Tables for 25

Task 235 ... Years 4 - 8

Summary

The story shell is familiar to students - the teacher says that he wants them to sit in certain size groups. But then he adds a condition - at least two boys in each group. Now we have the interesting problem of working out all the possible table arrangements for a class with a given number of boys and a given total number of students. A problem is always more interesting if it has more than one answer, and this one does. To find them all the problem has to be broken into parts and the strategy of try every possible case has to be applied. Then comes the What happens if...? questions and the possibility for each pair to investigate their own 'Tables For' problem.

Materials

6 'tables'
30 'students' - 15 in each of two colours

Content

basic arithmetic
complementary addition
multiples, factors and primes
problem solving strategies

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Mr. Edwards really did invent this task. His initial problem concerned his class of 25 with 14 boys. The missing table in the picture must have three boys and two girls and the complete solution could be recorded as (5, 3, 2, 2, 2) because only the boys need to be recorded. The number of girls is the complementary addition to 5 in each group and to 25 for the class.

Students usually realise early on that the 2 boys who must be at every table should be placed first. The other boys are can be placed anywhere after that. The reasoning goes something like this:

25 children and groups of 5 means five tables.
Ten boys must be used first and there will only be four left to add to the tables.
Once that is done the girls will go to empty places.
Then the problem must work out because only 25 seats have been allowed for at the tables.

So the key to the problem of finding all the ways to arrange the students is the number of ways the four extra boys can be distributed around the tables. Break the problem into parts:

All 4 to one table - no good because that would make 6 on that table, not 5.
3 to one table and 1 to another - which gives the solution (5, 3, 2, 2, 2)
2 to each of two tables - which gives the solution (4, 4, 2, 2, 2)
2 to one table and 1 to each of two other tables - which gives the solution (4, 3, 3, 2, 2)
1 to each of four tables - which gives the solution (3, 3, 3, 3, 2)

Now we have a way of reasoning that can be applied to any similar problem. And there are heaps of those because the:

number of students (and therefore the possible equal group sizes)
boy/girl ratio

can be varied. And the problem can be male-centric or female-centric.

If we consider a class size of 24 for example, the class could sit in groups of 2, 3, 4, 5, 6, 8. Then the number of girls in the class could be ... and the 'at least' statement could be ... well, for each group size what's the smallest it could be? ... the largest? ... how many arrangements for the highest and the lowest and how many for each of the numbers between. Now that's a serious investigation.

But suppose the class is 23 students. Then the class could sit in groups of ... 1 or 23. Then the number of boys in the class could be... well it doesn't matter much, because there really isn't much that can be asked. So what's different about the numbers 24 and 23?

Which other class sizes produce uninteresting results.
For the more interesting class sizes, which ones have only one way (apart from 1 and the number itself) to arrange groups?
Which have 2 or 3 or 4 or... ways to make groups?

And now we are experiencing multiples, factors and primes in a meaningful context.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

It seems appropriate to begin the lesson by using your own class, if it is the right number. That number can be any non-prime number. If your class is a prime number select the appropriate number of non-participating students to be your monitors. They can count for you and record details on the whiteboard. Once the students have the idea of how the problem works, introduce your friend Mr. Edwards who tried it in his class of 25 with 14 boys. What size groups would he ask for?

Give each pair Poly Plug, or other material, to represent the girls and boys, then ask them to solve Mr. Edwards problem. Torn paper is the easiest way to represent the tables. Continue with further What happens if...? questions as appropriate and consider the assessment potential of having every student select their own 'Tables For' problem, analysing it and presenting a report.

Every whole class investigation is included in the curriculum primarily to model aspects of the Working Mathematically process. Review the lesson by noting the students' interest in the problem as satisfying the first criteria of the process - First give me an interesting problem. Then review against the other elements of the process and in this case highlight the strategies of breaking the problem into parts and trying every possible case. This is dipping into the strategy toolbox. Ask also what tools have been selected from the skill toolbox.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 178, Tables For 25, which also includes a sample Assessment Project.

Visit Tables For 25 on Poly Plug & Tasks.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
Tables For 25 is not in any MWA kit. However it can be used to enrich the Number & Computation kits at Years 3/4 and Years 7/8.

Follow this link to Task Centre Home page.