Twizzling Your
Way Through the
Common Core:
A Focus on the Practice
Standards
KATM Fall Conference ~ 2011
Jerry Braun – USD489/Hays
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
2. Reason abstractly and
8.
Look
for
and
Express
Regularity
in
4.
2.
7.
Model
Reason
with
Abstractly
Mathematics
Make
and
Use
Quantitatively
ofand
Structure
3.
1.quantitatively
Construct
Make
Sense
Viable
of
Problems
Arguments
and
Persevere
5.
Use
Appropriate
Tools
Strategically
6.
Attend
to
Precision
Reasoning
and
Repeated
Reasoning
Applying
Making
Looking
sense
for
mathematics
and
oftools
using
quantities
to
solve
problems
to
relationships
solve
Critique
in
the
Them
Reasoning
ofand
Others
Using
available
topatterns
solve
problems
and in
Communicating
precisely
using
clear
Explaining
3.Solving
Construct
viable
arguments
and
Creating
patterns
from
calculations
arising
problem
problems.
solving.
everyday
Flexibility
life.
Making
using
sense
operations
ofin
the
Analyzing
Explaining
problems,
the
meaning
making
of
a in
problem
conjectures
and
and
recognizing
strengths
and
limitations
ofown
definitions
inthe
discussion
with
others
and
critiqueinthe
reasoning
of repeated
others
6. Attend to precision
1. Make sense of problems and
persevere in solving them
Grouping the
Standards for
Mathematical Practice
evaluating
the
reasonableness
of
results.
and
properties.
constructing
looking
for entry
arguments
points
to
based
its
on
data.
each.
reasoning.
Understanding
thesolution.
meaning
of
4. Modelto
with
mathematics
generalizations
and
shortcuts.
Listening
Monitoring
progress
the
arguments
and
changing
of
others
course
and if
mathematical
symbols
and
calculating
asking
necessary.
useful
Analyzing
questions
givens,
to determine
constraints,
efficiently
and
accurately.
Modeling and Using Tools
rightness.
relationships,
Justifying
and
goals.
conclusions
and
5. Use appropriate tools strategically
communicating them to others.
7. Look for and make use of structure
8. Look for and express regularity in
repeated reasoning
Seeing Structure and
Generalizing
Example Student Task
Your math partner turns to you and says:
I worked the problem 124 ÷ 8 on my paper
and got 15 r 4. But when I did it on my
calculator, it said 15.5. Which answer is right?
How would you respond to your
partner’s question?
4
Things to consider. . .
• Using the term “groups”, restate what 124 ÷ 8
means.
• In the context of the problem, are the quotients
15 r 4 and 15.5 the same or are they different?
• What do the quotients mean?
• What does it mean to have a remainder of 4
when you are dividing by 8?
• What does 0.5 mean?
• What could you do to convince your partner that
the answers are the same or different?
Teacher Actions
• Each person needs a Practice
Standards Sorting Mat and a set of
Teacher Action Sorting Cards.
• Read each Action Card and match it
with the correct Practice Standard.
Standards for
Mathematical Practice
1. Explaining the meaning of a problem and looking for entry points to its solution.
Monitoring progress and changing course if necessary. Analyzing givens, constraints,
relationships, and goals.
2. Making sense of quantities and relationships in problem solving. Flexibility in using
operations and properties.
3. Analyzing problems, making conjectures and constructing arguments based on data.
Listening to the arguments of others and asking useful questions to determine
rightness. Justifying conclusions and communicating them to others.
4. Applying mathematics to solve problems arising in everyday life. Making sense of the
results.
5. Using available tools to solve problems and recognizing the strengths and limitations
of each.
6. Communicating precisely using clear definitions in discussion with others and in own
reasoning. Understanding the meaning of mathematical symbols and calculating
efficiently and accurately.
7. Looking for and using patterns to solve problems.
8. Creating patterns from repeated calculations and evaluating the reasonableness of
generalizations and shortcuts.
Make sense of problems and persevere in
solving them.
• Ask clarifying questions, such as,
“Which part do you agree or disagree with?
• Suggest starting points, such as, “What answer do
YOU get when you do the division?”
• Provide appropriate tools for investigating the
problem.
8
Reason abstractly and quantitatively.
• Ask questions that focus the student’s thinking on
the operation, such as,
“What does it mean to have a remainder of 4 when
you are dividing by 8?”
• Ask questions that focus the student’s thinking on
the meaning of the numbers, such as, “What does
0.5 mean?”
9
Construct viable arguments and critique
the reasoning of others.
• Design tasks in a way that directly involves
mathematical argumentation.
• Ask questions that model the desired
thinking, such as, “Are you sure that the answers
are different? Can you convince me that they are
different? How can you explain to me what each
answer means?”
10
Model with mathematics.
Show students how to represent their thinking with
symbols, for example:
0.5 = 5 = 1 and
10
2
4
1
4 out of 8 is
=
8
2
11
Use appropriate tools strategically.
• Have a selection of tools (e.g. place value blocks,
calculators, graph paper) readily available for students
to use to represent their thinking.
• Ask students to use a drawing or a model to show
what they mean.
• Have students critique the uses of tools by asking
questions like, “What made the use of the calculator in
this problem confusing for this student? When would
using a calculator be helpful?”
12
Attend to precision.
• Ask students to describe the quotient and
remainder in words.
• Ask students to describe the decimal quotient in
words.
• Ask students to compare and contrast situations in
which you might decide to use a decimal quotient
instead of a remainder.
13
Look for and make use of structure.
• Ask questions that focus student thinking on the
relationship of the remainder to the divisor, i.e. 4 is
half of 8.
• Ask questions that focus student thinking on the
values of the digits in the quotient, i.e. in 15.5, the 5
to the right of the decimal point means 5 tenths.
14
Look for and express regularity
in repeated reasoning.
• Have students explore what happens with other
numbers that when divided by 8 leave a remainder
of 4, e.g. 12 ÷ 8
36 ÷ 8
804 ÷ 8
• Have students explore the remainders in other
division problems that have a quotient that ends in
.5, e.g. 9 divided by 6, 35 divided by 10, 126
divided by 12.
15
Teaching for Understanding
What is one half of one half divided by
one half?
Use your Twizzler to model your
understanding of this question.
1
What is
2
one whole
1
2
1
4
of
1
2
?
What
does it
mean?
What is
What do we mean when we ask,
“What is 6 ÷ 2?”
1
4
÷
1
2
?
How many
2’s will “fit”
into 6?
3
What is
1/4
1/2
1
4
÷
1
2
?
How many
halves will
“fit” into the
fourth?
1/2
1
2
How much of
the half will
“fit” into the
fourth?
Teacher Actions
• Get out your Practice Standards
Sorting Mat again.
• Brainstorm actions from this brief
“lesson” to match with an
appropriate Practice Standard.
Integrating the Standards for
Mathematical Practices
• Not “Problem Solving Fridays”
• Not “enrichment” for advanced students
• Most are in the process of arriving at an answer, not
necessarily in the answer itself
• Every lesson should seek to build student expertise in
both Content and Practice standards
The End!
• Thank you for a great session 
• Questions, Comments, Etc.??
– jbraun@usd489.com