Singapore Math, day one

I've just walked C. through his first day of Primary Mathematics 3A, including the 3A workbook.^*

Then I had him do one "Challenge Problem" from Intensive Practice 3A:

1. I am a three-digit number. All the three digits add up to 9. My tens digit is twice my hundreds digit and my ones digit is three times my tens digit. I have no zeros. Who am I?
(page 10)

He took a look at it and said, "That's hard!"

I thought so, too. As a matter of fact, I am currently doing word problems only slightly more advanced than this in Saxon Algebra 2, at the end of the book. [update: RJ and Cheng noted that the sum of the digits of their two-digit counting number was 9. If the digits were reversed, they found that the new number was 27 less than the original number. What was the original number? Saxon Algebra 2, Second Edition, p. 432, no. 1.]

I handed him the workbook to look at while I wrote the problem on separate sheets of graph paper for both of us to do. I figured it might be a good idea to get some practice drawing bar models.

A couple of seconds later C. said, "I have the answer."

He did. answer: 126

I pushed him to tell me how he got it, and finally he said (paraphrasing), "I asked what is 2 times, and then I realized it was 2, and then that was 6."

I believe this is called "making children's thinking processes visible," right? And, yes, this is the way kids talk when they're describing a mental process. Mental processes are slippery. When you get an answer fast, you don't necessarily remember -- or even know -- how you did it

What I take Christopher's report to mean is that he intuitively -- unconsciously -- understood immediately that the hundreds digit would have to be 1, then consciously asked himself what two times 1 was, had a eureka moment (I could tell this from his tone of voice) when he realized that 2 was put him in the running to end up with a grand total of 9, and from there semi-consciously realized that the ones digit was 6.

At least, that's the way I piece it together. Obviously I don't know how he came up with the answer five seconds after he read the problem.

This gives me hope. I was saying yesterday that Ed and I were hoping the fact that C. appears to be correctly guessing answers to fraction word problems means he's developed some number (& fraction) sense; today's Challenge problem seems to be evidence that he has.

I think I've mentioned before that "Math Dad," the parent here who is a veteran NY state math teacher, has told me, and I'm pretty close to quoting here, "It's too late for these kids. They are lost." He was speaking of Christopher's group, the kids who need effective instruction and a sound curriculum in order to succeed in accelerated math.

When I heard that I thought: over my dead body.

Maybe, just maybe, these past 3 years of slave labor and protracted Math War have produced some "growth" after all.


update

These are the three questions for which C. correctly guessed the answer. I'd forgotten that one of them is extremely simple algebra. Sigh.

Robin and Jim took cherries from a basket. Robin took 1/3 of the cherries and Jim took 1/6 of the cherries. What fraction of the cherries remain in the basket?
answer: 1/2

___ represents the number of magazines that Lina reads each week. Which of these represents the total number of magazines that Lina reads in 6 weeks?
choices:

6 + ____

6 x ____

____ + 6

( ____ + ____ ) x 6

[yes, I am horrified that I have a "rising" 8th grader who had to guess the answer to this question - though this does serve the purpose of illustrating why "Math Dad" goes around saying things like "It's too late for these kids."]

Penny had a bag of marbles. She gave one-third of them to Rebecca, and then left one-fourth of the remaining marbles to Jim. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with?
answer: 48

^* For newbies, C. is entering 8th grade and began taking Math A -- integrated algebra and geometry -- in the second semester of 7th.