Big Ideas in Mathematics for Grades 4-6
with Dr. Marian Small
Session 1 of 4
Focus on Number
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
18:15
Webinar Outline/Table of Contents
Clip Information & Key Ideas
Welcome and Using the Elluminate Live Tools
Introductions
Why Big Ideas in math are important
Alike and Different – how do you compare number pairs?
Big Idea: Classifying numbers helps us gain more insight into those
numbers
Purpose of Big Ideas
19:53
Marian’s 4 Sessions in this series
Outline of Marian’s 4 Math Webinar sessions
Time
1:55
8:43
12:45
17:08
Page 1 of 8
20:31
27:21
34:06
40:50
46:18
49:14
49:56
50:30
51:16
52:03
56:59
1:00:10
1:03:24
1:10:27
1:11:30
1:12:55
1:15:22
1:16:25
1:17:48
1:22:08
What are the Big Ideas in Number?
Which number does not belong?
Writing numbers in digit form using various numbers that were
given in word form
Using exactly 15 base ten blocks to represent a number, what
could the number be? How do you know?
Which set of objects is easier to count?
How could you represent 175 to show that it is 7 groups of 25?
Big Idea: You can represent numbers in a variety of ways. Each
representation of a number can focus on a different aspect of the
number.
How could you represent 175 as 17 tens and 5 more?
How do you phrase the questions about visual representation
without leading the students?
The questions are not asked until students have created their
representations
How does thinking of 138 and 173 in terms of 150 help you decide
which is greater?
A newspaper reports that about 150 people attended a meeting.
Exactly how many people do you think that might be?
Visualizing numbers on a number line
Fractions
Big Idea: What the “whole” is matters
Why does 2/3 of a set mean the same thing as 2/3 of a whole?
Writing fractions as decimals
Comparing fractions
What fraction does the green pattern block represent?
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
8:43
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
12:45
compare number
Alike and Different – how do you compare
Discuss:
Page 2 of 8
pairs
17:08
Big Idea
18:15
Purpose of Big
Ideas
19:53
Outline
20:31
Which number
does not belong
number pairs?
A: 30 and 40
B: 55 and 155
C: 98 and 102
Big Idea: Classifying numbers helps us
gain more insight into those numbers
Big Ideas are meant to:
• help you as a teacher see what you
are really going for
• Provide you with a teaching
framework- to see how outcomes are
connected.
• Give purpose to the activities you do
• Help students build connections
• Help students see the forest for the
trees
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
Which number does not belong?
6, 18, 27 or 90
Which Group (A, B or C) has numbers
that are more alike? Why
Note: no “wrong” answers – open up
discussion on how teachers see and
classily numbers
Great discussion on whether or not we
make the big ideas explicit. As
teachers, do we know our Big Ideas?
Do we share them with our students?
Should we? If so, what are some ways
we can make our Big Ideas more
obvious to students?
Do we think as math as a convergent or
a divergent subject?
What are possible reasons for choosing
any of the specified numbers?
27:21
Writing numbers
in digit form
Writing numbers in digit form using
various numbers that were given in word
form
When you read the number, some of the
words you say are: hundred, three, fifty,
twenty, thousand, six
What could the number be?
Possible answers:
26 350
53 621
121 653
(note, there are no “wrong” answers –
teachers can make knowledgeable
judgments on any of the numbers )
Have teachers try some of the given
numbers…. (see slide)
Discuss:
• Do the numbers you came up with
meet the criteria?
• Why do the written numbers have
so many digits, why are the
answers larger numbers?
• Discussion about place value, 6
words and 6 digits – do the number
of words have to be equal to the
number of digits?
• Patterns in place value system –
built on patterns and patterns make
our work more efficient
• As soon as we see “thousand” we
know at least 4 digits
• How does this activity vary from
“here’s a number, write the words”?
Page 3 of 8
34:06
Representing
numbers with
base 10 blocks
Using exactly 15 base ten blocks to
represent a number, what could the
number be? How do you know?
•
•
•
•
•
•
40:50
46:18
Counting with
visuals
Subgroups
Possible answers:
78: 7 tens and 8 ones
15: 15 ones
What is the difference between the
two columns of numbers below?
276
15
555
150
924
240
771
402
First row the digits add to 15 (the
number of blocks) these are more
“traditional answers”
Second row requires more thinking on
the student’s part (240 is actually 1
hundreds block and 14 ten blocks)
The second row, each number’s digits
add to 6, why is this? How is this
connected to patterns and place
value?
What do teachers think about having
these mathematical conversations with
students?
Have teachers come up with some
possible answers to the question on the
slide.
As a group, look at the answers
generated, what are the similarities and
differences in the answers?
(note, if there are not many answers
generated, you can show the list of
possible answers from the slide).
Marian left the following questions
unanswered:
In the second row (of the possible
answers), each number’s digits add to
6, why is this? How is this connected to
patterns and place value?
Which set of objects is easier to count?
Set A appears random, set B has the
items lined up in groups of 5 (or groups of
3, depending on how you look at it)
• Different ways to keep track of
items while counting
• Big Idea: Subgroups… to count
the number in a group, we often
create subgroups and count the
number of subgroups.
• How does this big idea of
subgroups relate to tally marks?
• Visual representations – how can
students see connections
between different ideas
(subgroups and tally marks)
• Sub grouping is related to place
value… one reason we re-group
ones into tens and tens into
hundreds is to make counting
easier/quicker
Discussion possibilities:
How could you represent 175 to show that
it is 7 groups of 25?
Discussion question for teachers: Why
might have Marian choose 175 as a
When objects are lined up in groups,
are they always easier to count?
Have teachers come up with examples
when it is easier to line up items and
when it does not matter. Why might it
not matter some times?
Does it matter if you have lots of objects
vs fewer
Page 4 of 8
Lots of ways to think about 175
• 7 groups of 25: Quarters
• 25 less than 200 :
49:14
Big Idea
starting number?
What other number patterns could we
look at?
Big Idea: You can represent numbers in
a variety of ways. Each representation of
a number can focus on a different aspect
of the number.
Important to let students know why we
ask them to represent numbers in
different ways. A different representation
might show students something they did
not see in the first representation.
49:56
Representing
numbers
50:30
Questioning
techniques
51:16
52:03
Emphasis on
visual
representation
benchmarking
How could you represent 175 as 17 tens
and 5 more?
Responses here would likely create totally
new visual representations
How do you phrase the questions about
visual representation without leading the
students?
• Ask students to represent 175 as
17 tens and 5 more
• Look at student work and have
students explain WHY they drew
what they drew
• When you see different student
responses, talk about them as a
class…. From the 175 as 17 tens
and 5 more diagrams, can you
also see that 175 is 7 groups of
25?
Perhaps the questions are a little leading,
but the questions are not asked until
students have created their
representations
With the new curriculum we emphasize
visual representation, so it is important to
help students understand why we do this
How does thinking of 138 and 173 in
terms of 150 help you decide which is
greater?
• Students may say that 173 is
larger because it has 7 tens
where 138 only has 3 tens
• However, how does thinking
about the two numbers in terms of
150 help determine which number
is larger?
• the Big Idea here is the idea of
benchmarking with familiar
numbers, 150
• When is benchmarking helpful,
what are some life examples
Have you tried these ideas with your
students? If so, what were the results,
did you feel like you were leading them?
If you have not tried some of these
ideas about visual representation, how
might you begin?
Discussion question:
Rules vs Benchmarks – when should
we use rules, when should we use
benchmarks? Is there a place for both
in the new math curriculum?
Page 5 of 8
•
56:59
estimation
where we benchmark?
Even when we get into decimals,
student can bench mark:
o 1.4 and 2.8
o 1.4 is less than 2
o 2.8 is more than 2
o So they know that 2.8 is
more than 1.4
A newspaper reports that about 150
people attended a meeting.
Exactly how many people do you think
that might be?
This question is the reverse of what we
often ask (148 people were at a meeting,
what number do you think the newspaper
would report)
1:00:10
Visualizing
numbers
With the first question, students are now
telling us what numbers they think are
“about 150”.
On a number line, draw two tick marks
and label with any numbers at all, and
then draw a line at the half-way mark
between your two dashes.
Allows for differentiation, some students
may choose large numbers, small
numbers, decimals, etc.
Discussion question:
What choices do we have when
estimating numbers, what biases might
be in the estimation? When might an
estimation preferred over a calculation?
How can this question allow for different
starting points for your students?
What are some answers you might
expect to see/hear in your classroom?
How can the various answers help
students with their own personal
strategies?
Great conversation about choosing two
numbers and then estimating a number
between them. Emphasizes that
benchmarks are helpful in determining the
value of a number
1:03:24
Fractions
Given the diagram below, what fractions
do you see?
Discussion question:
Do you have question from the
resources you are using (textbook, etc)
from which you could take a standard
question and turn it into a rich task?
Do you have colleagues you can work
with and share “rich tasks” resources?
Is this something of interest to you?
Do you see this as an area model (in
sevenths)?
Do you see this with the hexagon as a
whole (in sixths)?
Do you see this as a set (either 3 or 4
pieces as the total number of pieces)
Multiple correct ways to see this since you
Page 6 of 8
are not told what the whole is. This is a
rich task as it allows for multiple
viewpoints with multiple possibilities.
1:10:27
The big Picture
The big Picture: Fractions can represent
parts of regions, parts of sets, parts of
measures, division or ratios.
Big Idea: What the “whole” is matters
How could you adapt this example so it
meets the learner outcomes at your
grade level and the interest of your
students?
1:11:30
Set vs whole
Why does 2/3 of a set mean the same
thing as 2/3 of a whole?
Discussion question:
How can you visually show 2/3 of 7?
You can put it on an area model and
distribute the 7 counters around.
Multiplication and division of fractions
can be tough to think of visually,
especially if you are used to “the rules”
Discuss, what are some common
“rules” associated with the multiplication
of fractions?
Each section has 2 and 1/3 of a counter,
so in two sections you have 4 and 2/3
counters.
1:12:55
Fractions and
meaning of
division
Writing fractions as decimals. Why do we
divide the numerator by the denominator?
For example: 2/3
2 things, and you want to share it 3 ways
Possible suggestions:
Of = means multiply
Multiply numerators and they go on
top…. Multiply denominators and they
go on the bottom (new denominator)
How can we add meaning to these
rules? Should we throw out the rules
altogether?
Discussion question:
So, why do we divide the numerator by
the denominator?
Are there examples we can come up
with that might help with student
understanding?
1:15:22
Comparing
fractions
We need to show that the various
approaches to working with fractions are
all related. This shows how the division
meaning is related to the part of a set
meaning of fractions.
Which is more, 2/3 or 3/5?
Big idea: A fraction is not meaningful
without knowing what the whole is.
Discussion: Can a ¼ ever be more
than ½?
When we talk about comparing fractions
how can we step back and look at the
big ideas?
Possible ideas:
 ½ of a small pizza vs. ¼ of an extra
Page 7 of 8

1:16:25
Visualization of
fractions &
importance of
knowing the
whole
What fraction does the green pattern
block represent?
large pizza
½ of your allowance vs. ¼ of an
adult’s paycheck
If we asked students to take a given
shape (like the green triangle) and
create some kind of fraction, what might
their results look like? How can we use
their diagrams in our fraction
discussions?
It depends on what the whole is….. this is
one of many possibilities for the whole:
1:17:48
Closing Remarks
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.
marian@unb.ca
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
Page 8 of 8