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Adding All Nine
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!
Age 11 to 14
Challenge Level
Why do this problem?
In this problem students can work on both prime factorisation and area of rectangles.
Possible approach
Pose the first problem:
"Alison joins together nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm with no gaps and no overlaps, to form a rectangular quilt.
Can you find the dimensions of the finished quilt, and show how Alison fitted the squares together?"
Invite students to think on their own, then discuss in pairs how they might tackle the problem, and then share strategies in a class discussion.
Draw out useful ideas about factors with key questions such as:
"What must the area of the finished quilt be?"
"How do you know?"
"What can you say about rectangles with that area?"
"What are the possible dimensions of the quilt?"
Encourage students to test out different possible arrangements by sketching how the squares might fit into their proposed rectangle.
Once students have successfully found the first rectangle, challenge them with the second task:
"Alison wants to make a second quilt from ten squares with side lengths $3, 5, 6, 11, 17, 19, 22, 23, 24$ and $25$ cm. Can you find the dimensions of this quilt?"
"Alison joins together nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm with no gaps and no overlaps, to form a rectangular quilt.
Can you find the dimensions of the finished quilt, and show how Alison fitted the squares together?"
Invite students to think on their own, then discuss in pairs how they might tackle the problem, and then share strategies in a class discussion.
Draw out useful ideas about factors with key questions such as:
"What must the area of the finished quilt be?"
"How do you know?"
"What can you say about rectangles with that area?"
"What are the possible dimensions of the quilt?"
Encourage students to test out different possible arrangements by sketching how the squares might fit into their proposed rectangle.
Once students have successfully found the first rectangle, challenge them with the second task:
"Alison wants to make a second quilt from ten squares with side lengths $3, 5, 6, 11, 17, 19, 22, 23, 24$ and $25$ cm. Can you find the dimensions of this quilt?"
Possible support
You may wish to provide squared paper for students who want to make the nine squares and assemble them.
Possible extension
Dissecting squares and rectangles into smaller pieces is a very fruitful area for investigation. The problems Dissect and Building Gnomons might be of interest.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.