This problem is one of my favourites! I first came across this problem about the sums of consecutive numbers in Marilyn Burns‘ book About Teaching Mathematics: A K-8 Resource (Sausalito, California: Math Solutions) and I thought it was an intriguing question. Burns presents the problem to be solved this way:
“Ask the students, in their groups, to find all the ways to write the numbers from one to twenty-five as the sum of consecutive numbers. (For younger children, finding the sums for the numbers from one to fifteen may be sufficient.) Tell them that some of the numbers are impossible; challenge them to see if they can find the pattern of those numbers. Direct them to search for other patterns as well.” p. 58
I have done this problem over the last few years with quite a few classes in a range of grades (usually somewhere in the grade 3-8 span). For the older grades I have extended the problem, asking students to write sums for numbers up to 35.) I feel that it is not a problem to hurry the students through, and I often take more than one day with the task. What I like most about the problem is that it is full of patterns, and that, in finding the patterns, one “unlocks” the problem. It becomes so much easier to find and predict the various sums when one notices the patterns that are produced.
I also like this problem because we tend to teach patterning to students isolated from problems, and I think that there is something quite powerful about a problem that uses patterns to solve it!
The task of finding sums of consecutive numbers provides a good opportunity for students to develop some problem solving strategies. Burns suggests a list of useful problem-solving strategies, similar to lists proposed by other math authors, naming the major strategies useful to untangle problems:
- look for a pattern
- construct a table (chart)
- make an organized list
- draw a picture
- use objects
- guess and check
- work backward
- write an equation
- solve a simpler (or similar) problem
- make a model
I have found that students doing this problem tend to make use of a number of the above strategies including these: look for a pattern, make an organized list, use objets, guess and check, and work backward. Any time that students are engaged in problem solving It is important to discuss with them both the specific strategies they use to solve the problem and why those strategies were (or were not) effective choices. Additionally, this problem is simple enough on its most basic level that everyone has the chance to delve in and come up with some of the consecutive number sums. At the same time it is quite sophisticated and offers a challenge to the bright students in the class.
Burns offers ideas for extensions for this activity, too. For instance, she suggests students try to predict how many ways thirty-six can be written as the sum of consecutive numbers. Going further, she asks if a prediction can be made for any particular number.
So, I encourage you to give this one a try with your students — and it is OK if YOU do not have it all figured out ahead of time. Let your students know that you are solving the problem along with them.