Illustrative Mathematics and
Coherence in the Standards
MENA Common Core Conference
November 1-2, 2013
Ellen Whitesides
Institute for Mathematics and Education, University of Arizona
Director of Community Building, Illustrative Mathematics
Ellen Whitesides
Who do we have in the room?
Research suggests that there’s an
exceptionally strong relationship between
communal learning, collegiality, and
collective action (key aspects of
professional learning communities) and
changes in teacher practice and increases
in student learning.
Gulamhussein, Allison (2013) Teaching Teachers: Effective Professional Development in
an Era of High Stakes Accountability, Center for Public Education
Illustrativemathematics.org
Illustrative Mathematics is a discerning community of
educators dedicated to the coherent learning of
mathematics. We collaborate at
illustrativemathematics.org, sharing carefully vetted
resources for teachers and teacher leaders to give our
children an understanding of mathematics and skill in
using it. We provide expert guidance to states and
districts working to improve mathematics education.
Goals of Illustrative Mathematics
To illustrate standards with impeccably crafted tasks, videos,
lesson plans, and curriculum modules.
To be the premier source of freely available online mathematics
content for teachers, teacher leaders, assessment developers,
curriculum writers, and teacher educators.
To be a discerning professional community that creates content
and deploys expertise in multiple ways.
To engage individuals and small groups, with rare and needed
expertise, who are currently isolated in pockets across the
country.
To provide a space for teachers to share across classrooms and
help each other grow in teaching our children mathematics.
The Structure of the Common Core
The Structure of the Common Core
Each domain has standards organized into a small number of
clusters
The Structure of the Common Core
Illustrative Mathematics facilitates viewing standards across
grades to see a progression of mathematical ideas
The Structure of the Common Core
The Structure of the Common Core
Organized by conceptual category in high school
The Standards for Mathematical Practice
The Standards for Mathematical Practice
Slab of Soap
A slab of soap on one pan of a scale balances ¾ of a slab of soap
and a ¾ pound weight on the other pan. How much does the full
slab of soap weigh?
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
Commenting on Tasks and Rating them +1
Task Talks
Web Jams
A CCSSM Task Writing Community
Jammin’ on the gnarly standards, since 2012!
Progressions Documents
http://ime.math.arizona.edu/progressions/
Blog
http://commoncoretools.me/
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity
formed by 1 part when a whole is partitioned into b
equal parts; understand a fraction a/b as the quantity
formed by a parts of size 1/b.
0
b=3
⅓
1
4/3
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
What fraction does this diagram represent?
2/6 ?
2/3 ?
Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833
2?
What fraction does this diagram represent?
2/6
Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833
What fraction does this diagram represent?
2/3
Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833
What fraction does this diagram represent?
2/3
Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833
What fraction does this diagram represent?
2
Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833
What fraction does this diagram represent?
2
Based on Illustrative Mathematics task: http://www.illustrativemathematics.org/illustrations/833
Build fractions from unit fractions by applying and
extending previous understandings of operations on whole
numbers.
4.NF.3Understand a fraction a/b with a > 1 as a sum of
fractions 1/b.
a. Understand addition and subtraction of fractions as
joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the
same denominator in more than one way, recording each
decomposition by an equation. Justify decompositions, e.g.,
by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 +
1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
Add and subtract within 20.
1.OA.5 Relate counting to addition and subtraction
(e.g., by counting on 2 to add 2).
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
Relate addition and subtraction to length.
2.MD.6 Represent whole numbers as lengths from 0
on a number line diagram with equally spaced points
corresponding to the numbers 0, 1, 2, ..., and
represent whole-number sums and differences within
100 on a number line diagram.
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
What does understanding look like in
Algebra?
“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
mathematics, and may have a better chance to
succeed at a less familiar task such as expanding (a
+ b + c)(x + y)”
From Common Core Standards:
www.corestandards.org/the-standards/mathematics
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
MP 7: Look for and make use of structure
Mathematically proficient students
• look closely to discern a pattern or structure
• step back for an overview and shift
perspective
• see complicated things as single objects, or as
composed of several objects
© Institute for Mathematics & Education 2011
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
8
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
9
10
11
12
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
Expressions and
Equations
Geometry
Geometry
Modeling
© Copyright 2011 Institute for Mathematics and Education
The CCSSM: Pathway to Algebra
K
1
2
3
4
5
6
7
8
Number and Operations
Fractions
3
7
The Number
System
7
9
3
6
7
5
8
4
9
5
3
12
3
2
Arithmetic with Polynomials
and Rational Expressions
Expressions and
Equations
9
Operations and Algebraic Thinking
11
Seeing Structure in
Expressions
Number and Operations in Base Ten
6
10
Algebra
8
1
9
4
8
Creating Equations
Reasoning with Equations
and Inequalities
© Copyright 2011 Institute for Mathematics and Education
Algebra: Seeing Structure in
Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solve
problems
What does it look like to “see structure in
expressions”?
Interpret the structure of expressions
A-SSE.1 Interpret expressions that represent a
quantity in terms of its context.
a. Interpret parts of an expression, such as terms,
factors, and coefficients.
b. Interpret complicated expressions by viewing one
or more of their parts as a single entity. For example,
interpret P(1+r)n as the product of P and a factor not
depending on P.
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
Delivery Trucks
A company uses two different sized trucks to deliver sand.
The first truck can transport x cubic yards, and the second y
cubic yards. The first truck makes S trips to a job site, while
the second makes T trips. What do the following expressions
represent in practical terms?
a.
b.
c.
d.
S+T
x+y
xS + yT
xS+yT
S+T
If they do this in high school…
what needs to come before that high school work?
Apply and extend previous understandings of
arithmetic to algebraic expressions.
6.EE.4 Identify when two expressions are equivalent
(i.e., when the two expressions name the same
number regardless of which value is substituted into
them). For example, the expressions y + y + y and 3y
are equivalent because they name the same number
regardless of which number y stands for.
Use properties of operations to generate equivalent
expressions.
7.EE.2 Understand that rewriting an expression in
different forms in a problem context can shed light
on the problem and how the quantities in it are
related. For example, a + 0.05a = 1.05a means that
“increase by 5%” is the same as “multiply by 1.05.”
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
Writing expressions in different forms
for a purpose
Students used the following expressions to describe the number of tiles
needed to border a square pool with side length s feet. Demonstrate
these expressions in the figure:
s + s + s + s +4
4(s + 1)
2s + 2(s + 2)
Writing expressions in different forms
for a purpose
1
s
1
s + s + s + s +4
s+1
s
s+1
s
s
1
s+1
1
s+1
4(s + 1)
2s + 2(s + 2)
Compute fluently with multi-digit numbers and find
common factors and multiples.
6.NS.4 Find the greatest common factor of two
whole numbers less than or equal to 100 and the
least common multiple of two whole numbers less
than or equal to 12. Use the distributive property to
express a sum of two whole numbers 1–100 with a
common factor as a multiple of a sum of two whole
numbers with no common factor. For example,
express 36 + 8 as 4 (9 + 2).
The Common Core State Standards in Mathematics
K
1
2
3
4
5
Measurement and Data
6
7
Statistics and
Probability
Ratios and
Proportional
Relationships
CC
8
9
10
11
Statistics and Probability
F
Functions
Number and Operations
Fractions
Number and Operations in Base Ten
The Number
System
Number and Quantity
Algebra
Operations and Algebraic Thinking
12
Expressions and
Equations
Geometry
Geometry
Modeling
Finding factors for a purpose
Nina was finding multiples of 6. She said,
18 and 42 are both multiples of 6, and when I
add them, I also get a multiple of 6: 18+42=60.
Explain to Nina why adding two multiples
of 6 will always result in another multiple
of 6.
Finding factors for a purpose
Source: Primary Sources Report, Early Release, October 2013
Source: Primary Sources Report, Early Release, October 2013
Source: Primary Sources Report, Early Release, October 2013
Thank you!
Enjoy the conference.
ellen.whitesides@gmail.com