“The most complicated skill is to be simple.” Anonymous
The Simple Start
One of the joys of exploring mathematics is coming home to elementary ideas like addition and discovering a whole new world is waiting to be explored. Experiences like this has been a frequent occurrence as I have tried to read every blog entry from the incredible Marilyn Burns’ blog.
The post that made me stop reading and decide to play for quite some time is here, in which Marilyn follows up on a post about the number eleven. The genesis of the post was a comment from one of her readers, in which she noticed the sum of consecutive numbers five and six equal eleven. In addition, the difference of the squares is also equal to eleven, i.e.,
The question and Marilyn’s inquire if the difference of the squares of two consecutive numbers is always equal to the sum of the consecutive numbers. A process that is easy to check via an algebraic approach. Let n be any natural number. Then m, is the next natural number satisfying the relationship m=n+1. The sum of two consecutive numbers always yields an odd number which is easy to check since, m+n=(n+1)+n=2n+1.
To check the difference of these two consecutive numbers, m and n, consider
As shown in Marilyn’s entry, the difference of the two consecutive numbers is the odd number corresponding to the sum of the two consecutive numbers. This was an excellent exploration on Marilyn’s blog, and I thoroughly enjoyed the journey, especially her visual approach creating a colorful visual model.
The next step in this journey, then led me to the idea of considering a parameter, where the whole number value is arbitrary and what information might we uncover in this format. Again, let n be from the set of natural numbers, and let m be defined in the same manner, with a parameter value d (with d coming from the set of natural numbers as well) such that m=n+d. The sum is
where d=1 in our previous experience. Now let’s consider the difference of squares
Notice that the difference of these squares yields the exact case as Marilyn discussed in her blog, when d=1. Comparing the two results the sum of the two numbers is twice the original number plus the value of the parameter (i.e. how many integers away the second number is from the first), and in the difference of their squares it the same as the sum times the value of the product. So the ratio of the difference of squares and the sum of the two is equal to the value of the parameter, d, i.e.
Interestingly the ratio of the difference of their squares and the sum of the numbers is a constant once the parameter value is chosen, and it will also be a whole number since our field we pull n and d are whole numbers…that’s an unexpected and very cool conclusion.
For d=2, the purple line indicates the sum of the two numbers, the green line indicates the difference of squares, and the blue line indicates the ratio, which is 2 since d=2 in this case. When d=1, the green line and the dotted purple line are colinear.
The Deep End
This led me to a classic investigation I lead my Integrated Math 3 students through, which starts off with the question: What numbers can be written as a sum of two consecutive numbers?
Students usually have trouble getting started, understanding that two natural numbers, m and n, are said to be consecutive if they satisfy the relationship m=n+1. When students realize that adding one more to the previous number creates the condition for consecutive numbers, then addition of the two to yield the sum as 2n+1 always results in an odd number is met with grins and celebrations.
The next step in this is to pose the problem that leads to a difficult discovery and many opportunities for struggle. The question is: Given the input of a starting natural number and the number of consecutive integers, what is the resulting sum without having to perform the addition?
The question may take an example, or two, to catch on:
- Suppose we start at the number 10 and we had two consecutive numbers, the sum would be 10+11=21
- Suppose we start at 172 and we had four consecutive numbers then the sum would be 172+173+174+175=694.
- Suppose we start at 19293 and we had seven consecutive numbers then the sum would be 19293+19294+19295+19296+19297+19298+19299=135072.
So the question is asking for a function with the two inputs of starting location and number of consecutive terms, and outputs the sum as a result. This investigation was a ton of fun and was inspired by the book
I’ll leave the exploration as a fun journey to explore, but I love how the connection plays out in both exploring and the generalizing that arises in understanding the structure of the number system.
I love how simple ideas of addition and consecutive numbers can lead to such deep explorations of mathematics.
Here are some possible ideas for extending these activities:
- Expand the domain from natural numbers to all integer values and find what processes hold true and what other things generalizes.
- Explore the visual representation of this process, see what visualizations uncover.
- Ask a different question in these regards, but I’m not sure what that new question might be.
What curiosities do you have?
Let’s continue this conversation, like on Marilyn’s wonderful blog, a comment could start an entire new exploration.