The Sum of the First n Square Numbers

 One of the things I like to do as a teacher is provide visual proofs of why certain formulas work. A nice one that I use a lot in high school shows how to find the sum of the first n natural numbers:

I saw a method recently that shows a lovely visual approach to finding the sum of the first n square numbers and decided to share this with the students who come to our after school Math Contest Club. I use linking cubes to help me illustrate this.

First, I show them the sum of the first n square numbers and ask them to imagine that the last term is a general n by n square (and not a 3 by 3 square).

Then I make six copes of this sum:

Then I group them together: 

Then I get ready to put them together in two symmetric groups:

Then I join two of the sums together as shown to make an incomplete cuboid with a base of n by (n + 1):

Then I join the third sum to each of the blocks to create another incomplete cuboid with a height of (n+1):

Then I join these two blocks together to create a cuboid with dimensions n, (n+1) and (2n+1):

A simple division now allows me to get the sum of the first n square numbers:

This is a formula that I learnt to prove by induction but it is only when I have modelled it concretely that I fully understand why it works (especially the '2n+1' part and the '÷6' part).

What Do You Notice with Google Trends?

For the first lesson of any semester, I ask students to complete this sentence: 

Math is the study of...

I get a variety of responses with the most common one being 'numbers' although 'logic' and 'solving problems' often get mentioned. I then tell the students that: 

Math is the study of patterns.

I give examples of how it is sometimes patterns in number, or patterns in algebra, or patterns in shapes. I then say that it is sometimes patterns in data. Today, I showed students this image. I explained that I had typed in a phrase to Google Trends which then produced this graph to show how often people had searched that term since 2004. I then asked them to discuss the graph in pairs and to think about what they noticed:

Moving around the room, I could hear lots of students noticing that there were big spikes in the data at regular intervals. A lot of students surmised, correctly, that this spike was an annual occurrence. Some noticed that the biggest spike occurred in 2020. Others wondered what the numbers on the y-axis represented (they represent relative interest over time). Others wondered at what time of year the spikes occur. It is difficult to judge the exact month on this graph but as I was able to access the original graph on the Google Trends site, I could tell them that the large spikes occurred each August. One student noticed another almost imperceptible spike each year and I confirmed that this was each January. 

I then gave the class a minute to think about what phrase could produce data with this distinctive pattern. Some wondered if it could be connected to holidays but most suggested that it might be something to do with schools. In fact, the phrase that I used was 'Back to school'!

One student made a suggestion though that I wanted to explore further. He thought it might be something to do with allergies. Realising that there might be seasonal spikes for this search term, I thought it would be good to generate the graph there and then, and these are the two graphs superimposed:

We noticed that the pattern in this data is similar in that there is an annual spike but it is not as pronounced as the 'back to school' data. We also noticed the spikes are earlier than the 'back to school' data and that there was another small rise later in the year. Looking more closely, I could confirm that these spikes were in May and September: "Pollen season!" one student exclaimed!

It was a nice way to get all students to value and share their own thoughts and I was glad that I charted one student's suggestion to show how it was the similar and how it was different to the original data.