A few intro calculus ideas to help explain why we study basic properties of sums

My older son is doing a some review this summer in the Integrated CME Mathematics III book this summer. The topic in the 2nd chapter of the book is sequences and series. I thought it would be fun to show him where (at least some of) this math leads. So tonight we talked about some basic ideas in calculus.

First I introduced the topic and reviewed some of the basic ideas of sequences and series:

Now we used the ideas from the first part to find the area under the curve y = x by approximating with rectangles:

To wrap up we extended the idea to find the are under the curve y = x^2 from x = 0 to x = 2. It was fun to see that the basic ideas seemed to makes sense to him.

I was really happy with how this project went. Putting these ideas together to calculate the area under a curve – even a simple curve – is a big step. It might be fun to try a few more examples like these before moving on to the next chapter.

Another great problem from Matt Enlow’s collection

Today we talked about another problem from the amazing list of problems that Matt Enlow’s published a few weeks ago:

Project Problem

This is our second project from that collection. The first is here:

Sharing a neat problem from Matt Enlow with kids

For today’s problem I introduced the problem and asked the boys for their initial thoughts. My older son noticed an important property about the sum of 9 and consecutive integers. He explained the property that the sum of 11 consecutive numbers would have and then my younger son explained the similar property that the sum of 9 consecutive numbers would have:

Next we had to see if there were any special properties that the sum of 10 consecutive integers would have.

Once we had that property, my younger son was able to explain how you could use them to find a number that would work (though not necessarily the smallest one):

At the end of the last video we though that 495 would satisfy the conditions of the problem. Here we checked that it did and wondered if it was the smallest.

Finally, we checked to see if 495 was indeed the smallest positive number with the properties in the problem.

My older son thought that 0 would have worked, but working it out he saw that it didn’t.

After that, we saw that 495 was indeed the smallest.

Definitely a great problem – it is nice to hear the boys explain some basic ideas in number theory. It is also a nice problem because kids – well, at least my kids – often struggle to see the difference between “find the smallest” and “find an example” and this problem helps show that “find the smallest” requires a bit more work.

Introduction to geometric series

Earlier in the week we looked at a problem that involved the series 1 + 2 + 2^2 + 2^3 + \ldots + 2^n. Each kid had a different and interesting approach to summing that series. For today’s project I wanted to review each of their approaches and then explore a few other simple geometric series to see if we could use the same ideas to add them up, too.

We started with a review of my older son’s approach to the original series. His approach was essentially mathematical induction, which was pretty cool since we’ve never talked about mathematical induction. Oddly, though, telling him that his approach had a name seems to have confused him a little bit, and it takes a little while for him to remember what he did. That little bit of confusion made me happy that we decided to revisit this topic today:

 

Next I had my younger son explain his approach to summing the series. His approach came from noticing a connection to binary numbers when he first saw the series. That connection to binary is a really clever way to think about this sum:

 

With that introduction and review of their prior ideas, we decided to see if we could apply those ideas to a new series: 1 + 3 + 3^2 + \ldots + 3^n. My older son tries his approach first – the connection isn’t obvious, but then we compare a few of the sums to the powers of 3 and make some progress.

 

Now my younger son takes a shot a the new series with his “trinary” ideas.

 

Finally, we wrap up the project by looking at the series 1 + 9 + 9^2 + \ldots + 9^n At this point they’ve seen each of the two prior sums in two different ways and they are able to see how the prior ideas apply to the new series. They even speculate about what a general formula would be!

 

So, a fun little exploration. I’m happy that the boys had a chance to review their prior solutions, especially since some of the ideas weren’t quite in the front of their mind anymore. Hopefully that review was helpful even if it wasn’t intended to be the main point of the project. By the end of the discussion today they seemed much more comfortable applying their ideas to a few new (though obviously similar) series. It is fun to show them how two ideas that seem pretty different help you work through these problems.