Spring Training: Accessing Common
Core through the CA Math
Framework
March 19, 2015
Hilary Dito & Pam Tyson
Workshop Goals
• To examine the Common Core State Standards through
the lens of the CA Mathematics Framework at a
specific grade level
• To understand the major focus of a grade level
• To explore how the Mathematics Framework supports
learning for all students
Introductions
• Introductions
• Icebreaker Activity
– On a post-it, write one Learning experience that
you have had around the CCSS-M
Discuss with Your Group
1. Math is math – there is nothing
new in the CCSS-M
2. Why are we doing all of this crazy
stuff – let’s just get to the answer
3. CCSS-M is more focused that
previous sets of standards
4. CCSS-M is watered down – too
simplified
Overview of the Day’s Topics
• Introduction to Each Other and the
Mathematics Framework
• Critical Areas of Instruction in Secondary
Mathematics
• Walk Through
– A Grade Level Chapter
– Non-Grade Level Chapters
– The Appendices
• Taking This Information Back to Your Schools
Purpose of the Framework
• The Mathematics Framework for
California Public Schools:
Kindergarten Through Grade
Twelve is meant to guide teachers
in curriculum development and
instruction to ensure that all
students meet exceed the CA
CCSSM (p. 8)
http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp
Chapter One Introduction
Guiding Principles for Mathematics Programs in CA
1. Learning
Mathematical Ideas should be explored in ways that stimulate curiosity,
create enjoyment of mathematics and develop depth of understanding.
2. Teaching
An effective mathematics program is based on a carefully designed set of
content standards that are clear, specific, focused and articulated over time
as a coherent sequence.
3. Technology
Technology is an essential tool that should be used strategically in
mathematics education.
4. Equity
All students should have a high-quality mathematics program that prepares
them for college and careers.
5. Assessment
Assessment of student learning in mathematics should take many forms to
inform instruction and learning.
Chapter 2: Overview of
Standards Chapters
The standards call for learning mathematics
content in the context of real-world situations,
using mathematics to solve problems, and
developing “habits of mind” that foster
mastery of mathematics content as well as
mathematical understanding (p. 9).
Standards are Based on Three
Major Principles
1. Conceptual
understanding
2. Procedural skill
and fluency
3. Ability to apply
math to solve
problems
K-8 Areas To Understand Deeply
Grade
Key Areas of Focus in Mathematics
K-2
Place value, addition and subtraction – concepts,
skills, and problem solving
3-5
Multiplication and division of whole numbers and
fractions – concepts, skills, and problem solving
6
Ratios and proportional reasoning; early expressions
and equations
7
Ratios and proportional reasoning; arithmetic of
rational numbers
8
Linear algebra
Two Other Cross Grade Table in Framework Chapter 2 provide Exemplar
Standards Emphazing Understanding and Areas of Fluency
Standards Page(s)
Cluster
Content Domains Grades K - 12
HS Math Standards vs. Courses
• High schools standards are organized by conceptual themes
(not by grade level or course title).
• The Framework outlines the Courses
–
–
–
–
–
–
–
Algebra or Course 1
Geometry or Course 2
Algebra or Course 3
Precalculus
Statistics and Probability
Calculus
Advanced Placement Statistics and Probability
Common Core State Standards for Mathematics
Standards of Mathematical Practice
Two Types of Math Standards Across All Grades
• Grade-Level Content Standards
– K-8 grade-by-grade standards organized by domain
– 9-12 high school standards organized by conceptual
categories
• Standards for Mathematical Practice
– Describe mathematical “habits of mind”
– Standards for mathematical proficiency:
• reasoning, problem solving, modeling, decision
making, and engagement
– Connect with content standards in each grade
17
Standards of Mathematical Practice
Mathematical
Practices are crosscutting. They relate to
the content standards
at ALL grade-levels
Overview of A Grade-Level Chapter
•
•
•
•
Critical Areas of Instruction
Grade Content: Cluster-Level Emphases
Connecting Mathematical Practices & Content
Grade Learning by Domain
– Tables
– Example Problems
– Tips for Teaching
• Essential Learnings for the Next Grade
• Standards for the Grade
Break
THROUGH A GRADE LEVEL
Focus
Read pages 1-3 of the Grade Eight chapter of the
Framework
• What are the implications for classroom
instruction?
• Share one insight with a partner
Focus
WHAT STUDENTS LEARN IN GRADE EIGHT
Grade Eight Critical Areas of Instruction
In grade eight, instructional time should focus on three critical areas: (1)
formulating and reasoning about expressions and equations, including modeling
and association in bivariate data with a linear equation, and solving linear equations
and systems of linear equations; (2) grasping the concept of a function and using
functions to describe quantitative relationships; (3) analyzing two-and threedimensional space and figures using distance, angle, similarity, and congruence and
understanding and applying the Pythagorean Theorem.
Students also work toward fluency with solving simple sets of two equations with
two unknowns by inspection.
Focus
8th Grade Mathematics
8th Grade Mathematics– from PARCC Framework
Turn to the Content Standards Pages in the
framework chapter (Grade 8: pages 39-42)
• Highlight the Major Clusters green
How do the supporting clusters and the additional
clusters reinforce the major clusters?
Compare the Cluster-Level
Emphasis of other grades
• Read the cluster-level emphasis for
– 2nd Grade
– 11th Grade
How do the supporting clusters and the additional
clusters reinforce the major clusters?
How does
• grade 2 support the work in subsequent grades?
• grade 8 support the work in previous grades and
subsequent grades?
2nd Grade Mathematics
from PARCC Framework
Coherence
“Some of the connections in the standards
knit topics together at a single grade level.
Most connections are vertical, as the
standards support a progression of
increasing knowledge, skill, and
sophistication across the grades.”
CDE Mathematics Framework, Overview Chapter, page 3
Coherence
“Some of the connections in the standards
knit topics together at a single grade level.
Most connections are vertical, as the
standards support a progression of
increasing knowledge, skill, and
sophistication across the grades.”
CDE Mathematics Framework, Overview Chapter, page 3
Coherence: A LEARNING TRAJECTORY
6.RP.2 ▲ &
6.RP. 3a,b▲
7.RP.2▲
8.EE.5▲
● Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language
in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups
of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is
a rate of $5 per hamburger.” (6.RP.2 ▲)
● Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about
tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (6.RP.3a,b ▲)
o a. Make tables of equivalent ratios relating quantities with whole number measurements, find
missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to
compare ratios.
o b. Solve unit rate problems including those involving unit pricing and constant speed. For
example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in
35 hours? At what rate were lawns being mowed?
● Recognize and represent proportional relationships between quantities. (7.RP.2 ▲)
● Understand the connections between proportional relationships, lines, and linear equations. (8.EE.5 ▲)
8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in two different ways.
Example:
Almonds
(pounds)
Cost (dollars)
3
5
8
10
15
15.00
25.00
40.00
50.00
75.00
Graph the cost versus the number of pounds of
almonds. The number of pounds of almonds should be
on the horizontal axis and the cost of the almonds on
the vertical axis.
Use the graph to find the cost of 1 pound of almonds.
Explain how you got your answer.
8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in two different ways.
Classroom Connections
The concept of slope can be approached in its simplest form with directly
proportional quantities.
𝑦 = 𝑘𝑥, equivalently, 𝑦 𝑥 = 𝑘, where 𝑘 is a constant known as the constant of
proportionality.
In the case of almonds, the 𝑘 in an equation would represent the unit cost of
almonds.
Connecting to the Standards for Mathematical Practice
MP.1 Students are encouraged to attack the entire problem and make sense in
each step required.
MP.4 Students are modeling a very simple real-life cost situation.