Spring Training: Accessing Common Core through the CA Math

Framework

March 19, 2015Hilary 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 IntroductionGuiding Principles for Mathematics Programs in CA

1. LearningMathematical Ideas should be explored in ways that stimulate curiosity, create enjoyment of mathematics and develop depth of understanding.

2. TeachingAn 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. EquityAll students should have a high-quality mathematics program that prepares them for college and careers.

5. AssessmentAssessment 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 DeeplyGrade 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)

Domain

Standard

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

17

Common Core State Standards for Mathematics

• 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

Standards of Mathematical Practice

Two Types of Math Standards Across All Grades

Mathematical Practices are cross-

cutting. They relate to the content standards

at ALL grade-levels

Standards of Mathematical Practice

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

FocusWHAT 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 three-dimensional 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

CDE Mathematics Framework, Overview Chapter, page 3

“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.”

Coherence: A LEARNING TRAJECTORY

● 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 ▲)

6.RP.2 ▲ & 6.RP. 3a,b▲ 7.RP.2▲ 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.

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.

Example:

Almonds (pounds)

3 5 8 10 15

Cost (dollars) 15.00 25.00 40.00 50.00 75.00

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 = , equivalently, = , where is a constant known as the constant of 𝑘𝑥 𝑦 𝑥 𝑘 𝑘proportionality. In the case of almonds, the in an equation would represent the unit cost of 𝑘almonds.

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.