Why do this problem?
helps to extend learners' familiarity with factors as well as providing interesting work on multiplication tables. They can make hypotheses, and test them in a simple and easily understood environment. It provides an opportunity to introduce the idea of a letter standing for "any number".
It can also add to their ideas of pattern and design.
If learners have not done many table patterns on hundred squares you could start by doing some of these. It is important that they do not see these as a series of straight lines, so it is advisable to get them to do the three times and four times tables before they embark on the two and five times. They should be asked why the tables of twos and fives make straight lines. Of course the word
"factor" might pop up here and there.
If plenty of work has already been done on in this area ask what makes straight vertical lines and why. Next give out unnumbered
$10$ grids (on this sheet or on squared paper) and get different learners to do
different numbers so all are done, asking them to predict what sort of pattern they think they will get. It is advisable to suggest putting crosses on the appropriate squares as colouring them in can take a long time!
After this introduce the different grids. They can be found on this sheet. Learners could work in pairs to share each other's findings and also discuss their ideas with a partner.
There are two photocopiable "problem sheets" for learners to identify the tables and grids. These have more on them than the ones given in the problem. This sheet (A) has clearly defined grids and so is much easier than this one (B) which is a real challenge. In sheet B the sizes of the grids are more ambiguous and which may, therefore, have more than one answer. Squared paper is useful in investigating these.
At the end of the lesson learners should be asked to say what makes the different patterns. The factors of the grid number should be discussed and also the idea of one more than and one less than. It is possible here to introduce or practise using a letter to stand for "any number".
You could also discuss what makes an interesting all-over pattern.
What kind of pattern do you think you are going to get?
Which tables will make vertical lines?
Why do you think this is?
Which tables will make diagonal lines?
Why do you think this is?
What kind of grid do you need to make a pattern of checks?
What do you think makes an interesting all-over pattern?
If they finish early learners could make some similar "puzzles" on different grids for others to do.
Those who are really confident with the grids up to ten could predict what they will get and then explore eleven and twelve grids (which can be drawn on squared paper). They could also be expected to express their findings in algebraic terms where possible.
Suggest concentrating on the patterns on hundred squares and $10$ grids.