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[Music] Welcome to Representations
of Part-Whole Relationships.

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My name is Sorsha Mulroe.

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I'm a math support teacher at
Bryant Woods Elementary School.

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The lesson today involved some
second graders in a classroom

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of a first-year teacher who
I am working with this year.

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I was working on part-whole
relationships with respect to fractions,

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and I also was thinking about
the different representations

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that I wanted to expose the students to.

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It made sense to start with a
sharing, equal sharing activity.

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In looking at the fractions Practice
Guide, it really helped me think

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through the details of
my lesson planning.

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And, in particular with recommendation
one, you're starting with something

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that we think that students
already know, with fair sharing,

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but starting with that and thinking
about the numbers that we choose,

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starting with those even
numbers and then moving on.

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The guide really helped me think
about starting with a certain number

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of objects and then moving
onto one object being shared.

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So I presented them with a story
problem where they had to share,

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and I was careful to choose the number
so that it wouldn't be difficult

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for them to come up with
an even or equal number

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of shares among three friends.

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I really thought they'd use the half
sheet of paper more for drawing rather

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than going right into number, that
they would draw cookies and draw people

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and assign cookies to each person.

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Some of them did draw some dots
after they worked with counters

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to show their equal parts, but
for the most part I was surprised.

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All of them actually went with
numbers, and they used a fraction bar

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to represent their thinking.

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I could tell that I could really move
on because they really had a good grasp

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of the fair sharing and were able
to extend that in some cases,

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and I thought, "We have used fair
sharing using several objects divided

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among people, and now I think
we can move on to one object

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and still use fair sharing there."

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What I intended to do with that activity
was to go into not just labeling

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but then to look at how many
fractional pieces made the whole.

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What was different with
that challenge is

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that they normally don't have
difficulty when they are given one whole

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and sharing with two people.

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But as the numbers got larger,
I had to guide them through.

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And they were able to get it, but
that was challenging for them.

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I anticipated that they were going to
have trouble with those odd numbers.

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By telling them this one
whole square I presented them

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with was a three inch
by three inch square.

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And then I had rulers there and
I didn't say to use the ruler,

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but I was hoping that they'd go to that.

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And both of them did, but they
didn't know how to use the ruler

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to evenly divide the one whole.

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So I had to sort of put the
ruler side by side with the whole

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and guide them through, "Well if you
look at how the ruler is divided,

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how could that help you also
make the slices in the whole?"

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I wanted to look a little
bit more at naming fractions,

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specifically what we call the parts once
we divide a whole into certain parts

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and also how we write
the fractional part.

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I wanted to get into five-fifths
made one whole, if I had three-fifths

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that that would be three-fifths
and labeling it that way.

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But I wanted to go back and also
have them think about, given a part,

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what that whole might look like,
because we didn't really even get

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into the fact that, depending
on what the whole was,

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that one-half might be different or
look different depending on the whole.

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We didn't get as far as that because
I noticed that when they offered

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that they knew something about
numerator and denominator

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that I would just ask a little bit more
about what they knew, and that told me

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to sort of hold off on
getting further with my plans.

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I wanted to really focus on what
the numerator and denominator meant

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and the relationship among
those two parts in a fraction.

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We started with the Cuisenaire
rods and talking about one whole

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and then the number of pieces
needed to make one whole;

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whether it was the red
pieces or the orange pieces,

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trying to help them see the
relationship between the number

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of pieces needed being the
parts or the denominator,

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and then what that numerator was in
relationship to those number of pieces.

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I really want them to focus on
the numerator and denominator

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and that understanding with
unit fractions before we go

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into non-unit fractions.

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So I would begin there again for next
time, and then I think once they got

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that I would go into how many fifths
make one whole, and maybe leading them

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on to identifying equivalency
with halves and wholes.

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So there is all that to
explore before we can even move

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on into equivalency in
comparing and ordering.

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Something that I am working on with
all of our teachers here is the role

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of mathematical discourse in a
math classroom and really looking

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at not just the questions I ask so
that the students respond to me,

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but the questions that the
students ask of each other,

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the comments that they make to
each other, and the questions

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that they ask of themselves.

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And I'm also working on trying to be a
better facilitator of that discourse,

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because my first inclination is
sort of to jump in and rescue,

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but I'm trying to move away from that

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and really having them explain their
thinking to me in their own words,

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instead of giving them the
vocabulary or the ideas.

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I see my role as a math support teacher

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as really developing the
teacher capacity to think

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about their lesson planning
in particular,

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to anticipate what students might
already know or offer in their lesson,

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and then to plan ahead for that to
guide their instructional planning.

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To learn more about Representations
of Part-Whole Relationships,

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