Always a Multiple?
Why do this problem?
This problem uses a number trick which will intrigue students, and then uses this curiosity as a context to introduce a useful algebraic technique which can be applied to a wide variety of related problems. By switching between a numerical and an algebraic representation, students can gain a clearer understanding of our place value system.
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 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 use different representations 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 the Number Jumbler 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 support
Diagonal Sums provokes a need to use place value to solve the problem, and could be a good foundation for this activity.
Possible extension
These problems can all be solved using similar techniques:
Special Numbers
Think of Two Numbers
Legs Eleven
Puzzling Place Value
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