Transcript Instructional Leadership Cadre Math 6th * 12th

Instructional Leadership Math
Cadre
Kindergarten - 5th Grade
SHIFT 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
Mike McCallum
The Importance of
Coherence in Math
Coherence

Math should make sense
Math should make sense
• Within a grade level
• Across many grade levels

A progression of learning

Coherence supports focus

Use supporting material to teach major content
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
Draw a scaled picture graph
and a scaled bar graph to
represent a data set with
several categories. Solve oneand two- step “how many
more” and “how many less”
problems using information
presented in scaled bar graphs.
3.MD.3
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
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
CCSS
Grade 4
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.
Grade 5
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.
Grade 6
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.
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
Manipulatives
or Tools
Real-Life
Situations
Oral/Written
Language
Pictures/
Graphs
Written
Symbols
Problem
Solving
Structures
WRITE A SIMPLE
ADDITION/
SUBTRACTION
WORD PROBLEM.
Result Unknown
Change Unknown
Start Unknown
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
Addition
&
Subtraction
Add to
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
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
Three red apples and two
green apples are on the table. How many
apples are
on the table?
3+2=?
Five apples are on the table.
Three are red and the rest are green. How
many apples
are 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 + 0
5 = 1 + 4, 5 = 4 + 1
5 = 2 + 3, 5 = 3 + 2
Difference Unknown
Bigger Unknown
Smaller Unknown
(“How many more?” version):
Lucy has two apples. Julie has five apples. How
many
more 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 than
Julie. 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
Take From
Put Together/
Take Apart
Compare
Unknown Product
Group Size Unknown
Number of Groups Unknown
3x6=?
3 x ? = 18, and 18 ÷ 3 = ?
? x 6 = 18, and 18 ÷ 6 = ?
There are 3 bags with 6 plums in each bag.
How many plums are there in all?
If 18 plums are shared equally into 3 bags,
then how many plums will be in each bag?
Multiplication & Division
Equal Groups
Arrays, Area
Compare
General
If 18 plums are to be packed 6 to a bag, then
how many bags are needed?
Measurement example. You have 18 inches of
Measurement example. You need 3 lengths of Measurement example. You have 18 inches of
string, which you will cut into pieces that are 6
string, each 6 inches long. How much string will string, which you will cut into 3 equal pieces.
inches long. How many pieces of string will you
you need altogether?
How long will each piece of string be?
have?
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?
A blue hat costs $6. A red hat costs 3 times as
much as the blue hat. How much does the red
hat cost?
If 18 apples are arranged into 3
equal 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?
A red hat costs $18 and that is
3 times as much as a blue hat costs. How
much does a blue hat cost?
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?
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
Measurement example. A rubber band was 6
Measurement example. A rubber band is 6 cm
rubber band is stretched to be
cm long at first. Now it is stretched to be 18 cm
long. How long will the rubber band be when
18 cm long and that is 3 times as long as it was
long. How many times as long is the rubber
it is stretched to be 3 times as long?
at first. How long was the rubber band at first?
band now as it was at first?
axb=?
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
Level 0
Emergent
Level 1
Initial
1-10
Level 2
Child cannot produce fnws 1-10
Child can produce fnws 1-10
Cannot tell number word after in range 1-10
Intermediate Child can produce fnws 1-10
1-10
Can 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 -10
Can tell number word after numbers in range 1-10
Level 4
Facile
1-30
Child can produce fnws 1-20
Can tell number word after numbers in range 1-30
Level 5
Facile
1-100
Child can produce fnws 1-100
Can 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
FORWARD NUMBER WORD SEQUENCE (fnws)/ NUMBER WORD AFTER (nwa)
Level 0
Emergent
Child cannot produce bnws 10-1
Level 1
Initial
10-1
Level 2
Intermediate
10-1
Level 3
Facile
10-1
Child can produce 10-1
Can tell number word before numbers in range 1-10 without dropping back
Level 4
Facile
0-1
Child can produce bnws 30-1
Can tell number word before numbers in range 1-20 without dropping back
Level 5
Facile
100-1
Child can produce bnws 100-1
Can tell number word before numbers in range 1-100
Level 6
Facile
1-1,000
Child can produce bnws 1-1,000 on and off the decade
Can tell number word 1,10,100 before in range 1-1,000
Level 7
Facile
1-1,000,000
Child can produce bnws
Cannot tell number before in range 1-10
Child can produce bnws 10-1
Can tell number word before in range 1-10, but drops back to generate a running count
Child can produce bnws 1-1,000,000* on and off the decade
Can tell number word 1, 10 100, 1000 before in range 1-1,000,000
BACKWARDS NUMBER WORD SEQUENCE (bnws) / NUMBER WORD BEFORE (nwb)
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, 34
PREREQUISITE 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
3 – 5 Goals

To develop number stands

Number sense

To develop fluency with small
numbers

Place Value

Fluency

Subitizing

Properties

Making tens

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
__________________.

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?
How did you decide to _______________?
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
Stage
0
Emergent
Child 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
Perceptual
Child 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
Figurative
Child 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 on
Stage
4
Intermediate #
Stage
5
Stage
6
Child 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
Child uses most efficient counting strategy to solve +/- problems. (counting up to, counting back
from, counting back to.) Understands part/whole
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.)
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)
Level 1
Level 2
Level 3
Initial Concept
Child 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.
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)
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)
Learning Framework in Number
BASE TEN - UNDERSTANDING SKILL LEVELS
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

Less than _____

Odd

Greater than _____

Multiples of 2


Multiple of 5
Our number is between
_____ and _____

Multiple of 10

Closer to _____ than _____

Single digit

10 more is _____

Two-digit

10 less is _____

Three-digit
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
• How many?
• Which has bigger
• What if I added 2 more dots?
Additional Ideas:
Taking it to the Next Level:
• Making Ten Memory
• Screen some
• Making Ten with Dice
• Add two frames
• Go Fish
The Empty Number Line
Visual representation for recording thinking
strategies during mental computation

Jump to given numbers

Solve Addition & Subtraction: 2-digit,
3-digit

Need to have a secure understanding of
numbers to 100

Solve Multiplication & Division
problems

Counting on and back

Solve word problems

Recall of addition and subtraction for
numbers to 10

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 Properties
 Repeated Addition
 Skip
Counting
Multiplicative Properties
 Area/Area
 Partial Product
 Box method
 Decomposing
Learning Framework in Number
EARLY MULTIPLICATION/DIVISION LEVELS
Level
0
Emergent
Grouping
Cannot describe or make equal groups or shares
Uses 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
1
Initial Grouping
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
Level
5
Counts composite units using repeated addition or subtraction where items are
Repeated abstractunseen. Uses the composite unit the specified number of times. (4 x 6 is 4 + 4 is 8
Composite
and another 4 +4 = 8. 8 + 8 = 16, 16 + 4 = 20 ad 20 + 4 = 24.
Grouping
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
Area Model
300 + 20 + 4
X 6
24
120
1,800
6x4
x
300
6
1,800
6 X 20
6 x 300
20
4
120 24
1,944
1,800 + 120 + 24 = 1,944
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 – [email protected]
Kim Glow – [email protected]
Math
Common Core
in the Media
Common Core in the Media