### Calendar Capers

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

# What Numbers Can We Make Now?

### Why do this problem?

This problem offers students the opportunity to consider the underlying structure behind multiples and remainders, as well as leading to some very nice generalisations and justifications.

### Possible approach

This problem should be attempted after working on What Numbers Can We Make?

Start by showing the interactivity from the problem, and clicking on 'New Numbers' several times:
"This interactivity can generate lots of different sets of bags like the set we worked on last lesson. Later on I'm going to generate a set of bags and ask you what is special about the total when I choose three, four, five, six... 99 or 100 numbers. To prepare a strategy for answering these questions, here are some bags to get you started."

Display this image (available as a PowerPoint). Then arrange the class in pairs or small groups, and allocate one or two sets of bags to each.

"In a while, you'll need to be able to explain to the rest of the class what happens when you add together three, four, five, six... 99, 100 numbers from your set of bags, and how you worked it out."

While groups are working, circulate and listen for any useful insights to bring out in the whole class discussion later. If anyone finishes their set of bags early, they can apply their strategy to someone else's set.

Bring the class together, and invite groups to share what they found. Then allow groups a few minutes to discuss a general strategy for answering the questions generated by the interactivity.

Finally, display the interactivity again. Generate new questions, and invite the groups to use their strategy to work out what happens for three, four, five, six, 99 or 100 numbers. Check their answers, and then repeat, giving each group a chance to have a go at answering a 99 or 100 question.

You could finish off by asking the final question from the problem:
"If the bags contained 3s, 7s, 11s and 15s, can you describe a quick way to check whether it is possible to choose 30 numbers that will add up to 412?"

### Key questions

If I choose 5 numbers that are each one more than a multiple of 5, what is special about their total? Why?

### Possible extension

There are a few related problems that students could work on next:

### Possible support

Begin by asking students to explore what happens when they add numbers chosen from a set of bags containing 2s, 4s, 6s and 8s.

They could then consider what happens when they add numbers chosen from a set of bags containing 1s, 11s, 21s and 31s.

Can they explain their findings?