CCSS within the Teaching Framework:Mathematics 3-6

Rochester City School DistrictAugust 9, 2012

August 10, 2012

August 20, 2012

August 27, 2012

Today’s Agenda 8:30-9:30 (Full Group)

Overview and Change of Direction

Overview of Danielson Rubric

9:45-10:45 CCSS Sample Questions from NYSED – June 2012

10:45 – 11:30 Modules

Correlation charts

11:30-12:30 Lunch

12:30-2:30 (Grade Level Groups) Emphases in Common Core Standards for Mathematical Content

NBT / Tasks

2:30–3:00 Accessing Resources on Rochester Curriculum and the RCSD Math Department’s web page

3:00-3:30 Evaluation and Feedback

When not knowing Math can cost you $15,000…

http://www.youtube.com/watch?v=BbX44YSsQ2I

Math Sample

Questions

CCSS Sample Questions from NYSEDJune 2012

o How do you think students would interpret these questions?

o What are your impressions of the questions?

o What would your students think of these questions?

Answer each of the sample questions.

Match each question to one of the standards.

Discuss the following:

NYS Common Core Standards addressed in the sample questions:3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷3, 6 × 6 = ?3.OA.5. Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6= 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2= 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

4.OA.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 4.NF.4c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

5.NBT.3a. Read, write, and compare decimals to thousandths.a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).5.NBT.2. 2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10.

6.RP.3c. 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.c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

So…o What are your impressions of the questions?

o What would your students think of these questions?

o How do you think students would interpret these questions?

Now let’s see how well we matched the grade level

standards to each question…

NYS Common Core Standards addressed in the sample questions:

3.OA.4, 3.OA.5, 3.OA.6

4.OA.3, 4.NF.4c

5.NBT.3a and 5.NBT.2

6.RP.3c

Commentary: This question aligns with CCLS 3.OA.4, 3.OA.5, and 3.OA.6 and assesses the student’s ability to determine the unknown whole numbers in multiplication and division equations. The question also assesses the student’s ability to apply properties of operations as strategies to multiply or divide and to understand division as an unknown factor problem.

Commentary: This question is aligned with CCLS 4.NF.4c and 4.OA.3. It assesses a student’s ability to multiply a fraction by a whole number and to solve a multi-step word problem using addition and subtraction of whole numbers.

Commentary: This question aligns with CCLS 5.NBT.3a and 5.NBT.2 because it requires students to write decimals in equivalent forms and apply understanding of place value.

Commentary: This question aligns to CCLS 6.RP.3c because it assesses a student’s ability to work with percents, namely, finding a percent of a quantity in a contextual situation.

How to Use Different Aspects of the Sample Questions

Interpret the way the standards are conceptualized in each question.

Note the multiple ways the standard is assessed throughout the sample questions.

Take note of numbers (e.g., fractions instead of whole numbers) used in the samples.

Pay attention to the strong distractors in each multiple‐choice question.

Don’t consider these questions to be the only way the standard will be assessed.

Don’t assume that the sample questions represent a mini‐version of future state assessments.

Understanding Math Sample Questions

Multiple Choice

Sample multiple‐choice math questions are designed to assess CCSS math standards and incorporate both standards and math practices in real‐world applications. Math multiple‐choice questions assess procedural and conceptual standards. Unlike questions on past math assessments, many require the use of multiple skills and concepts. Answer choices are also different from those on past assessments. Within the sample questions, all distractors will be based on plausible missteps.

Understanding Math Sample Questions

Short Constructed Response

Math short constructed response questions are similar to past 2‐point questions, asking students to complete a task and show their work. Like multiple‐choice questions, short constructed response questions will often require multiple steps, the application of multiple math skills, and real‐world applications. Many of the short constructed response questions will cover conceptual and application standards.

Understanding MathSample Questions

Extended Constructed Response

Math extended constructed response questions are similar to past 3‐point questions, asking students to show their work in completing two or more tasks or one more extensive problem. Extended constructed response questions allow students to show their understanding of math procedures, conceptual understanding, and application.

For additional information and sample questions, follow this link:

http://www.p12.nysed.gov/apda/common-core-sample-questions/ 

 

NYS Curriculum

Modules

NYS Curriculum

Modules

RochesterCurriculum

Common Core State Standards

NYS Common Core Curriculum Maps in Mathematics

Rochester Curriculum Main Page

Grade 3 Correlation

Chart

Grade 4 Correlation Chart

Grade 5 Correlation Chart

Grade 6

As of today, the state has not released the Common Core Curriculum map for Grade 6, so no correlation chart is available at this time.

The Rochester Curriculum will provide the framework for Grade 6 to begin the school year.

As new information becomes available, Administrators and Grade 6 teachers will be informed.

Lunch 11:30 – 12:30

Afternoon Agenda

12:30-2:30 (Grade Level Groups)

Emphases in Common Core Standards for Mathematical Content

NBT / Tasks

2:30–3:00

Accessing Resources on Rochester Curriculum and the RCSD Math Department’s web page

3:00-3:30 Evaluation and Feedback

Emphases in Common Core Standards for Mathematical

Content

• Not all of the content in a given grade is emphasized equally in the standards.

Emphases in Common Core Standards for Mathematical Content

Kindergarten – High School March 12, 2012

• The list of content standards for each grade is not a flat, one‐dimensional checklist; this is by design.

• Some clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to master, and/or their importance to future mathematics or the demands of college and career readiness.

• In addition, an intense focus on the most critical material at each grade allows depth in learning, which is carried out through the Standards for Mathematical Practice.

Emphases in Common Core Standards for Mathematical Content

Kindergarten – High School March 12, 2012

Explanation of terms used:

Major clusters – areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%).

Supporting clusters – rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%).

Additional Clusters – expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).

Grade 3Major Supporting Additional

Operations and Algebraic Thinking• Represent and solve problems using

multiplication and division.• Understand the properties of

multiplication and the relationship between multiplication and division.

• Multiply and divide within 100.• Solve problems involving the four

operations, and identify and explain patterns in arithmetic.

Number and Operations – Fractions• Develop understanding of fractions

as numbers.Measurement and Data• Solve problems involving

measurement and estimation of intervals of time, liquid volumes, and masses of objects.

• Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Geometry• Reason with

shapes and their attributes. (1)

Measurement and Data• Represent and

interpret data. (2)

Number and Operations in Base Ten• Use place value

understanding and properties of operations to perform multi-digit arithmetic

Measurement and Data• Geometric

measurement: Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

1 Work should be positioned in support of area measurement and understanding of fractions. 2 Students multiple and divide to solve problems using information presented in scaled bar graphs. Pictographs and scaled bar graphs are a visually appealing context for one‐ and two‐step word problems.

Grade 4Major Supporting Additional

Operations and Algebraic Thinking• Use the four operations with whole

numbers to solve problems.Number and Operations in Base Ten• Generalize place value

understanding for multi‐digit whole numbers.

• Use place value understanding and properties of operations to perform multi‐digit arithmetic.

Number and Operations – Fractions• Extend understanding of fraction

equivalence and ordering.• Build fractions from unit fractions by

applying and extending previous understandings of operations on whole numbers.

• Understand decimal notation for fractions, and compare decimal fractions.

Operations and Algebraic Thinking• Gain familiarity

with factors and multiples. (3)

Measurement and Data• Solve problems

involving measurements and conversion of measurements from a larger unit to a smaller unit.

• Represent and interpret data.(4)

Operations and Algebraic Thinking • Generate and

analyze patterns.Measurement and Data• Geometric

measurement: understand concepts of angles and measure angles.

Geometry• Draw and identify

lines and angles, and classify shapes by properties of their lines and angles.

3 Work in this cluster supports students’ work with multi‐digit arithmetic as well as their work with fraction equivalence. 4 The standard in this cluster requires students to use a line plot to display measurements in fractions of a unit and to solve problems involving addition and subtraction of fractions, connecting it directly to the Number and Operations – Fractions clusters.

Grade 5Major Supporting Additional

Number and Operations in Base Ten• Understand the place value system. • Perform operations with multi‐digit

whole numbers and with decimals to hundredths.

Number and Operations – Fractions• Use equivalent fractions as a

strategy to add and subtract fractions.

• Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Measurement and Data• Geometric measurement:

understand concepts of volume and relate volume to multiplication and to addition.

Measurement and Data• Represent and

interpret data. (5)

• Convert like measurement units within a given measurement system. (6)

Operations and Algebraic Thinking • Write and interpret

numerical expressions.

• Analyze patterns and relationships.

Geometry• Graph points on

the coordinate plane to solve real-world and mathematical problems.

• Classify two-dimensional figures into categories based on their properties.

5 The standard in this cluster provides an opportunity for solving real‐world problems with operations on fractions, connecting directly to both number and Operations – Fractions clusters. 6 Work in these standards supports computation with decimals. For example, converting 5 cm to .05 m involves computation with decimals to hundredths.

Grade 6Major Supporting Additional

Ratios and Proportional Relationships• Understand ratio concepts and use

ratio reasoning to solve problems.The Number System• Apply and extend previous

understandings of numbers to the system of rational numbers.

• Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

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. (7)

Statistics and Probability • Develop

understanding of statistical variability.

• Summarize and describe distributions.

The Number System • Compute fluently

with multi‐digit numbers and find common factors and multiples.

7 In this cluster, students work on problems with areas of triangles and volumes of right rectangular prisms, which connects to work in the Expressions and Equations domain. In addition, another standard within this cluster asks students to draw polygons in the coordinate plane, which supports work with the coordinate plane in the Number System domain.

Grade Level Tasks

NBT

Please read the Overview in the K-5, Number and Operations In Base Ten learning progression document.

Then please read the section for your grade level:

• Grade 3-4 – pages 11-14

• Grade 5-6 – pages 16-19

Discussion to follow...

Complete the tasks for your grade level.

Note the standards that are addressed in your task.

Be prepared to share one of your tasks with the larger group, considering the following questions:

o What are your impressions of the task?o What would your students think of this task?o How do you think students would interpret

this task?o How does the task address the standards?o What mathematical content was needed to

complete the task?

Task Analysis

Grade 3 CCSS Domain:Number and Operations

in Base Ten

Grade 4 CCSS Domain:Number and Operations

in Base Ten

Grade 5 CCSS Domain:Number and Operations

in Base Ten

Grade 6 CCSS

Domain:Number System

Accessing Rochester Curriculum

Accessing Rochester Curriculum

Accessing Rochester Curriculum

Accessing Rochester Curriculum

Accessing Rochester CurriculumUnit Page 1

Accessing Rochester CurriculumUnit Page 2

How to:NYS P-12 Common Core Resources

www.rcsdk12.org Click Departments

Click Mathematics

Click Common Core Math Resources

Thank you for your time and attention today…

Have a wonderful year!!