Overview of the K-5 CCSS Common Core State Standards mathematics by Judith T. Brendel Math Supervisor, Pascack Valley Regional HS District [email protected]

Overview of the K-5 CCSS Common Core State Standards

mathematics

by Judith T. BrendelMath Supervisor, Pascack Valley Regional HS District

[email protected]

THINGS we shouldn’t forget!

ACTIVITY: Use the ruler on handout to measure a writing tool you have with you.

What decisions did you make? :

1- What should I measure (shorter than the ruler)?

2- Should I use inches or centimeters?

3. Where should I start on the ruler?

4- Should I round up or down? How accurate do I need to be? What should I round to?

CCSS: Common CORE State standards

NOT TO BE CONFUSED WITH

CCC STANDARDS; NJ’s present

Core-Curriculum Content Standards

But, don’t throw away what you already have!

Shocking but true?1985: 3,800,000 Kindergarten students

1998: 2,810,000 High school graduates

1998: 1,843,000 College Freshmen

2002: 1,292,000 College Graduates

2002: 1,50,000 STEM majors

2006: 1,200 PhD’s in mathematics

Let me know a little bitHow familiar you are with the CCSS?

Closed fist – I know nothing.

1 finger – I know they exist.

5 fingers – I could be standing up here doing this presentation.

Our plan for these 2 days:

• CONTENT

• ORGANIZATION

• NEW DESIGN

• IMPACT on STUDENT LEARNING

• IMPACT on how STUDENTS are ASSESSED

• NEEDS and RESOURCES

Be Pro-activePrepare for the change

• Timeline: implementation & assessment

• Transition Phase - Today’s Reality

• Last Year vs CCSS expectations

• Mathematical Practices

• Plan: Curriculum, Pacing, Assessments

• Align to texts

• OE Questions, Projects, Resources

Measure standards that are rigorous, globally competitive, and consistent across the states.

New Jersey’s choiceThe assessment consortium PARCC

(Partnership for Assessment of Readiness for College and Careers)

New assessments will replace current state NCLB tests (NJASK, HSPA) in 2014-2015.

WHAT’S NEW

What We’ve Learnedduring the past decades:

Absolutely, students must know the basics, but knowing basics not enough.

Students learning math with understandings is essential to enable students to solve problems.

Students learn math primarily by doing math rather than just by listening and memorizing.

What they’ve Learnedduring the past decades:

absolutely students must know the basics, but knowing basics not enough

students learning math w/understandings is essential to enable students to solve problems

students learn math primarily by doing math rather than just by listening and memorizing

“IS” OVER

“OF”

WINDOWS doors WINDOWS doorsYOU JUST MAKE A BOX! R x T

= D

.64 is POINT 64

When you multiply you

always get a BIGGER

number!

What we should do!Start with a problem (division w/one-digit divisors) Give to students and see what they do with it. Let them work in pairs. Let them talk about it.

Paulo has 39 patches from states that he and his relatives have visited. He wants to arrange the patches on a board in 3 rows. How many patches will be in each row? (from Dr. Janet Caldwell)

Characteristics of the CCSS

Fewer and more rigorous

Aligned with college and career expectations

Internationally benchmarked

Rigorous content and applications of higher-order skills – rigor a depth, not complexity

Research-based

Common, coherent, fair and teachable

COMMON ADVANTAGESFEWER standards means that teachers and students

will have more time to focus on each knowledge & skill.

CLEAR standards means the parents, teachers and the general public will find it easier to understand the expectations for students

HIGHER standards means that all students will be fully prepared for the future, whether they choose to further their education or immediately enter the workforce.

COMMON standards means that schools can compare student test scores state-by-state or state to the national average.

Big Things to Notice K-8Number is emphasized in K-5.

Algebra is imbedded in number until grade-6.

Less is expected in statistics and probability early; more at later grades.

Geometry is included in every grade but is limited in scope in K-8.

Length is included in grades 1-2, both other types of measurement are postponed until grade-3.

Some topics change grades from current NJ CCCs.

GRADE-K focusThe Common Core Standards emphasize that instructional time in Kindergarten should focus on two critical areas:

1.Representing and comparing whole numbers, initially with sets of objects; and

2.Describing shapes and space.

https://sites.google.com/site/pvrsdportal/kindergarten-curriculum

Grade-K questions ? What about the calendar?

What about patterns?

What about telling time?

GRADE-1 focusinstructional time should focus on four critical areas:

1.developing understanding of addition, subtraction, and strategies for addition and subtraction within 20;

2.developing understanding of whole number relationships and place value, including grouping in tens and ones;

3.developing understanding of linear measurement and measuring lengths as iterating length units; and

4. reasoning about attributes of, and composing and decomposing geometric shapes.

Grade-1 What’s new? New topics: what should I do?

Describe these shapes without talking about color.

Students need to know how to add 25 + 17. Ask kids what they’d do to find out how much 17 + 25 is.

What do you think they might do?


1.Extending understanding of base-10 notation (multi-digits to 1,000)

2.Building fluency with addition and subtraction

3.Use standard units of measure (cm., inch; use ruler)

4.Describe and analyze shapes (sides, angles; compose, decompose; 2D and 3D) … building a foundation for area, volume, congruency, similarity, symmetry in later grades.

TIMELINE Curriculum Assessments

2011-12 K-2 New (none)

6-8,11 NJASK, HSPA

* CCC database but Only CCSS questions

2012-13 K-2, 6-8 (none), NJASK3-5, HS New NJASK*, HSPA*

2013-14 6-8 New NJASK*K-5, HS NJASK*, HSPA*

2014-15 K-11 NEW (PARCC)(This year’s 8th graders)


1.developing understanding of multiplication and division and strategies for multiplication and division within 100;

2.developing understanding of fractions, especially unit fractions (fractions with numerator 1);

3.developing understanding of the structure of rectangular arrays and of area; and

4.describing and analyzing two-dimensional shapes.

Grade-3 before/after NJASK?

Multiplication facts by memory (all of them for NJASK … Do these all year long.)

Fractions on number line (do after NJASK)

Equivalent fractions (do after NJASK)

Time (see new CCSS 3.MD.1.) see – symbol

Areas of rectilinear figures (like an ‘L’ shape w/right angles).

Left Out of Grade-3 Decimals

Estimation

Congruence, symmetry, circles

Coordinate geometry, transformations

Patterns, functions

Probability, discrete math

First time students use metric measuring

Ex. What might students do? 7 x 28

We need to do more mental math to help students gain number sense … working on pencil-paper procedures for computation will not help students gain good number sense.

Ex. Pose a problem. Jose has 3 boxes of 24 crayons, how many does he have in all.

What’s Different in Grade-3?

Fractions Across GradesGrade-1

taking squares/circles and dividing them up.

Grade-2 Know equal shares don’t need to be the

same shape to be the same size

Grade-3unit numeratorsfractions on the number line

Grade-4should focus on three critical areas:

1.developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends;

2.developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; and

3.understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

What are we doing? GRADE-4

FRACTIONS, FACTIONS, fractions

Don’t’ change till after NJASK

Fill in gaps fall 2012 from grades K – 3

Add new topics:

number sentences w/variables (start using letters for unknowns)

factors and multiples

Grade-4: Do AFTER NJASK

multiply 1 digit by 3 digits (425 x 6)

divide 4 digits by 1 digit ( 427 ÷ 3)

equivalent fractions (2/3 and 4/6)

decimal notation for fractions (2/5 = 0.40)

multiply fraction by whole number (2/3 x 6 )

Leave for next year:

Convert units to smaller ones (2 ft = 24 inches)

measure angles

What’s omitted in grade-4?

Negative numbers

Estimation

Congruence,

Symmetry,

Transformations

Grade-5Should focus on three critical areas:

1.developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions);

2.extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and

3.developing understanding of volume.

Gr.5 – Fractions, Division, Decimals

(This is going to be hard!)

Fill in gaps fall 2012 from grades K-4

Add new topics

Whole number exponents for powers of 10 ( 321 = 3 x 102 + 2 x 101 + 1 or 100 )

Multiply fractions and mixed numbers (½ x 4 ½ )

Divide unit fractions and whole numbers (3 ÷ ½ )

Converting little units into big 24” = ___ ft.

Line plot with fractional measures (a big group)

Volume

Multiplcation and Division Across Grades

Grade-2 Intro to multiplication briefly

Grade-3 When do you use it? (Understand and master facts) 1 digit x multiples of 10 4 x 60 = 240

Grade-4 Use multiplication to compare (twice as big, three times a big) and 1 digit mult. up to 4 digits, 2 digits x 2 digits, 2 digits x 3 digits

Grade-5 Fluent with multiplication

Grade-6 Fluent with division

Not explicitly in CCSSZero as an additive identity element

Ordinals

Calendar-related skills

Mode

Recognizing when an estimate is appropriate, and understand usefulness of an estimate.

Know approximate equivalents between standard and metric systems (one kilometer is about 6/10 of a mile)

Solve problems involving different units of measure within a measurement system (4’3” plus 7’10” = 12’1”)

New Assessments (PARCC)

K-2 ASSESSMENTS

Will not be done electronically

May be optional for districts

GRADES 3-8, 11

Will have all/part electronically

Results back to teachers in a few weeks

Technology-Availability Survey to districts this fall to collect data about each school.

New Assessments (PARCC)

25%content

50%content

75%content

Allcontent

Formative diagnostic

tool

Formative performance based (?)

Formative(machine &

hand-scored)

Summative end-yr results

before end yr

Optional Optional Required (?)

required

Whatever is in the CCSS for each grade-level K-8 may be assessed.

Quarterly assessments are being considered

Detailed results to be received by teacher w/in a few weeks.These will all be ‘secured’ tests (Districts will have testing ‘windows’ instead of same date for all districts and schools.)

New Assessments

Computer adaptive assessments with routers and appropriate items for stage-2

Four-week testing windows

Computer score-able constructed response items

Item banks for Formative, Benchmark AND Summative assessments

Student, class, school, district, state and nation results three days after the window closes

Whatever is in the CCSS for each grade-level K-8 may be assessed.

Quarterly assessments are being considered

The Dream !

Grade K-8, 9-12 math is divided into categories

K to 8 High School

Grade Conceptual Category

Domain Domain

Cluster Cluster

Standards Standards

Is the HSFORMAT different?

CONCEPTUAL CATEGORY• DOMAIN• CLUSTER• Standards

+ H.S. STEM

K-8 math is divided into 4 DOMAINS

OA Operations & Algebraic Thinking

NB Number & Operations in Base 10 NBF Number & Operations - Fractions

MD Measurement and Data

G Geometry

What does it look like? (Grade-2)

DOMAIN: 2.0A Operations & Algebraic thinking

CLUSTER: ADD & SUBTRACT WITHIN 20

Standard 2.0A.3. Fluently add and subtract within 20 using mental strategies.

By the end of grade 2, know from memory all sums of two one-digit numbers

2+2 9+9 8+9 5+7

See 1.OA.6 for a list of mental strategies.

What does it look like? Grade-K

DOMAIN: Operations & Algebraic thinking

CLUSTER K.OA: Understand addition as putting together & adding to, and understand subtraction as taking apart & taking from.

Standard K.0A.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, expressions, or equations.

Standard K.OA.2 Solve addition & subtraction word problems, … add and subtract with 10

Standard K.OA.3 Fluently add and subtract within 5.

What does it look like? Grade-1


CLUSTER: 1.OA Represent and solve problems involving addition and subtraction

Standard 1.0A.1 Use addition & subtraction within 20 to solve word problems involving putting together, taking apart, and comparing with unknowns … by using objects, drawings & equations with a symbol for the unknown to represent the problem.

Standard K.OA.2 Solve word problems that call for addition of three whole numbers … sum is < 20.



CLUSTER: 2.OA Represent and solve problems involving addition and subtraction

Standard 2.0A.1 Use addition & subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns … by using objects, drawings & equations with a symbol for the unknown to represent the problem.



CLUSTER: 3.OA Represent and solve problems involving multiplication and division.

Standard 3.0A.1 Interpret products of whole numbers, e.g. interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.

Standard 3.0A.2 Interpret whole-number quotients

Standard 3.0A.3 Use x and ÷ within 100 …

Standard 3.0A.4 Determine the unknown ..

8 x ? = 48 5 = ÷ 3, 6 x 6 =



CLUSTER: 4.OA Use the four operations with whole numbers to solve problems

Standard 4.0A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 … Represent verbal statements of multiplicative comparisons as mutiplication equations.

Standard 4.0A.2 Multiply or divide to solve word...

Standard 3.0A.3 Solve multi-step word problems … whole numbers … four operations … Assess the reasonableness of answers.


DOMAIN:Operations & Algebraic thinking

CLUSTER: 5.OA Write and interpret numerical expressions.

Standard 5.0A.1 Use parenthesis, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. (new to gr. 5)

Standard 5.0A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

Easy to remember!

4.0A

3.MD

5.G

2.NBT

4.NF

Number & Operations Base 10Number & Operations Fractions

Operations & Algebraic ThinkingMeasurement and Data

AN OVERVIEW

WHAT has been added? What has been moved?

Grade- 1 (add/remove?)ADD: Move symbols = < > from gr.-3 to gr. 1 1.NBT.3

ADD: Zero as the identity element (e.g. 7 + 0 = 7) Zero as an additive identity element is not explicitly articulated in the CCSS at any grade. 1.OA.3

ADD: Calculator (use of a calculator is explicitly included only in the CCSS for Mathematical Practice. 1.NBT.4)

ADD: Weight (lb., gram, kilogram), Capacity (pint, quart, liter), Temperature (degrees Celsius, degrees Farenheit)

ADD: Data generated from chance devices 1.MD.4

ADD: Use simple shapes to make designs, patterns, …

Grade- 2 (add/remove)ADD: Determine whether a whole number is

ODD or EVEN from grade-3 to 2.OA.3

ADD: Compare two 3-Digit numbers based on meanings of hundreds tens and ones digits using > < = symbols to record comparisons 2.NB.4

REMOVE

Move actual using of symbols > < = for first time to grade-1.NBT.3

Grade- 3 (add/remove)ADD: Fluently multiply and divide within 100,

… 3.0A.7

ADD: Move memorize the multiplication table from grade-4 to grade-3. By end of grade-3, know from memory all products of two one-digit numbers.

REMOVE

Move determining whether a whole number is odd or even from grade-3 to grade-2.A.3

Move actual using of symbols > < = from grade-3 to grade-1.NBT.3

Grade- 4 (add/remove)ADD: Move from grade-5: Find all factor pairs of

whole numbers in the range. Recognize 1-100. 4.OA.4

ADD: Move from grade-5: Recognize angles as geometric shapes when … understand concept of angle measurement. 4.MD.5

ADD: Move from grade-5: Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 4.MD.6

Move memorize the multiplication table from grade-4 to grade-3. By end of grade-3, know from memory all products of two one-digit numbers.

Grade- 5 (add/remove) ADD: EXPLAIN patterns in numbers of zeros … 5.NBT.2 (moved

from grade-6; note this includes exponents)

ADD: + - x ÷ decimals to hundredths using concrete models .. 5.NBT.7 (moved x ÷ of decimals from grade-6)

ADD: Solve real world problems with multiplication of fractions and mixed numbers 5.NF.6 (moved from grade-6)

ADD: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions 5.NF.7 (moved from grade-6)

Move from grade-5 to 4: Find all factor pairs of whole numbers in the range. 1-100. Recognize angles as geometric shapes when … understand concept of angle measurement. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 4.OA.4, 4MD.5, 4.MD.6

Potential Challenges

When you TRANSITION to the COMMON CORE in mathematics

See handout (tan/gray)

CCSS (June 16,2010) Grade/Course Change

2 – 8

Algebra

Geometry

SMALL-GROUP ACTIVITIES

CUT THE CAKE

FIND THE PERIMETER

(Algebra Tiles: black =“x” yellow = “1”)

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 precision7. Look for and make use of structure.8. Look for and express regularity in

repeated reasoning.

MATHEMATICAL PRACTICES

REASONING and SENSE

MAKING

Why PROBLEM-BASED Interactive learning

Motivates new learning

Recalls prior learning

Engages students in thinking about math

Requires students to communicate

Introduces or develops math representations

Fosters reasoning skills

As The Crow Flies (** Activity)

CCSS takes what seems so usual, just a bit further.

School

Your house

Friend’s House

As The Crow Flies (activity)CCSS takes what seems so usual, just a bit further.

School

Your house

Friend’s House

As The Crow Flies (activity)

CCSS takes what seems so usual, just a bit further.

School

Your house

Friend’s House

Grade-5 reasoning example

Students write an expression for calculations given in words such as: Divide 144 by 12, and then subtract 7/8.

They write (144 ÷ 12) – 7/8.

Students recognize that 0.5 x (300 ÷ 15) is

1⁄2 of (300 ÷ 15) without calculating the quotient.

Grade 4.NF.2.

Compare two fractions ….Fractions with common denominators may be

compared using the numerators as a guide:

2/6 3/6 4/6 smallest to largest

Fractions with common numerators may be compared and ordered using the denominators as a guide.

1/10 1/5 1/3 smallest to largest

NOT JUST MEMORIZE, understand, visualize!

Grade-3 MODELINGModels help build understanding of the

commutative property:

Example: 3 x 6 = 6 x 3 OR 3 x 2 = 2 x 3

In the following diagram it may not be obvious that 2 groups of 3 is the same as 3 groups of 2. A student may need to count to verify this.

Grade 2- ReasoningThis standard calls for students to solve one- and two-

step problems using drawings, objects and equations. Students can use place value blocks or hundreds charts, or create drawings of place value blocks or number lines to support their work. Two step-problems include situations where students have to add and subtract within the same problem.

Example: In the morning there are 25 students in the cafeteria. 18 more students come in. After a few minutes, some students leave. If there are 14 students still in the cafeteria, how many students left the cafeteria? Write an equation for your problem.

Do and Understand !(new handouts)

The Question Changes

Processes are Open Ended

WATER CONTAINERS

YOU CHOOSE

INTERNET RELAY CHAT

POSSIBLE FLOOD

ICE CREAM MELT (Pre Calculus)

REASONING & SENSE MAKING

http://www.nctm.org

Not the CORE onlyThe Common Core State Standards

identify mathematics that ALL students should understand and be able to do.

The Common Core State Standards do NOT delineate ALL of the mathematics that students should know and be able to do.

There is more to the apple than the core!

Bob Riehs, NJDOE Sept. 2011

BRING for TOMORROWYour Textbooks

Curriculum guide or whatever you use

Laptop if you can access the Internet in school with it.

QUESTIONS?

http://www.corestandards.org

[email protected]

http://www.state.nj.us/education/aps/cccs/math

http://www.corestandards.org/

mailto:[email protected]

http://www.state.nj.us/education/aps/cccs/math

KEEP IN TOUCHJUDITH T. BRENDEL

[email protected]

http://www.pascack.k12.nj.us

[curriculum][mathematics][Workshop: K-5 CCSS Standards]

mailto:[email protected]

http://www.pascack.k12.nj.us/

Documents

Overview of the K-5 CCSS Common Core State Standards mathematics by Judith T. Brendel Math Supervisor, Pascack Valley Regional HS District [email protected]