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Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Digit Sum

What is the sum of all the digits in all the integers from one to one million?

Always a Multiple?

Age 11 to 14 Challenge Level:

Why do this problem?

This problem introduces students to a useful technique (representing numbers algebraically according to their place value) for solving a wide variety of related problems.


Possible approach

This printable worksheet may be useful: Always a Multiple.

"Think of a two-digit number and write it down."
"Reverse the digits and add your answer to your original number."
"What answers did you get?"
Collect a few students' answers together and write them up on the board.

"Does anyone notice anything interesting?" "Multiples of 11."
"Does anyone have an answer that isn't a multiple of 11?"
"With your partner, without trying all possible two-digit numbers, try to find a convincing explanation why it will always work."

Give students some time to explore the problem. While they are working, circulate and listen for useful insights. Then bring the class together and share ideas.

If Alison's and Charlie's explanations from the video aren't offered, demonstrate them or show the video.

Students could also be invited to work backwards - for example, what two-digit numbers can be reversed and added together to give 154 (a multiple of 11)?

"These methods can be used for lots of similar number tricks. Here are a few more. Work with your partner to figure out what each trick does, and then adapt the methods to explain why the tricks work."

If appropriate, bring the class together to share explanations for why each trick works, or ask them to present their clear explanations on a poster to display.

Finally, challenge students to devise their own number tricks using similar structures, and to test them out on each other.

You may wish to show this online 'mind reader' activity to see if your students can explain how it works. Perhaps you could show it at the start of the lesson (without an explanation) and then again at the end of the lesson once they have the tools to deconstruct it.


Possible extension

These problems can all be solved using similar techniques:

Special Numbers
Think of Two Numbers
Legs Eleven
Puzzling Place Value

Possible support

Diagonal Sums provokes a need to use place value to solve the problem, and could be a good foundation for this activity.