Trail Games Years K - 7

Summary Trail Games - a numbered board with a start and finish which players wind their way around - provide opportunities to develop social skills such as following rules and taking turns as well as mathematical skills such as counting, complementary addition, difference and probability experiences. What happens if questions extend the possibilities of these games, especially in the area of probability, and lead the activity into Maths300 Lesson 183, Snakes & Ladders. Suitable for threading. Materials One Poly Plug per pair Poly Plug 13 Trail (PDF) Poly Plug 25 Trail (PDF) One dice per pair One calculator each

Procedure The activity Fill The Board includes teachers' comments and children's work in classes from K to 6. These indicate the extent to which probability, data collection and data interpretation activities can be developed from a simple game. You might like to use this activity first because trail games develop from it by asking What happens if we put a number grid under the red board?. The activity will also encourage data collection skills that will be useful in Trail Games. Depending on the level of your children you can begin with either Poly Plug 13 Trail or Poly Plug 25 Trail. Children push out all the red plugs from one board and place it on top of the sheet. In the first the children sit opposite each other and use one plug of the appropriate colour to count their way in a spiral from 1 to END. First to END wins and it is not necessary to roll the exact number to END, that is, the last dice roll might be more than the count needed to reach END, but reaching END completes the game anyway. Players begin with their plugs off the board. Movement is by taking turns to roll the dice and moving according to the number that turns up. In the second the players sit beside each other and use different coloured plugs to count their way from 1 to FINISH. First to FINISH wins and it is not necessary to roll the exact number to END. Players begin with their plugs off the board. Movement is by taking turns to roll the dice and moving according to the number that turns up.

Content 1:1 correspondence addition facts beyond 10 addition facts to 10 complementary addition counting data: collecting, recording, displaying data: describing & comparing with statistics data: interpretation likely, less likely and unlikely events estimating number mathematical conversation number line - ordering, operations probability

Children can keep a running total of their dice rolls on a calculator or a written record of the actual numbers rolled.

• Using the calculator will support children still developing counting and addition skills.
• A written record builds a collection of data about the outcomes of the experiment of rolling a dice.

Whichever trail is being used, look for opportunities to ask mathematical questions such as:

• How many more do you need to reach ...?
• How many steps are you away from ...?
• What dice rolls will let you win the game from here?
• How many moves would be unused if you roll ... now and end the game?
• What is your chance of winning the game on your next roll?
• Can ... catch up to you, or pass you on the next roll? What numbers would make that happen?
• How many steps would it take for ... to catch up to you?
• What is the difference between the number you are on and the number ... is on?
• What is the smallest number of rolls that it would take to end the game? What could those rolls be?
• What is the largest number of rolls it would take to end the game? What could those rolls be?

• What is the most likely number of rolls to end the game?

Also, at the appropriate level, it could include discussion of the Expected Number of rolls. Opposite dice faces sum to 7, so there are three sets of seven on a dice. Therefore we could expect a total of 7 moves in every two rolls. On the 25 Trail that means we could expect 2 rolls = 7 moves, 4 rolls = 14 moves, 6 rolls = 21 moves, 7 rolls = 24.5 moves and so conclude that the Expected Number of rolls is 7 rolls 'and a little bit'.

Another way to calculate the Expected Number of moves is to reason that because all faces are equally likely, in six rolls we could expect one each of 1, 2, 3, 4, 5, 6. So we could expect a total of 21 moves after 6 rolls, or 3.5 moves per roll. Calculating this way leads to the same result of 7 rolls 'and a little bit' to end the game.

These are theoretical calculations, so it is interesting to compare, again at the appropriate level, the variation between the short term results and this Expected Value, as well as considering how closely (or otherwise) the Expected Value and the mean, median and mode move towards each other (converge) as the amount of data increases.

The investigation can be extended further by asking:

• What happens if you must roll to exactly finish the game?
• What happens if you must roll a 6 to start?
• What happens if we change the type of dice?

Snakes & Ladders

Still using the 25 Trail, ask children to plug in the numbers shown here. The yellow plug will cover the number so it can't be seen, but it can still be known. Now we have made a ladder that goes from 10 straight up to 24. If you land exactly on the bottom of the ladder, you must finish your move by going straight to the top. It doesn't work for any other yellow plug on the ladder. They just count as a normal number. Ask children to predict the number of moves the game will take now, then play to collect, organise and analyse data as above. What is the lowest number of moves to finish this game? What is the highest number of moves to finish this game? What dice rolls must happen to make you land on 10 and climb the ladder? What is the probability of landing on 10? What happens if we change the position of the ladder? Can we place a ladder so that the smallest number of moves is less than 3? Can we place a ladder so that the smallest number of moves is exactly 4, 5, ... moves? What happens if we add another ladder - perhaps a short one with only two rungs (plugs)?

Explore these special trail games further by putting in the red plugs 23, 17, 13, 9, 2. Again the numbers will no longer show, but they will be known. This time we have added a snake that curls its way down from the head at 23 to the tail at 2. If you land exactly on the head of the snake you must finish your move by sliding all the way to its tail. The snake can't get you if you land on any other part of it. The other parts just count as a normal number. Investigation of this variation is guided by the same sorts of questions as above and What happens if we started with a bigger board, say, 1 - 50 or 1 - 100? Teachers whose schools are members of Maths300 can continue interest in this investigation by using Maths300 Lesson 183, Snakes & Ladders. The software from this lesson can be used to create an on-screen model of this board and the computer can then play thousands of games and show the statistics associated with the results. The children can also use the software to design and test their own board with snakes and ladders in different places, and to test any size board up to 10 x 10. Any board the children design can also be printed and played on by hand.

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