1. Patterns, Primes & Purposeful Practice
Ideas and activities to engage
Jonathan Hall: Lead Practitioner
School: Leeds City Academy
Website: MathsBot.com

2. Some of my resources MathsBot.com FormTimeIdeas.com FlashMaths.co.uk
StudyMaths.co.uk

3. 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.

4. Calculate 8.5² (x + 0.5)² = (x + 0.5)(x + 0.5)

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

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

7. 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?

8. Factor towers by @Jeremy_Denton
Thanks to Steve Lyon and Mike for the idea.

9. 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

10. 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?

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

12. 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.
(it actually divides by 21 and 22 as well)

13. How many factors does 2520 have?
2 3 x x 5 x 7 = 2520
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 x 2 = 48 factors.

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

15. Square numbers in the Ulam Spiral

16. Triangle numbers in the Ulam Spiral

17. Primes in a Ulam Spiral starting at 41

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

19. 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 ≤ 2n.
“Chebyshev said it, but I'll say it again; There's always a prime between n and 2n.”

20. A prime factor trick (7)(11)(13)x = 1001x = 1000x + x
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 = 1001x = 1000x + x

21. 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 = = 82.
8² + 2² = = 68.
6² + 8² = = 100.
1² + 0² + 0² = 1.

22. 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?

23. Directed Number
Negative Patterns

24. 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.

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

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

27. 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.

28. 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?

29. Sierpinski Triangle

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

31. Chaos Game
mathsbot.com/activities/chaos

32. Recent Additions: ‘Virtual Manipulatives’
MathsBot.com

33. ‘Interactive Tools’
MathsBot.com

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

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

36. Thanks for listening Jonathan Hall: Lead Practitioner
School: Leeds City Academy
Website: MathsBot.com

37. 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!?