students to realize that there are 3 groups of cars within the 315. Then we deter-
mined that the students would need to realize that the is an important piece of
information.
KAREN: Well, I’d say, “What’s important in that sentence?” Hopefully
they’d say of the cars are silver.
JOHN: At this point, they might infer what fraction is not silver. Or actually
how many cars are not silver.
MARTA: Yes, you may want to stop at that point and say, “If of the cars
are silver, how many ninths, are not silver? The black and the white to-
gether make how many ninths? That’s , that is almost 1 whole, so, you
hardly have any silver cars at all.”
KAREN: That’s where the visualization comes in, to compare the silver cars
with the nonsilver cars.
MARTA: You don’t have a number but you can already visualize this little
tiny group and then these two other groups. Now it is time to go to the next
sentence.
JOHN: There were more cars made, they were black and white. We need to
know how many of each.Introducing Representations
At this point, we discussed why it might be important to suggest that students use
a physical model to represent the problem. In this case, we decided that different-
colored cubes would be an appropriate model.
MARTA: They might use groups of 5 cubes, 3 black and 2 white, to repre-
sent the relationship between the number of black and white cars. You
might ask, “If you have 6 black cubes, how many whites would you have?”
That’s one way to model or represent the black and white cars.
JOHN: I wonder if at that point when it is time to solve it, the students will
try to build a chart with a row for each set of 5 cars [see Figure 23–1] or if
they’ll be able to work directly with the number of nonsilver cars. Are they
going to try to go each step? You know, go: 3 black, 2 white; 6 black, 4
white; 9 black, 6 white?
MARTA: I hope not. With the blocks and the chart, I am hoping that they
will see the ratio as a constant. Every time you get 3 more black cars, you’ll
get 2 more white ones and vice versa. And you will always be able to see
groups of 2 and groups of 3 no matter how large your numbers get.
JOHN: For every 3 black cars, 2 are white. That means of the remaining
nonsilver cars are black.3
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9Planning Guided Math Groups