Common Core Mathematics and Next Generation Science Sue Gendron March 21, 2012

Common Core Common Core Mathematics and Mathematics and Next Generation Next Generation

ScienceScienceSue GendronSue Gendron

March 21, 2012March 21, 2012

Rigor -Require fluency, Rigor -Require fluency, application, and deep application, and deep

understandingunderstanding Conceptual understanding Conceptual understanding – solving – solving short conceptual problems, applying short conceptual problems, applying math in new situations, and math in new situations, and speaking about their understandingspeaking about their understanding

Procedural skill and fluency - Procedural skill and fluency - speed speed and accuracy in calculation.and accuracy in calculation.

Application - Application - “real world” situations “real world” situations

ReasoningReasoning

Invite Exploration of important Invite Exploration of important mathematical conceptsmathematical concepts

Allow students to solidify and make Allow students to solidify and make connectionsconnections

Make connections and develop Make connections and develop coherent framework for mathematical coherent framework for mathematical ideasideas

Problem formulation, problem solving Problem formulation, problem solving and mathematical reasoningand mathematical reasoning

ReasoningReasoning

More than one solutionMore than one solution

Development of all students’ Development of all students’ disposition to do mathdisposition to do math

Mathematically Mathematically proficient studentsproficient students

Make conjecturesMake conjectures

Build logical progressions to Build logical progressions to explore the truth of their explore the truth of their conjecturesconjectures

Justify and communicate their Justify and communicate their conclusionsconclusions

Respond to argumentsRespond to arguments

Which number does Which number does not belong? Why?not belong? Why?

4 16 36 48 64 814 16 36 48 64 81

From: Math for All: Differentiating Instruction, Grades 3-5, Dacey and LynchFrom: Math for All: Differentiating Instruction, Grades 3-5, Dacey and Lynch

Procedural FluencyProcedural Fluency

Knowledgeable about proceduresKnowledgeable about procedures

Know when and how to use themKnow when and how to use them

Skill in performing procedures Skill in performing procedures flexibly, accurately, efficiently and flexibly, accurately, efficiently and with understandingwith understanding

Using Mathematical Using Mathematical DiscourseDiscourse

Five Reasons Talk is CriticalFive Reasons Talk is Critical

Reveal student understanding Reveal student understanding and misunderstandingand misunderstanding

Support robust learning by Support robust learning by boosting memoryboosting memory

Support deeper reasoningSupport deeper reasoning

Support language developmentSupport language development

Support social skillsSupport social skills

The Structure is The Structure is the Standardsthe Standards

http://commoncoretools.me/2012/02/16/the-structure-is-the-standards

Why is paying attention to the structure Why is paying attention to the structure important?important?

The standards are meant to be a blueprint The standards are meant to be a blueprint for math instruction that is more focused for math instruction that is more focused and coherentand coherent

The focus and coherence in this blueprint is The focus and coherence in this blueprint is largely in the way the standards progress largely in the way the standards progress from each other, coordinate with each other from each other, coordinate with each other and most importantly cluster together into and most importantly cluster together into coherent bodies of knowledge.coherent bodies of knowledge.




The focus and coherence in The focus and coherence in this blueprint is largely in the this blueprint is largely in the way the standards way the standards progressprogress from each other, from each other, coordinatecoordinate with each other and most with each other and most importantly importantly cluster cluster together together into into coherent bodies of coherent bodies of knowledgeknowledge..


The natural distribution of The natural distribution of prior knowledge in classrooms prior knowledge in classrooms should not prompt abandoning should not prompt abandoning instruction in grade level instruction in grade level content, but should prompt content, but should prompt explicit attention to explicit attention to connecting grade level content connecting grade level content to content from prior learning.to content from prior learning.


It is the nature of It is the nature of mathematics that much mathematics that much new learning is about new learning is about extending knowledge extending knowledge from prior learning to from prior learning to new situations.new situations.

Grade 6 FocusGrade 6 Focus

(1) connecting ratio and rate to whole (1) connecting ratio and rate to whole number multiplication and division and using number multiplication and division and using concepts of ratio and rate to solve problems; concepts of ratio and rate to solve problems;

(2) completing understanding of division of (2) completing understanding of division of fractions and extending the notion of fractions and extending the notion of number to the system of rational numbers, number to the system of rational numbers, which includes negative numbers;which includes negative numbers;

(3) writing, interpreting, and using (3) writing, interpreting, and using expressions and equations; and expressions and equations; and

(4) developing understanding of statistical (4) developing understanding of statistical thinkingthinking..

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Fluency Fluency ExpectationsExpectations

6.NS.2 Students fluently divide multidigit numbers using the standard algorithm. This is the culminating standard for several years’ worth of work with division of whole numbers.

6.NS.3 Students fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation. This is the culminating standard for several years’ worth of work relating to the domains of NBT, OA, and NF.

6.NS.1 Students interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions. This completes the extension of operations to fractions.

Examples of Opportunities for Connections Among Standards,

Clusters, or Domains Students’ work with ratios and proportional relationships (6.RP) can be combined with their work in representing quantitative relationships between dependent and independent variables (6.EE.9).

Plotting rational numbers in the coordinate plane (6.NS.8) is part of analyzing proportional relationships (6.RP.3a, 7.RP.2) and will become important for studying linear equations (8.EE.8) and graphs of functions (8.F).12

Students use their skill in recognizing common factors (6.NS.4) to rewrite expressions (6.EE.3).

Examples of Opportunities for Connections Among Standards,

Clusters, or Domains Writing, reading, evaluating, and transforming variable expressions (6.EE.1−4) and solving equations and inequalities (6.EE.7–8) can be combined with use of the volume formulas V = lwh and V = bh (6.G.2).

Working with data sets can connect to estimation and mental computation. For example, in a situation where there are 20 different numbers that are all between 8 and 10, one might quickly estimate the sum of the numbers as 9 × 20 = 180.

Examples of Opportunities for In-Depth Focus

6.RP.3 When students work toward meeting this standard, they use a range of reasoning and representations to analyze proportional relationships.

6.NS.1 This is a culminating standard for extending multiplication and division to fractions.

6.NS.8 When students work with rational numbers in the coordinate plane to solve problems, they combine and consolidate elements from the other standards in this cluster.

Examples of Opportunities for In-Depth Focus

6.EE.3 By applying properties of operations to generate equivalent expressions, students use properties of operations that they are familiar with from previous grades’ work with numbers — generalizing arithmetic in the process.

6.EE.7 When students write equations of the form x + p = q and px = q to solve real-world and mathematical problems, they draw on meanings of operations that they are familiar with from previous grades’ work. They also begin to learn algebraic approaches to solving problems.13

Connecting Mathematical Content and Mathematical

Practices Reading and transforming expressions involves seeing and making use of structure (MP.7).

The sequence of steps in the solution of an equation is a logical argument that students can construct and critique (MP.3).

Thinking about the point (1,r) in a graph of a proportional relationship with unit rate r involves reasoning abstractly and quantitatively (MP.2).


Practices Area, surface area, and volume present modeling opportunities (MP.4).

Students think with precision (MP.6) and reason quantitatively (MP.2) when they use variables to represent numbers and write expressions and equations to solve a problem (6.EE.6–7).

Working with data gives students an opportunity to use appropriate tools strategically (MP.5). For example, spreadsheets can be powerful for working with a data set with dozens or hundreds of data points.

Grade 6

Grade Seven FocusGrade Seven Focus

(1) Developing understanding of and (1) Developing understanding of and applying proportional relationships; applying proportional relationships;

(2) Developing understanding of operations (2) Developing understanding of operations with rational numbers and working with with rational numbers and working with expressions and linear equations; expressions and linear equations;

(3) Solving problems involving scale (3) Solving problems involving scale drawings and informal geometric drawings and informal geometric constructions, and working with two- and constructions, and working with two- and three-dimensional shapes to solve three-dimensional shapes to solve problems involving area, surface area, and problems involving area, surface area, and volume; and volume; and

(4) Drawing inferences about populations (4) Drawing inferences about populations based on samples.based on samples.

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7.EE.3 Students solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. This work is the culmination of many progressions of learning in arithmetic, problem solving, and mathematical practices.

7.EE.4 In solving word problems leading to one-variable equations of the form px + q = r and p(x + q) = r, students solve the equations fluently. This will require fluency with rational number arithmetic (7.NS.1–3), as well as fluency to some extent with applying properties operations to rewrite linear expressions with rational coefficients (7.EE.1).


7.NS.1–2 Adding, subtracting, multiplying, and dividing rational numbers is the culmination of numerical work with the four basic operations. The number system will continue to develop in grade 8, expanding to become the real numbers by the introduction of irrational numbers, and will develop further in high school, expanding to become the complex numbers with the introduction of imaginary numbers. Because there are no specific standards for rational number arithmetic in later grades and because so much other work in grade 7 depends on rational number arithmetic (see below), fluency with rational number arithmetic should be the goal in grade 7.

Opportunities for Connections Among Standards, Clusters, or

Domains

Students use proportional reasoning when they analyze scale drawings (7.G.1).

Students use proportional reasoning and percentages when they extrapolate from random samples and use probability (7.SP.6, 8).

Opportunities for in Depth Focus

7.RP.2 Students in grade 7 grow in their ability to recognize, represent, and analyze proportional relationships in various ways, including by using tables, graphs, and equations.

7.NS.3 When students work toward meeting this standard (which is closely connected to 7.NS.1 and 7.NS.2), they consolidate their skill and understanding of addition, subtraction, multiplication, and division of rational numbers.

7.EE.3 This is a major capstone standard for arithmetic and its applications.


7.EE.4 Work toward meeting this standard builds on the work that led to meeting 6.EE.7 and prepares students for the work that will lead to meeting 8.EE.7.

7.G.6 Work toward meeting this standard draws together grades 3–6 work with geometric measurement.


Practices When students compare arithmetic and algebraic solutions to the same problem (7.EE.4a), they are identifying correspondences between different approaches (MP.1).

Solving an equation such as 4 = 8(x – 1⁄2) requires students to see and make use of structure (MP.7), temporarily viewing x – 1⁄2 as a single entity.


Practices When students notice when given geometric conditions determine a unique triangle, more than one triangle, or no triangle (7.G.2), they have an opportunity to construct viable arguments and critique the reasoning of others (MP.3). Such problems also present opportunities for using appropriate tools strategically (MP.5).

Proportional relationships present opportunities for modeling (MP.4). For example, the number of people who live in an apartment building might be taken as proportional to the number of stories in the building for modeling purposes.

Grade 7

Grade 8 FocusGrade 8 Focus

(1) formulating and reasoning about expressions (1) formulating and reasoning about expressions and equations, including modeling an and equations, including modeling an association in bivariate data with a linear association in bivariate data with a linear equation, and solving linear equations and equation, and solving linear equations and systems of linear equations;systems of linear equations;

((2) grasping the concept of a function and using 2) grasping the concept of a function and using functions to describe quantitative relationships; functions to describe quantitative relationships;

(3) analyzing two- and three-dimensional space (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and and figures using distance, angle, similarity, and congruence, and understanding and applying congruence, and understanding and applying the Pythagorean Theorem.the Pythagorean Theorem.


8.EE.7 Students have been working informally with one-variable linear equations since as early as kindergarten. This important line of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions and cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers.


8.G.9 When students learn to solve problems involving volumes of cones, cylinders, and spheres — together with their previous grade 7 work in angle measure, area, surface area, and volume (7.G.4–6) — they will have acquired a well-developed set of geometric measurement skills. These skills, along with proportional reasoning (7.RP) and multistep numerical problem solving (7.EE.3), can be combined and used in flexible ways as part of modeling during high school — not to mention after high school for college and careers.16

Opportunities for Connections Among Standards, Clusters, or

Domains

Students’ work with proportional relationships, lines, linear equations, and linear functions can be enhanced by working with scatter plots and linear models of association in bivariate measurement data (8.SP.1–3).

Work with the number system in this grade (8.NS.1–2) is intimately related to work with radicals (8.EE.2), and both of these may be connected to the Pythagorean Theorem (8.G, second cluster) as well as to volume problems (8.G.9), e.g., in which a cube has known volume but unknown edge lengths.


8.EE.5 When students work toward meeting this standard, they build on grades 6–7 work with proportions and position themselves for grade 8 work with functions and the equation of a line.

8.EE.7 This is a culminating standard for solving one-variable linear equations.

8.EE.8 When students work toward meeting this standard, they build on what they know about two-variable linear equations, and they enlarge the varieties of real-world and mathematical problems they can solve.


8.F.2 Work toward meeting this standard

repositions previous work with tables and

graphs in the new context of input/output rules.

8.G.7 The Pythagorean Theorem is useful in

practical problems, relates to grade-level work

in irrational numbers, and plays an important

role mathematically in coordinate geometry in

high school.


Practices When students convert a fraction such as 1/7 to a decimal, they might notice that they are repeating the same calculations and conclude that the decimal repeats. Similarly, by repeatedly checking whether points are on a line through (1,2) with slope 3, students might abstract the equation of the line in the form (y − 2)/(x − 1) = 3. In both examples, students look for and express regularity in repeated reasoning (MP.8).


Practices The Pythagorean Theorem can provide opportunities for students to construct viable arguments and critique the reasoning of others (e.g., if a student in the class seems to be confusing the theorem with its converse) (MP.3).

Solving an equation such as 3(x – 1⁄2) = x + 2 requires students to see and make use of structure (MP.7).

Much of the mathematics in grade 8 lends itself to modeling (MP.4). For example, standard 8.F.4 involves modeling linear relationships with functions.

Grade 8

Teaching ChannelTeaching Channel

http://www.teachingchannel.org/videos/surface-area-lesson?fd=1




One Hundred Plus One Hundred Plus ToolsTools

One hundred Plus tools One hundred Plus tools on Stem for teacherson Stem for teachers http://www.livebinders.com/play/play_or_edit?id=126258

http://www.livebinders.com/play/play_or_edit?id=126258



MSP Project Drew MSP Project Drew Kravin & Phil Kravin & Phil

GonzalesGonzales

https://sites.google.com/a/wccusd.net/mcc-wccusd1/home/resources/lessons






Student Student Achievement Achievement

PartnersPartnershttp://www.achievethecore.org

http://www.achievethecore.org/

http://www.achievethecore.org/

Smarter Claims for the Mathematics Summative

Assessment Overall Claim for Grades 3–8

“Students can demonstrate progress toward college and career readiness in mathematics.”

Overall Claim for Grade 11

“Students can demonstrate college and career readiness in mathematics.”


Assessment Claim #1 – Concepts & Procedures

“Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.”


Assessment – Claim 1 Grade 6

Ratios and Proportional relationships

- Understand ratio concepts and use ratio reasoning to solve problems.

The Number System - Apply and extend previous understandings of multiplication and division to divide fractions by fractions. - Compute fluently with multi-digit numbers and find common factors and multiples. - Apply and extend previous understandings of numbers to the system of rational num



Expressions and Equations

- Apply and extend previous understandings of arithmetic to algebraic expressions.

- Reason about and solve one-variable equations and inequalities.

- Represent and analyze quantitative relationships between dependent and independent

variables



Geometry

- Solve real-world and mathematical problems involving area, surface area, and volume.

Statistics and Probability

- Develop understanding of statistical variability.

- Summarize and describe distributions.


Assessment – Claim 1

Grade 7

Ratios and Proportional relationships

- Analyze proportional relationships and use them to solve real-world and mathematical problems.

The Number System

- Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.



Grade 7

Expressions and Equations

- Use properties of operations to generate equivalent expressions.

- Solve real-life and mathematical problems using numerical and algebraic expressions and

equations.

Geometry

- Draw, construct and describe geometrical figures and describe the relationships between them.

- Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.



Grade 7

Statistics and Probability - Use random sampling to draw inferences about a population. - Draw informal comparative inferences about two populations. - Investigate chance processes and develop, use, and evaluate probability models. The Number System

- Know that there are numbers that are not rational, and approximate them by rational numbers.



Grade 7

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software.

- Understand and apply the Pythagorean theorem.

- Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.

Statistics and Probability

- Investigate patterns of association in bivariate data.


Assessment Claim #1 – Concepts & Procedures

“Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.”

Claim #2 – Problem Solving

“Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.”

Claim 2Claim 2

Common Core Standards for Mathematical Practice

- Make sense of problems and persevere in solving them.

- Use appropriate tools strategically.

- Look for and make use of structure.

- Look for and express regularity in repeated reasoning.

Claim 3Claim 3

Construct viable arguments and critique the reasoning of others.

Attend to precision.

Claim 4Claim 4

Common Core Standards for Mathematical Practice

- Reason abstractly and quantitatively.

- Model with mathematics.

- Use appropriate tools strategically.


Assessment Claim #3 – Communicating Reasoning

“Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.”

Claim #4 – Modeling and Data Analysis

“Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.”

The Framework and The Framework and Standards will Standards will

Impart a coherent and sharpened focus on the core Impart a coherent and sharpened focus on the core ideas of the major fields ideas of the major fields

Take into consideration the knowledge and skills Take into consideration the knowledge and skills required for science literacy, college readiness, and for required for science literacy, college readiness, and for pursuing further study in STEM fields pursuing further study in STEM fields

Integrate conceptual knowledge and science Integrate conceptual knowledge and science practices practices

Base decisions on evidence—to the degree possible—Base decisions on evidence—to the degree possible—as well as on professional judgment as well as on professional judgment

Reflect the expectations that high-performing Reflect the expectations that high-performing countries hold for students countries hold for students

Designing Designing Learning TargetsLearning TargetsWhat does the standard mean?What does the standard mean?

Kid-ify language of the standardKid-ify language of the standard

Identify the pre-requisite Identify the pre-requisite knowledge/skillsknowledge/skills

Find place in Learning ProgressionsFind place in Learning Progressions

Make is demonstrableMake is demonstrable

Ratio and Proportional Ratio and Proportional relationshiprelationship

Grade 6 Grade 7 Grade 8

Understand ratio concepts and use ratio reasoning to solve problems.

analyze proportional relationships and use them to solve real-world and mathematical problems.

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Why is it Why is it important to important to

teach ratio and teach ratio and proportional proportional reasoning?reasoning?

Exploring Ratio and Exploring Ratio and RelationshipsRelationships

What are the definitions for rate, ratio What are the definitions for rate, ratio and proportional relationships?and proportional relationships?

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A A ratio ratio is a pair of non-negative numbers, is a pair of non-negative numbers, A A : : B, B, which are not both 0.which are not both 0.

A rate is expressed in terms of a unit that is A rate is expressed in terms of a unit that is derived from the units of the two quantities derived from the units of the two quantities (such as m/s, which is derived from meters (such as m/s, which is derived from meters and seconds). and seconds).

A A proportional relationship proportional relationship is a collection of is a collection of pairs of numbers pairs of numbers

that are in equivalent ratios. that are in equivalent ratios.

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Ratio and Ratio and Proportional Proportional RelationshipsRelationshipsWhat are the key focus areas within the What are the key focus areas within the

domain?domain?

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Ratio and Ratio and Proportional Proportional RelationshipsRelationshipsWhat are the key focus areas within the What are the key focus areas within the

domain?domain?Ratios, rates, proportional relationships, Ratios, rates, proportional relationships, and percent and percent

Recognizing and describing ratios, rates, Recognizing and describing ratios, rates, and proportional relationships and proportional relationships

Representing ratios, collections of Representing ratios, collections of equivalent ratios, rates, and proportional equivalent ratios, rates, and proportional relationships relationships

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Ratio and Ratio and Proportional Proportional RelationshipsRelationshipsWhat are the priorities for 6What are the priorities for 6thth grade grade

areas of focus – areas of focus –

What are the priorities for grade 7?What are the priorities for grade 7?

What problem solving strategies are What problem solving strategies are important at each grade?important at each grade?

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Statistics and ProbabilityStatistics and ProbabilityGrade 6 Grade 7 Grade 8•Develop understanding of statistical variability.

• Summarize and describe distributions.

Use random sampling to draw inferences about a population.

• Draw informal comparative inferences about two populations.

• Investigate chance processes and develop, use, and evaluate probability models.

Investigate patterns of association in bivariate data.

100100

The Number SystemThe Number SystemGrade 6 Grade 7 Grade 8Apply and extend previous understandings of multiplication and division to divide fractions by fractions.• Compute fluently with multi-digit numbers and find common factors and multiplesApply and extend previous understandings of numbers to the system of rational numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

101101

Expressions and EquationsExpressions and EquationsGrade 6 Grade 7 Grade 8

•Apply and extend previous understandings of arithmetic to algebraic expressions.• Reason about and solve one-variable equations and inequalities.• Represent and analyze quantitative relationships between dependent and independent variables.

•Use properties of operations to generate equivalent expressions.

• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

•Work with radicals and integer exponents.

• Understand the connections between proportional relationships, lines, and linear equations.

• Analyze and solve linear equations and pairs of simultaneous linear equations.

102102

GeometryGeometryGrade 6 Grade 7 Grade 8

draw, construct and describe geometrical figures and describe the relationships between them.

• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand congruence and similarity using physical models, transparencies, or geometry software.

• Understand and apply the Pythagorean theorem.

• Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.103103

Statistics and ProbabilityStatistics and ProbabilityGrade 6 Grade 7 Grade 8•Develop understanding of statistical variability.

• Summarize and describe distributions.

Use random sampling to draw inferences about a population.

• Draw informal comparative inferences about two populations.

• Investigate chance processes and develop, use, and evaluate probability models.

Investigate patterns of association in bivariate data.

104104

Documents

Common Core Mathematics and Next Generation Science Sue Gendron March 21, 2012