tried to stay with us, but when we got to the ratio piece, her thinking started
to break down.
JOHN: So that was when you used the cubes to represent that situation?
KAREN: Yeah. I made the cubes into a physical model, matching the black
and white cubes with the black and white cars in the situation. I modeled
the groups of 5 several times, and had the students visualize that the groups
would continue until there were 280 cars. I think we had 4 groups of 5 out
at that point. One student wanted to keep separating the groups, emphasiz-
ing that you can dividethem. Once again, it was Gloria who noticed that
that just wouldn’t work. She brought up that the total had to be 280 black
and white cars.
MARTA: Ahh... that’s really important.
JOHN: So she remembered to maintain that quantity.
KAREN: Yeah, she said if you divide the 56, you’re not going to end up with
280 cars. Then I led the conversation so that we started talking like we did
in our conversation, like we’ve got this group of 5, and there are 2 white and
3 black. And we started putting the groups together like that, using the
cubes to make rows of 3 black cars and 2 white cars. Then they realized that
we have to multiply.
MARTA: So, I heard you mention our earlier conversation. Did our con-
versation about having the 3 and the 2 help you sort of scaffold for the
kids?
KAREN: Yes, because I knew where we needed to go when the thinking was
getting off track. When they got stuck, I had a plan to pull them back by
using the cubes to model the problem.
JOHN: Nice. So, when you do this problem again, with my class or with one
of your tutors that goes into other classes, what can we do to help it go more
smoothly? What would you do differently?
KAREN: I felt ready! They pulled out what they knew, and that went very
well. They are comfortable with the KWC, whether they want to do it or
not. But, this was a problem with so much information that even the reluc-
tant ones were like, “OK, I need to do the chart.”
MARTA: This was a great problem for showing them the necessity of the
KWC chart.
KAREN: The biggest problem was that a couple students just wanted to
read the whole problem at once without figuring out what the problem
was asking. But you can’t just do something to 315 and come up with an
answer. This problem has a lot of steps. I’m not sure what other strate-
gies to use to help those students understand what they are trying to
solve.
Planning Guided Math Groups