WEBVTT

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[Music] Welcome to Cross-Multiply?

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Not So Fast!

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Ro: I am James Ro, I teach
eighth-grade pre-algebra and geometry

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at Elkridge Landing Middle School.

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Price: Hi, I am Jackie Price,
and I am the special educator.

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We have an inclusion classroom with
students with a variety of needs that we

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as teachers together work to
support while we teach math.

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Ro: Today's lesson was geared towards
attacking proportions in different ways.

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We teach 80-minute classes, and so
for today we had three distinct parts

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that we wanted to accomplish.

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First, in our warm-up, what we tried to
do is we tried to get them to think--

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just basically, building
strategies or looking at unit rate

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in solving for proportions.

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We wanted them not to gravitate towards
the cross-multiplication method;

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we wanted them to kind
of start from the basics.

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After the warm-up, what we
wanted to do is we wanted to kind

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of discuss the various
strategies that will build up

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and eventually get to
cross- multiplication.

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After that we wanted them to
get some real-life experience

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about how this would work.

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Price: For the warm-up we had this
scenario that they were at Chili's

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and they had certain numbers
of bowls of salsa and people;

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so we said that two bowls of
salsa could serve four people.

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So we used that as our scenario, and
then we had questions that went with it.

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So we asked the students to use the
blocks as a visual representation

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to show us how they would solve the
problem outside of cross-multiplying.

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And they worked in pairs
to complete the activity.

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Ro: Conceptually, I guess there
were a couple of different ways

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that we saw students
attack with these blocks.

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The first problem was set up so that
it had a very integral relationship,

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meaning they could just simply
keep adding those groups

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until they got the answer.

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And I saw many groups do
that with the first problem.

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Price: With the one pair
that I was working with,

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they were kind of struggling with
how to really break it apart,

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and because they are so
used to the procedural steps

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of following cross-multiplying that it
was hard for them to go outside of that

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and think about another strategy.

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And so we really broke it down into the
unit rate of how many people per bowl,

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and then we kind of set up a chart.

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But I really had to ask
them questions and say,

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"So if we had this many
people, how many bowls?"

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Ro: If you put them in a restaurant
and you told them to do the same thing,

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I think they would get it more quickly.

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I could see that kind of tension
between what you learn in the classroom

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and what's generally
taught about numbers,

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and I think that's why we had a
lot of students who just gravitated

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"cross-multiply, cross-multiply"
rather than think outside the box.

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Ro: You can use cross-multiplication
for any kind of proportion problem.

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There are more efficient ways of
solving using some build-up strategies

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or unit rate strategies.

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I know I tell them, you know,
"We might give you many options;

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you kind of choose which one
that you understand the best."

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We wanted them to partner
up and practice problems

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where they would practice various
strategies, and we had them do it:

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one person from the pair
do one set of problems

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and the other one the
other set of problems.

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And then they teach it to each other,

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so maybe they see different
ways of being efficient.

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Price: Especially being in an inclusion
setting, we definitely have students

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who are not always going to get
it the one way that we show them.

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Ro: We try to give opportunities

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where students really bounce
off each other's ideas.

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We really want them to talk with each
other, see what they are thinking,

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and then explaining it out loud.

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Ro: The versatility of the
way the fractions are used,

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I think that's hard for kids to grasp.

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In elementary school you are taught
that a fraction is a part over a whole,

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and when you get to ratios you are
comparing two different quantities

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of the same unit, and in a rate
you are doing two different units.

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When they see a ratio it's
kind of hard to visualize it

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in the same way they have seen
fractions before as a part over a whole.

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One great example is
slope-- rise over run--

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and students can get the
concept of rise over run,

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but I think some students are like,
"Why do we put it as a fraction?"

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I think that's where we have to really
tie in that relationship in a fraction,

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how they are all really kind of the same
thing, just expressed in different ways.

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Ro: I think one of the focuses this year
is really bring what they do out there

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when they go home, and making
that come to life rather

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than what they are just
learning in the classroom,

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you know if you were to build a cafe .

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The first part was purchasing a lot,
and it was really to compare unit rates.

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So right off the bat, I didn't
want to make them think, "Okay,

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everything is a proportion;
everything is a proportion,

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where I set it up and cross-multiply."

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And I noticed some people
were trying to cross-multiply,

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but that wasn't going to work.

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When you are comparing two
companies, you know, how do you tell

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which one you are going to choose?

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Usually the cheaper amount of money per
square foot was what we were hinting at.

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The second step was a mixture problem--
mixture ratio of how to make mortar

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with sand and cement with
the ratio of three to one.

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If we purchased nine tons of sand,
how many tons of cement will we need

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in order to make a proper
mixture for the mortar?

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So after that we do a
little scale drawing.

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On the map, from one point to another,

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if the scale is two centimeters
equals five miles, and you measure it,

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and it's eight centimeters,
how far you are going to drive?

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And then within that question,
we pose a challenge problem.

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Let's say your Volvo
drives 17 miles per gallon--

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how many gallons are you going to use
driving to the store and then back?

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Ro: I think today's class
was a good example

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about what they were thinking about,
and that helps us as teachers whether

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to maybe next class go over a concept
or spend a little bit more time

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on this concept, rather
than just moving on ahead.

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So it gives us a good gauge
of how they are thinking

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and what they see from their eyes.

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To learn more about ratio,
rate, and proportion,

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please explore the additional resources
on the Doing What Works website.

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[Music]