There are 30 kids in the cafeteria and 23 more kids come in for lunch. How many kids are in the cafeteria now?
She began using manipulatives at first, but then decided to use her paper instead. She first wrote out 30 - 23. Then she drew a line between the 3 and the 2 and wrote "50" underneath it, and a line between the 0 and the 3 and wrote "3" underneath it. Then she added the "50" and the "3" to get 53. This really showed how she was able to break the number apart into tens and ones and add them together separately.
When she wrote it out on her paper, it looked something like this:
In this way, she grouped her groups of tens together (3 + 2 to make 5, which she knew represented 50) and her ones together (0 + 3).
Another strategy she could have used would be to use manipulatives to demonstrate the addition by having one color (red) representing tens and another color (blue) representing ones. Then she could add the red manipulatives together and the blue manipulatives together to represent each place value of the answer : 53.
She could also use a counting technique of drawing out ten sticks on her paper and then adding them together. She could then draw out ones on her paper and add those together. Then she could add the two answers together.
Both of these methods are similar to the method she utilized, but are just slightly different ways of approaching it.
It's crazy to me that so many students are using thins expanded notation to add or subtract, but I saw this expansion in my 2nd grade GED setting last fall. Because my CT was very creative and implemented a lot of neat scaffolding and differentiation techniques, I just assumed she was the only teacher teaching this strategy. I had no idea it was so popular! It has been my experience that this strategy is very helpful for students and as well as teachers. Seeing the students expand the problem, allows us to have a larger sample to look at of student work. If they make mistakes, it's much easier to pinpoint what it is because there are several steps to look at.
ReplyDeleteAlso, they seemed to find it much easier to add the tens together. I feel this is because it's a "nice" number; no regrouping required! Breaking it down by tens+tens and ones + ones seems to help the students because the ones are more than likely going to be "quick fact" recall and the tens are "nice" numbers as mentioned above.
I find it interesting when I see students use methods I never thought of using when in school. I do like how she kept the numbers together, yet expanded them. The only thing she didn't do was rewrite the problem, which I think is neat because she was able to recognize what she was doing by just connecting the values and then adding.
ReplyDeleteI think the colored manipulatives, for me at least, would have been challenging. I say this only because I would have to remember that they represent a certain value and not just one. I think it'll be interesting to see how she uses her strategy as the problems progress to bigger numbers or become more complex.
That's a good point you make, Jillean, that breaking up the number into the tens place and the ones place is perhaps easier because addition of single-digit numbers is more likely to be a quick fact for the students than addition of two-digit numbers. I know that during some of my math lessons, when I introduced the concept of subtracting three-digit numbers, some of the students suddenly became wide-eyed and reluctant to begin the problem-solving process. With Kate's ideas of other ways that the problem could have been solved, I can see how a teacher could shape the student's thinking so that over time the use of more abstract problem-solving processes is possible. Much like what we are asked to do at the end of play teach presentations, determining the most and least abstract problem-solving methods could be a way to help students move their thinking from the direct modeling approach to more inventive strategies. For example, Kate's idea of using colored manipulatives to first represent the problem could over time be faded to the more abstract method of using dash marks to represent the problem. Finally, the most abstract means of representing the problem, that is, using the individual numbers, could be used once understanding of the other means of representation was ascertained.
ReplyDeleteKate,
ReplyDeleteI was really impressed by your student's strategy. I find the way she solved the problem very creative, and I could see how it would make the problem easier to solve. My only concern would be would she be confused when it came time to carry a one if there was a chance for that to happen.
I'd never heard of addition like that, so thank you for teaching us something new.