Mathematical habits of mind
• Noticing patterns and regularity
•
•
•
•
•
•
Making connections
Seeking generalizations
Abstracting the essence
Looking for alternatives
Inclination to argument & proof
etc.
• Mathematical Concepts
Number
Measurement
Symmetry
Probability
Order
System
Representation
Error / Uncertainty
Proportion
Relationships
Function
Problem solving
Quantification
Ratio
Behavior
Pattern
Truth
Change
Prediction
• Mathematical Understandings
Our number system maintains order and is rich with
patterns.
Mathematicians quantify data in order to establish realworld probabilities.
All measurement involves error and uncertainty.
Nanci Smith
The Role of Teacher
Questions
• To engage students with important
mathematical tasks
• To scaffold student learning without taking
the thinking away from students
• To help students pull the mathematics out
of their experiences, generalize what they
have learned, and make the mathematics
explicit
Multiplication Understandings
• Multiplication is an operation that works
with different “things.”
• Multiplication has several different
meanings in different contexts including:
– Repeated addition
– Total number based on groups and how
many in each group
– Area
Nanci Smith
1
Number Understandings
• There are several ways to think of any number
including product of primes, or factors.
• Any number is a member of one or more number
systems each of which has clearly defined
properties (including basic operations).
• Numbers represent quantities that can be
compared as ratios, percents, and fractions.
• A number can be written in different forms.
• Operations and procedures have specific
meanings, and follow specific patterns relating to
one another.
Nanci Smith
Systems of Equations
Systems
Interdependence
Commonality
Solutions
Quantity
Equation
Comparison
Relationship
Analysis
Application
Model
Systems of equations may have common solutions.
Analysis of systems can be used to discover interdependence.
Real world situations can be modeled by systems and analyzed
to find optimum relationships.
Nanci Smith
CRITICAL COMPONENTS
OF EDM
• identify beginning, developing, and secure goalscompare to standards (consider the secure goals
when assessing and grading)
• use a variety of ungraded on-going
assessments – exit cards, double entry
journals, math boxes, white boards, etc.
• regularly play the games – be sure there is
time for EVERYONE to participate, not just
those who finish early
• discuss, pre-teach and use the vocabulary –
students create posters for a Math Word Wall
• keep moving – not everything needs to be
mastered (beginning, developing, secure)
JUDICIOUSLY CHOOSE SELECTIVELY ABANDON!
Judy Rex
Judy Rex
2
DOUBLE ENTRY JOURNAL
DOUBLE ENTRY JOURNAL
(Advanced)
(Basic)
CONTENT
Note Taking
RESPONSE
Sense Making
•
•
•
•
•
•
•
•
•
•
•
•
Key phrases
Important words
Main ideas
Puzzling passages
Summaries
Powerful passages
Key parts
Etc.
My
Appointment Clock
•
•
•
•
How to use ideas
Why an idea is important
Questions
Meaning of key words,
passages
Predictions
Reactions
Comments on style
Etc.
CONTENT
RESPONSE
Key passages
Key vocabulary
Organizing
concepts
Key principles
Key patterns
Why ideas are
important
Author’
Author’s
development of
elements
How parts and
whole relate
Assumptions of
author
Key questions
ANOTHER VOICE
Teacher
Author
Expert in field
Character
Satirist
Political cartoonist
Etc.
MAX
Round the Clock Learning Buddies
Make an appointment with 12 different people – one for
each hour on the clock. Be sure you both record the
appointment on your clocks. Only make the appointment if
there is an open slot at that hour on both of your clocks.
Tape this paper inside a notebook, or to
something that you will
bring to class each day.
Judy Rex
3
B
I
Write a story
problem in which
you use 2 items and
the sum is 48.
N
Write an addition
word problem
based on something Write an addition
that has happened word problem using
to you.
2 items.
Write a subtraction
word problem in
which you start with
Write a subtraction 112 and you
story problem using subtract two
2 items.
different items.
G
Write a story
problem in which
you have $25 and
you go to Walmart.
Do not spend all of
your money.
Write a story
problem in which
Write an addition
you use 2 items and word problem using
the difference is 48. 2 items.
O
The Red Contract
Key Skills: Graphing and Measuring
Key Concepts: Relative Sizes
Note to User: This is a Grade 3 math contract for students below grade level in these skills
Write a word
problem in which
you use 3 items and
the sum is 112.
red
the p
e to sho
Com work ay
th
m a M ond
on nd
a
y
sda
Tue
Write an addition
problem in the
situation where you
go to Toys R Us.
Us
s o e th
yo lve e d
an ur fo the omin
p
yo d th lde rob oes
ur en r. D lem to
an w
sw rite raw s in
er
s
Write a subtraction
story problem in
Write a story
Write a story
Write a subtraction
which you start with problem in which
problem in which
word problem about Write an addition
98 and you subtract you use 2 items and you use 3 items and something that
word problem using
two different items. the sum is 105.
the sum is 87.
happened at home. 2 items.
an aph e
r
n
.
h
g
t
ig
nt
a
e s n g ri
D a l o s in e p p h k e
r
u u
im r b l g r a m a , o
a n a p e u r e u r e n in g
t
p t o h
e a t y T in it
cr G e ed. , pa l of
ov ng de
pr w i o
a p d ra m
Write a subtraction Write a subtraction
problem about you story problem using
and your friends.
2 items.
It's your Birthday!
Write a word
Write a word
problem telling us
problem in which
how much money
you use 3 items and you got for your
the sum is 98.
birthday.
Write a story
problem in which
you use 2 items and
the difference is
105.
Write a word
problem in which
Write a story
you go to the mall
Write an addition
problem in which
and spend money.
word problem using you use 2 items and You must start with
2 items.
the sum is 76.
$50.
Write an addition
story problem about
something that
happened at school.
Write a subtraction
story problem in
which you start with
87 and you subtract
two different items.
The Green Contract
Co
mu mp
Re ltip lete
on cor licat the
the d yo ion dom
wa ur cha ino
ll c ans llen es
ha w e ge
rt
rs .
e the
Solv raph
tg
grea in your n
ca
tery
mys er. You
on
fold
eone ’d
math ith som
u
if yo
w
r
Work en team r answe
re
u
g
yo
r
The
eck
ache
. Ch
Like ith the te
W
Extend
Make a
group story
or one of
your own –
that uses
measurem
ent and at
least one
graph. Turn
it into a
book at the
author
center
ring
measu until
at the
er
Work hing cent red
ap
the
and gr complete
you
work
Read
Apply
Extend
Alexand
er Who
Used to
be Rich
Last
Sunday
or Ten
Kids, No
Pets
Complete the
math
madness
book that
goes with the
story you
read.
Now, make
a math
madness
book based
on your
story about
kids and
pets or
money that
comes and
goes.
Directions
are at the
author
center
Key Skills: Graphing and Measuring
Key Concepts: Relative Sizes
Note to User: This is a Grade 3 math contract for students advanced in these skills
the
e to
Com blue
p
sho
ork
th w n
a
o
m
or
Do
y
a
sd
Tw a t
Tue ursday
mu o-d imed
g
Th
rnin
te s
pe ltip igit
mo
er lica
to
f
mo tio
nit n.
or U s
ea
a
n d in to
F i c e o o l te rn th e e e
a
t
p l s c h p a k e th r
a te a
r a
ou e . M a r
a k o f re fo
m p h d c m s a te
n e
.
a
g r h a o b l s s m lv e
a p p r c la s o
gr
to
an aph e
r h t.
n
g
t
ig
n
a
e s n g ri
D a l o s in e p p h k e
r
u u
im r b l g r a m a , o
a n a p e u r e u r e n in g
t
p t o h
e a t y T in it
cr G e ed. , pa l of
ov ng de
pr w i o
a p d ra m
ring
measu until
at the
er
Work hing cent een
ap
e gr
and gr mplete th
you co
work
Apply
Work with a
friend to
graph the size
of at least 6
things on the
list of “10
terrific things.”
Label each
thing with how
you know the
size
The Blue Contract
Note to User: This is a Grade 3 math contract for students at or near grade level in these skills
W
o rk
p r th e
of oble ev
m e
o
th e u r m s o n n u
e x a th n p a m b
e
p
to o rt b o o g e 7 re d
au of
k
1
dit the . Us
w o yo
e
rk u r d a y
Read
How big
is a foot?
e the
Solv raph
tg
grea in your
eck
tery
mys lder. Ch
fo
ith a
r
math swers w teache
r an ith the
u
o
Y
en w
th
,
y
budd
Key Skills: Graphing and Measuring
Key Concepts: Relative Sizes
the
e to th
Comen ma
p
gre
sho y
work onda
M
on nd
a
ay
d
Fri
Fin
d
B af
Pr oard riend
ob
m
Pa lem ath and
ge s 1 w do
ith
7
M
R a th 1 o 1 0 o
“no eme bo f ou n
m o m be ok. r
re
r
rule tha the
n4
”
ex C o
on tens mple
74 gra ion te t
of phin pro he
o
b
g
Us ur m on lems
p
a
m o e a p th b a g e
au nito eer ook
.
dit r t
w o you o
rk . r
ring
measu until
at the
er
Work hing cent blue
ap
e
and gr mplete th
you co work
e the
Solv ystery
hm
ou
grap lder. Y
eone
ur fo
som ou’d
in yo
with
y
work e team if
can
e blu like.
th
n
o
Read
Apply
Extend
Dinosaur
Before
Dark or
Airport
Control
Research a kind
of dinosaur or
airplane. Figure
out how big it is.
Graph its size on
graph paper or
on the blacktop
outside our
room. Label it by
name and size
Make a book
in which you
combine
math and
dinosaurs or
airplanes, or
something
else big. It
can be a
number fact
book, a
counting
book, or a
problem
book.
Instructions
are at the
author
center
4
5th Grade Division
Know:
1. M ultiplication facts (prerequisite)
2. How to divide
3. Vocabulary: factors, product, dividend, divisor, quotient
4. Not all division will be even.
Understand:
1. D ivision looks for and creates equal groups, or equal “chunks.”
2. T h ere are many ways to do any mathematical operation.
3. M ultiplication and Division have an inverse relationship.
Do:
1. E x p lain how multiplication and division are related.
2. L o ng division with and without remainders.
Intro: Explain double entry journals.
Tell me about multiplication. If I didn’t know a multiplication fact, how could I
figure it out (drawing?) note: looking for equal groups
Starting in 1st grade you learned that addition and subtraction are inverse
operations. What does that mean? Fact families for addition (2 + 3 = 5 so 5 – 3
= 2) Lead to related facts for multiplication and division.
In division we are looking for how many groups I can get out of a number. For
example, if 7 groups of 4 is 28, how many groups of 4 can I get out of 28?
7 4 = 28 28 ÷ 4 = 7
White Board Pre-assessment:
Try: 39 ÷ 3 = 13
135 ÷ 5 = 27
298 ÷ 7 = 42 r. 4
Instruction: 351 ÷ 3 = 117 1 1 1
÷ 5 = 22 r. 1 453 ÷ 11 = 41 r. 2
Talking point: Looking for groups of the divisor. Give other vocabulary for notes.
What are easy numbers to use for the number of groups? 2, 5, 10, 100. If you
know 5, can you know 50? Etc.
Nanci
Smith
What Do You Know About Division?
Pick one of the following RAFT lines to complete. Your
product needs to show what you now know and understand
about division.
Division RAFT
ROLE
AUDIENCE
FORMAT
Multiplication
Division
Poem
TOPIC
We’re
related!
Equal Groups
Dividend
Pictures and I’m coming
captions
out of you
A division
Students
List
How to solve
problem
me
Remainder Equal Groups
Friendly
How come I
letter
don’t fit?
Whichever line you choose, be sure your work clearly shows
that you understand division! Use correct vocabulary. You
may use a problem if it will help you explain what you mean.
Multiplying by 3 and 6!
•
•
•
•
•
•
Play Multiplication Memory card game
(Kinesthetic, interpersonal).
Make a picture book of multiplication facts for 3
and/or 6 (visual/spatial).
Make up a song about (or of) the multiplication
facts for 3 and/or 6 (musical).
Write a diary entry about the 3 and 6
multiplication facts. What are they? How can
you remember them? If you forget one, how
could you figure it out? (Intrapersonal / verbal
linguistic)
Write a story that involves multiplication by 3
and 6 (verbal linguistic).
Show as many different models of multiplication
by 3 and 6 of which you can think. How is
multiplying by 6 related to multiplying by 3?
(Logical / Mathematical)
5
Algebra RAFT
Angle Relationship RAFT
Role
Audience
Format
Topic
Role
Audience
Format
Topic
One vertical
angle
Opposite
vertical angle
Poem
It’s like looking
in a mirror
Coefficient
Variable
Email
We belong
together
Interior
(exterior)
angle
Acute angle
Alternate
Invitation to
interior
a family
(exterior) angle
reunon
My separated
twin
Scale /
Balance
Students
Advice
column
Keep me in mind
when solving an
equation
Missing angle
Variable
Humans
Monologue All that I can be
An angle less
than 180°
Supplementary angle
Variable
**Angles
Humans
Algebra
students
Public
Instruction How and why to
isolate me
manual
Passionate Why you really
plea
do need me!
Wanted
poster
Wanted: My
complement
Persuasive Together, we’re a
straight angle
speech
Video
See, we’re
everywhere!
Multiplication Think Dots
• Struggling to Basic Level
It’s easy to remember how to multiply by 0 or 1! Tell how to remember.
Jamie says that multiplying by 10 just adds a 0 to the number. Bryan
doesn’t understand this, because any number plus 0 is the same number.
Explain what Jamie means, and why her trick can work.
Explain how multiplying by 2 can help with multiplying by 4 and 8. Give at
least 3 examples.
We never studied the 7 multiplication facts. Explain why we didn’t need to.
Jorge and his ____ friends each have _____ trading cards. How many
trading cards do they have all together? Show the answer to your problem
by drawing an array or another picture. Roll a number cube to determine the
numbers for each blank.
What is _____ X _____? Find as many ways to show your answer as
possible.
Algebra
Multiplication Think Dots
• Middle to High Level
There are many ways to remember multiplication facts. Start with 0 and go through 10 and tell
how to remember how to multiply by each number. For example, how do you remember how
to multiply by 0? By 1? By 2? Etc.
There are many patterns in the multiplication chart. One of the patterns deals with pairs of
numbers, for example, multiplying by 3 and multiplying by 6 or multiplying by 5 and
multiplying by 10. What other pairs of numbers have this same pattern? What is the pattern?
Russell says that 7 X 6 is 42. Kadi says that he can’t know that because we didn’t study the 7
multiplication facts. Russell says he didn’t need to, and he is right. How might Russell know
his answer is correct? Give 2 different explanations.
Max says that he can find the answer to a number times 16 simply by knowing how to multiply
by 2. Explain how Max can figure it out, and give at least two examples.
Alicia and her ____ friends each have _____ necklaces. How many necklaces do they have all
together? Show the answer to your problem by drawing an array or another picture. Roll a
number cube to determine the numbers for each blank.
What is _____ X _____? Find as many ways to show your answer as possible.
6
Describe how you would
1 3
+
5 5
solve
or roll
the die to determine your
Explain the difference
between adding and
multiplying fractions,
own fractions.
Compare and contrast
Create a word problem
these two problems:
that can be solved by
1 2 11
+ =
3 5 15
+
Nanci Smith
and
(Or roll the fraction die to
1 1
+
3 2
determine your fractions.)
Describe how people use
Model the problem
fractions every day.
___ + ___ .
Roll the fraction die to
Nanci Smith
determine which fractions
to add.
Describe how you would
solve
2 3 1
+ +
13 7 91
or roll
Explain why you need
a common denominator
the die to determine your
when adding fractions,
own fractions.
But not when multiplying.
Can common denominators
Compare and contrast
ever be used when dividing
these two problems:
fractions?
1 1
3 1
+ and +
3 2
7 7
Create an interesting and
challenging word problem
Nanci Smith
A carpet-layer has 2 yards
that can be solved by
of carpet. He needs 4 feet
___ + ____ - ____.
of carpet. What fraction of
Roll the fraction die to
his carpet will he use? How
determine your fractions.
do you know you are correct?
Diagram and explain the
solution to ___ + ___ + ___.
Kindergarten
Counting
Task 1: Find a way to count & show how
many people are in our class today. How
did you get your answer?
Task 2: Find a way to show how many people are in our
class. How many are absent today? How many are
here today? How do you know?
Task 3: Find a way to show how many boys are in our
class today. How many boys are absent today? How
many girls are here today? How many girls are absent
today? Prove you are right.
Roll the fraction die to
determine your fractions.
7
Adding Fractions
Green Group
Use Cuisinaire rods or fraction circles
to model simple fraction addition
problems. Begin with common
denominators and work up to
denominators with common factors
such as 3 and 6.
Explain the pitfalls and hurrahs of
adding fractions by making a
picture book.
Red Group
Use Venn diagrams to model LCMs.
Explain how this process can be
used to find common
denominators. Use the method on
more challenging addition
problems.
Write a manual on how to add
fractions. It must include why a
common denominator is needed,
and at least three different ways
to find it.
Blue Group
Manipulatives such as Cuisinaire
rods and fraction circles will be
available as a resource for the
group. Students use factor trees
and lists of multiples to find
common denominators. Using
this approach, pairs and triplets
of fractions are rewritten using
common denominators. End by
adding several different problem
of increasing challenge and
length.
Suzie says that adding fractions is
like a game: you just need to
know the rules. Write game
instructions explaining the rules
of adding fractions.
Nanci Smith
8