Transitioning to the
Common Core State
Standards – Mathematics
Pam Hutchison
pam.ucdmp@gmail.com
AGENDA
Party Flags
 Overview of CCSS-M

Standards for Mathematical Practice
 Standards for Mathematical Content

Word Problems and Model Drawing
 Math Facts



Quick review – Multiplication and Division Facts
Area Models, Multiplication, and Division
Expectations
We are each responsible for our own learning
and for the learning of the group.
 We respect each others learning styles and
work together to make this time successful for
everyone.
 We value the opinions and knowledge of all
participants.

Erica is putting up lines of colored flags for a party.
The flags are all the same size and are spaced equally along the line.
1. Calculate the length of the sides of each flag, and the space between flags.
Show all your work clearly.
2. How long will a line of n flags be?
Write down a formula to show how long a line of n flags would be.
CaCCSS-M
 Find a partner
 Decide who is “A” and who is “B”
 At the signal, “A” takes 30 seconds to talk
 Then at the signal, switch, “B” takes 30 seconds to
talk.
“What do you know about the CaCCSS-M?”
CaCCSS-M
“What do you know about the CaCCSS-M?”
Using the fingers on one hand, please show me how
much you know about the CaCCSS-M
National Math Advisory Panel
Final Report
“This Panel, diverse in experience,
expertise, and philosophy, agrees
broadly that the delivery system in
mathematics education—the system
that translates mathematical knowledge
into value and ability for the next
generation — is broken and must be
fixed.”
(2008, p. xiii)
Common Core State Standards
Developed through
Council of Chief State School Officers
and
National Governors Association
Common Core State Standards
How are the CCSS different?
The CCSS are reverse engineered from an
analysis of what students need to be
college and career ready.
The design principals were focus and
coherence. (No more mile-wide inch
deep laundry lists of standards)
How are the CCSS different?
Real life applications and
mathematical modeling are essential.
How are the CCSS different?
The CCSS in Mathematics have two sections:
Standards for Mathematical CONTENT
and
Standards for Mathematical PRACTICE


The Standards for Mathematical Content
are what students should know.
The Standards for Mathematical Practice
are what students should do.
 Mathematical “Habits of Mind”
Standards for
Mathematical
Practice
Mathematical
Practice
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving
them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the
reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated
reasoning.
1. Make sense of problems and persevere in
solving them
6. Attend to precision
OVERARCHING HABITS OF MIND
CCSS Mathematical Practices
REASONING AND EXPLAINING
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the
reasoning of others
MODELING AND USING TOOLS
4. Model with mathematics
5. Use appropriate tools strategically
SEEING STRUCTURE AND
GENERALIZING
7. Look for and make use of structure
8. Look for and express regularity in repeated
reasoning
CCSS Mathematical Practices

Cut apart the Eight Standards for Mathematical
Practice (SMPs)

Look over each Taxedo image and decide which
image goes with which practice
 The
more frequently a word is used, the larger the
image

Using the Standards for Mathematical Practice
handout…did you get them right?
 Glue

the Practice title to the appropriate image.
What did you notice about the SMPs?
Reflection



How are these practices similar to what you
are already doing when you teach?
How are they different?
What do you need to do to make these a
daily part of your classroom practice?
Supporting the SMP’s
Summary
 Questions to Develop Mathematical
Thinking
Common Core State Standards Flip Book
 Compiled from a variety of resources,
including CCSS, Arizona DOE, Ohio DOE
and North Carolina DOE
 http://katm.org/wp/wp-content/uploads/
flipbooks

Standards for
Mathematical
Content
Content Standards


Are a balanced combination of procedure
and understanding.
Stressing conceptual understanding of key
concepts and ideas
Content Standards

Continually returning to organizing
structures to structure ideas
place value
 properties of operations


These supply the basis for procedures
and algorithms for base 10 and lead
into procedures for fractions and
algebra
“Understand”
means that students can…
 Explain the concept with mathematical
reasoning, including
Concrete illustrations
 Mathematical representations
 Example applications

Organization K-8

Domains

Larger groups of related standards. Standards
from different domains may be closely related.
Domains K-5
Counting and Cardinality (Kindergarten
only)
 Operations and Algebraic Thinking
 Number and Operations in Base Ten
 Number and Operations-Fractions
(Starts in 3rd Grade)
 Measurement and Data
 Geometry

Organization K-8

Clusters


Groups of related standards. Standards from
different clusters may be closely related.
Standards
Defines what students should understand and be
able to do.
 Numbered

5th Grade – CCSS-M
Look through the CCSS-M
 What has changed?

What’s missing?
 What’s still there but what they are asking for is
different?
 What’s the same?

Word Problems
and Model Drawing
Model Drawing


A strategy used to help students
understand and solve word problems
Pictorial stage in the learning sequence of
concrete – pictorial – abstract

Develops visual-thinking capabilities and
algebraic thinking.
Steps to Model Drawing
1) Read the entire problem, “visualizing”
the problem conceptually
2) Decide and write down (label) who
and/or what the problem is about
3) Rewrite the question in sentence form
leaving a space for the answer.
4) Draw the unit bars that you’ll eventually
adjust as you construct the visual image of
the problem
H
Steps to Model Drawing
5) Chunk the problem, adjust the unit
bars to reflect the information in the
problem, and fill in the question mark.
6) Correctly compute and solve the
problem.
7) Write the answer in the sentence and
make sure the answer makes sense.
Missing Numbers 1
Mutt and Jeff both have money. Mutt has
$34 more than Jeff. If Jeff has $72, how
much money do they have altogether?
H
Missing Numbers 2
Mary has 94 crayons. Ernie has 28
crayons less than Mary but 16 crayons
more than Shauna. How many crayons
does Shauna have?
Missing Numbers 3
Bill has 12 more than three times the
number of baseball cards Chris has. Bill
has 42 more cards than Chris. How many
baseball cards does Chris have? How
many baseball cards does Bill have?
Missing Numbers 4
Amy, Betty, and Carla have a total of 67
marbles. Amy has 4 more than Betty.
Betty has three times as many as Carla.
How many marbles does each person
have?
Representation

Getting students to focus on the
relationships and NOT the numbers!
Computation
Teaching for Understanding
Telling students a procedure for solving
computation problems and having them
practice repeatedly
rarely results in fluency
Because we rarely talk about how and
why the procedure works.
Teaching for Understanding


Students do need to learn procedures for
solving computation problems
But emphasis (at earliest possible age) should
be on why they are performing certain
procedure
Learning Progression
Concrete
 Representational
 Abstract
Research
Students who learn rules before they learn
concepts tend to score significantly lower
than do students who learn concepts first
 Initial rote learning of a concept can create
interference to later meaningful learning

Fact Fluency

Institute of Educational Sciences Practice
Guide “Assisting Students Struggling with
Mathematics: Response to Intervention for
Elementary and Middle Schools”

Recommends approximately 10 minutes per
day building fact fluency
Fact Fluency


The intent IS NOT to administer basic fact
tests!
Teachers need to build basic fact strategy
lessons for conceptual development, which
builds fluency.
Fact Fluency
Fact fluency must be based on an
understanding of operations and
thinking strategies.
 Students must

Construct visual representations to
develop conceptual understanding.
 Connect facts to those they know
 Use mathematics properties and
relationships to make associations

Multiplication
Multiplication

3x2
3
groups of 2
 Repeated
Addition
2+2+2
Multiplication
3

rows of 2
This is called an “array” or an “area model”
Advantages of Arrays
as a Model

Models the language of multiplication
4 groups of 6
or
4 rows of 6
or
6+6+6+6
Advantages of Arrays
as a Model
Students can clearly see the difference
betweenfactors (the sides of the array) and
the product (the area of the array)
7 units
4 units

28 squares
Advantages of Arrays

Commutative Property of Multiplication
4x6
=
6x4
Advantages of Arrays

Associative Property of Multiplication
(4 x 3) x 2
=
4 x (3 x 2)
Advantages of Arrays

Distributive Property
3(5 + 2)
=
3x5+3x2
Advantages of Arrays
as a Model

They can be used to support students
in learning facts by breaking problem
into smaller, known problems

For example, 7 x 8
8
5
7
3
35 + 21 = 56
4
7
8
4
28 + 28 = 56
Teaching Multiplication Facts
1st group
Group 1
Repeated addition
 Skip counting
 Drawing arrays and counting
 Connect to prior knowledge

Build to automaticity
Multiplication

3x2
3
groups of 2
2
4
6
Multiplication

3x2
3
groups of 2
2+2+2
Multiplying by 2
Doubles Facts
 3 + 3
 2 x 3
5+5
 2 x 5

Multiplying by 4
Doubling
 2 x 3 (2 groups of 3)
 4 x 3 (4 groups of 3)
2 x 5 (2 groups of 5)
 4 x 5 (4 groups of 5)

Multiplying by 3
Doubles, then add on
 2 x 3 (2 groups of 3)
 3 x 3 (3 groups of 3)
2 x 5 (2 groups of 5)
 3 x 5 (3 groups of 5)

Teaching Multiplication Facts
Group 1
Group 2
Group 2

Building on what they already know


Breaking apart areas into smaller known areas
Distributive property
Build to automaticity
Breaking Apart
7
4
Teaching Multiplication Facts
Group 1
Group 3
Group 2
Group 3

Commutative property
Build to automaticity
Teaching Multiplication Facts
Group 1
Group 2
Group 3
Group 4
Group 4

Building on what they already know


Breaking apart areas into smaller known areas
Distributive property
Build to automaticity
Connecting
Multiplication
and
Division
Division
 What
does 6
 Repeated
2 mean?
subtraction
6
-2 1 group
4
-2 2 groups
2
-2 3 groups
0
3 groups
Measurement Division
I
have 21¢ to buy candies with. If each
gumdrop costs 3¢, how many gumdrops
can I buy?
Fair Share Division
 Mr.
Gomez has 12 cupcakes. He wants
to put the cupcakes into 4 boxes so that
there’s the same number in each box.
How many cupcakes can go in each
box?
Difference in counting?

Measurement

4 for you, 4 for you, 4 for you
 And


so on
Like measuring out an amount
Fair Share

1 for you, 1 for you, 1 for you, 1 for you

2 for you, 2 for you, 2 for you, 2 for you
 And

so on
Like dealing cards
Measurement Division
 What
6
does 6
2 mean?
split into groups of 2
Fair Share Division
 What
6
does 6
2 mean?
split evenly into 2 groups
Models for Division
 Repeated
subtraction
 Groups
 Finding
the number in each group
 Finding the number of groups
 Arrays
– finding the missing side
Repeated Subtraction

21 ÷ 3
1 group for you
 1 group for you
 1 group for you
 1 group for you
 1 group for you
 1 group for you
 1 group for you

21 – 3 = 18 left
18 – 3 = 15 left
15 – 3 = 12 left
12 – 3 = 9 left
9 – 3 = 6 left
6 – 3 = 3 left
3 – 3 = 0 left
7 groups of 3
Skip Counting
30 ÷ 5

5
1

10
2

15
3

20
4

25
5

30
6
Using Arrays
5
3
?
15
6
?
4
24
Using Arrays
4
?
7
28
6)
8
48
Connection to Multiplication

42 ÷ 7 =
think “7 x _?_ = 42”

Fact Families
Practicing Facts
Triangle Flash Cards
56
Flash Card Practice
Facts I Know
Quickly
Facts I Can
Figure Out
Quickly
Facts I Am
Still Learning
Create 1
representation
for each fact
Create 2
representations
for each fact
Assessing Facts
Fluency Assessments
20-25 facts
 2 colors of pencils (or pens)

After 60 seconds, call switch. Students
change the color of the pencil they are using.
 Give students another 60-90 seconds
 If students finish before time to stop,
continue to write and solve your own fact
problems

Advantages

All students get to finish!

Let’s you assess both fluency and accuracy.
Area Models and
Multiplication
Using Arrays to Multiply
23
x 4
80
12
92
4 rows of 20 = 80
4 rows of 3 = 12
Patterns in Multiplication
Groups of 4
groups
1’s
1
4
2
8
3
12
.
.
.
.
.
.
10’s
40
80
120
100’s
400
800
1200
.
.
.
.
.
.
5
20
200
2000
8
32
320
3200
Using Arrays to Multiply

Use Base 10 blocks and an area model to
solve the following:
21
x 13
Multiplying and Arrays
21
13
21
x 13
Partial Products
21
x 13
200
10
60
3
273
(10  20)
(10  1)
(3  20)
(3  1)
14
31
Partial Products
31
x 14
4
120
10
300
434
(4  1)
(4  30)
(10  1)
(10  30)
Pictorial Representation
84
x 57
80
50
+
7
+
4
50  80
4,000
50  4
200
7  80
560
74
28
Pictorial Representation
37
x 94
30
90
+
4
+
7
90  30
2,700
90  7
630
4  30
120
47
28
Pictorial Representation
347
x 68
300 + 40 +
60 18,000 2,400
7
420
+
8 2,400
320
56
Decimals
0.4
0.4 x 0.6
0.6
Try It!

0.3 x 0.8

0.6 x 3

2 x 1.4

1.4 x 3.1
Patterns in Multiplication
Start with 3
6
x6
.6
.06
.006
3
18
1.8
.18
.018
.3
1.8
.18
.018
.0018
.03
.18
.018
.0018
.003
.018
.0018
Fractions
2  3
3
5