Download Big Ideas in Math 4-6 Conversation Guide Session 3

Big Ideas in Mathematics for Grades 4-6
with Dr. Marian Small
Session 3 of 4
Focus on Patterns & Relations and Statistics & Probability
Overview of Learning Opportunity: This recorded webinar will focus on such
questions as:

What are the big ideas you should bring to students’ attention when
teaching Math in Division 2?

How do they link to curriculum?

How do teachers ask those critical questions that help students see those
ideas?

How can thinking about big ideas help shape lessons?
This conversation guide is intended for Professional Learning Communities,
instructional leaders or as a self-paced study to help guide instruction,
conversations and reflections on using the big ideas in math as a means to
deepen student mathematical understanding.
*Please visit the following link to access the handouts for this session
http://erlc.wikispaces.com/Big+Ideas+in+Math+4+-+6
Webinar Outline/Table of Contents
Time
Clip Information & Key Ideas
0:00
Recap from previous session
What are the Big Ideas in Patterns & Relations and Statistics & Probability?
2:08
Would you…
5:47
22 x 13
10:34
What about division?
11:15
Which representation?
17:07
And what procedure?
24:06
What about estimating?
27:32
I added...
31:03
I subtracted...
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31:07
33:39
34:17
34:49
39:33
45:56
48:51
53:07
1:02:33
1:04:39
1:10:41
The product is...
I wonder...
Patterns
Multiplication table
Look at this
Picture a 100 chart
How else could you...
Variables and equations
Data
Which graph would you use?
How do these graphs differ in the impression they give?
Closing Remarks
End of the webinar
This “Conversation Guide” provides an overview of the webinar with ideas for
continuing the conversation as well as questions for extended learning. This
guide may also assist with self-paced study when viewing the archived webinar.
Time
Code
Clip
Introductions
Clip Information, Key Points &
Suggested Activities to use this
Webinar for your own PD Sessions.
Introductions
 Introduce yourself
 Ask participants to introduce
themselves
*Session handouts are available at
Questions for Extended Learning
Opportunities
http://erlc.wikispaces.com/Big+Ideas+i
n+Math+4+-+6
2:08
Agenda
Outline of Marian’s 4 Math Webinar
sessions
• Session 1– A focus on number
• Session 2 – A focus on operations
• Session 3 – A focus on patterns and
relations and statistics and probability
• Session 4 – A focus on shape and
space
Would you…
As a teacher, would you calculate
532 – 99 the same way you would
calculate 532 – 111?
Marian left a thought for us: should our
students calculate these two questions
the same way? What do you think?
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5:47
22 x 13
Discussion question: would you prefer
the method on the left or the visual
method on the right? When the
teachers in the clips discuss their
preference and why they prefer a
specific method, do any of their
comments speak to you?
Are there pros and cons to both
methods?
10:34
What about
division?
Division:
• what it means and
• how to calculate
Marian commented that division is tricky
because it builds on the other three
operations. What does she mean by
this? As a group, can we come up with
examples as to how most case in
addition and subtraction all fit in to
division?
Is it important/helpful for students to see
how the other three operations fit into
division?
11:15
Which
representation?
Which picture best shows what 72 ÷ 3
means?
Marian continues to show us multiple
representations. As you can see from
the teachers’ responses different people
preferred a different representation.
Are there additional ways we can
represent 72 divided by three?
or
Discussion question: how can we
increase our inclusion of multiple
representations in our daily classroom
work? Do we think this is necessary?
Note: rectangles represent 10, small
squares represent 1
If we think it is important, are there
ways we can collect and share student
samples so that we all have a larger
pool of resources to draw from?
Teachers’ responses:
Page 3 of 8
17:07
And what
procedure?
What do each of the procedures below
help a student see better than the other
procedures?
Discussion question: which method do
you commonly use in class?
Why do you think you use your
preferred method?
Do you think we should try all three
methods in our classes? Or do you
think multiple strategies might lead to
confusion?
Discussion question: when you look at
the three methods, do any of them
emphasize process over
understanding? Do any of them
emphasize understanding over
process? Does it matter what we
emphasize in our classrooms?
Note: larger images are available in the
presentation.
24:06
27:32
What about
estimating?
I added...
It’s no longer just about rounding rules
I added
 3 numbers.
 One is little.
 One is close to double the other.
 The sum is 5000.
 What could the numbers be?
31:03
I subtracted...
I subtracted
 a number from 3000.
 The result had the digits 3 and 4 in it.
 What could the subtraction have
been?
31:07
The product is...
The product of two numbers is almost
400.
What might the numbers be?
This example is considered an open
question. The possibilities are infinite.
This question combines the ideas of
adding and multiplication and
estimation.
Is there a way to bring in the vague
descriptions or how could be open this
up so it didn't necessarily have to be
exactly 5000? Is that a good idea or
might we have too many possibilities?
This is an example of a double
question. First we have to find which
two numbers could give us the exact
product of 400. Then we have to adjust
those numbers so that the product is
almost 400.
Discussion question: what are some
answers you might get from your
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students? Where can you see some of
your classroom discussions going
based on your students answers?
33:39
I wonder...



34:17
Patterns
34:49
Multiplication
table
I divided _3_ by 4.
The answer was a 3 digit number.
Tell me anything about _3_ that
you’re sure of.
Big idea: there are a variety of
appropriate ways to estimate sums,
differences, products and quotients
depending on the numbers involved and
the context.
WNCP Pattern Outcomes
 Grade 4- focus on identifying patterns
and relationships in tables and charts
 Grade 5- focus on pattern rules to
predict
 Grade 6- focus on table of value
relationships
Big idea: many geometric attributes,
measurements and calculations
involving numbers are simplified by
using patterns.
What is a pattern in the table that might
help you figure out 12x5?
Big idea: patterns represent identified
regularities. There is always an element
of repetition, whether the same item
repeats, or whether a “transformation”,
for example adding one, repeats.
In this section of the clip, a teacher
volunteered a strategy she saw and
Marian shared her strategy. Did you
come up with a strategy different from
either of these? Did seeing these
strategies help you come up with one of
your own?
How can sharing and discussing pros
and cons of strategies help students
develop strategies that work for them?
39:33
Look at this
To show the final digits of the multiples of
8:
Marian noted that many of these
multiples result in visual patterns, and
the six star is also the four star, In fact,
the following multiples all have the
same shape:
•
•
•
•
6 and 4
9 and 1
8 and 2
Any two numbers that add up to 10
Do we believe this? Why does this
Page 5 of 8
happen?
Big idea: the mathematical structure of
a pattern can be represented in a
variety of ways.
To show the final digits of the multiples of
6:
Teacher sample in the clip:
45:56
Picture a 100
chart
Marian noted that patterns in which
students continue to add the same
amount each time, are the most
important patterns they will work on in
grades 7 -11. Why might this be the
case?
Note: this visual is larger in the
presentation.
48:51
How else could
you...
How else could you represent the pattern
2, 5, 8, 11,….
Discussion idea: Can we think of
examples in which pattern work leads to
algebra?
Big idea: some ways of displaying data
highlight patterns.
These visual representations are a
strong foundational piece for algebra.
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A visual representation really helps
students to create a formula in division
three.
Side note, Formula: y = 3n-1
53:07
Variables and
equations
•
•
•
Grade 4- problems as equations with
unknowns; solve one-step equations
Grade 5- single-variable, one-step
equations
Grade 6- generalizations written using
variables; model equality preservation
Big idea: algebra is a way to represent
and explain mathematical relationships
and to describe and analyze change.
Big idea: using variables is a way to
efficiently and generally describe the
relationships that can also be
described.
Discussion idea: what do these two big
ideas mean to us? What are some
possible strategies we could use to help
students understand these big ideas?
1:02:33
Data
•
•
•
1:04:39
Grade 4- many-to-one
correspondence for pictographs and
bar graphs
Grade 5- double bar graphs
Grade 6- line graphs
Which graph
would you use?
In this clip but there is a discussion
about bar graphs versus pictographs,
and about the scale used.
A.
B.
C.
D.
1:10:41
How do these
graphs differ in
the impression
they give?
Big idea: graphs are powerful data
displays since visual displays quickly
reveal information about data.
bar graph with scale of 2
bar graph with scale of 5
pictograph with scale of 2
pictograph with scale of 5
How do these differ in the impression they
give?
Discussion idea: what types of graphs
do we most commonly used in our
classrooms? Do we ask students why
we prefer these types of graphs? Is the
“why” a useful discussion?
The way we display data, really affects
the impression we get from that data.
Discussion idea: does anyone have an
example of an extremely biased graph
we could show our students? It might
be a great learning experience to try to
get students to buy into the misleading
graph and then dissect the graph
together.
Alternately, if you are working with a
group of teachers, perhaps you could
bring in an extremely biased (or subtly
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biased) graph and dissect the graph
with the teachers.
1:13:24
Closing Remarks
Note: Larger graphs are available in the
presentation
Closing Remarks
Write a word or two that stood out for you
from today’s session.
Reflect on your learning – What is one
thing you will try in your classroom to
engage students in math’s big ideas?
Contact Info.
[email protected]
End of the webinar
Resources
Additional Resources
Share additional handouts with
participants
http://erlc.wikispaces.com/Big%20Ideas%
20in%20Math%204%20-%206
If participants are interested in Marian’s
PowerPoint slides or the archives from
the 4 sessions, they can be found on
the ERLC wiki
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