Patterns Primes Purposeful Practice Ideas and activities to

Patterns, Primes & Purposeful Practice Ideas and activities to engage. Jonathan Hall: Lead Practitioner School: Leeds City Academy Twitter: @Study. Maths Website: Maths. Bot. com

Some of my resources Maths. Bot. com Form. Time. Ideas. com Flash. Maths. co. uk Study. Maths. co. uk

Patterns, Primes & Purposeful Practice Workshop Aims: • Look into how purposeful practice can be embedded into every day teaching through interesting mathematical ideas and activities. • To share a collection of resources which are simple to deliver and require minimal preparation. • Not “try to make maths interesting” but highlight how interesting mathematics can be.

Calculate 8. 5² (x + 0. 5)² = (x + 0. 5) = x² + 0. 5 x + 0. 25 = x² + x + 0. 25 = x(x + 1) + 0. 25

Calculate 98² = (98) = (98 + 2)(98 – 2) + 2² = (100)(96) + 4 = 9604 In general: x² = (x + a)(x – a) + a²

Factor Skyscrapers Build a skyscraper on each number. It's height should represent the number of different factors the number has.

Probing Questions • Which columns contains the tallest towers? • Which number has the smallest tower? • What is special about the numbers with 2 factors? • Is one a prime number? • Do the prime numbers form a pattern? • What is special about the numbers with an odd number of factors?

Factor towers by @Jeremy_Denton Thanks to Steve Lyon and Mike Anderson @STEMLearning. UK for the idea.

With 6 columns, all primes except 2 and 3 occur in columns 1 and 5. Triangle numbers never end in 2, 4, 7 or 9. Sieve Of Eratosthenes

Deficient, perfect and abundant • A number is deficient if the sum of its factors (other than itself) is less than the number. • A number is perfect if the sum of its factors (other than itself) is equal to the number. • A number is abundant if the sum of its factors (other than itself) is greater than the number. What’s the first perfect number? What’s the first abundant number?

Number Factors Sum of factors (except itself) Deficient, perfect or abundant? 1 1 0 Deficient 2 3 4 5 1, 2 1, 3 1, 2, 4 1, 5 1 1 3 1 Deficient 6 7 8 9 10 11 12 1, 2, 3, 6 1, 7 1, 2, 4, 8 1, 3, 9 1, 2, 5, 10 1, 11 1, 2, 3, 4, 6, 12 6 1 7 4 8 1 16 Perfect Deficient Deficient Abundant

Prime Factors What's the smallest positive integer which is divisible by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10? 2 x 3 x 2 x 5 x 7 x 2 x 3 = 2520 Extension: Find the smallest positive integer divisible by all the integers from 1 to 20. 232792560 (it actually divides by 21 and 22 as well)

How many factors does 2520 have? Each factor can have: 0, 1, 2 or 3 occurrences of the factor “ 2”. 0, 1, or 2 occurrences of the factor “ 3”. 0 or 1 occurrences of the factor “ 5”. 0 or 1 occurrences of the factor “ 7”. So 2520 has 4 x 3 x 2 = 48 factors.

HCF/LCM Pyramids Rich Templates

Ulam Spiral The Ulam spiral is constructed by writing integers in a spiral arrangement and then highlighting the prime numbers. Uam Spiral

Square numbers in the Ulam Spiral

Triangle numbers in the Ulam Spiral

Primes in a Ulam Spiral starting at 41

Eventually, the diagonal of primes is finally broken. Oddly enough, on row 41.

Goldbach’s conjecture Goldbach's conjecture is one of the oldest unsolved problems in number theory and all of mathematics. It states: “Every even integer greater than 2 can be expressed as the sum of two primes. ” Bertrand’s theorem Bertrand's theorem states that for each n > 1 there is a prime p such that n < p ≤ 2 n. “Chebyshev said it, but I'll say it again; There's always a prime between n and 2 n. ”

A prime factor trick • Choose any positive integer less than 1000. • Multiply it by 7. • Multiply your answer by 11. • Finally, multiply your answer by 13. • What do you notice? • Why does this work? (7)(11)(13)x = 1001 x = 1000 x + x

Happy Numbers • Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1, or it loops in a cycle. • Numbers for which this process ends in 1 are happy numbers, while those that get stuck in a loop are unhappy numbers. Is 19 a happy number? 1² + 9² = 1 + 81 = 82. 8² + 2² = 64 + 4 = 68. 6² + 8² = 36 + 64 = 100. 1² + 0² = 1.

Probing Questions • Find all the happy numbers between 1 and 20. • Unhappy numbers will always get stuck in a repeating sequence, can you find it? • What do you notice about all the numbers in a sequence if the number is happy? • What do you notice about all the numbers in a sequence if the number is sad? • If you rearrange the digits of a number, i. e. 19 becomes 91, does the happiness of the number change? • How many happy prime numbers are there below 100?

Directed Number Negative Patterns

The four 4's challenge • Try to make all the numbers from 1 to 20 using exactly four 4's. • Any of the standard operations +, -, × ÷ and brackets may be used. • For example: (4 + 4) ÷ (4 + 4) = 1. • If you get stuck, you are allowed to concatenate digits or use the square root symbol, e. g. 44 ÷ 4 + √ 4 = 13.

Fill each of the tiles with the digits 1 to 9. Each digit can only be used once. Four Ops Puzzle

Fill each of the tiles with the digits 1 to 9. Each digit can only be used once. Four Ops Puzzle

Kaprekar's routine • Step 1: Write down any three digits. Make sure at least 1 digit is different from the rest. • Step 2: Write down the largest number you can make from these digits. • Step 3: Write down the smallest number you can make from these digits, leading zeros allowed. (E. g. 024) • Step 4: Subtract the smallest from the largest number. • Step 5: Go back to step 2 until you have a very good reason to stop.

495 • Can you find ‘Kaprekar’s Constant’ for 4 digits? • What happens if you try it with 2 digits? • What is the largest cycle you can make?

Sierpinski Triangle

Odd and Evens with Pascal mathsbot. com/activities/pascal

Chaos Game mathsbot. com/activities/chaos

Recent Additions: ‘Virtual Manipulatives’ Maths. Bot. com

‘Interactive Tools’ Maths. Bot. com

Recommended Reading Fascination of Numbers W. J. Reichmann Professor Stewart's Cabinet of The Magic of Pineapples Mathematical Curiosities William Emeny Ian Stewart

Websites • • MEDIAN Don Steward: donsteward. blogspot. co. uk NRICH: nrich. maths. org Mr Barton Maths: mrbartonmaths. com/teachers Resourceaholic: resourceaholic. com

Thanks for listening Jonathan Hall: Lead Practitioner School: Leeds City Academy Twitter: @Study. Maths Website: Maths. Bot. com

A few questions to keep you busy until the next session… Which square numbers can’t you make by adding two prime numbers together? How many ways can 100 be expressed as the product of three positive integers? Take any prime number greater than 3 , square it and subtract one. What do you notice? Explain why. How many zeros are at the end of 200!?