## 1. Introduce & play the game
## 2. Explore the game with counters
## 3. Challenge
## 4. Make & test hypotheses
Allow the children time to explore. Some may not need the materials. As each pair finds a solution, ask them:
This encourages those who used counters to make use of tables facts, and those who used tables facts to make use of counters. Suggest that there might be another solution.
Eventually, the class will develop this list of possible class sizes: {6, 26, 46, 66, ...}
Students usually readily make the hypothesis that the 'skip' in the pattern is the product of the two group sizes in the question, ie: the product of 4 and 5 in the challenge above is 20.
Encourage the students to question and explore whether this would happen with other group sizes, for example: - groups of 3 with 2 remainders and groups of 5 with 4 remainders
- groups of 6 with 3 remainders and groups of 7 with 5 remainders
As this exploration continues it is likely that students will find every case they try does fit the hypothesis. This is because students seem to naturally try group sizes which have no common factor. So, if it does turn out that all cases work, encourage students to test their hypothesis on a case like: - groups of 6 with 5 remainders and groups of 3 with 2 remainders.
The lists of relevant class sizes for the problem in step 5 is: {5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, ...}and the list of common class sizes for these two sets is:{5, 11, 17, 23, 29, ...}
The 'skip' in this pattern is 6, and 6 is NOT the product of 6 and 3, so the earlier hypothesis has failed. Now the hunt begins for a new hypothesis to cover the case of factors in common between the group sizes, and perhaps an hypothesis to cover all cases.
Again allow sufficient time for exploration and testing. There is no reason why this investigation needs to be confined to one timetabled class lesson. - Investigation Guide 1 and Investigation Guide 2 will assist this exploration.
- You might also like to use the Extra Investigation Guides.
## ConclusionWhen students have extensively investigated this problem, it is appropriate that they should have opportunity to publish their results. This could be in the form of a class wall display, or a report from each student which could count towards an assessment portfolio. If the investigation guides have been used, these will provide a sound background for such a report.
A sample student report is provided. It links to the Extra Investigation Guide used by the student's teacher. ## ExtensionWhen students have developed an hypothesis to cover predicting the 'skip' number in the pattern if the group sizes are known, raise the question of being able to predict thefirst number in the sequence of 'class
sizes' which would work for BOTH sets of group sizes and
remainders. For example, the Challenge problem was:
- when we made groups of 4 there were 2 children remaining, and
- when we made groups of 5 there was 1 child remaining.
The simplest positive answer to this question is a class size of 6. Is there a way of predicting that 6 from the numbers in the problem?
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