Educating Risa

Our addition of a “problem of the day” question to our math routine continues to go very well. We had a super discussion about today’s question—the version Daegan created to try to stump Mom required some careful thinking and strategy! Here’s what I put on the board for him, which he immediately took a logical and ordered approach to solving:

After listing the numbers in which the digit 2 occurred, from smallest to largest, he then counted them:

And found their were 19 such numbers, but in one of them, 22, you write the digit 2 twice. So the answer to the problem of the day is 20.

Now it was time for Daegan’s favourite part, in which he creates a similar question in an attempt to stump Mom. He was giggling and quite proud of his “tricky question.” He said, “I’ve stumped you…and I’ve even stumped myself!”

 

I began by talking about the differences between his question and the one I gave him. Counting and saying are different than writing—you don’t say “2” when you say “12 / twelve”, and I noted his representation of “1 to 1000” using the dash—I didn’t realize he knew that convention! I started my solution as Daegan did, working from smallest to largest, and then looking for patterns. Here’s my entire, very messy-looking solution—I’ll explain it bit by bit below:

I started by listing the numbers in which we’d say “2” where the digit 2 was in the ones place: 2, 22, 32, 42, 52, 62, 72, 82, 92. I then started a new column for the hundreds. At the 200s, I realized I’d need to do them separately, as you say “two” at the beginning of each number in the 200s. I continued on with the 300s, and through to the 900s, making a grid. I then counted 9 sayings of “two” in each column, and 9 columns, giving me 81 times I say “two”. (We haven’t done multiplication yet, but Daegan easily followed this when I explained it as fast skip counting by 9s).

I then went back to look at the 200s. From 200-299 there are 100 digits. I realized that for some of these numbers, you say “two” twice—like 202, 222, 232, 242…. So I found 100 times when you say “two” at the beginning of a 200s-number, and 9 times you say “two” again in the ones place.

It was then a simple matter of adding 81, 100, and 9 together to yield the final answer. Again, we’re just getting to addition with regrouping in our curriculum, but no matter—with base ten blocks Daegan was able to follow the addition on the board:

So the answer was 190. Great question, kiddo!

One thing I am finding is that the “problem of the day” is an ideal time to introduce new math terminology or symbolism. Instead of a lesson and explanation and several practice problems, the focus on ONE problem really allows Daegan to grasp the symbols / vocabulary involved. This was the case when we did the question about mean, median and mode, and yesterday I gave him an algebra question involving the “greater than” sign, something again that he has not seen a whole lot.  Once he got clear that “the alligator eats the most fish”, Daegan was able to do the question below using a “guess and check” strategy. (“The alligator eats the most fish” is the way I was taught this sign in grade school. Imagine the greater than sign, >, as an open mouth. The alligator wants to eat the larger number).

F > P
P > S
F + P + S = 6

What the the values of F, P and S? 

So if you try this approach at home, don’t restrict yourself to “grade level” or “age level” questions. Go ahead an introduce some materials that stretch your student, and lead to lots of discussion. You don’t want these questions to be too hard and induce frustration, but you certainly don’t want them to be too easy either. This is a nice, safe place to get mathematical language, symbols and other conventions clarified—something that underlies many of the difficulties of students I have taught or tutored in math.