NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Shifts that Support Teaching,
Learning, and Monitoring
Student Progress in Mathematics
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Session Objective
• Understand the role of the eight Standards for
Mathematical Practice in Common Core aligned
instruction.
• Recognize evidence of the Standards for
Mathematical Practice in classroom instruction.
• Refine tools for grading and communicating progress
with content and practice standards to familes.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Group Tasks
• Read the MP and A Story of Ratios examples
• As a group, discuss:
1. Restating the MP in student friendly terms
2. What student behaviors does the MP elicit?
3. Teaching practices that support the MP’s
4. Teaching practices that suppress the MP’s
• Chart the results of your discussion
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Sample Chart
MP.1: Make sense of problems and persevere in solving
them
Student Friendly : Don’t give up on tricky problems
Content: Multiple approaches to problem solving, toolbox of
strategies from which to draw
Engagement: Persistence, risk-taking
Support
•
•
•
•
xxxxx
xxxxx
xxxxx
xxxxx
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Suppress
•
•
•
•
xxxxx
xxxxx
xxxxx
xxxxx
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
MP 1: Make sense of problems and persevere in
solving them.
When presented with a problem I can make a plan,
carry out my plan, and evaluate its success.
Before
During
Explain the problem to
Persevere
myself
•
Have I solved a problem like
this before?
Organize information
•
•
•
•
What is the question I need to
answer?
What is given? What is not given?
What are the relationships between
known and unknown quantities?
What prior knowledge do I have to
help me?
Monitor my work
After
Check
Change my plan if it
isn’t working out
Ask myself “Does this
make sense?”
http://www.insidemathematics.org/index.php/standard-2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Is my answer correct?
How do my representations
connect to my algorithm?
•
•
Evaluate
•
•
•
•
What worked?
What didn’t work?
What other strategies were
used?
How was my solution similar to
or different from my peers?
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP1 in action – Grade 7
Jake Disston works with his middle school students to deepen their
understanding of the connections between different representations of
functions—graphs, t-charts, and equations. In this clip, he begins the
period by challenging his students to work at their tables to group the cards
they have in front of them, creating as many different groupings as they
can, and devising language to describe each group. Disston's colleague
Jesse Ragent then asks the students to do a "matching game." He passes
out sets of tables and equations to the students, and challenges the
students to "find a triple"-- an equation, a table, and a graph that all make
up a "family, triple, or set."
http://www.insidemathematics.org/index.php/standard-1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP 2: Reason abstractly and quantitatively.
I can use reasoning habits to help me contextualize
and decontextualize problems.
Contextualize
I can take numbers and put
them in a real-world context.
Decontextualize
I can take numbers out of context
and work mathematically with them.
Reasoning habits include: 1) creating an understandable representation of
the problem solved, 2) considering the units involved, 3) attending to the
meaning of the quantities, and 4) using properties to help solve problems.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP 2 in action – Grades 5/6
Fran Dickinson leads a lesson on numeric patterning, helping students to
investigate a numeric pattern and to generalize what they see happening as
the pattern grows. In this clip, Dickinson describes the importance of
individual think time before he asks his students to discuss the relative
strength of two different approaches to a patterning task. One pair
discusses the numbers within the sample strategy, and Dickinson repeats
back their conversation to the whole group, telling his students, “I’ve heard
two really good questions about Learner B’s strategy. One was, what are all
these 3’s? and Kelcey’s question was, what about this 4? Where’s the 4
coming from?” This clip is also indicative of standard 1 (make sense of
problems and persevere in solving them).
http://www.insidemathematics.org/index.php/standard-2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
MP 3: Construct viable arguments and critique
the reasoning of others.
I can make conjectures and evaluate the
appropriateness of peer’s thinking.
I can construct, justify, and
communicate argument by
 considering context
 using examples and nonexamples
 using objects, drawings,
diagrams and actions
© 2012 Common Core, Inc. All rights reserved. commoncore.org
I can critique the reasoning
of others by




listening
comparing arguments
identifying flawed logic
asking questions to clarify
or improve arguments
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP3 in action – 6th grade
Jacob Disston leads a lesson on connections between ideas about equations,
inequalities, and expressions, helping students to use mathematical
vocabulary for a purpose to describe, discuss, and work with these symbol
strings. In this clip, his students have grouped themselves by the common
features of the symbol strings. Disston then asks the students to explain
their groupings, saying “Oliver, you had another whole category. Who are
you standing with? … Why are you guys saying you guys are similar? You
want to tell us?” He involves the whole class in evaluating the groups’
justifications. This clip is also indicative of standard 1 (make sense of
problems and persevere in solving them) and standard 6 (attend to
precision).
http://www.insidemathematics.org/index.php/standard-3
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
MP 4: Model with mathematics
I can recognize math in everyday life and use math I
know to real world problems.
I can
 make assumptions and
estimate to make complex
problems easier
 identify important quantities
and use tools to show their
relationships
 evaluate my answer and make
changes if needed
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Represent Math Using





Concrete models
Pictures
Real-world situations
Oral language
Symbols
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP4 in action – Grade 7/8
Cecilio Dimas leads a lesson on making comparisons between three different
financial plans, helping students use multiple representations of
mathematical problems: verbal, tabular, graphical, and algebraic
generalization. In this clip, Dimas connects to the prior day’s lesson, in
which the class “started a conversation about the economic status of our
world [and] about making responsible decisions when we’re spending our
money.” His students share that they were to represent various DVD rental
plans using verbal and tabular representations.
http://www.insidemathematics.org/index.php/standard-4
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP 5: Use appropriate tools strategically
I know when to use certain tools to help me explore
math and deepen my understanding.
I have a math toolbox and I
 know HOW to use math tools
 know WHEN to use math tools
 can reason: “Was the tool I used
appropriate for the situation?”
 can reason: “Did the tool allow
me to find the correct answer?”
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP5 in action – Grade 9/10
Cathy Humphreys leads an extended exploration of a proof of the properties of
quadrilaterals, helping students learn to investigate, formulate, conjecture,
justify, and ultimately prove mathematical theorems. In this clip,
Humphreys circulates around groups of students as they use rulers,
protractors, and sheets of paper to make different kinds of quadrilaterals
and examine and “guarantee” their properties. A group of students debates
whether or not one student’s assertion about the properties of a trapezoid
hold in all circumstances, using the kite “sticks” to illustrate their points.
9th/10th grade first video This clip is also indicative of standard 1 (make
sense of problems and persevere in solving them).
http://www.insidemathematics.org/index.php/standard-5
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP 6: Attend to precision.
I can use precision when solving problems and
communicating my ideas.
Problem Solving
Communicating
 I can calculate accurately.
 I can calculate efficiently.
 I can SPEAK, READ,
WRITE, and LISTEN
mathematically.
 My answer matches what
the problem asked me to
do – estimate or find the
exact answer.
 I can correctly use
 math symbols
 math vocabulary
 units of measure
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP6 in action – Grades 9/10
Cathy Humphreys leads an extended exploration of a proof of the properties of
quadrilaterals, helping students learn to investigate, formulate, conjecture,
justify, and ultimately prove mathematical theorems. In this clip, she
orients students to the task and explains how they are to communicate
their ideas to one another in their group work. This clip is also indicative
ofstandard 1 (make sense of problems and persevere in solving them).
http://www.insidemathematics.org/index.php/standard-6
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
MP 7: Look for and make use of structure.
I can step back and see the connections between
concepts and look closely to see the patterns.
Algebra
For Example:
Students recognize this expression as
5 minus a positive number times a
square and use that to realize that its
value cannot be more than 5 for any
real number x and y.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Geometry
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP7 in action – Grades 9 - 12
Carlos Cabana works with his high school English language learning students
on algebraic reasoning and multiple representations around parabolas. In
this clip, his students work together in a group, clarifying each other's
process and thinking. The female students clarify accurate steps for the
male student. They discuss how to use the x- and y-intercepts to support
their process. They step back and ask themselves what they are seeking in
their work.
http://www.insidemathematics.org/index.php/standard-7
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP 8: Look for and express regularity in
repeated reasoning.
I notice when calculations are repeated. Then, I can
find more efficient methods.
Noticing the regularity in the way terms cancel when expanding
(x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1)
might lead them to the general formula for the sum of a
geometric series. As they work to solve a problem,
mathematically proficient students maintain oversight of the
process, while attending to the details. They continually evaluate
the reasonableness of their intermediate results.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP8 in action – Grade 8
Jesse Ragent asks the students to do a "matching game." He passes out sets of tables
and equations to the students, and challenges the students to "find a triple"-- an
equation, a table, and a graph that all make up a "family, triple, or set." He asks
students to consider distinctions, characteristics, attributes as they make their
grouping decisions. Students jump right in and start the matching quite eagerly.
They are justifying their matches in different ways. Some are using the math
vocabulary from the prior whole class discussion, while others are just matching
points from the graph with the t-chart, or checking to see if an equation will
generate the points on the chart or graph. Either way, students are making
connections among these different representations in ways that are meaningful to
them. In the clips excerpted it seems that the students are listening to one another
and letting the mathematics sway the argument rather than the force of a
personality.
http://www.insidemathematics.org/index.php/standard-8
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
MP Sorting Activity
Use the posters to sort the behaviors/actions into
Mathematics Practice groups.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Chalk Talk
What does this mean for Common Core aligned
teaching and learning at my school/district?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Student Progress
Discuss with your table group:
What does/doesn’t the report card tell you about
the student as a mathematician?
What questions does this raise for you?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Revisions
What are 3-5 characteristics of a high quality tool
for reporting student progress with Common Core
aligned mathematics?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Moving Forward
• What do we need to do?
• What do they need to do?
• What do I need to do to help us/them get there?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
http://commoncore.americaachieves.org/module/10
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Biggest Takeaway
Turn and Talk:
• What questions were answered for you?
• What new questions have surfaced?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios