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# What Numbers Can We Make?

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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.

Display the image of the four bags (available as a PowerPoint slide).

*Alternatively, you could start with this image, with 7s, 10s, 13s and 16s.*

"Here are some bags containing lots of $1$s, $4$s, $7$s and $10$s. I'd like you to choose any three numbers from the bags and add them together."

Clarify that they can reuse numbers if they like, then collect some of their answers on the board.

"What do you notice?" *"They're all multiples of three".*

"Try to find three numbers from the bags that don't add up to a multiple of three. If you can't do it, see if you can come up with an explanation of why it's not possible."

Give students some time to work in pairs. While they are working, circulate and listen to their ideas. Then bring the class together to share insights.

If the class have not developed a useful representation of their own, you could introduce Charlie's and/or Alison's representation from the problem.

"Next, I'd like you to explore what happens if you select four numbers from the bag, or five, or six, or... In a while I'm going to pick a large number at random. Your challenge will be to tell me what is special about the total if I selected that many numbers from the set of bags."

Give students time to work in pairs on the task. Then bring the class together and choose a large number such as $99$ or $100$: "What would be special about the total if I added $99$ numbers chosen from the bags?" Make sure students explain their reasoning with reference to the structure of the problem rather than just by spotting a pattern.

What's the smallest number I can make?

What's the next smallest?

What is special about the numbers that make up each set of bags?

Begin by asking students to explore what happens when they add two, three, four... numbers chosen from a set of bags containing $2$s, $4$s, $6$s and $8$s. Can they explain their findings?

Take Three from Five and What Numbers Can We Make Now? are suitable follow-up problems.

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!

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?