D to count cubes • Worked out patterns so we could do this for any sized chessboard or cube nxn square contains: n 2+ (n-1)2+ (n+2)2 +…. . + 22 +12 cubes nxnxn cube contains: n 3+ (n-1)3+ (n+2)3 +…. . + 23 +13 cubes What is the pattern here? What if there were 4 dimensions?

We hope you have enjoyed exploring systematically with us! What questions do you have? Any unanswered questions can be written down and emailed to “Ask the Ri Masterclass Team” using this email: masterclasses@ri. ac. uk We don’t know all the answers instantly, but we will find out and get back to you before the next Masterclass. Any comments you have about what you enjoyed or what you’d like to do more of can be written on a post-it note and handed in. rigb. org @Ri_Science

What else can you do to extend your knowledge of working systematically? 1 step, 2 step https: //nrich. maths. org/1019 Summing Consecutive Numbers https: //nrich. maths. org/507 Shady Symmetry https: //nrich. maths. org/1868 Try these as extra activities in class, or try them at home…

Extra activity: Being Systematic How many triangles? How many triangles can you count? 16 + 7 + 3 + 1 =27

Extra activity: Being Systematic How many triangles? 16 of area 1 16 of area 2 8 of area 4 4 of area 8 Total: 16 + 8 + 4 = 44 How many triangles are hidden in the pattern? What strategy might you use to count them all to ensure you don’t miss any out? Problem taken from: www. transum. org