I’m not entirely sure why but I’ve been spending a lot of time recently thinking about different ways to build up number sense. About a week ago I started a chapter on similar triangles with my older son, and the problems in that chapter have helped me gain a better understanding of the importance of building “algebra sense” (for lack of a better phrase) too. I’m surprised how many opportunities there are to focus on both of these topics now that I’m actively paying attention to them. An odd coincidence today made me want to write up the conversations I had with my kids this morning.
But first I want to back up to a coincidence from yesterday.
As I mentioned above I’ve been studying similar triangles with my older son for a week or so. The bit of math that seems to be giving him the most difficulty isn’t the geometry, it is working with the ratios that arise in the problems about similar trirangles. Here’s one of the problems we worked through yesterday just to give an example of the ratios that come up in these problems:
I felt that it would be good to review some of the algebra behind these ratio equations before finishing the similar triangle section and found several sets of practice problems on Khan Academy that provided more or less exactly the review I was looking for. Here’s one set for example:
Khan Academy Ratio problems
Although people have widely differing views about Khan Academy, I think one nice advantage that it has is that the problem sections are great for this type of review.
Interestingly last night Steven Strogatz posted this picture on twitter:
The similarity between the homework I wanted my son to do and the homework assignment in Strogatz’s tweet got me thinking about context. The motivation to learn more about geometry was enough for my son to understand the purpose of the Khan Academy problems. Actually, he even asked to do more. I don’t know the context of the other homework assignment, but do think that without proper context that assignment could seem quite dull. This coincidence from yesterday reminds me to be careful to be clear about why I’m asking the boys to do the homework I give them.
Now on to today . . . .
This morning my younger son and I were talking about palindromes (section 6.5 in Art of Problem Solving’s Introduction to Number Theory book). We began with several simple examples – numbers like 11, 454, 34543 – and then he stopped me:
kid: “I know a long list of palindromes.”
me: “what is it?”
kid: [ writes the first 4 rows of Pascal’s triangle on the board ]
This example is definitely a fun one for looking at palindromes, but it also turns out to be a great one for building on number sense. The connection I wanted to focus on was how the rows related to powers of 11, and how that connection seems to break down in the row: 1 5 10 10 5 1.
My first question to him was whether or not this specific row was a palindrome. He surprised me by saying that although the number you get by putting all of the terms together, namely 15101051, was not a palindrome, you could get a palindrome you looked only at the last digits, so 150051. Interesting observation. We’ll have to return to this topic later when we talk about modular arithmetic!
My next question for him was about the powers of 11. Starting at , the powers of 11 are 1, 11, 121, 1331, 14641, and 161,051. Why did we lose the connection to Pascal’s triangle when we computed ? This led to a wonderful conversation about place value and eventually to showing why we did not actually lose the connection to Pascal’s triangle at all. Really fun, and I think a neat way to talk through place value while getting in a little arithmetic practice, too.
Later in the morning my older son got tripped up on this problem from the 2006 AMC 8:
2006 AMC 8 problem 24
The problem has a really lucky connection to palindromes since an important observation in solving it is that one number is equal to another number multiplied by 101. Talking through this problem also led to a good conversation about place value. Luckily the notes from the conversation about Pascal’s triangle and place value happened to still be up on the board when this second conversation took place.
Seeing some of the earlier work that was on the board my older son said that he thought you could make the row 1 5 10 10 5 1 into a palindrome by working in base 11. Ha – another unexpected response, but also now a wide open door to talk a little about what I’m calling “algebra sense.”
We quickly reviewed the place value conversation I had with my younger son about how the rows connect to powers of 11, but then looked at what happens in base 11. Surprise – powers of 12!! Don’t think he saw that coming 🙂 Now maybe 5 to 10 minutes of conversation about what the polynomials and look like and we’ve quite unexpectedly done some neat work that helps build up familiarity with algebra and algebraic expressions.
So a fun morning. As I have the goal of working on number sense in the back of my mind, I’m excited to see all of opportunities that come up to work on it. Algebra sense, too, but Strogatz’s post from yesterday reminds me to be extra careful about context. It is fun to take advantage of the lucky times like this morning when that context appears almost by magic!