1. You are going to work in pairs to produce a Maths board game

2.  Part One  Design a board game base on substitution  Part Two  Use the game to look at sequences

3.  Design a board game base on substitution Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win Your game must be based on substitution

4.  Design a board game base on substitution Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win This is a very basic example of a possible game.

5.  Design a board game base on substitution Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win If you start by rolling a “2”, you will land here.

6.  Design a board game base on substitution Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win You then need to substitute n = 2 into the expression given in the square

7.  Design a board game base on substitution Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win If n = 2 then n – 3 will be 2 – 3 which is -1 You would have to move back by 1 space. It is then the other players turn.

8.  Design a board game base on substitution Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win Notice that not all the squares are substitutions – but most are.

9.  Design a board game base on substitution Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win Your turn

10.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win Lets look at all the possible results from the dice

11.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win It would be sensible to work sequentially

12.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total

13.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total2 + 3

14.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total5

15.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total54 + 3

16.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total57

17.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total576 + 3

18.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total579

19.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total579111315

20.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total579111315 This is a linear sequence

21.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total579111315 It goes (up) by the same amount each time +2

22.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 total579111315 We say the difference is + 2 +2

23.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n total579111315 This tells us that the basic sequence is 2n in other words, it is based on the 2 times table

24.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n24681012 total579111315 Lets look at the two times table

25.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n24681012 total579111315 So, the first part of our rule is 2n but that’s not the end because our sequence is slightly different – we need a correction factor

26.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n24681012 total579111315 How do we get from here

27.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n24681012 total579111315 How do we get from here To here

28.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n24681012 total579111315 How do we get from here To here We +3

29.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n24681012 2n+1579111315 This gives us our final rule 2n + 3

30.  Look at the first square Move forwards by 2n + 3 Move forwards by n - 3 Move backwards by 2n + 3 Move forwards by 4 squares Move forwards by -n + 3 Move backwards by n + 7 Move forwards by 2n + 3 Go back to the start. Throw a four to win Start Win n123456 2n24681012 2n+3579111315 This gives us our final rule 2n + 3 Use your game to show that this always works.