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Summing Consecutive Numbers
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?
Always the Same
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
Fibs
The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?
Age 11 to 14
Challenge Level
Why do this problem?
Many students are accustomed to using number patterns in order to generalise. This problem offers an alternative approach, challenging students to consider multiple ways of looking at the structure of the problem. The powerful insights from these multiple approaches can help us to derive general formulae, and can lead to students' appreciation of the equivalence of different algebraic expressions.Possible approach
Display this
image of a full size $5$ box of chocolates, and ask students to
work out how many chocolates there are, without speaking or writing
anything down. Compare solutions and share approaches.
Mention that mathematicians like to find efficient methods
which can be used not only for simple cases but also when the
numbers involved are very large. Explain that the pictures of
Penny's, Tom's and Matthew's partially-eaten chocolates could be
used by a mathematician as a starting point for finding an
efficient method for counting the total number of chocolates.
Click here for
the three pictures.
Hand out these chocolate
box templates and ask students to show how the images of the
partially-eaten chocolates can be used to calculate the
total.
Ask students to report back, explaining the methods which have
emerged.
Then ask students to use all three methods, along with any
methods they devised for themselves, to work out the number of
chocolates in a size $10$ box, and verify that all methods
agree.
Challenge students to express each method for finding the number of chocolates in any size of box, perhaps introducing some algebra and the idea of a size $n$ box if appropriate.
Bring the class together to share findings. Compare the different "formulae" which have emerged, and ask students to explain why they are equivalent.
Key questions
How does each image help you to count the total number of
chocolates quickly?
Can you demonstrate the equivalence of different algebraic
expressions?
Possible extension
The problems Summing Squares and Picture Story lead to formulae for some intriguing sequences through analysis of the structure of the contexts.Possible support
The problem Seven Squares gives lots of simple contexts where formulae emerge by looking at structure rather than number sequences.
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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.