Instructional Leadership Math CadreKindergarten - 5th GradeSHIFT 2: COHERENCE

THE PROE CENTER

Multi-Tiered System of Support (MTSS/RtI)

Statewide System of Support

Priority

Focus

Foundational

Focus Areas:- Continuous

Improvement Process (Rising Star)

- Common Core ELA- Common Core Math- Teacher Evaluation- Balanced Assessment

Commitments

Today’s Outcomes

Use the Standards for Mathematical Practice while problem solving

Define coherence in the Common Core Identify coherence within the standards Utilize coherent problem solving structures and instructional

strategies that build conceptual understanding Explore resources that build coherence Plan for implementation

Today’s Agenda

Review Shift 1: Focus Digging Deeper with the Standards for Mathematical Practice Shift 2: Coherence Designing Instruction to Support Diverse Learners:

Modes of Representation Problem Solving Structures Number Talks Problem Solving Strategies

Instructional resources Planning for Implementation

Shift 1: Focus

REVIEW

Focusing on Solving Problems

USING THE STANDARDS FOR MATHEMATICAL PRACTICE

Solving Problems

Standards for Mathematical Practice

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 precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Debriefing the Activity

Discuss:

1) The content

2) The math practices

Shift 2: Coherence

Coherence

Math should make sense.

Mike McCallum

The Importance of Coherence in Math

Coherence

Math should make sense

A progression of learning

Coherence supports focus

Use supporting material to teach major content

Math should make sense• Within a grade level• Across many grade levels

Coherence WITHIN a grade level

The standards within a grade level strategically allow: Instruction that reinforces major content and

utilizes supporting standards

Important to remember: Meaningful introduction to topics so that skills

complement one another

Coherence WITHIN a grade level

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. 2.MD.5

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two- step “how many more” and “how many less” problems using information presented in scaled bar graphs.

3.MD.3

Geometric measurement: understand concepts of area and relate area to multiplication and addition. 3.MD.3rd cluster

Make a line plot to display a data set of measurements in fractions of a unit ( ½, ¼, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. 4.MD.4

Coherence ACROSS grade levels

Students apply skills from previous grade levels to learn new topics in their current grade level

Meaningful math progressions reflect this, building knowledge across the grade levels

Coherence ACROSS grade levels

One of several staircases to algebra designed in the OA domain.

Coherence ACROSS grade levels

4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

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

6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

Grade 4

Grade 5

Grade 6

CCSS

Coherence Activity

Learning Progression:

A pathway or stream students travel as they progress toward mastery of the skills.

These streams contain carefully sequenced building blocks (content standards) that

support students’ progression towards mastery.

Benefits of Progressions/Streams

Enables teachers to build instructional sequences

Provide a framework to systematically implement effective formative assessment

Common Core Progression Streams

1. Counting and Cardinality (K)

2. Algebraic Thinking (K-HS)

3. Number and Quantity

4. Geometry

5. Functions

6. Statistics and Probability

7. High School Modeling

Visual Map

Activity:Progression Jigsaw

1) Read the intro for your designated Progression

2) Read your grade level in the designated Progression, highlighting key skills for your grade level

3) Create a flow-chart of skills, visually depicting how the skills in each grade level build on one another in developing conceptual understanding within the Progression (domain)

4) Creatively share your learning progression with the rest of the group

Modes of Representation

Modes of Representation

Manipulativesor Tools

Pictures/ Graphs

Written Symbols

Oral/Written

Language

Real-Life Situations

Problem Solving Structures

WRITE A SIMPLE ADDITION/ SUBTRACTION WORD PROBLEM.

Addition & Subtraction  Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies

are on the grass now?2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over

to the first two?2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then

there were five bunnies. How many bunnies were on the grass before?

? + 3 = 5

Take From

Five apples were on the table. I ate two apples. How many apples are on the table

now?5 – 2 = ?

Five apples were on the table. I ate some apples.

Then there were three apples. How many apples did I eat?

5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How

many apples were on the table before?? – 2 = 3

  Total Unknown Addend Unknown Both Addends Unknown

Put Together/ Take Apart

Three red apples and twogreen apples are on the table. How many

apples areon the table?

3 + 2 = ?

Five apples are on the table.Three are red and the rest are green. How

many applesare green?

3 + ? = 5, 5 – 3 = ?

Grandma has five lowers.How many can she put in her

red vase and how many in her blue vase?5 = 0 + 5, 5 = 5 + 05 = 1 + 4, 5 = 4 + 15 = 2 + 3, 5 = 3 + 2

  Difference Unknown Bigger Unknown Smaller Unknown

Compare

(“How many more?” version):Lucy has two apples. Julie has five apples.

How manymore apples does Julie have than Lucy?

(“How many fewer?” version):Lucy has two apples. Julie has five apples.

How many fewer apples does Lucy have than Julie?

2 + ? = 5, 5 – 2 = ?

(Version with “more”):Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie

have?(Version with “fewer”):

Lucy has 3 fewer apples thanJulie. Lucy has two apples.

How many apples does Julie have?2 + 3 = ?, 3 + 2 = ?

(Version with “more”):Julie has three more apples than Lucy. Julie

has five apples. How many apples does Lucy have?

(Version with “fewer”):Lucy has 3 fewer apples than

Julie. Julie has five apples.How many apples does Lucy have?

5 – 3 = ?, ? + 3 = 5

Multiplication & Division

 

Unknown Product Group Size UnknownNumber of Groups

Unknown3 x 6 = ? 3 x ? = 18, and 18 ÷ 3 = ? ? x 6 = 18, and 18 ÷ 6 = ?

Equal Groups

There are 3 bags with 6 plums in each bag. How many plums are there in all?

 Measurement example. You need 3 lengths

of string, each 6 inches long. How much string will you need altogether?

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?

 Measurement example. You have 18 inches

of string, which you will cut into 3 equal pieces. How long will each piece of string

be?

If 18 plums are to be packed 6 to a bag, then how many bags are needed?

 Measurement example. You have 18 inches of string, which you will cut into pieces that

are 6 inches long. How many pieces of string will you have?

Arrays, Area

There are 3 rows of apples with 6 apples in each row. How many apples are there?

 Area example. What is the area of a 3 cm by

6 cm rectangle?

If 18 apples are arranged into 3equal rows, how many apples will be in each

row? 

Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long,

how long is a side next to it?

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?

 Area example. A rectangle has area 18

square centimeters. If one side is 6 cm long, how long is a side next to it?

Compare

A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the

red hat cost? 

Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?

A red hat costs $18 and that is3 times as much as a blue hat costs. How

much does a blue hat cost? 

Measurement example. Arubber band is stretched to be

18 cm long and that is 3 times as long as it was at first. How long was the rubber band

at first?

A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat

cost as the blue hat? 

Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18

cm long. How many times as long is the rubber band now as it was at first?

General a x b = ? a x ? = p, ad p ÷ a = ? ? x b = o, and p ÷ b = ?

Problem Solving Structures

WHICH PROBLEM SOLVING STRUCTURE DID YOU USE?

The Importance of Mental Math

BUILDING DEEP CONCEPTUAL UNDERSTANDING

The Role of Mental Math

Encourages students to build on number relationships

Forces students to rely on what they know and understand about numbers

Strengthens students understanding of place value system

Instructional Strategies

LEARNING NUMBER AND QUANTITY

FORWARD NUMBER WORD SEQUENCE (fnws)/ NUMBER WORD AFTER (nwa)

Level 0 Emergent Child cannot produce fnws 1-10

Level 1 Initial1-10

Child can produce fnws 1-10Cannot tell number word after in range 1-10

Level 2 Intermediate1-10

Child can produce fnws 1-10Can tell number word after in range 1-10 but needs to count from 1 to do so

Level 3 Facile 1-10

Child can produce fnws -10Can tell number word after numbers in range 1-10

Level 4 Facile 1-30

Child can produce fnws 1-20Can tell number word after numbers in range 1-30

Level 5 Facile 1-100

Child can produce fnws 1-100Can tell number word after numbers in range 1-100

Level 6 Facile 1-1,000

Child can produce fnws 1-1000 by 1s, 10s, 100s, 1,000s on and off the decades, Can tell nwa for 1s, 10s 100s, 1000s after any number in range 1 – 1,000,000

BACKWARDS NUMBER WORD SEQUENCE (bnws) / NUMBER WORD BEFORE (nwb)

Level 0 Emergent Child cannot produce bnws 10-1

Level 1

Initial10-1

Child can produce bnwsCannot tell number before in range 1-10

Level 2

Intermediate 10-1

Child can produce bnws 10-1 Can tell number word before in range 1-10, but drops back to generate a running count

Level 3

Facile10-1

Child can produce 10-1Can tell number word before numbers in range 1-10 without dropping back

Level 4

Facile0-1

Child can produce bnws 30-1Can tell number word before numbers in range 1-20 without dropping back

Level 5

Facile100-1

Child can produce bnws 100-1Can tell number word before numbers in range 1-100

Level 6

Facile1-1,000

Child can produce bnws 1-1,000 on and off the decadeCan tell number word 1,10,100 before in range 1-1,000

Level 7

Facile1-1,000,000

Child can produce bnws 1-1,000,000* on and off the decadeCan tell number word 1, 10 100, 1000 before in range 1-1,000,000

Learning Framework in Number NUMERAL IDENTIFICATION (#ID) LEVELS

Level 0 Emergent Child cannot name all numerals in the range 1-10

Level 1 1 to 10 Child can name all numerals in range 1-10

Level 2 1 to 20 Child can name numerals in range 1-20

Level 3 1 to 100 Child can name and order one and two digit numbers

Level 4 1 to 1,000 Child can name all numbers in the range of 100 to 1,000

Level 5 1 to 1,000,000

Child can name and write all numbers in the range of 1,000 to 1,000,000

Number Words and Numerals

Student objective: knows fnws of number words (rote counting)

Number naming Number recognition Sequencing numbers Ordering numbers

Sequencing vs. Ordering

Sequencing: children rely on an auditory and visual structure to put a set of numbers in ascending or descending order (by 1s,2s, 5s, 10s) PREREQUISITE FOR ORDERING, ROUNDING, OR ESTIMATING NUMBERS

Ordering numbers: placing a set of numbers in ascending or descending order that do not have a visual, imaginary, or auditory pattern; Ex: 17, 23, 34PREREQUISITE FOR ROUNDING AND ESTIMATING

PREREQUISITE FOR KNOWING WHETHER COMPUTATION IS REASONABLE

PREREQUISITE FOR UNDERSTANDING PLACE VALUE CONCEPTS

Counting and Early Arithmetic

1:1 Counting correspondence Establishing how many in a collection Relative position of number Relative magnitude of number

Groupings

Includes the skills of: Subitizing Recognizing spatial patterns Making and recognizing temporal patterns Making and recognizing finger patterns

Instructional Strategies

NUMBER TALKS

Number Talks

CHANGING THE WAY WE ENGAGE IN MATHEMATICS INSTRUCTION IN THE CLASSROOM

EMBEDDING THE STANDARDS FOR MATHEMATICAL PRACTICE

THOUGHT BUBBLES

Number Talks

Builds mathematical proficiency

Promotes mental math

Focuses children on numerical relationships

Goals of Number Talks

K – 2 Goals To develop number stands To develop fluency with

small numbers Subitizing Making tens

3 – 5 Goals Number sense Place Value Fluency Properties Connecting mathematical

ideas

Process

Students sit on the floor near the board Teacher presents a problem on the board Students think quietly to themselves as to how to

complete the problem Students give a “private” thumbs-up when they can

figure it out Students share ways of figuring out problems – whole

group

Teacher’s Roles

The Facilitator

Guide students to ponder and discuss examples

Ask open-ended questions

Number “Talking”

Student Prompts I agree with ____________because

______________

I do not understand ______________. Can you explain this again?

I disagree with ______________ because __________________.

How did you decide to _______________?

Facilitating Questions How did you think about that?

How did you figure it out?

What did you do next?

Did someone solve it in a different way?

What strategies seem to be efficient, quick, simple?

Who would like to share their thinking?

Start with Five Small Steps

1. Start with smaller/easier problems to elicit thinking from multiple perspectives

2. Be prepared to offer a strategy from a previous student

3. It is ok to put a student’s strategy on the backburner

4. Limit number talks to 5 – 15 minutes

5. Be patient with yourself and your students

Instructional Strategies

COMPUTATION WITH ADDITION AND SUBTRACTION

Learning Framework in Number (SEAL) STAGES OF EARLY ARITHMETIC LEARNING

Stage0

EmergentChild is learning the counting sequence and developing one-to-one correspondence in counting, but cannot yet count visible items accurately. (disorganized count, skips some, counts some twice, etc.)

Stage 1

PerceptualChild can count visible items but can only deal with adding visible quantities, not screened items. Child may solve simple screened problems in finger range by re-presenting with fingers and counting forward from 1 three times to solve. (Watch the fingers)

Stage 2

FigurativeChild can deal with screened addition by using a number sequence logic, counting forward from 1. (Early 2 – fingers/touches, Mid 2 – spatial patterns, High 2 – mental counts)

Stage 3

Counting onChild solves addition by counting on, subtraction by counting back and missing addend problems by counting up or counting back. They understand cardinality of number and think of numbers as composite groups as well as units

Stage4

Intermediate #Child uses most efficient counting strategy to solve +/- problems. (counting up to, counting back from, counting back to.) Understands part/whole

Stage 5

Facile #Child solves addition/subtraction problems by choosing from a full range of non-counting by one strategies including doubles, think ten, partitioning, using known facts, etc. (A rule of thumb is that the child must show evidence of at least three different strategies.)

Stage 6

Advanced #

Child extends and applies knowledge of addition and subtraction to solve a range of tasks, including multi-digit addition and subtraction, by focusing in on the relationships between numbers and operations and generalized number sequences/relations. (commutativity, associativity, relationship between +/-, growing number patterns, etc)

Learning Framework in Number BASE TEN - UNDERSTANDING SKILL LEVELS

Level 1 Initial ConceptChild doesn’t see ten as a composite unit. Focuses on ten individual items that makeup the ten. On +/- tasks involving tens and ones, child counts forward or backward by ones.

Level 2 Intermediate Concept of Ten

Ten is seen as a unit comprised of ten ones. Child is dependent on representations of units of ten (Craft sticks bundles, dot strips) The child can perform = - tasks involving tens and ones when these tasks are presented with materials, (Could add 10s and 1s, such as 4, 14, 24, 34, etc. if dot strips and dots/bundles, base ten materials)

Level 3 Concept of Ten

Child can mentally solve +/- tasks using knowledge of tens and ones (additive property, place value property without using materials or representations of materials. 9penil/paper, “visual” chalkboard)

Understanding Quantity Composing & Decomposing Number

Number Bonds

Part-Part-Whole

Quack Attack, Shake & Spill Cube Trains

Daily Warm Up / Number of the Day

Number of the Day (using ten-frames)

Ex: Show different ways to make 83 using two addends

Teacher records expressions on board

Students slide ten-frames into columns to show two addends

Challenge: build 83 with 3 addends

Properties Checklist Even

Odd

Multiples of 2

Multiple of 5

Multiple of 10

Single digit

Two-digit

Three-digit

Less than _____

Greater than _____

Our number is between _____ and _____

Closer to _____ than _____

10 more is _____

10 less is _____

Reasoning with Ten Frames

Helps children:

Visualize number

Develop strategies for mental computation

Master math facts

Internalize place value concepts

*Concepts are learned best in context that makes them imaginable

Making Ten

Additional Ideas:

• Making Ten Memory

• Making Ten with Dice

• Go Fish

• How many?• Which has bigger• What if I added 2 more dots?

Taking it to the Next Level:

• Screen some

• Add two frames

The Empty Number Line

Visual representation for recording thinking strategies during mental computation

Need to have a secure understanding of numbers to 100

Counting on and back

Recall of addition and subtraction for numbers to 10

Jump to given numbers

Solve Addition & Subtraction: 2-digit, 3-digit

Solve Multiplication & Division problems

Solve word problems

Understanding place value

Instructional Strategies

FOR MULTIPLICATION

Multiplication/Division

Standard algorithms are process driven Typically, students will mimic steps without

understanding the process

To build conceptual understanding – begin with developmentally appropriate steps

Moving from additive thinking to multiplicative thinking

Draw it out!

Kids go from thinking 8 things to thinking 8 groups of something

Additive Properties

Additive PropertiesRepeated Addition

Skip Counting

Multiplicative PropertiesArea/AreaPartial ProductBox methodDecomposing

Learning Framework in Number EARLY MULTIPLICATION/DIVISION LEVELS

Level 0

Emergent Grouping Cannot describe or make equal groups or shares

Level 1

Initial GroupingUses perceptual counting (by ones) to establish the numerosity of a collection of equal groups to share items into groups of equal size or to share items into an equal number of groups.

Level 2

Perceptual Counting in multiples

Uses a multiplicative counting strategy (Skip counting) to count visible items arranged in equal groups, to share items into groups of equal size of to share items into an equal number of groups

Level 3

Figurative Composite

Uses multiplicative counting strategy (skip counting0 to cunt items arranged in equal groups where items are unseen (but group markers are shown)

Level 4

Repeated abstract- Composite Grouping

Counts composite units using repeated addition or subtraction where items are unseen. Uses the composite unit the specified number of times. (4 x 6 is 4 + 4 is 8 and another 4 +4 = 8. 8 + 8 = 16, 16 + 4 = 20 ad 20 + 4 = 24.

Level 5

Multiplication as Operations

Can recall and derive many facts for multiplication and division ( 8 x 6 = 48, 7 x 9 is like 7 x 10 - .

Array

Visual representation of rows and columns for multiplication computation

4 x 3 = 12

Array Game

Pairs of students receive a grid, 2 dice and 2 different colored crayons/markers

Each child rolls the dice and then creates an array that matches the number pair rolled

Goal: fill the grid If a child’s array does not fit in the remaining space

on the grid they must pass and receive a strike. 3 strikes and the child is out

Area Model: 3rd, 4th, 5th

Pictorial representation of the Partial products method

Capitalizes on student’s understanding of place value

After sufficient practice with partial products, the standard algorithm makes much more sense

If one bag of fertilizer can cover 16 sq meters, how many bags will he need to cover the entire garden?

Partial Products 324 300 + 20 + 4

X 6

24 6 x 4

120 6 X 20

1,800 6 x 300

1,944

Area Model

x 300 20 4

6

1,800 + 120 + 24 = 1,944

1,800 120 24

Common Core Resources

Online Resources

K-5 Math Teaching Resources

Edcite

StraightAce

Planning for Next Steps

TAKING IT BACK TO MY CLASSROOM

SHARING WITH MY COLLEAGUES

Comments…

Questions…

Concerns…

Cindy Dollman – cdollman@peoriaroe48.netKim Glow – kglow@peoriaroe48.net

Math Common Core in the Media

Common Core in the Media