Play With Your Math: Summing Numbers from 1 to 100
In a recent session, we discussed how we might use this task in grades 4 to 8. In a problem-solving classroom, I would not hesitate to give it to any of these grades.
Consider Ontario, grade 5, for a minute. “Know thy curriculum” is our constant message.
I would attach this problem to this overall expectation.
We are working with whole numbers, and we are reading, representing and comparing them, although this isn’t the strongest curriculum link.
Luckily, the curriculum is made up of content and processes (the mathematics itself, and ways of thinking about and doing mathematics.
Here are the process expectations for grade 5:
We can comfortably say we are interested in how kids represent this problem, and how they think about it. Problems like this one, where we are exploring the structure of the number system, and searching for insight, are well-suited to discussions of strategies, reasoning, and ways of representing.
The hidden deep structure of the problem may be out of reach of kids. Here is what a young Gauss found:
I would not expect young kids to get all the way to the algebraic formula for sums, that Gauss found. I would expect them to find patterns like the one Gauss found. We are novices, not mathematicians, but we are learning to think, and to recognize patterns and get comfortable representing our thinking.
I have no student samples at this moment, but here are a few things adults did on this problem.
Nifty, simple, elegant, Gaussian. I would not, however, expect this to just “spring up” with just the question I gave.
Kids need scaffolding, and kids need tools to help them think about interesting mathematics.
Hundreds squares are the thinking tool needed here to find patterns, and to find a “way in” to this problem.
Start in the top left corner, and attach that number to the bottom right corner. Go across, continuing to make pairs. Notice anything?
A few other combinations are possible, like making hundreds, and then adding the 50 at the end.
Some kids will total the first row, and get 55, and then multiply by 10. Course correction is needed: “why is the total of the second row much bigger than the first row?”
You may want to consider a guided math group for kids who are struggling. A good teacher move would be to explicitly make connections by finding patterns in the hundreds chart with them. Also consider carefully who to deploy calculators to, if not everyone.
To avoid cognitive overload, you could chop it in half: total the numbers from 1 to 50.
With higher grades, you may want to issue the challenge/extension:
Can you total the numbers to 1000? How does knowing the total of 1 to 100 help?
For grades 7 and 8 we can guide towards the algebraic structure, even if the exact formula is out of reach. (Some kids will probably find it though- many grade 8s I have known would)
Teacher Moves To Consolidate the Problem
Consider whether your kids are ready to have the curtain pulled back and see the formula for sum of a series:
[n(n+1)] / 2
If they are, great. If not, sequence the sharing of the student work such that you can look at the diversity of ways of approaching the problem. Kids are inspired by seeing others’ thinking.
Pulling The Structure of This Problem Down to Primary Grades
For grade one, I might consider a story frame such as:
A caterpillar eats one leaf the first day, two leaves the second day, three leaves the third day. Continue the pattern up to 10 days. How many leaves has the caterpillar eaten?
(With apologies to Eric Carle)
Here are grade two expectations about number:
I wondered if we extended the caterpillar problem to 20, if that would work. Alternatively, we could create a new story frame that requires kids to sum from 1 to 20. I think giving a “slice” of the hundreds square would work here.
It would be very possible to give a picture of one dollar coins in the growing pattern: 1, 2, 3… and then having kids total the money.
Switching context from caterpillars to money does not change the structure of the problem, although seeing it as a visual pattern might change how kids “attack” it.
Summary
Problems involving sums of series of numbers are interesting. Consider how you would deploy them across all grades. This problem is not just Gauss’- it is accessible to all of us, depending on what context we choose (how we dress up or disguise the structure of the problem, or not), and what teacher moves we make. It’s also included in Alex Bellos new book of puzzles and problems, which you should buy.
