Here's a fly-on-the-wall view of an introductory lesson (shh; don't let them know you're in the room):
Teacher: Suppose we choose a number from 1 to 100. We'll call that number n. We often use the letter n in math to stand for any number. Someone pick a number for n--
Student: 38!
Good enough. So if n is 38, what's n + 10? 38 + 10, right? Which is--
Students: 48.
That's right. Okay, let's make a table and try it using some other numbers for n:
n n + 10 Result
--- ------- ----
38 38 + 10 48
17 17 + 10 27
90 90 + 10 100
45 45 + 10 55
Looks good. Okay, what patterns do you see? How does n change when you add 10?
Students: The ones digit stays the same.
Yeah? Always, or only most of the time?
Students, a bit hesitantly, because you always have to watch out for trick questions: Always...always so far, anyway.
That's right. Can you think of a number n where the ones digit would change after you add 10?
Students: several suggestions, all of them withdrawn upon further reflection.
...Why doesn't it change?
Student: The number 10 has 0 in the ones column, so it doesn't change the ones.
Another student: Oh, and when you add ten on the hundreds board you just go down to the next row, so if you're in the threes column you stay in the threes column...[We use the hundred board a lot; one is pictured here.]
What happens with the tens? The tens go up? Good; by how much? By one? Always, or only sometimes?
Students, less hesitantly than before: Always.
How do you know? So, okay, let's put the rule into words: When you add 10 to a number n, the ones digit stays the same but the tens digit goes up by one.
Nice job! Okay, let's try it again, only this time we'll look at what happens when you add 11 to n.
n n + 11 Result
--- ------- ----
12 12 + 11 23
28 28 + 11 39
77 77 + 11 88
Student, bursting to be the first: I know, I know! I know the rule! It's the tens digit goes up and the ones digit goes up too!
Student, bursting to be the second: Yeah! It's the tens digit goes up and the ones digit goes up too!
By how much? Let's say it as a rule.
Students, cautiously: It goes up by one in the tens column and one in the ones column.
Always, or just sometimes?
n-2 or n-3 students, where n is the total population of the class: Always.
Two or three students: Sometimes.
Why sometimes?
2 or 3 students: Because what happens when the number is in the nines? Say you add 11 to a number like 59...
2 or 3 more students: Ohhh!
Let's extend the table--
59 59 + 11 70
69 69 + 11 80
n/2 students: It goes up two in the tens!
The other n/2 students: And it goes down to 0 in the ones.
Okay, let;s make the rule. Help me out here:
[And so we develop the rule: When you add 11 to a number n, the tens digit usually goes up one and the ones digit goes up one as well, EXCEPT that when the ones digit is 9, the tens digit goes up by 2 and the ones digit goes back to 0. We talk about why this might be the case--and then out go the students to work on developing rules for n+1, or n+19, or n-2, or perhaps even n x 5...]
Okay, class is over for the day. You can come down from the wall now! Aren't you glad none of the kids brought flyswatters today??
[Edited to add: I should note that this lesson is adapted from a set of activities in a new book by math education guru Marilyn Burns. In 2008, I spoke at a national conference of math teachers. I was disappointed to discover that I was scheduled at the same time as Marilyn, which was disappointing for two reasons...first, I didn't get to hear her, and second, hardly anybody was left to come to my workshop...
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