Why do this problem?
is useful for helping learners to identify and explain number patterns. It is one that can help the more reluctant see that numbers and mathematics can be absorbing. It also has strong design and art links.
You could start by showing the group some examples of various styles of calendars and encourage them to talk about the ways the months are laid out. Then show the class the picture of the calendar in the problem and invite them to talk in pairs about any number patterns they see in it. Can they offer any explanations? Next, show the second arrangement of the twenty eight squares. What number
patterns can they see now?
You could then give them all a piece of $1$ cm squared paper and suggest they explore different arrangements of the twenty eight squares with the numbers arranged in some logical order. In the same way, they can look for any patterns of numbers that appear. If they work in pairs they will be able to talk through their ideas with a partner.
At the end of the lesson the group should come together again to show their various designs and discuss the number patterns they have found. Are there number patterns that keep on appearing? Are there some that have only been found once? Can they suggest why this should be? Can they explain why the patterns occur in each case?
How else could you arrange the squares?
How will you write in the sequence of numbers?
Can you explain why this pattern of numbers occurs?
Learners could move on from the calendar situation and arrange other numbers of squares in an interesting way. They could be asked "What would be a good new number to try?" and "Why?".
The help needed will depend on whether the learner is finding the numbers or alternative arrangements of the squares difficult. You could suggest arranging the numbers in the original calendar rectangle starting in different corners before trying other arrangements of the squares.