Tag algebra

Introductory U-substitution

We’ve been doing a little bit of work in the integration chapter of our calculus book. Today the topic was u-substitution. I was expecting the topic to produce a great deal of confusion, but things went pretty smoothly. There will be plenty of time for confusing integration problems later, I guess, ha!

When my son got home from school tonight I had him try out three problems from the u-substitution section. The first two are straightforward and the third is a bit of a surprise.

Here’s the first one: \int (1 - 2y)^{13} dy

For the next problem I chose a trig integral: \int \cos(x)^4 \sin(x) dx

Finally, I chose a problem that isn’t obviously a u-substitution problem when you look at it: \int x \sqrt{x - 1} dx

Again, I was worried that we’d get off to a bit of a rocky start, but thinks went well today. Going to have him work through a bunch of problems tomorrow to get a little more u-substitution practice.


An exploration of Euler’s method

My older son is studying calculus and my younger son is studying algebra this year. ¬†I’d run across some problems on Euler’s method looking at old BC calculus exams, but we are still a long way from talking about Euler’s method in the calculus course. ¬†However, as my younger son begins to study lines, I thought it might be fun to do some visuals on slope fields and just touch on Euler’s method as a way of talking about slope.

Here’s how I introduced the topic:

Next I gave them a neat visual example -> wind speed and direction on the earth

Now I moved to Mathematica to talk about slope fields in general. The two specific slope fields we looked at here were (in calculus language) dy/dx = y and dy/dx = x.

Finally I had the boys choose their own fields and try to describe them ahead of time:

Definitely a fun project. The mathematical idea behind Euler’s method isn’t that hard and kids can understand the concept pretty easily. I really had a lot of fun exploring the ideas of slope fields with the boys this morning.

Playing around with Newton’s method

The next section in my son’s Calculus book is Newton’s method. I think it is a neat topic, but I chose to do a high level overview today because I wanted my younger son to join in. He’s learning algebra this year and I think (obviously) that the calculus details would be both over his head and not interesting to him.

We have looked at Newton’s method before in this project:

Exploring Newton’s method with kids

and I used the Mathematica code from A. Peter Young at UC Santa Cruz in this project, too. That code can be found here:

The page from A. Peter Young at U.C. Santa Cruz that gave me the Newton’s method code for Mathematica

So, here’s the high level overview I gave for Newton’s method and, more generally, the problem of finding roots of equations.

One fun thing that came up at the end of this video is that my older son noticed that Newton’s method might not always find the root you were hoping to find.

In the next part of the project we explored the idea my older son brought up at the end of the last video -> Are there cases where Newton’s method might not work as expected?

Next we looked at the function f(x) = x^2 - 4. We used Newton’s method with an initial guess of x = 3 to try to find approximations to the root x = 2.

Finally, we explored Newton’s method for complex numbers. This part was just for fun and to explore a few pretty pictures.

A beginning calculus example

My older son is studying introductory calculus this year and so far (roughly the first two weeks) has been mostly learning about limits and derivative rules.

I wanted to try out a simple max / min problem just so he could get a peek at an application of some of the ideas he’s studying. I picked the standard problem about building a fence.

Here’s the first part -> what is the largest area enclosed by a rectangular fence if the perimeter is 40 feet?

Here’s the second part of the problem – now the rectangular fence has one side up against an existing wall. What’s the the maximum area now?

What a kid learning calculus can look like

My older son is working through Stewart’s Calculus book this year (edition 3). Right now he is working on one of the introductory derivative sections. Today I picked to problems from that section and asked him to work through them live.

Here’s the first problem -> a problem about finding a line perpendicular to a parabola:

Here’s the second problem -> a problem about a piecewise defined function. I was using this problem to introduce the concept of “differentiable”. Although we had not discussed that definition ahead of this problem, he was able to reason through what it meant.

I think this is a really nice example of a kid working through a challenging problem.

What a kid learning geometry can look like

My younger son is working through a bit more of Art of Problem Solving’s Introduction to Geometry book this summer. Yesterday he came across a problem that have him a lot of trouble.

The problem asks you to prove that the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides.

Yesterday he worked through the solution in the book – today I wanted to talk through the problem with him. We started by introducing the problem and having my son talk through a few of the ideas that gave him trouble:

Next he talked through the first part of the solution that he learned from the book. We talked through a few steps of the algebra, but there were still a few things that weren’t clear to him.

Now we dove into some of the algebraic ideas that he was struggling with. One main point for him here, I think, was labeling the important unknowns in the problem.

For the last part, I wrote and he talked. I did this because I wanted him to be able to refer to some of our prior work. The nice thing here was that he was able to recognize the main algebraic connection that allowed him to finish the proof.

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.

An equation with roots of sqrt(5) + sqrt(7)

My older son is working thorugh the Integrated CME Project Mathematics III book this summer. Last week he came across a pretty interesting problem in the first chapter of the book.

That chapter is about polynomials and the question was to find a polynomial with integer coefficients having a root of \sqrt{5} + \sqrt{7}. The follow up to that question was to find a polynomial with integer coefficients having a root of 3 + \sqrt{5} + \sqrt{7}.

His original solution to the problem as actually terrific. His first thought was to guess that the solution would be a quadratic with second root \sqrt{5} - \sqrt{7}. That didn’t work but it gave him some new ideas and he found his way to the solution.

Following his solution, we talked about several different ways to solve the problem. Earlier this week we revisited the problem – I wanted to make sure the ideas hadn’t slipped out of his mind.

Here’s how he approached the first part:

Here’s the second part:

Finally, we went to Mathematica to check that the polynomials that he found do, indeed, have the correct numbers as roots.

I like this problem a lot. It is a great way for kids learning algebra to see polynomials in a slightly different light. They also learn that solutions with square roots are not automatically associated with quadratics!

A quick look at remainders

My older son was learning about the polynomial remainder theorem yesterday and then the Theorem of the Day twitter account tweeted about the theorem:

I took it as a sign that we should review remainders. My younger son doesn’t have a lot of experience with polynomials, so I wanted the main focus of today’s project to be on remainders when dividing integers. Here’s how we got started:

Next we looked at remainders in different bases to see what was the same and what was different:

Now we looked at the relationship between divisibility rules and remainders

Two wrap up, we looked at polynomials. Obviously this part is not meant to be comprehensive as my younger son isn’t that familiar with polynomials. What I was trying to do here was just give a simple overview of the remainder theorem for polynomials, and show that it wasn’t really that different than what we’d just looked at for numbers.

It was definitely a fun surprise to see the polynomial remainder theorem show up in two totally different places yesterday. Hopefully this review of remainders today was a nice exercise for the kids and helped my older son see a connection between division with integers and division with polynomials.

Playing with Polynomials

We’ll be doing a little bit of review work in the Integrated CME Project III book. Today my son came across an interesting problem about trying to (sort of) match two polynomials. He came up with a nice solution this morning and we talked about the problem when he got home from school today.

The problem goes like this:

Find a polynomial that agrees with x^3 - x at x = 1, 2, and 3, and has a value of 0 at x = 4.

Here’s my son talking through his solution:

After he finished his explanation, I showed him my solution to the problem:

To wrap up we went to Mathematical to look at both solutions and also so that I could show him a little surprise:

So, a nice start to this review project. It’ll be fun to work through the book over the summer.