DOUBLE SIDED COUNTERS Jonathan Hall Study Maths Lead

DOUBLE SIDED COUNTERS Jonathan Hall @Study. Maths Lead @La. Salle. Ed Lead Practitioner @Leeds. City. Way Maths. Bot. com

Some of my stuff…

Since the last Sheffield Maths. Conf… Some of my stuff…

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

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

https: //mathsbot. com/manipulatives/double. Sided. Counters

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

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/double. Sided. Counters

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/double. Sided. Counters

Odd integers Show that the sum of the odd integers from 1 to n is a square number. Sequence 1 1 + 3 + 5 Sum 1 4 9 1 + 3 + 5 + 7 + 9 1 + 3 + …. + (2 n – 1) 16 25 n² https: //mathsbot. com/manipulatives/double. Sided. Counters

Triangular numbers Show that the sum of two consecutive triangle numbers is a square number. Sequence 1 + 3 3 + 6 6 + 10 Sum 4 9 16 10 + 15 15 + 21 25 36 https: //mathsbot. com/manipulatives/double. Sided. Counters

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

Venn Diagrams https: //mathsbot. com/manipulatives/double. Sided. Counters

Venn Diagrams n – 3 Reds (n – 3) Yellows https: //mathsbot. com/manipulatives/double. Sided. Counters

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/double. Sided. Counters

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.

Simpson’s Paradox Which box has the greatest chance of picking a yellow counter? https: //mathsbot. com/manipulatives/double. Sided. Counters

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

Frogs and Toads Frogs Toads Slides Jumps Total 1 1 2 1 3 2 5 2 2 4 4 8 3 2 5 6 11 f t f + t f × t (f + t) + ft https: //mathsbot. com/manipulatives/double. Sided. Counters

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

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/double. Sided. Counters