Fill The Board Years 1 - 6

Summary Fill The Board is simple to set up, offers quality learning at whichever year level it is used and, almost as a natural consequence, the children think of further investigations they would like to try. It simply involves rolling dice and plugging yellow/blue plugs into the red board until it is full. Scroll down to see all the Fill The Board information, or you can print this page and read it in the sun, or you can 'jump' to any section using these links: Procedure Teacher Responses Coolgardie CAPS School, Years K & 1 Herdsman's Cove Primary School, Years 1 & 2 Goodwood Primary School, Year 3/4 Jo Vertigan Margate Primary School, Year 4 Anne Briggs Smithton Primary School, Year 6 Toni Popowski Children's Reports Materials One Poly Plug set per pair One dice per pair Recording paper

You are invited to Fill The Board and submit any exciting responses from your class. NRICH Contribution The well known University of Cambridge site NRICH has developed a variety of challenges based on Fill The Board, and each has its own software application that runs in Flash. They call their activity All Change and we highly recommend exploring it. The software will run on your electronic white board.

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

• Visit the Home Page for more Background.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

Acknowledgement

This activity developed from a thought recorded on page 20, Prime Number, Volume 17, No. 1, March 2002, Mathematical Association of Victoria. It has been trialed by teachers of several grade levels in several schools and thanks is offered to them all. Particular responses from the feedback are listed below.

Perhaps most importantly the investigation provides a simple context in which children are willing to make and test hypotheses and justify, explain and publish their results as a mathematician would.

Extensions

1. How many experiments do we need to do to be nearly certain that the most likely number of rolls is...?
2. Once we decide the most likely number of rolls can this information help us predict the result for filling 2, 3, 4... boards?
3. Is there a pattern in the most likely number of rolls for 1, 2, 3, 4, 5, ... boards?
4. What happens if we use two dice to fill the board?

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Teacher Responses

• The kids enjoyed the activity. The suggested reasons were tactile, guesstimating/ checking aspect, co-operative rather than competitive.
• It was applicable to a large range of kids. Many teachers reported that every child could be involved and wanted to talk with each other and the teacher about their results
• Children quickly saw there were limits to the problem. The least number of rolls had to be 5 (four sixes and another roll) and the most had to be 25 (rolling 25 ones in a row).
• Some children, especially younger ones guessed wildly the first time, but their recording showed they learned from repeated trials and their predictions become more accurate.
• Data from the first set of trials began informed and reflective class discussion about, for example, the best statistic to use to indicate what was most likely, eg:
  • 6 because that turned up most.
  • 9 because we had results of 5, 6, 7, 8, 9, 10 , 11, 12, 13 and 9 is in the middle.
  • 7·8 because that is the average.
  How many trials was enough to be confident about the likely chosen number?
• One class examined the possible rolls to fill the board under the headings of Impossible, Improbable and Possible.
• In some classes the students suggested other Chance investigations as a result of this experience.
• Young kids needed the experience just to develop skills like tallying. For many, just keeping track of the number of rolls was a challenge, but teachers found that as the game was repeated children became more efficient and learnt from each other. One teacher reported that it was a particular child who suggested the four vertical strokes and a fifth diagonal slash approach and the others picked up on it.
• Older students, especially those who were taught using a Working Mathematically approach, where soon posing their own problems, eg: What if we used more boards? or What if we used more dice?
• One teacher took the tally/total approach a step further by asking the children to write each total on a card as they finished a trial. The cards were all the same size and when collected and organised they became a bar graph.

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Coolgardie Christian Aboriginal Parent-directed School (CAPS), Year K/1

The children were initially told that they could plug in anywhere on their turn. However, after one round of the game, one group had played so that they organised their plugs into yellow and blue sets. The moment was celebrated and guess what ... next time round everyone thought it was a good idea.

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Herdsman's Cove Primary School, Year 1/2

The main 'aha' moment was our own! We realised how little the children knew about the concept of chance!

• learning how to tally independently
• remembering that we were recording the number of throws, not the number on the dice
• sharing the jobs - dice roller, plug operator and recorder
• finding the interesting part of the activity - comparing their prediction to the number of throws it took

During the game children explain orally why they chose their predictions:

• 8 was the right guess last time.
• Because 7 is a good number.
• Because it's getting lower and lower.
• Predict 8 because that is in between the others.
• I chose a different one because it keeps on changing.
• Whichever number we choose we will get the other one.
• I chose it because I wanted a different number.
• We could put 6 'cause that's what we have done twice before.
• We probably won't keep rolling 5s.
• It's just luck from rolling the dice.

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Fill The Board Round Guess Rolls Tally No. of Rolls 1       2       3       4       5       6       7       8       9       10

Goodwood Primary School, Grade 3/4 The children were mostly enthusiastic about engaging in the activity. The one hold out - who is always difficult to engage - chose to work on the computer while the rest of us got stuck into it. He stayed aloof for the first session but then got stuck into it with the best of them when he saw that it was non-threatening. Another really great aspect was that the only person left unpartnered was a boy he normally fights with but he accepted the pairing in order to take part. I worked with them initially to set the ground rules but after that I left them to it with minimal interaction from me. They are still working together on the activity and there is some positive flow-on happening now in the rest of the day in and out of class.

Fill The Board is a good opportunity to further develop cooperative skills in that partners can be opposite gender, another grade, someone you don't work with often and so on.

Directions needed to be explicit, eg:

You'll be using both boards. Remove the red plugs and store them in the bag. Keep the red frame in front of you. Remove the blue and yellow plugs and store their frame in the bag. Line the plugs up above your red mat so you can both reach them.

Decide who will roll the dice first, the other partner can then choose whether they will be blue or yellow plugs.

Some were then fascinated to discover that each time they filled a board they both ended up with roughly half the plugs which will lead on to further activities in chance and data. Questions about how many times you might expect to turn up a 6 when you roll are still to be investigated.

The question about how many times you can expect to roll a 6 developed out of the indignation of some students with their colleagues who were particularly adept at manipulating the dice. This led to a discussion about 'a fair roll of the dice' and to a session on finding a method that all would use and could trust.

The end result was to use a container (we're using empty film canisters), put your hand over the end and shake it up and down and then turn the container over on the desk before you remove the container. Since the container is opaque you don't know what you've rolled until you lift the container off. (This also stopped the dice from ending up all over the room which was a further bonus).

Flowing on from the above is a request from some children to invent their own dice rolling machine which they will do as a free activity.

I came up with the form when some of the children had difficulty remembering or working out how many rolls it took to fill the board. We're using tally marks (four vertical strokes and a fifth diagonal slash) to keep track of the rolls but when we start collecting information about how many times a number is rolled we'll write down the number on the dice instead. We collected the first lot of class information as shown here. It took only one round for the children to figure out what was a reasonable guess. There was excited conversation as they pointed out why 30 rolls would be too much and why 5 would probably not be enough.

Fill The Board   Guess 1 Rolls Guess 2 Rolls T & B 30 9 8 8 R & BJ 5 5 9 8 S & J 7 8 8 8 C & B 8 8 10 7 J & A 25 9 8 _ P & J 7 7 6 6 J & D _ _ 9 7

We also noted from the data 'what accurate guessers' some children appeared to be which contributed to the conversation about 'a fair roll of the dice' and manipulating data or 'making things come out the way you want them to'.

We've just completed another 10 rounds using 'fair go' conditions and are now starting work with the data collected. The children have ordered their 10 rolls from lowest to highest and are finding out what is their most frequent number (mode). We have a few places to explore from here including collating all data from 100 rounds (10 pairs of students) and comparing it to the pairs data. We have yet to look at mean and median but will look do so in the next sessions.

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Margate Primary School, Year 4

Our class worked in pairs to play Fill The Board. We made predictions each time before we played the game. These are our comments: They were mostly 7. No one chose below 5. People chose different numbers. There were no numbers over 15. The first time some people chose high numbers and then after that no one went over 10. Sometimes they chose the same number, sometimes they changed.

We also worked out the average of our predictions. It was 8·3 The average of our actual dice throws was 7·4. The highest possible number of throws would be 25. The lowest number of throws would be 5.

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Smithton Primary School, Year 6

Teacher Observations Every child on task Engaged all children ---> wide range in class Great discussions Feelings of success Co-operative activity Good way to consolidate understanding in Chance & Data Good connections made: I wouldn't predict 4 because that's impossible. You could only roll four sixes and that's only 24. There's 25 gaps! One boy checking another boy's work: You can't have that 'cause there are seven plugs in a row and the highest number on a dice is six.

Areas Covered Counting Recording Prediction Pattern Different learning styles - kinaesthetic Concrete ---> symbol Average Mode, median Rules Recording data Interpreting data Graphing (including IT) Sample size, population Co-operative learning

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Children's Reports

Year 1/2 Brendan shows us how his class made a graph of the class results by using squares of card. He also explains his reasoning for choosing 8 as the most likely number of rolls.	Year 3 T.G. and D.S. explore the results for 1, 2, 3, 4 & 6 dice. It is interesting that they managed to make the prediction aspect into a win/lose game for themselves.
	Year 6 Jessica's report based on the Smithton investigation sheet above. She also gives some feedback about the task. ... Return to link list above.

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