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Double Sided Counters

Double Sided Counters. Jonathan Hall @StudyMaths Maths Lead @ LaSalleEd Lead Practitioner @ LeedsCityWay MathsBot.com. Some of my stuff…. Since the last Sheffield MathsConf …. Some of my stuff…. Hotel Doors – Try it in your head first. There are 12 closed doors in a hotel corridor.

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Double Sided Counters

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  1. Double Sided Counters Jonathan Hall @StudyMaths Maths Lead @LaSalleEd Lead Practitioner @LeedsCityWay MathsBot.com

  2. Some of my stuff…

  3. Since the last Sheffield MathsConf… Some of my stuff…

  4. Hotel Doors – Try it in your head first. There are 12 closed doors in a hotel corridor. Every 2nd door is then changed. (opened if closed or vice versa) Then every 3rd door is changed. Then every 4th door is changed. Then every 5th door is changed. Finally, every 6th door is changed. Which doors are now open? Open Closed https://mathsbot.com/manipulatives/doubleSidedCounters

  5. 1 2 3 4 5 6 7 8 9 10 11 12 Initial 2nd door 3rd door 4th door 5th door 6th door https://mathsbot.com/manipulatives/doubleSidedCounters

  6. https://mathsbot.com/manipulatives/doubleSidedCounters

  7. Sequences How many counters are in the next pattern? How many counters are in the 10th pattern? How many counters are in the nth pattern? https://mathsbot.com/manipulatives/doubleSidedCounters

  8. Sequences How many counters are in the next pattern? How many counters are in the 10th pattern? How many counters are in the nth pattern? https://mathsbot.com/manipulatives/doubleSidedCounters

  9. Zero Pairs Field axiom - Existence of additive inverses. Every element x within a field is invertible for addition. The additive inverse for x is unique, and denoted by -x. https://mathsbot.com/manipulatives/doubleSidedCounters

  10. Sequences/Directed Number • Work out the sum of 4 rows. • Work out the sum of 10 rows. • Work out the sum of n rows. 1, -1, 2, -2, 3, -3, 4, -4, … If n is odd: (n + 1) ÷ 2 If n is even: -n ÷ 2 https://mathsbot.com/manipulatives/doubleSidedCounters

  11. Odd integers Show that the sum of the odd integers from 1 to n is a square number. https://mathsbot.com/manipulatives/doubleSidedCounters

  12. Triangular numbers Show that the sum of two consecutive triangle numbers is a square number. https://mathsbot.com/manipulatives/doubleSidedCounters

  13. Fibonacci Show that the sum of the 1st and 5th term in a Fibonacci sequence is 3 times larger than the 3rd term. https://mathsbot.com/manipulatives/doubleSidedCounters

  14. Venn Diagrams Place some counters such that: • In ‘A’ the probability of picking a yellow counter is . • In ‘B’ the probability of picking a red counter . • What is the smallest number of counters you can do this with? Why? • What happens if you change the probabilities to instead? • What about ? https://mathsbot.com/manipulatives/doubleSidedCounters

  15. Venn Diagrams https://mathsbot.com/manipulatives/doubleSidedCounters

  16. Venn Diagrams https://mathsbot.com/manipulatives/doubleSidedCounters

  17. Venn Diagrams https://mathsbot.com/manipulatives/doubleSidedCounters

  18. Venn Diagrams https://mathsbot.com/manipulatives/doubleSidedCounters

  19. Venn Diagrams (n > 2) n – 3 Reds (n – 3) Yellows https://mathsbot.com/manipulatives/doubleSidedCounters

  20. To replace or not to replace? The are 2 red and 2 yellow counters in a bag. Game 1: A counter is chosen at random then replaced. A second counter is then chosen at random. Game 2: A counter is chosen at random and not replaced. A second counter is then chosen at random. You win if you get two counters of the same colour. Which game do you play? https://mathsbot.com/manipulatives/doubleSidedCounters

  21. To replace or not to replace? Game 1: P(win) is 1/4 + 1/4 = 2/4 Game 2: P(win) is 1/6 + 1/6 = 2/6 Investigate what happens if you change the initial number of counters.

  22. Simpson’s Paradox Which box has the greatest chance of picking a yellow counter? = = = = https://mathsbot.com/manipulatives/doubleSidedCounters

  23. Averages and Algebra (2y + 3r) + (y + 4r) + (3y + 2r) = 6y + 9r = 3(2y + 3r) The mean is 2y + 3r https://mathsbot.com/manipulatives/doubleSidedCounters

  24. Frogs and Toads https://mathsbot.com/manipulatives/doubleSidedCounters

  25. A game of reds and yellows Can you predict what colour the final counter will be? https://mathsbot.com/manipulatives/doubleSidedCounters

  26. In summary, they can be used for… • Directed number • Ratio • Sequences and nth term • Proof • Averages • Collecting like terms • Factorising • Venn Diagrams • Probability • Tree Diagrams • Factors, Multiples and Primes • Square and Triangle numbers • Long Division • Modelling Problems https://mathsbot.com/manipulatives/doubleSidedCounters

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