Divide and rule
A) You haven't described the approaches that you have tried. Maths is a conversation and, through that interchange, understanding is born. However, this approach can have you biting your tongue, as you want to tell them the right answer. Meaningful investigation and the ensuing discussion also elicits understanding - I don't mean doing coursework to satisfy curriculum guidelines, but because you and your students would find it interesting and fun.
A good beginning, as pupils will have met division before, is to stimulate that conversation. Pupils will work on the exercise in pairs. Hand out 12-long sticks of interlocking cubes to each pair of pupils. Write "12 V 3", and ask them to show you what they think this means to them by creating a picture with their cubes for the answer.
Don't be afraid of incorrect solutions (for example one stick of 3 cubes and one of 9, which might indicate subtraction). Also, stress that they must keep their picture secret until they are told to display it. This is important as pupils often rearrange their cubes to look like their neighbours. The arrangements might be the following: Choose pairs of pupils who have either of these arrangements and get them to explain why they chose them. What number do they suggest should be put in for the answer to "12 V 3"? When guiding the discussion, ask them how this relates to multiplication tables.
When I first tried this with support staff, they said that the two arrangements represented "12 divided by 3" and "12 divided into 3" - a very subtle difference in the language.
Using matching colours, show them what the block length looks like when it is divided by three, as in the picture.
Hand out some more cubes and ask pupils to recreate the picture. Next, have some questions ready for them on the board and ask them to show the answer using a single stick of coloured blocks, sectioned as in the last demonstration. (Questions that you might put up are 8 V 2, 16 V 4, 15 V 5, 24 V 3. Ask them to make some up.) Next, hand out a pre-measured paper strip (30cm is ideal) to each pupil, with a ruler, a calculator, a pair of scissors and a pencil. Tell them that their task is to fold the paper into five equal pieces along the long side.
They can use anything that they think might help.
Folding something equally into five is really quite difficult. It might be that one of your pupils realises that they can measure the strip and then divide the length by five to work out where each fold should be. Discuss with them how they completed the task and why it is easier to measure first.
In the final part of the lesson, have the original problem on the board, with pictures of the two solutions. Ask them in what situations they might use division. For homework, can they each create a word question that would lead to the sum 12 V 3. Encourage them to ask their parents, and perhaps offer a reward for the most unusual, the most innovative and so forth.
The work from this lesson can be extended to questions that involve remainders, and leads nicely into fractions and what they mean.
Q) In what way is a round drain-hole cover better than a rectangle one or a square one?
A) This is down to the geometry. If you pick up the lid of a round drain-hole cover, you can't push it down the hole. But if you take the rectangle or square (an equilateral rectangle) cover and rotate the lid so that the narrow side on the cover faces the wider opening, this can be dropped inside the drain.
* On the TES maths forum, "dydx" introduced a really nice topic for probability, concerning Bob Geldof, KBE, and the Live8 concert. You can find this for the June 13 posting under the heading "An interesting probability question... with some relevance Sir Bob". It is about using texting on your phone and the chances of gaining a ticket. Worth a visit.
Go to www.tes.co.uksectionstaffroom and click on "Mathematics"