Division BoxesTask 210 ... Years 4  10SummaryThis fascinating puzzle involves divisibility tests. However, it's not just about being presented with a number and asked if it is divisibility by X. The rules on the card require 3 divisibility tests each time. For a number to 'pass the test' all three tests must be true. There are many solutions, so success at the first level  3 digit numbers  is easy. But what happens if we use 4 digit numbers and 4 corresponding divisibility tests? Or 5, or 6, or...? Now the investigation is opening up. Perhaps the most wonderful thing about the investigation is that there actually is a solution for 10 digit numbers! 
Materials
Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
This task starts simply but through the challenge of finding all the solutions for a given number of boxes it turns into an extended investigation. The main content knowledge is tests for divisibility. However, the investigation involves many aspects of the investigative process such as using different strategies and particular tools such as tree diagrams and the (possible) use of a computer simulation. With just one box, the number of solutions is 9. The digit 0 being the exception.For two boxes, the first digit has to be divisible by 1 and the 2digit number divisible by 2. This produces the following 41 solutions  and some noticeable patterns to assist the search.
Including the third box means the first digit must be divisible by 1, the first two digits by 2 and the whole three digit number by 3.Students may well know one test for divisibility by 3 is that the sum of the digits must add to a multiple of 3. As the problem proceeds, knowledge of divisibility tests for other numbers will be useful. They are listed below, but this need to know could be a good reason to visit the Web. Entering 'divisibility tests' into a search engine produces many useful results. For the three box case, one strategy might be to start with the 41 solutions above and add a third digit so that the whole 3digit number is divisible by '3'. For example:
A tree diagram is a systematic visual representation of the strategy of testing every possible combination, and may be more appropriate for learners with a visual intelligence preference. Continue, with 2 at the top of the tree:
A major benefit for students of this systematic search is firstly seeing a viable problem solving method, but also realising the need to be very systematic and rigorous in checking each possibility. Divisibility Tests

Whole Class InvestigationTasks 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. 
Begin the class by asking pairs to tear one piece of paper in half. Then each person tears their half into five pieces. The ten pieces are then numbered from 1  10. You will also need 10 large cards numbered the same so you can start the investigation with a fish bowl demonstration on the floor or at a central table. Use the large cards to make 1, 2, 3 4, ... digit numbers as appropriate. Perhaps starting by making 40 and asking Is this number divisible by 10? introduce a discussion about the meaning of divisibility and what the students already know about it. Record key points on the board and set a few exercises for students to create and record numbers divisible by 3. Return to the fish bowl and introduce the challenge on the card. Write the rules on the board and invite students to record any solutions underneath them. After a while ask:
For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 146, Division Boxes, which includes companion software that helps to explore the problem. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Division Boxes task is an integral part of:
The Division Boxes lesson is not included in any kit, but can be used to enrich Number & Computation kits in Years 5 & 6 and Years 9 & 10. 