Transcript 6th - 12th Grade - The PROE Center

Instructional Leadership Math
Cadre
6th – 12th 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 that build conceptual
understanding

Describe instructional strategies for computation

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

Supporting Diverse Learners: Modes of Representation

Problem Solving Structures

Coherent instructional resources

Common Core Math in the Media
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
Building Conceptual Understanding
Adding Modes of Representation
45
x 24
45
x 24
Problem
Solving
Structures
WRITE A SIMPLE
MULTIPLICATION/
DIVISION 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?
Instructional
Strategies
FOR
MULTIPLICATION
Additive Properties
 Repeated
 Skip
Addition
Counting
Distributive Property
 Area/Area
 Partial
Product
 Box method
 Decomposing
Advanced Strategies
 Double
and Halve
 (Standard
Algorithm)
Coherence in
Action
VIEWS YOU
CAN USE
Dan Meyer
Math Class
Needs a
Make-over
Dan Meyer Resources
Dan Meyer creates
Coherence for
teachers, too.
Three-Act
Math Tasks
on Livebinders
Math
Common Core
in the Media
Common Core in the Media
Planning for
Next Steps
TAKING IT BACK TO
MY CLASSROOM
SHARING WITH MY
COLLEAGUES
Comments…
Questions…
Concerns…
Cindy Dollman – [email protected]
Joe Delinski – [email protected]
Kim Glow – [email protected]