Dividing Fractions

Ran across an basic fraction division problem on an old AMC 8 today.

Problem #12 from the 1989 AMC 8 (AJHME).

The problem is easy to state:

Evaluate \frac{ 1 - 1/3 }{1 - 1/2}

When my son dove in to the problem he immediately remembered the “flip the denominator” rule. That rule makes pretty much takes care of solving the problem. BUT, after he got the answer I asked him if he knew why the rule was true. He said that he didn’t know.

So, we spent the last couple of minutes in the video talking about why it was true:

When I first introduced fraction division to him, we looked at flipping the denominator and also a second approach involving snap cubes. I guess the “flipping the denominator” rule is a lot easier to remember, but I still remember the fun we had making this video 🙂

Two fraction problems – what a difference!

My younger son was trying out some old AMC 8 problems this morning, and ran across this problem from the 1985 AMC 8 (or the AJHME as it was called then):

What number is the middle of 1/5 and 1/3?

He wanted to talk about this question for our math movie today. Great. He worked through the question fairly quickly and mentioned that one of the reasons the solution wasn’t that hard was that both 3 and 5 were odd.

That observation led me to ask what fraction was in between 1/3 and 1/2. It turned out that this question was much more challenging!

here is the work split into two videos:

What a difference a seemingly small change makes!

Fawn Nguyen’s fraction puzzle

Saw this old blog post from Fawn Nguyen making the rounds again this week:

Drawing Rectangles instead of Writing Equations

and decided to give the problem a try with the kids this morning.

Here’s the problem: In a town, 3/7 of the men are married to 2/3 of the women. What fraction of the people in the town are married?

I did the problem with pairs of snap cubes instead of marriage. Here’s now it went:

/

After they finished their solution, I showed them Fawn’s solution. My younger son was a little confused about adding fractions versus what’s going on in this problem. Hopefully that confusion was straightened out by the end of the video.

/

So, another nice problem from Fawn. It is fun to be able to talk through a non-standard fraction problem like this one with the boys.

Talking through Christopher Long’s neat probability problem with kids part 2

Yesterday we talked a bit about this really neat probability problem that Christopher Long posted on Twitter:

Here’s a link to that conversation:

Talking through Christopher Long’s probability problem with kids (part 1)

We finished up the conversation today.

Yesterday we ended after talking about why the probability that two randomly chosen integers will share no common divisors is 6 / \pi^2. Today we revisited that conversation and also discussed why picking two random positive integers is actually a little bit of a hard thing to do:

 

Next I wanted to move on to discuss the last part of the solution to the original probability problem, but a question about fractions came up and we had a short conversation about which fraction was larger 36 / 81 (which arose in the problem from the approximation that \pi = 3, or the fraction 36 / \pi^4. So, this strange probability problem gave us a surprising way to talk about fractions 🙂

 

After the short conversation about fractions we returned to the problem. We now know that the probability that the two randomly selected pairs of integers will both be relatively prime is 36 / \pi^4. But what about the probability that both pairs will have a common divisor of 2?

The answer to this question is a little subtle, but it is the key to solving the original problem. Thinking about this question led to a great conversation about primes and what GCD means.

 

Now that we had a little bit better of an understanding about primes and GCD, we dove into how to think about the problem where the two pairs of integers share a common divisor of 2. In this video we talk about why the probability that two randomly selected integers will have a the greatest common divisor of 2 is 1/4 of the probability that they are relatively prime.

 

Finally, we arrive at the solution to the original problem: the probability is 2/5. Surprise!! I bunch of powers of \pi all cancel at the end. Super fun problem.

 

The original problem is obviously way too difficult for kids, but walking them through the solution was really fun. Along the way we got to talk about fun concepts like infinity, fractions, primes, divisibility, and even some really advanced topics like how does \pi show up in this problem?

I definitely wouldn’t do a project like this one too often, but once in a while an advanced problem actually has some pretty neat stuff for kids hiding inside of it.

Odds of winning the US Open

Since we are doing a summer project on counting and probability I thought it would be neat to talk a little bit about odds in sports. We decided to look at the odds for a few of the players to win the US Open golf tournament for today’s project.

I grabbed the posted odds from the European sports aggregator site Oddschecker and asked the kids if they understood what the odds meant. I wasn’t expecting them to understand odds completely, but they did have some interesting ideas:

 

After a brief introduction to odds, we took a quick look at what it meant that Jordan Spieth had odds of 2 to 1 to win today. My youngest son thought these odds meant he would win half of the time. My oldest son was a little confused by how sports betting worked, but I tried to simplify the ideas by just talking about the amount of money you start with and the amount you end with.

 

After working through Jordan Spieth’s odds of winning, we tried to understand the odds for a few of the other players. Dustin Johnson, for example, had odds of 11 to 4 – a tiny bit harder to understand than 2 to 1.

 

Now we switched to looking at the odds of finishing in the top 5. One of the nice things about Oddschecker is that you can find odds for lots of unusual bets. The listed odds for Jordan Spieth finishing in the top 5 today were 1 to 5. I wanted to look at this case since the high and low numbers are reversed from the first example that we looked through.

This section went a little long because of a little confusion about what 1/6th is as a percent. I’m not sure where the confusion came from, but we spent the last 4 minutes of the video sorting out how to write 1/6th as a percent.

 

We started this last section by finishing up the talk about 1/6th and then moved on to talking about both the odds and percentages of the other players finishing in the top 5. Here we encountered some more fractions that we needed to convert to percentages and the work we did on 1/6th seemed to help.

 

So a fun project that is somewhat related to the summer project that I’m working through with the boys. Hopefully next time they hear about odds they’ll have a little bit better understanding of how they work. Also, I’m happy that we ran across some of the confusion about fractions and percents. It has been a while since I covered percents with them and the struggles today gave me a nice reminder to spend a little time reviewing percents at some point this summer.

Practicing fractions while learning some introductory geometry

Saw a nice introductory geometry problem in our Prealgebra book today. It is problem #7 on the 2005 AMC 8

2005 AMC 8 problem number 7

Here’s the problem:

Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point?

I like a couple of different things about this problem. First, there are multiple ways to solve it, and each provides a little different insight into the geometric situation. Second, it provides a nice opportunity for a little fraction review, since you’ll likely encounter adding, multiplying, and maybe even dividing fractions in the solution.

Here’s my son’s approach to the problem:

His initial solution involved using two different triangles to find the length. I was interested to see if he could do it with just one triangle. This new solution involves drawing in two new lines, or maybe just rearranging the initial picture. I wanted to walk through this second solution because I think it is instructuve, but a little harder for a kid learning geometry to see.

It took a while, but we got there. One of the stumbling blocks was understanding what happened to the distances as we moved some of the triangles around.

I really love problems like this one. It gives a great opportunity to cover a new topic from a few different angles and also gives you an opportunity to sneak in a little review of an old topic. Definitely a fun morning.

David Wees’s Fraction exercise

Saw this tweet from David Wees yesterday:

Seemed like a really interesting exercise, so I gave it a shot with the boys this morning. I like the simplicity of the task – here are some shapes, shade in 1/4 and shade in 1/3 of each shape. Definitely a great way to get kids talking about math.

A feel that my kids approach math in two different ways. My older son loves to calculate and my younger son’s approach is much less about calculation and more about feel. You’ll see that difference come through in their approach to the questions in this exercise.

First, my older son shading in 1/4 on the 4 shapes. His approach to chopping up the right triangle was pretty interesting:

Second, my younger son shading in 1/4. His approach in the triangle is also really interesting, though one clear difference from my older son is that he’s not calculating. His symmetry argument on the square was nice to hear, too:

Next up, my older son shading in 1/3. This task was a bit harder for both kids. My older son things that part of the difficulty comes from 3 being prime. His approach on the triangle, the square, and the diamond is pretty neat – he realizes that you can cut a triangle into three equal areas by dividing one side by three, and then extends that idea to the two quadrilaterals.

[also, on publication, youtube isn’t generating a thumbnail image for this video. The video seems to work fine, though, so hopefully we’ll get a non-grey thumbnail sometime soon.]

Finally, my younger son talking about 1/3. He thinks the additional difficulty in this part comes from the fact that 3 is not a power of 2. His feel for math comes through in his explanation about how to cut a triangle into three pieces of equal area. He knows there’s a way to do it and, as above, explains it without calculating:

On the Rhombus, he knows that it is easy to cut a square into thirds and figures that by first bending the rhombus into a square, cutting it, and then bending it back into a rhombus, you’ll cut the rhombus into thirds. That’s a wonderful piece of mathematical reasoning for a young kid 🙂

His explanations of the how to chop up the circle and the square into thirds are neat, too.

So, a really neat exercise for kids, and one that is really easy to implement, too. I’m always really excited to hear kids talking about math. With this exercise, you get kids thinking a little about fractions, a little about geometry, and even a little bit about symmetry. Really fun morning.

Square roots day 2 – approximating the square root of 2

One of the examples in my son’s Prealgebra book today was prove that \sqrt{2} is less than 2. We were having a pretty good discussion about the ideas in this example, so I thought it would be fun to see if we could go a little deeper. Since we just talked about continued fractions last weekend, I was hoping that end up being able to find something to say that was much more accurate than just “less than 2.”

Our initial discussion of the problem is here:

 

Next up was the beginning of looking at \sqrt{2} as a continued fraction. We’ve spent very little time on this subject, so it is still new to him and we had to go slowly through the process. Luckily the continued fraction starts to repeat fairly quickly.

 

We finished up by figuring out some of the fractions that approximate \sqrt{2}. This exercise was why I wanted to go down the path of calculating the continued fraction. First off, we’ll see some of the fractions that we saw already in part 1. Second, we’ll find a couple better approximations, which is neat. Third, we’ll get to see directly that these fractions are nearly equal to 2 when you square them. AND, we get lots of good fraction practice in the process. Yes!

 

Stuart Price and Joshua Bowman’s PIth roots of unity exercise

Saw this amazing post about the \pi^{th} roots of unity yesterday and wanted to use it for a quick extension of our \pi day activity:

Unfortunately, kids up at 11:00 pm, dog up at 1:00 am, cat up at 4:00 am and then everyone up at 6:00 am let to “one of those nights” . . . . So, instead of making use of a really great exercise, I sort of totally butchered it – but it is the idea that counts, right 🙂 Despite stumbling through our project this morning, I can’t recommend Price’s post and Bowman’s Desmos program enough.

For the kids to understand the project a little better, I wanted to do a quick introduction to the complex plan and how the roots of unity show up on the unit circle. We’ve talked a little bit about i before, so the ideas here aren’t totally new to the kids, but a quick re-introduction seemed appropriate:

 

Next up was a reminder of some of the rational approximate to \pi that we found yesterday in our activity inspired by Evelyn Lamb:

The fractions that we found that approximate \pi are 22/7, 333/106, and 355/113. We reviewed these fractions and also what the similar approximations to 2\pi would be.

 

With the background out of the way, we moved on to Bowman’s Desmos activity. First I just like the kids play around with it and see if they could find a situation in which we nearly had a regular polygon using powers of the \pi^{th} roots of unity. This was a fun “what do you notice” exercise.

Also, sorry for the extra blue screen – don’t know what happened to the camera here. Double also, ignore all of my talking for the first minute, please . . . . I was tired, confused, and incoherent.

 

Finally, having found the number 44 as a case where the dots where nearly equally spaced and having seen that this approximation was the same number we saw in the numerator of our 44/7 approximate for 2\pi, we looked to see if we’d see something interesting at 666 and 710. Right around 2:00 is the “wow” moment.

 

So, a fun project showing a geometric representation of some continued fraction approximations for \pi. Definitely one I’d like to have a 2nd, non-exhausted chance at, but oh well. Hopefully the awesome work of Stuart Price and Joshua Bowman shines through over my several stumbles in this project.

Celebrating Pi day with Evelyn Lamb’s idea

Last week Evelyn Lamb wrote a nice piece about \pi and continued fractions.

Since we’ve talked a little bit about continued fractions in the past, this seemed like a great way to celebrate \pi day. We started with a quick reminder about continued fractions:

 

After the quick introduction, we used my high school teacher’s fun continued fraction technique – Split, Flip, and Rat – to calculate the continued fraction for \sqrt{2}. This exercise gives you a great opportunity to talk with kids about fractions and decimals.

 

Next up was today’s activity – the continued fraction for \pi! Unfortunately, for this continued fraction split, flip, and rat doesn’t work so well. Nonetheless, we do get to have a good discussion about decimals while calculating the first two pieces of the continued fraction for \pi.

 

To calculate a few more parts of the continued fraction we went to Wolfram Alpha. Turned out to be a pretty neat way (and obviously a much quicker way) to see the next few numbers in the continued fraction. Again, we got to have a great discussion about decimals and reciprocals.

 

Now, having found a few terms in the continued fraction, we went and looked at what fractions other than 22/7 were good approximations to \pi. Happy 333/106 day everyone 🙂

 

Finaly (and sorry for the camera screw up on this one), I wanted to show a different continued fraction for \pi. In a previous video my younger son thought that we’d find a pattern in the continued fraction for \pi. We didn’t in the first one that we looked at, but there are indeed continued fractions for \pi that do have amazingly simple patterns.

 

So, a fun little project for pi day. A great opportunity to review lots of arithmetic in the context of learning about continued fractions and \pi.