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You will need a piece of squared paper for this activity. If you don't have any, you can get a sheet here.
Write the ones digits of the numbers in the two times table from $1 \times 2$ up to $10 \times 2$ in a line. Leave some room at the top of the paper, and some space to the left and right, a bit like this:
This problem involves learners noticing patterns, and making and proving conjectures. It is, incidentally, also an engaging way of revising multiplication tables! In addition, it is also very useful for encouraging learners to predict what they think they will find out.
You could introduce the investigation by inviting the class to help you record the ones digits of the two times table on the board. You may like to pause with just that row of digits written up, and invite learners to comment on anything they notice. Can they explain the pattern of digits?
Complete the ones digits of the three times table in a similar way, directly underneath the two times table, so that learners get a feel for the task. Encourage them to work in pairs from there so that they are able to talk through their ideas with a partner. Have plenty of squared and plain paper available. (Squared paper can be found here.)
Once they have completed the ones digits of the four and five times tables, draw everyone together again and ask whether they notice any patterns. Invite them to talk to a partner about their noticings, and then encourage everyone to share them with the whole group. Say that mathematicians aren't satisfied with noticing patterns, although that is a great start, we want to know why the patterns occur. Set the class off again, inviting them to choose a pattern they have noticed and try to explain why it occurs.
As they work in pairs, listen out for those children who are drawing on what they know about the number system to explain a noticing. You could warn a few pairs that you'd like them to share their explanations with the whole class in the final plenary.
If you are able, you could extend this task further to look at the zero, one, six, seven, eight, nine and tens times tables, as is suggested.
Some learners will find it helpful to use a standard $10 \times 10$ 'table-square' to help with the tables. The resulting ones-numbers can then be transferred to a plain sheet of squared paper. This ready-made sheet might help.
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?