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[Music] Welcome to Connecting
Mathematical Ideas to Notation.

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I am Sybilla Beckmann, Josiah Meigs'
Distinguished Teaching Professor

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at the University of Georgia.

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I was a member of the panel for
the Problem Solving guide as well

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as a member of the panel for the
Response to Intervention guide.

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Problem solving can be
an excellent vehicle

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for connecting mathematical
ideas to mathematical notation.

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Let me illustrate that
with this example:

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Three submarine sandwiches are
shared equally among four students.

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How much of a submarine
sandwich does one student get?

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Students might solve this problem by
drawing three submarine sandwiches.

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If they are going to share those
sandwiches equally among four friends,

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they might cut each sandwich
up into four equal parts

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and then they might give one person one
part from each of the three sandwiches.

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So at this point students
might realize that, oh,

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each person will get three parts,

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each of which is one-fourth
of the submarine sandwich.

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This means that the three subs that have
been divided equally among four shares,

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which we can think of as three
divided by four, can also be expressed

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as three parts, each of
which is one-fourth of a sub

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and that is the fraction three-fourths--
that those two things are equal.

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So we have two pieces of notation
and two mathematical ideas.

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One is division, three divided by four.

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And the other is fractions; in
this case, we have three parts,

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each of which is a fourth, that's
the fraction three-fourths.

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And those two things are equal.

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The problem-solving situation
is an opportunity

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to develop a real mathematical
idea, and that is a connection

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between division and fractions.

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A problem-solving situation can bring
such an idea to the fore and allow

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for the notation and mathematical
ideas to be discussed in the classroom.

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Problem solving can be an excellent
venue for comparing different ways

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of thinking about a situation,
and it can be used

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to extend students' thinking.

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It can help students go from
something that they understand and know

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and can state in their own words
to bring them perhaps to the cusp

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of some new knowledge that they
are ready to start to develop,

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but perhaps haven't quite
yet appreciated.

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Let's take a look at this problem:
Carla and Jessica each have some money.

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Carla has $11 more than Jessica,
altogether they have $85.

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How much does Carla have?

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How much does Jessica have?

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One student might solve this
problem using a strip diagram.

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The student draws two strips that stand
for the amount of money that Jessica has

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and then the amount of
money that Carla has,

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also is that same amount
plus another $11.

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And we see how together it is $85.

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And the student might then reason that
if they take away that $11 from the $85,

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then the remaining amount, which is $74,

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must be split equally among those
two parts that are the same size.

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So the student then divides
74 by 2 and gets 37,

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and therefore Jessica must have had
$37 and then Carla will have $11 more.

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And the student also checks the work
that, yes, indeed, that does fit

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and those do add up to $85.

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Now another student might use
a guess-and-check strategy

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and just try some numbers for
Jessica and make Carla's $11 more

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and then check what the
total is in each case.

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Both of those ways of reasoning lead to
a correct answer, and both of those ways

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of reasoning can be used to
discuss another strategy,

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which is to use algebra.

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If we think of Jessica's
strip there, that rectangle,

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as standing for J dollars,
then we also see

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that Carla's strip stands for J + $11.

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If we add all of those parts together,
we will have J + J + another 11,

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and that's supposed to equal $85.

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Now, not only that, not only do we get
the equation from the strip diagram,

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but also the way of reasoning

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with the strip diagram actually
parallels algebraic reasoning.

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If we look at this equation
2J + 11 = 85,

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the next step might be
subtract 11 from both sides.

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And we see that reflected in
the strip diagram by seeing

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that we have those 2J pieces or those
2 rectangular pieces that are unlabeled

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after we subtract $11 from the $85, 2J
= 74, and we now divide both sides by 2,

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is also reflected in the strip diagram.

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The reasoning parallels, the algebraic
reasoning parallels what is being done

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with the strip diagram, and therefore
we could use the strip diagram

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as a transition point into
that more symbolic algebra.

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Problems can be an opportunity
to take that next step to go

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into the more advanced way of reasoning

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and to extend students' thinking beyond
what they are already comfortable with

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and to teach them a new way
of thinking about something.

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To learn more about Connecting
Mathematical Ideas to Notation,

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

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[Music]