Month: January 2015

Going back to Similar Triangles

My own history of learning math is a little strange. When I was in 5th grade we lived in Europe and I taught myself Algebra out of of an Italian algebra book. My elementary school didn’t really know what to do with me in 6th grade, so I ended up learning algebra again from a tutor from the University of Nebraska at Omaha. In 7th grade me Junior High also didn’t know what to do with me, and I was place in – you guessed it – algebra.

I don’t remember much from that class in jr. high. The teacher’s name was Mr. Stribley. The class was out in one of the portable classrooms that were constructed after the school was hit by a tornado in 1975. My girlfriend in high school was also in the class, though I doubt she knew my name then.

Anyway, I’ve always wondered whether or not that three year tour of algebra was useful or not. If nothing else, though, it left me with a reasonably good base in basic algebra!

I bring this little bit of history up because last week I noticed that my son was struggling a little with similar triangles. We covered the similar triangles chapter in our book a few months ago and maybe we just haven’t returned to the topic enough in the interim so the topic wasn’t fresh in his mind. Likely a much better explanation is that the ideas didn’t really sink in the first time around and I failed to notice. But, thinking about how to best learn the topic and also thinking about my own history of learning math, I decided that it would be worth spending a little time going back over that chapter again. Not as a review, but just starting from scratch. Hopefully this idea isn’t a huge mistake, but we have already had some good discussions about similar triangles, so I think the re-do is going to be useful.

Today we worked through a really challenging example in the section about “Side / Angle / Side” similarity. For me anyway, this type of similarity is the most difficult to see in problems. The picture for this particular problem makes the similarity especially hard to see because the two similar triangles are not oriented the same way. Our first time through the problem this morning probably took 30 minutes. The videos below are the 2nd time through. I thought that talking through the problem a second time would help him digest a few of the ideas in his mind. It takes most of the first video to get our arms around the picture:

 

Once we have a good picture, the problem becomes much easier to solve:

Definitely a hard problem, but definitely a good lesson in similar triangles. Hopefully this second time through the section on similar triangles will help him build up a good base in geometry.

A challenging number pattern problem

My younger son struggled with a number pattern problem from an old MOEMS test today. I enjoyed talking through it with him tonight because it was interesting to see how he approached the pattern in the numbers once he saw it – his approach was quite a bit different that what I was expecting.

Here’s an introduction to the problem and our initial talk that gets us on the path that surprised me:

 

So, my surprise in the last video is that he wanted to go to the end of a row and subtract a certain amount to get back to the beginning. I thought it would be interesting to see if he could see that you could also add 1 to the square at the end of the last row. This idea was hard for him to see, but eventually we got there.

 

At the end of the last video we talked about how the odd numbers relate to the perfect squares. The sequence of rows in the original problem hints at the relationship, though for me, at least, the connection doesn’t jump off the page. To get a better sense of that relationship we went to our kitchen table and looked at the relationship using snap cubes:

 

So, a fun little project starting from an old math contest problem. Ultimately the lesson I’m hoping to convey with my son here is about looking for patterns. The connection between arithmetic and geometry in the last part is also something that I hope he finds interesting. I always find it fun when geometry helps us understand arithmetic a little better.

A nice divisibility rule problem from James Tanton

Saw this fun puzzle from James Tanton today:

By luck, my younger son and I just finished the section about divisibility rules in our Introduction to Number Theory book, so this was a well-timed puzzle.

When I got home tonight we looked at the puzzle together – starting with a quick review of the divisibility rule for 11. It was also nice to see that my son was able to count the number of 5 digit palindromes since I thought that we might have to spent a little extra time on that part:

 

Next we took a first crack at counting the number of 5 digit palindromes that are divisible by 11. We got really unlucky when we were filming this video and the camera’s memory got filled up about 2 minutes in the first time through. Sorry if this one seems like it starts too quickly – if it seems that way it was because we were starting over 😦

The unlucky start start aside, though, this video shows why I think there’s a great project in here for kids – so much secret little algebra and arithmetic practice hiding in this problem. Lots of interesting patterns, too!

 

The next part was counting the number of 5 digit palindromes that have the alternating sum of their digits equal to 11. There’s a fun little pattern on this part, but seeing it involves going through several different examples. Lots of great algebra and arithmetic practice, though, and I was really happy to see that my son stayed engaged all the way through – even taking a couple of guesses at the patterns.

 

Now we were in the home stretch! In this part we count the number of palindromes with alternating sum equal to zero. The surprise in this section is that the pattern is similar to the one we saw in the last video. Why are the patterns for summing to 11 and summing to 0 related? Hmmm . . . is there more going on here than James Tanton is letting on?

 

The last step was checking when the alternating sum was equal to -11. As my younger son guessed, we find a similar pattern to the case when the alternating sum was equal to 22. Fun!

 

So, we find that a total of 6 + 35 + 35 + 6 = 82 5-digit palindromes that are divisible by 11. As I said above, the timing of James Tanton’s tweet couldn’t have been better since we just talked about the divisibility rule for 11. With that rule in our back pocket, this project became a really great example of how to put that rule to work and also get lots of fun arithmetic and algebra practice. Another great problem from James Tanton!

I like this problem

We were driving back from Boston this morning, so we didn’t have our normal morning math class. Instead I spent about an hour talking through some of the problems from the 2010 AMC 10a. Sort of a different approach to math contest problems – my son would read the problem out loud and we’d talk about each one for a few minutes.

This one caught my attention:

Problem #13 from the 2010 AMC 10 A

Here’s the problem in full:

“Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop?”

\mathrm{(A)} \ 80t + 100(\frac{8}{3} -t) = 250 \qquad \mathrm{(B)} \ 80t = 250 \qquad \mathrm{(C)} \ 100t = 250

\mathrm{(D)} \ 90t = 250 \qquad \mathrm{(E)} \ 80(\frac{8}{3} -t) + 100t = 250

I realize the difficulties that can arise from the phrasing “could be used” but putting that aside for now, I really liked the discussion that arose from this particular problem.

Two things that I enjoyed especially were:

(1) For lack of a better word, I’ll call it math fluency. The conversation about the five answer choices helped me understand that my son does not see equations like these in the same way that I do. Perhaps not the exact right comparison, but I thought it was similar to watching someone who is just learning a language reading along with a native speaker.

I steered the conversation to talking about the 3 simple equations first. I made this decision to avoid having the conversation stop once he recognized (a) was the right answer, but it proved useful from the fluency side, too. Working through the three easier equations helped me get a better feel for what he thought the equations were saying. This part of the conversation was actually made a little easier since the solutions to the three straightforward equations are large (compared to 3 hours) and that makes it not so hard to see that these times cannot be correct answer to the question.

(2) The two remaining complicated equations were really confusing to my son, and finding a way to talk through them without just telling him what every term meant was a challenge for me. I decided to ask him where he thought the 1/3’s came from in those two equations. In retrospect I’m not sure how I arrived at this question for him, but thinking about those fractions did seem trigger some new ideas for him. Eventually he was able to see that (a) did indeed represent the correct answer.

This conversation today was unusual since we didn’t have any way to write down what we were talking about – we really just talked through the problems. Talking through the problems gave me a different view of his understanding of math – more from a reading comprehension side than a math comprehension side in this setting. The lesson for me from today is to look more carefully at the fluency side and pay more attention to each of the kid’s first impressions on encountering a problem.

. . . and the 5th lesson – the Pythagorean Theorem!

[sorry for the rushed feel of this one, as I mentioned in the last post I’m at a Starbucks and have to be home in 20 min!]

The last post talked about one problem with four lessons:

One Problem with 4 Lessons

This morning I talked through a 5th lesson from this problem. That lesson comes courtesy of this amazing video from Numberphile and Harvard math professor Barry Mazur:

 

So, perhaps the greatest surprise of all from the problem we were talking through last week – Problem #7 from the 2008 AMC 10 A – is that it helps you prove the Pythagorean theorem in a clever way.

The first step was reviewing how area scales when you increase the size of a 2 dimensional object:

 

Next we moved on to checking if area seemed to scale the same way for triangles:

 

Finally the punch line – how the combination of the original problem and the idea of how area scales proves the Pythagorean theorem:

 

I’m really happy about these two projects, but the 2nd one makes me really happy for a couple of reasons. First, the ideas are easy enough that I could talk through them with both kids. Yesterday’s lessons, while important, require a bit more background in algebra and geometry than my younger son has right now.

The second reason this lesson makes me happy builds on the first – the combination of the relatively easy ideas and the amazing result – the Pythagorean Theorem! – is a great example of mathematical reasoning. “Hard” theorems do not necessarily require “hard” ideas!

One problem with four lessons

A few days ago I wrote this post:

Pythagorean Triangles and Differences of Squares

about this problem:

Problem #7 from the 2008 AMC 10 A

Yesterday I used the problem to show 4 different things that we could learn from it. This morning we looked at a 5th. Have to write up those two projects quickly – I’m at a Starbucks and have to be back home in a hour!

The first piece of the project was reviewing the problem itself and how we solved it. My son chose the solution that used similar triangles:

 

Having talked through the similar triangles solution, we moved on to talking through an algebraic solution. Although I didn’t film it, my son’s original attempt at a solution (the one that gave him trouble) was an algebraic solution that resulted in the need to find the solutions of a 4th degree polynomial. Our algebraic solution here is a little less complicated, thankfully!

 

Next up, an interesting lesson – without really realizing it, in solving this problem we’ve learned how to construct square roots with a compass and straight edge (assuming we are given two lengths). This lesson is a nice little surprise!

 

Finally, connecting the constructing of square roots back to algebra allows us to see a picture for the “arithmetic mean / geometric mean” inequality. We approach this inequality geometrically first, and then algebraically.

 

 

It was not at all obvious to me that there were so many lessons hiding in this problem. Thinking about it now, it really is amazing how much this one little problem has to offer, and the biggest surprise is the next blog piece 🙂

One of my all time favorites – The McNugget problem!!

This morning my younger son had some trouble with a problem on an old MOEMs test that is similar to the famous Chicken McNugget problem. Similar enough, actually, that I decided to put off our second day of divisibility rules until tomorrow in order to spend our time today talking about this problem.

Given that we just finished up a section on modular arithmetic, there is a bit of a connection to what we’ve been studying, but I mostly just wanted to talk McNuggets!

First up was talking through his approach to the MOEMs problem this morning. In this video we do get to the answer by checking every number until we get a bunch in a row. I wanted to let him talk through this approach just to show him that he really could solve this problem on his own.

 

Next up was an approach to the problem that was a little more systematic and also brought in some ideas about pattern recognition. When we write out our grid of numbers my younger son notices that above 13 you can just keep adding 3’s to one of the rows and get all of the remaining numbers.

 

The last part of talking about the MOEMs problem was bringing in modular arithmetic. The ideas here help us see why we could just keep adding 3’s in the last step:

 

Finally, the punchline – the Chicken McNugget problem. This one is a tiny bit more difficult because there are 3 sizes rather than just the 2 from the MOEMs problem, but we do manage to get to the end using basically the same ideas we talked about earlier in the project.

 

So, an accidentally fun morning with one of my all time favorite little math puzzles 🙂

Fun trapezoid coincidence on twitter tonight

Earlier tonight I saw this great question on twitter from Wendy Menard:

Dave Radcliffe posted a link to his really clever solution:

The discussion following Dave’s tweet was about why the triangles ACD and BCD in his picture have the same height (their areas are the same because the have the same base and the same height, but as Dave says in the discussion, the fact that the heights are the same isn’t obvious).

Well . . . funny enough before I saw Wendy Menard’s post tonight my son and I were talking through problem 20 from the 2008 AMC 10a:

Problem #20 from the 2008 AMC 10 A

Here’s the problem: Trapezoid ABCD has bases \overline{AB} and \overline{CD} and diagonals intersecting at K. Suppose that AB = 9, DC = 12, and the area of \triangle AKD is 24. What is the area of trapezoid ABCD?

Trapezoid

Working through the AMC 10 problem you’ll see that triangles AKD and BKC have the same area. That fact gives a little insight into why the two shaded triangles in Dave’s picture have the same area (and why the heights of triangles ABD and BCD are the same) -> ABCD in Dave’s picture is a trapezoid because angle DAB is 72 degrees!

How fun that two problems brought to our attention almost randomly have really similar ideas that help you get to the solution!

Pythagorean triangles and differences of squares

This is the second short (in words) post for today.

Problem #7 from the 2008 AMC 10 gave my son a little trouble this morning, but talking about it led to a fun conversation about geometry and algebra. Here’s the problem:

Problem #7 from the 2008 AMC 10 A

After talking about the solution to this problem, I scrapped the actual lesson for today and talked a bit about some special Pythagorean triangles. Then I gave my son this problem as a challenge:

 

He was able to solve the problem in the last video by finding a pattern in the side lengths. Next I challenged him to find a different pattern. This part was a little bit of a struggle, but he did eventually find a different pattern that connects the side lengths:

 

Finally, I wanted to use the pattern that we found in the second video to find some new triangles. We found the next couple of triangles in the pattern and that showed him, I think, that this new pattern could be pretty useful.

 

This was an interesting little project for me. I guess there’s no way to know what patterns that people will find easy to see and what ones that they will find hard to see, but I do think looking for patterns that you don’t see initially is an important skill in problem solving. I’ll be on the lookout for similar project connecting geometry, algebra, and patterns in the future.

Introduction to divisibility rules

This is the first of two short blog posts today.

My younger son and I started the section on divisibility rules in our Introduction to Number Theory book. I don’t remember the context, but we have talked a little bit about divisibility rules before. He knows some of the rules, but now we are going to learn how to understand these rules through the lens of modular arithmetic.

Last night we talked about divisibility rules without even looking at the book. I just wanted to hear what he had to say:

 

Today we talked a little more in depth about some of the basic rules – namely divisibility by 2, 5, and 10. He seemed to be able to understand the ideas and gave a really nice explanation of why the divisibility rules for these numbers work. It was fun to hear his explanations (despite my stumbling explanation of the problem that we were working on . . . .):

 

I remember being fascinated by these divisibility rules as a kid, though I’m sure that I just learned the rules without really understanding why they worked. Learning the ideas behind these rules isn’t too complicated, though, and hopefully helps build up number sense and a little bit of sense about place value, too. Definitely a fun little project.