Transcript Slide 1

CCSS Mathematics
Grades 6-8
Curriculum Review Week 2012
I can understand that CCSS math
standards require a different type of
teaching, which also requires a different
type of assessing.
 I can understand that this different way of
teaching and assessing has an impact on
the work that we will be doing this week,
as we create our Baseline and Benchmark
Assessments.

Today’s Learning Targets
teachers & students:

Focus- Teach less, go deeper

Coherence- Build on previous learning and
connect concepts

Rigor- Balance of conceptual
understanding, procedural skill and
fluency, and application of skills in
problem solving situations
Math Shifts
When planning, ask
“What task can I give that will build
student understanding?”
rather than
“How can I explain clearly so they will
understand?”
Grayson Wheatley, NCCTM, 2002
Understanding mathematics
 Asking a student to understand something means asking a
teacher to assess whether the student has understood it.
But what does mathematical understanding look like?
 One hallmark of mathematical understanding is the ability
to justify, in a way appropriate to the student’s
mathematical maturity, why a particular mathematical
statement is true or where a mathematical rule comes
from. There is a world of difference between a student who
can summon a mnemonic device to expand a product such
as (a + b)(x + y) and a student who can explain where the
mnemonic comes from. The student who can explain the
rule understands the mathematics, and may have a better
chance to succeed at a less familiar task such as expanding
(a + b + c)(x + y).
 Mathematical understanding and procedural skill are
equally important, and both are assessable using
mathematical tasks of sufficient richness.
Definitions
http://www.corestandards.org/
 The
“Why” behind the mathstudents need to know why and
be able to explain why!
 Develop the concept then teach
the procedure that goes along
with that.
Concrete  Representational  Abstract
Understanding the Math
Timeline for Common Core
Mathematics Implementation
Common Core State Standards Adopted June, 2010
Year
Standards To Be Taught
Standards To Be Assessed
2011 – 2012
2003 NCSCOS
2003 NCSCOS
2012 – 2013
CCSS
CCSS (NC)
2013 – 2014
CCSS
CCSS (NC)
2014 – 2015
CCSS
CCSS (SBAC)
Let’s look at a familiar
problem…
Which of the following represents
a.
b.
c.
d.
?
Same problem- a new
twist…
For numbers 1a – 1d, state whether or not each figure has
of its whole shaded.
1a.
ο Yes
ο No
1b.
ο Yes
ο No
ο Yes
ο No
1c.
1d.
ο Yes
ο No
For numbers 1a – 1d, state whether or not each figure has
of its whole shaded?
1a.
ο Yes
ο No
1b.
ο Yes
ο No
ο Yes
ο No
1c.
1d.
ο Yes
ο No
This item is worth 0 – 2
points depending on
the responses. What
series of the yes and
no responses would
give a student:
2 points? 1 point? 0 points?
“Turn and Talk”
For numbers 1a - 1d, state whether or not each
figure has of its whole shaded.
2 points: YNYN
1 point: YNNN, YYNN, YYYN
0 point: YYYY, YNNY, NNNN, NNYY, NYYN, NYNN
NYYY, NYNY, NNYN, NNNY, YYNY, YNYY
Teaching for Understanding
Through Cognitive Demand
The kind and
level of thinking
required of
students to
successfully
engage with and
solve a task.
Cognitive Demand Spectrum
PROCEDURES
PROCEDURES
DOING
MEMORIZATION
WITHOUT
WITH
MATHEMATICS
CONNECTIONS
CONNECTIONS
(TO
(TO
UNDERSTANDING, UNDERSTANDING,
MEANING, OR
MEANING,
CONCEPTS)
OR CONCEPTS)
Tasks that require
memorized procedures in
routine ways
Tasks that require
engagement with concepts,
and stimulate students to
make connections to
meaning, representation,
and other mathematical
ideas
Procedures Without
Connections
Convert the fraction
to a decimal and a
percent.
Memorization
What are the decimal and
percent equivalents for the
fractions:
and
Procedures With
Connections
Using a 10x10 grid,
identify
the decimal and percent
equivalents of
Level 1: Recalling and Recognizing
Student is able to recall routine facts or
knowledge and can recognize shape, symbols,
attributes and other qualities.
Level 2: Using Procedures
Student uses or applies procedures and
techniques to arrive at solutions or answers.
Depth of Knowledge
Level 3: Explaining and Concluding
Student reasons and derives conclusions.
Student explains reasoning and processes.
Student communicates procedures and findings.
Level 4: Making Connections, Extending and
Justifying:
Student makes connections between different
concepts and strands of mathematics. Extends
and builds on knowledge to a situation to arrive
at a conclusion. Student uses reason and logic
to prove and justify conclusions.
Depth of Knowledge
“WHERE”
THE
MATHEMATICS
WORK
Problem
Solving
DOING
MATH
Computational
& Procedural
Skills
Conceptual
Understanding
“HOW”
THE
MATHEMATICS
WORK
“WHY”
THE
MATHEMATICS
WORK
Balance Defined
Let’s do
some Mathematics
• Shade 6 small squares in a 4 x 10
rectangle.
• Using the rectangle,
explain how to
determine
each of the following:
a. the fractional part of the area shaded.
b. the decimal part of the area shaded.
c. the percent of the area that is shaded.
Doing Mathematics
Different Objects, Same Ratios
Trucks and Boxes
On which cards is the ratio of trucks to boxes the same?
Given one card, students are to select a card on which the ratio
of the two types of objects is the same. This task moves
students to numeric approach rather than a visual one and
introduces the notion of ratios as rates. A unit rate is depicted
on a card that shows exactly one of either of the two types of
objects. For example the card with three boxes and one truck
provides one unit rate. A unit rate for the other ratio is not
shown. What would it be?
Solve this problem without using any numeric
algorithms such as cross-products. This could include
pictures or counters, but there is no prescribed
method.
Two camps of scouts are having pizza parties. The
Bear Camp ordered enough so that every 3 campers
will have 2 pizzas. The leader of the Raccoons
ordered enough so that there would be 3 pizzas for
every 5 campers. Did the Bear or the Raccoon
campers have more pizza to eat?
A bird flew 20 miles in 100 minutes at a constant speed. At that
speed:
(a) How long would it take the bird to fly 6 miles? Explain.
(b) How far would the bird fly in 15 minutes? Explain.
(c) How fast is the bird flying in miles per hour?
(d) What is the bird’s pace in minutes per mile?
(e) Explain the meaning of the answers for parts c and d?
How would using the table help?

Assessment Specifications Summary is in
your CRW folders and on our ISS
Common Core wiki to guide you as you
create your Baseline and Benchmarks.
Assessments
 Let’s
put our brains to work,
creating quality baselines and
benchmarks that reflect the
teaching and understanding of
the CCSS!! 
Thank You!!
Given: x2 – 9
Which is true?
I. x = 3
II. x = -3
III. x2 = 9
a. I only
c. I and III only
e. none
b. I and II only
d. I, II, and III
Given: (a – b)(a2 – b2) = 0,
which is true?
I. a = b
II. a = -b
III. a2 = b2
a. I only
c. I and III only
b. I and II only
d. III only
http://mathnotations.blogspot.com/2007/07/understanding-algebra-conceptually.html
A high level task…
Has no predictable pathway
 Requires student to access and use
relevant knowledge
 Requires student to explain
 Has the potential to engage student in
higher level thinking

Soup and Beans
Consider these two
Measurement Problems
Carpeting Task

Martha is re-carpeting
her bedroom, which is
15 feet wide. How many
square feet of carpet will
she need to purchase,
and how much
baseboard will she need
to run around the edge
of the carpet? Explain
your thinking.
The Fencing Task

Ms Brown’s class will
raise rabbits for their
science fair. They have
24 feet of fencing with
which to build a
rectangular pen. If Ms.
Brown’s students want
their rabbits to have as
much room as possible,
how long should each of
the sides of the pen be?
Explain your thinking.
For each problem, what kind of thinking is required?
How are they alike? How are they different?
Carpeting Task vs. Fencing Task
Alike
Both require Area
and perimeter
calculations
 Both require students
to “explain your
thinking”
 Both are word
problems, set in a
“real world” context

Not Alike



Fencing requires a
systematic approach
Fencing leads to
generalization and
justification
The “thinking in
Fencing is complex –
requires more than
applying a
memorized formula
Technology and Testing
Content of the North Carolina
assessments is aligned to the CCSS-M;
however, the technology will not be as
sophisticated as in assessments created
by the Smarter Balanced Assessment
Consortium (SBAC).
www.smarterbalanced.org
www.ncdpi.wikispaces.net
DPI Contact Information
Kitty Rutherford
Elementary Mathematics Consultant
919-807-3934
[email protected]
Amy Scrinzi
Elementary Mathematics Consultant
919-807-3839
[email protected]
Robin Barbour
Middle Grades Mathematics Consultant
919-807-3841
[email protected]
Johannah Maynor
Secondary Mathematics Consultant
919-807-3842
[email protected]
Barbara Bissell
K – 12 Mathematics Section Chief
919-807-3838
[email protected]
Susan Hart
Program Assistant
919-807-3846
[email protected]
Let’s do some MORE math!
a) Pick
any integer greater than 10: __________
b) Square the number you picked: __________
c) Find the product of the integer that is one more than
your original number and the integer that is one less
than your original number: ___________
d) Compare your result in part a to your result in part b.
What do you notice?
e) Repeat steps a – c three or four times, using a different
positive integer as your starting number each time.
f) Create an algebraic equation to represent what you see
happening, but for ANY starting number.
g) Turn to someone sitting near you and explain as clearly
as you can what your equation represents.
Place Value

Traditional worksheet vs Common Core
worksheet

What do you notice?
Mathematical Language
Growing Staircase
Building staircase
Blocks
Pattern
Total blocks for
nth step
Total blocks for a
staircase with n
number of steps
Base of a
staircase
Top step of staircase
Step vs. stage
Generalize
Conjecture
Explicit or
general/closed
formula or function
Recursive formula or
function
Justify/prove
Growing Staircases