From Rational Number to Algebra:
Building a Landscape
Mathematics in the City, Summer Institute
August 13 – 18, 2012
Today’s Agenda
§  Book Groups (table)
§  Warm-Up
§  Investigation: Kitten Food
ü  Launch (Joel)
ü  Small Group Work (partner)
§  Analyzing Students at Work (partner)
§  Video: Math Congress
§  Number Strings: Try, Analyze, Write (table)
§  Mini-Investigations (partner)
Book Groups…..
First, without opening your book or your
journals, identify anything that you
remember from the reading last night.
Then, open your books/notes/journals
and give everyone a chance to share:
§  Ideas that resonated
§  New ideas that are not yet clear
§  Lingering questions
§  Doubts or wonderings
Launching an Investigation
Math Congress
Planning for the Math Congress
§  Before the congress, study all of the
posters and choose a few to use with
the whole class.
§  The goal of the congress is not to
share all posters (not a show-and-tell),
but to focus the conversation around
a big idea.
§  Choose posters that support and
develop that big idea.
§  Be prepared to highlight the big ideas
and important strategies
(e.g., equivalence).
Looking at Student Work
§  What do these students
appear to understand?
§  What strategy do they use,
and how do you know?
§  What question(s) might you
ask this group if you were
conferring with them?
Looking at Student Work
Students
What do
Describe
What
they appear
their
questions do
to
strategy.
you have
understand?
about them
or for them?
Together with your partner, make a table
like this one on chart paper. Then use it to
plan a Math Congress. Who should
present? In what order? What are you
“going for”?
Conferring with Students
Conferring with Students
§
§
§
§
§  Move around the room, trying to
get a sense of the variety of
strategies that students are using.
Make note of which groups appear to be
using the different strategies.
Choose a few groups to confer with as
they work.
Start by simply observing, then think about
how you want to support them.
Try to identify where these students are on
the Landscape of Learning.
Conferring with Students
Your questions might fall into one of three
categories:
Questions
about
strategy
Questions
about
representation
Questions
about big
ideas
• How did you decide to start?
• What did you try so far?
• Was that helpful?
• I want to understand this drawing.
• How is this chart helping you?
• What does this 1/5 in your drawing represent?
• Is this 0.8 the same as 4 out of 5 pieces? How do
you know? Could you convince all of us?
Conferring with Students: An Example
§  Watch as Corinne and Laila work on the
Kitten Food problem
§  What do they appear to understand?
§  What do they appear to be struggling
with?
§  Take notes on:
ü  what Joel says,
ü  what questions he asks,
ü  and when he says nothing at all
Bob’s Birdseed
Bob sells a five-pound bag for $8.
But other amounts sure would be helpful if
you have a very small or very large bird.
He also sells the following weights in prepackaged bags:
1/4, 3/4, 1, 2, 3, 8, 13, 21, 34, 90
The prices depend on the weight only and
there are no discounts for larger bags.
Make a big chart that Bob could use.
Include other amounts that his customer
might want. Explain how you found these
prices.
Mathematics as Sense-Making
§  There are 3 “rubber
bands” in this method.
What are they?
§  What do you think the
teacher was trying to
help the students with?
§  What big ideas
underlie this “trick?”
§  What could you add
to this poster to explain
why the “rubber band
method” seems to
work?
Mathematics as Sense-Making
3(x + 2)
4x
§  Let’s say you are a high school
student who has been taught
that the “bottom number” tells
you the number of pieces and
the “top number” tells you the
number to shade. How does
that help you here?
§  What other ways of thinking
about a fraction could help you
make sense of this expression?
§  What big ideas about fractions
would help you reason here?
Day 2 Homework
§  Read Chapter 2 and 3 from Young
Mathematicians at Work: Constructing
Fractions, Decimals and Percents (the
green book)
§  Be prepared to share your responses,
questions and wonderings with the
group tomorrow morning.
karaimm@gmail.com
Kara Imm