# 2012 CCSA

```Mathematics Rigor with
Common Core State Standards
CCSA Conference
March, 2012
Kitty Rutherford kitty.rutherford@dpi.nc.gov
Robin Barbour robin.barbour@dpi.nc.gov
www.corestandards.org
Critical Areas
Focal Points
Critical Area
Level
Overview
Mathematical
Practices
3/16/2016 • page 4
Domain
Standards
Cluster
High School Themes
•
•
•
•
•
•
Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
High School Standards Notation
Perform operations on matrices and use matrices in applications.
6. (+) Use matrices to represent and manipulate data, e.g., to represent
payoffs of incidence relationship in a network.
11. Explain why the x-coordinates of the points where the graphs of the
equations y = f(x) and y =g(x intersect are the solutions of the equations f(x)
= g(x); find the solutions approximately, e.g., using technology to graph the
functions, make tables of values, or find successive approximations. Include
cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions.★
K-8 Domains
Domains
Counting and Cardinality
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
Number and Operations - Fractions
Ratios and Proportional Relationships
The Number System
Expressions and Equations
Statistics and Probability
Functions
K
1
2
3
4
5
6
7
8
Standards for Mathematical Practices
1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
Mathematical practices describe the
habits of mind of mathematically
proficient students…
• Who is doing the talking?
• Who is doing the thinking?
• Who is doing the math?
3/16/2016 • page 12
• What rectangles can be made with a
perimeter of 30 units? Which rectangle
gives you the greatest area? How do you
know?
• What do you notice about the relationship
between area and perimeter?
Compared to….
5
10
What is the area of this rectangle?
What is the perimeter of this
rectangle?
3/16/2016 • page 15
Make sense of
problems and
preserve in solving
them.
3/16/2016 • page 16
“What task can I give that will
build student
understanding?”
rather than
“How can I explain clearly so
they will understand?”
Grayson Wheatley, NCCTM, 2002
3/16/2016 • page 17
How Teachers Implemented
Making Connections Math Problems
Types of Math Problems Presented
100
100
90
90
84
80
80
77
69
70
70
61
60
59
60
57
54
52
50
50
41
40
40
30
30
48
46
37
31
24
20
17
16
15
20
16
18
20
19
13
10
10
8
0
0
0
Australia
Czech Republic
Using Procedures
Making Connections
Hong Kong
Japan
Netherlands
United States
Australia
Czech Republic
Using Procedures
Making Connections
Hong Kong
Japan
Netherlands
United States
Lesson Comparison
United States and Japan
The emphasis on skill acquisition is
The emphasis on understanding is
evident in the steps most common in U.S. evident in the steps of a typical Japanese
classrooms
lesson
•Teacher instructs students in concept or
skill
•Teacher poses a thought provoking
problem
•Teacher solves example problems with
class
•Students and teachers explore the
problem
•Students practice on their own while
teacher assists individual students
•Various students present ideas or
solutions to the class
•Teacher summarizes the class solutions
•Students solve similar problems
19
What do you see?
40
10
30
4
2
4
20
Predict some additional data.
40
10
30
4
2
4
21
How close were you?
40
10
30
20
4
2
4
3
22
All the numbers – so?
45
25
15
40
4
3
2
4
10
30
20
2
4
3
23
Where are you?
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
45
25
15
40
4
3
2
4
Merry-go-Round
Water Slide
Fun House
10
30
20
2
4
3
24
Fill in the blanks.
Ride
???
???
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
45
25
15
40
4
3
2
4
Merry-go-Round
Water Slide
Fun House
10
30
20
2
4
3
25
The Amusement Park
Ride
Time Tickets
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
45
25
15
40
4
3
2
4
Merry-go-Round
Water Slide
Fun House
10
30
20
2
4
3
26
The Amusement Park
The 4th and 2nd graders in your school are going
on a trip to the Amusement Park. Each 4th
grader is going to be a buddy to a 2nd grader.
Your buddy for the trip has never been to an
amusement park before. Your buddy want to
go on as many different rides as possible.
However, there may not be enough time to go
on every ride and you may not have enough
tickets to go on every ride.
27
The Amusement Park
The bus will drop you off at 10:00 a.m. and
pick you up at 1:00 p.m. Each student will
get 20 tickets for rides.
Use the information in the chart to write a
letter to your buddy and create a plan for a
fun day at the amusement park for you
28
Standards for Mathematical Practices
1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
Timeline
Common Core
Mathematics Implementation
Common Core State Standards Adopted June, 2010
Year
Standards To Be Taught Standards To Be Assessed
2011 – 2012
2003 NCSCOS
2003 NCSCOS
2012 – 2013
CCSS
CCSS (NC)
2013 – 2014
CCSS
CCSS (NC)
2014 – 2015
CCSS
CCSS (SBAC)
Mathematics Claims
The Smarter Balanced Assessment
Consortium has released a document
outlining four claims about what
mathematically proficient students can do.
The claims are a synthesis of the
Standards for Mathematical Practice, and
form the guiding principles to be used in
creating assessments.
Mathematics Claim 1 &amp; 2
• Students can explain and apply
mathematical concepts and carry out
mathematical procedures with
precision and fluency.
• Students can frame and solve a range
of complex problems in pure and
applied mathematics.
Mathematics Claim 3 &amp; 4
• Students can clearly and precisely
construct viable arguments to support their
own reasoning and to critique the
reasoning of others.
• Students can analyze complex, real-world
scenarios and can use mathematical
models to interpret and solve problems.
http://www.k12.wa.us/smarter/
Which of the following represents 2/5?
a.
b.
c.
d.
For numbers 1a – 1d, state whether or not each
figure has 2/5 of its whole shaded.
ο Yes
ο No
ο Yes
ο No
1c.
ο Yes
ο No
1d.
ο Yes
ο No
1a.
1b.
Scoring Rubric
Responses to this item will receive 0 – 2 points, based
upon the following:
2 points: YNYN
1 point: YNNN, YYNN, YYYN
0 point: YYYY, YNNY, NNNN, NNYY, NYYN,
NYNN, NYYY, NYNY, NNYN, NNNY, YYNY,
YNYY
RIGOR
Conceptual
Understanding
Application
Skills and
Procedures
Rigor through Standards
Rigor though Standards
6th Grade Critical Area:
Students use the meaning of fractions, the meanings of
multiplication and division, and the relationship
between multiplication and division to understand and
explain why the procedures for dividing fractions
make sense. Students use these operations to solve
problems.
Content Acceleration
• Skipping material will create gaps in learning which
jeopardizes foundational content needed to
maximize the likelihood of success in High
School Mathematics
• Compacting 3 years of content into 2 is supported
by research; 2 years into 1 is considered too
challenging
• Considering high school courses is essential when
making middle school recommendations
High School Courses in Middle School
Option 1:
100 %
Math 1 Standards
Option 2:
100 % 6th grade
Math 1 Standards
At A Glance
Instructional Implications High School
• Integer Exponents (8.EE.1)
• Multiplication and Division with Scientific
Notation (8.EE.4)
• Solving Systems by Substitution (8.EE.8)
• Volume of Pyramids, Cones and Spheres
(7.G.6, 8.G.9)
• Surface Area of Pyramids (6.G.4, 7.G.6)
3/16/2016 • page 48
At A Glance
Instructional Implications Math One
• Angles (7.G.5, 8.G.5)
• Using Pythagorean Theorem in 3-D Figures
(8.G.7)
• Mean Absolute Deviation (6.SP.5c)
• Two-way Tables (8.SP.4)
• Qualitative Graphs (8.F.5)
• Graphing Proportional Relationships
(7.RP.2a, b, c, d, 8.EE.5)
3/16/2016 • page 49
www.ncdpi.wikispaces.net
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www.ncdpi.wikispaces.net
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Crosswalk Documents
The crosswalks reflect a comparison
between the Common Core State Standards
and the North Carolina Standard Course of
Study. They inform educators about how the
current standards align with the CCSS
Standards.
3/16/2016 • page 53
Mathematics Crosswalk
CAUTION!!
CONTENT APPEARING TO BE
THE SAME MAY ACTUALLY BE
DIFFERENT!!
The CCSS Requires CLOSE Reading!!!
3/16/2016 • page 55
Unpacking Documents
The purpose of the Unpacking Documents is to
increase student achievement by ensuring
educators understand the new standards.
The “unpacking” of the standards done in these
documents is an effort to answer a simple question
“What does this standard mean that a student
must know and be able to do?” and to ensure the
description is helpful, specific and comprehensive
for educators.
3/16/2016 • page 56
Unpacking – At a Glance
3/16/2016 • page 57
Unpacking – Standards for
Mathematical Practice
3/16/2016 • page 58
Unpacked Content
3/16/2016 • page 59
Common Core Glossary
Table 1. Common multiplication and division situations
K-5 Units
Students learn mathematics by exploring
mathematically-rich tasks and sharing strategies,
ideas, and approaches with one another.
(practices)
K Adding and Subtraction
1st Exploring Two-Digit Numbers
2nd Two- &amp; Three-Digit Addition &amp; Subtraction
3rd Unit on Area and Perimeter
4th Fractions
5th Fractions
3/16/2016 • page 63
Format of the Lessons
The phases of the lesson:
•
•
•
•
•
Engage - Brief opening activity
Explore - Mathematically-rich task
Explain - Discussion of task and concepts
Elaborate - Follow-up activity
Evaluate - description of formative and
summative assessments
3/16/2016 • page 64
Information
Presidential Awards for Excellence
in
Mathematics and Science Teaching
www.paemst.org
Nomination
Application
Elementary Teachers
Grades K - 6
April 1, 2012
May 1, 2012
Secondary Teachers
Grades 7 - 12
April 1, 2013
May 1, 2013
Year
Who Can Apply
2012
2013
Elementary Mathematics
• 18-hour Graduate program (6 courses)
• Participating Universities
–
–
–
–
–
–
–
East Carolina University
Appalachian State University
NC State University
UNC Chapel Hill
UNC Charlotte
UNC Greensboro
UNC Wilmington
• Dr. Sid Rachlin (rachlins@ecu.edu)
Webinars
Archived Webinars:
- November 17th: CCSS and Math I Standards
- January 10th: K-12 Getting Started:
Organization Tools and Instructional Planning
Model
- February 9th: Making Mathematics
Accessible (K-12)
- March 8th: K – 2 Assessment and Calendar
Time
http://illustrativemathematics.org/
http://commoncoretools.wordpress.com
K, Counting and Cardinality;
K–2, Operations and Algebraic Thinking
K, Counting and Cardinality; K–5, Operations and Algebraic
Thinking
K–3, Categorical Data; Grades 2–5, Measurement Data*
(data part of the Measurement and
Data Progression)
3-5, Number and Operations – Fractions
6-7, Ratio and Proportional Relationships
6-8, Progression for Statistics and Probability
6–8, Expressions and Equations
QUESTIONS