Focusing on the Mathematical Practices
of the Common Core
Grades 6
–
8
Day 1
Professional Development
Facilitator Handbook

SAMPLER

Materials Checklist
Item
ISBN
Special
Instructions
Participant
Facilitator
Quantity
Consumable
Computer
No
te
1
X
X
1 each
No
Projector
Note
2
X
1 total
No
Speakers
X
1 set
No
PowerPoint
X
1 total
No
Participant Workbook
X
X
1 each
Yes
Chart paper/graph paper
Note
3
X
X
1 pad
Yes
Colored markers
X
X
1 for the facilitator and
1 per table
No
Manipulatives (Tile Problem)
Personal paper copy of the
Common Core State
Standards for Mathematics
Note
4
X
1 total
No
Electronic copy of the
Common Core State
Standards for Mathematics
Note
5
X
1 total
No
Video: Juanita “Nita” Copley,
“B
alance of Procedure and
Understanding”
Note
6
X
1 total
No
Video: Phil Daro, “Standards
for Mathematical Practice”
Note
7
X
1 total
No
Video: Phil Daro, “Developing
Mathematical Expertise and
Building Character”
Note
8
X
1 total
No
Video: John V
an de Walle,
“Teacher Workshop”
Note
9
X
1 total
No
Video: “
Video Case Study:
The Tile Problem with
Commentary”
Note
10
X
1 total
No
Pens or pencil
X
1 each
Yes
Quotes on perseverance
Note
11
X
2 sets of 12
No
Note
1
The school district ne
eds to provide a training room with a computer for the facilitator. Prior to the workshop, the facilitator should obtain
information about how to log in to the system. The facilitator should also confirm that PowerPoint is installed on the comput
er and tha
t the
computer has Internet access. The facilitator should install QuickTime Player on his or her computer. To download the applica
tion, visit
http://www.apple.com/quicktime.
For optimal performance, the facilitator should copy the entire presentation fol
der onto his or her hard drive. All videos need to be in the same
location as the PowerPoint file. For example, if the PowerPoint file is on the desktop, all of the video files also need to b
e on the desktop. The
facilitator should test the PowerPoint file
on the computer her or she is using for the presentation, including all videos, before beginning the
training.
If the facilitator wishes to use his or her personal computer during the workshop, he or she needs to obtain permission from
the district’s
Te
chnology Department.
Note
2
The district should provide a projector if one is available. If a projector is not available, make arrangements for one to be
brought to the
presentation site.
Note
3
Request that the district provide chart paper, because it is
very difficult to travel with. Otherwise, the facilitator should purchase the chart
paper prior to arrival.
Note
4
The Common Core State Standards are in the warehouse and will be shipped to the workshop site prior to the training. Please c
heck with
custo
merservice@pearson.com to ensure materials have been shipped. Please contact the school to ensure materials have arrived.
Note
5
It is a good idea for the facilitator to have an electronic copy of the CCSSM on a flash drive that he or she brings to the t
ra
ining. The
facilitator should download the PDF file named CCSSI_Math Standards.pdf from the Additional Resources folder to his or her co
mputer.
Note
6
The file name for the video titled “Balance of Procedures and Understanding” is Copley Interview.mov.
No
te
7
The file name for the video titled “Standards for Mathematical Practice” is CCSS_Mathematical_Practice.mov.
Note
8
The file name for the video titled “Developing Mathematical Expertise and Building Character” is CCSS_Expertise_Character.mov
.
Note
9
The
file name for the video titled “Teacher Workshop” is VanDeWalle.mov.
Note
10
The file name for the video titled “
Video Case Study: The Tile Problem with Commentary”
is PhilDaro_TilePrb_Commentary.mov.
Note
11
Find twelve quotes on perseverance either from
a personal book or from the Internet. One example is a Web site called the Quote Garden
located at http://www.quotegarden.com/perseverance.html.
Preparation and Background
Content Information
Note to Facilitator:
Please check if the state or distri
ct where you will be leading the workshop has created their own state documents. States such as Arizona and
New York have already released state

specific versions of the Common Core State Standards (CCSS). You should become familiar with the state

specific
documents and adapt the workshop materials to use the state

specific CCSS.
*Please note that even though this workshop has a prerequisite, districts may not have fulfilled that requirement. If they ha
ve, adjust the content
accordingly.
“The Common Cor
e State Standards Initiative is a state

led effort coordinated by the National Governors Association Center for Best Practices
(NGA Center) and the Council of Chief State School Officers (CCSSO)” (Common Core State Standards Initiative 2010b).
These stan
dards were developed for three reasons. One is to provide consistency across states. A set of common standards allows for con
sistent
and quality education across all 50 states. Secondly, they align with international standards. In order to compete in globa
l markets, students in the
United States cannot lag behind their peers in other countries. The Common Core State Standards (CCSS) are benchmarked agains
t international
standards so that students can compete in a global economy. Lastly, the standards help p
repare students for college and work. Colleges and
universities expect students to read complex texts independently, and employers look for workers who have the skill set to so
lve problems and the
ability to integrate new knowledge. Elementary and secondar
y education needs to prepare students to be ready for these challenges.
The Common Core State Standards Initiative (2010b) states the following:
The standards were developed in collaboration with teachers, school administrators, and experts, to
provide
a clear and consistent framework to prepare our children for college and the workforce. The NGA
Center and CCSSO received initial feedback on the draft standards from national organizations
representing, but not limited to, teachers, postsecondary educato
rs (including community colleges), civil
rights groups, English language learners, and students with disabilities. Following the initial round of
feedback, the draft standards were opened for public comment, receiving nearly 10,000 responses.
The standard
s are informed by the highest, most effective models from states across the country and
countries around the world, and provide teachers and parents with a common understanding of what
students are expected to learn. Consistent standards will provide appro
priate benchmarks for all students,
regardless of where they live.
These standards define the knowledge and skills students should have within their K
–
12 education
careers so that they will graduate high school able to succeed in entry

level, credit

beari
ng academic
college courses and in workforce training programs. The standards:
Are aligned with college and work expectations;
Are clear, understandable and consistent;
Include rigorous content and application of knowledge through high

order skills;
Build
upon strengths and lessons of current state standards;
Are informed by other top performing countries, so that all students are prepared to succeed in our
global economy and society; and
Are evidence

based.
“The Standards for Mathematical Practice describ
e varieties of expertise that mathematics educators at all levels should seek to develop in their
students. These practices rest on important ’processes and proficiencies‘ with longstanding importance in mathematics educati
on. The first of
these are the NC
TM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second
are the strands of mathematical proficiency specified in the National Research Council’s [NCR] report
Adding It Up
: adaptive reasoning,
strategic
competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (
skill in carrying
out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habit
ual inclination to see mathematics as sensible,
useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy)” (Common Core State Standards Initiative 2
010a, 6).
The chart on the next page includes additional resources to read and vie
w in order to have the background knowledge to lead the discussions
outlined in this workshop.
Read the following prior to the workshop:
Work through and take notes on the following prior to the
workshop:
Adding It Up
:
Helping Children L
earn Mathematics
o
http://www.nap.edu/catalog.php?record_id=9822#toc
A Nation Accountable: Twenty

five Years After A Nation at Risk
(U.S.
Department of Education)
o
http://www2.ed.gov/rschstat/research/pubs/accountabl e/index.
html
All problems and problem set
s
Companion document for the Comparing Quantities problem set
Companion document for the Tile Problem
Copley Interview.mov
CCSS_Mathematical_Practice.mov
CCSS_Expertise_Character.mov
VanDeWalle.mov
PhilDaro_TilePrb_Commentary.mov
Familiarize yourself with
the following:
Gather the following from state

specific Web sites:
Highlights From PISA 2009: Performance of U.S. 15

Year Old Students
in Reading, Mathematics, and Science Literacy in an International
Context
http://nces.ed.gov/pubs2011/2011004.pdf
A Na
tion At Risk: The Imperative for Educational Reform
http://www2.ed.gov/pubs/NatAtRisk/index.html
Curriculum and Evaluation Standards for Mathematics
Education
http://www.ericdigests.org/pre

9215/evaluation.htm
Principles and Standards for School Mathemat
ics
http://standards.nctm.org/
No Child Left Behind information
http://www2.ed.gov/nclb/overview/intro/edpicks.jhtml?src=ln
Adding It Up: State Challenges for Increasing College Access and
Success
http://www.cpec.ca.gov/CompleteReports/External Document
s/Addi ng_I
t_Up.pdf
Race to the Top Program Executive Summary
http://www2.ed.gov/programs/racetothetop/executi ve

summary.pdf
Common Core State Standards
http://www.corestandards.org/
Trends in International Mathematics and Science Study (TIMSS) 1999
Video
Study
http://www.llri.org/html/TIMSS/index.htm
Fordham Institute Comparison
http://www.edexcellence.net/index.cfm/news_the

state

of

state

standards

and

the

common

core

in

2010
State

specific versions of the CCSS. (See Arizona and New York for
examples.)
Major changes in practice
Common Core State Standards state

specific implementation timeline
Appropriate assessment information from either the Partnership for
Assessment of Readiness for College and Careers (PARCC) or the
SMARTER Balanced Assessment Cons
ortium (SBAC)
Workshop Information
Big Ideas
Mathematical proficiency is more than “getting the answer.” It includes the process of using mathematical concepts effectivel
y as
identified in the Standards for Mathematical Practice.
Developing a student’s
ability to use the Standards for Mathematical Practice helps to cement the student’s understanding of
mathematical processes and proficiencies.
The mathematical practices are consistent for all grade levels, even though they manifest themselves differently
as students grow
in mathematical maturity.
Teachers can embed mathematical practices into daily routines
through teacher modeling, task selection, relevant mathematical
discourse, lesson structure, and so forth.
The Standards for Mathematical Practice pro
vide students with the opportunity to demonstrate their expertise as mathematical
thinkers and learners in an intentional and public way.
Strategically chosen tasks provide teachers with opportunities to model and teach the mathematical practices. Teacher
s must
practice and teach modeling with mathematics; it cannot be an unspoken expectation.
Essential Quest
ions
What are the Standards for Mathematical Practice, and why are they important for mathematical proficiency?
What does “understanding” in mathemat
ics mean, and how do the Standards for Mathematical Practice lead to understanding?
How can strategies and classroom routines help students develop the mathematical practices of mature mathematical thinkers an
d
learners?
How can teachers use problem solv
ing to promote the Standards for Mathematical Practice?
Assessments of Participants’ Learning during the Workshop
Reflection: Rank the Standards
Reflection: 3

2

1 Activity
Reflection: Symbolic Prompts
Review: Objectives
Essential Questions Review
Assess
ment Back in the School/Classroom
Implement a list of changes to participants’ classrooms that expands students’ use of mathematical practices.
Outcomes
Participants will be able to
connect the Standards for Mathematical Practice to the National C
ouncil of Teachers of Mathematics’ (NCTM) Process Standards
and Proficiencies as detailed in
Adding It Up: Helping Children Learn Mathematics
;
identify a structure for collaboration and use of the eight mathematical practices;
connect current practice and
articulate the changes needed to implement the Standards for Mathematical Practice;
articulate ways to routinely promote and assess the mathematical practices;
describe how specific mathematical practices are embedded in the Standards for Mathematical Cont
ent;
identify the attributes of a rich, instructional, problem

based approach and how it can support access to the Standards for
Mathematical Practice;
identify sub

performance tasks as a means for providing students the opportunity to routinely demonstrat
e the eight mathematical
practices; and
connect the analysis of student work to ongoing support of the Standards for Mathematical Practice.
Facilitator Goals
Improve participants’ success in implementing the Standards for Mathematical Practice.
Provide pa
rticipants with specific approach suggestions that will increase students’ use of the mathematical practices.
Encourage participants to engage in the mathematical practices as a model for their students.
Support participants’ skills in formatively assessin
g the mathematical practices through observation.
Guide participants in encouraging all students (including English language learners (ELLs), special education, and other
populations) to engage in the mathematical practices.
Section 2: Teaching the Standa
rds for Mathematical Practice (Slides 16
–
31)
Time: 110 minutes
Essential Question
How can strategies and classroom routines help students develop the mathematical practices of mature mathematical thinkers an
d
learners?
Learning Objectives
Explain how to d
evelop the Standards for Mathematical Practice through choice of task, discourse in the classroom, assigned
homework, and daily classroom routines.
Identify features of participants’ instruction that facilitate the development of the Standards for Mathemat
ical Practice.
Create a list of changes needed in participants’ classrooms that will expand students’ use of the Standards for Mathematical
Practice.
Materials per Section
Computer
Projector
PowerPoint presentation
Participant Workbook, pages 16
–
24
CCSSM d
ocument, pages 6
–
8
Video: Phil Daro, “Developing Mathematical Expertise and Building Character”
Chart paper
Colored markers (each group should have a different color)
Pens or pencils
Topic
Presentation Points
Presentation Preview
Supporting Students as
They Develop
the Mathematical Practices
Display Slide 16.
Introduce this section by making the point that “teaching the Standards for
Mathematical Practice” means supporting the development of students as
practitioners of the discipline of mathematics.
Ask
participants to reflect on the question on the slide, ‘How do you
currently support students in becoming effective mathematics
practitioners?”
Ask for volunteers to share their thoughts.
Topic
Presentation Points
Presentation Preview
Display Slide 17.
Have participants turn to page 18 in the Part
icipants Workbook.
In the space provided, have participants individually brainstorm the answer
to the question on the slide in the space provided. “How can teachers
support students as they develop the mathematical practices?”
After participants have creat
ed their personal lists, facilitate a discussion
about the different ways teachers can support students as they develop
the mathematical practices. Chart participants’ responses.
Note:
o
Examples of participant responses may include the following:
–
Provide co
ncrete materials.
–
Provide real

world contexts.
–
Require the use of multiple representations.
–
Ask the questions, “Does this make sense?” and “Why?”
o
Use participants’ background knowledge to make direct connections
between instructional strategies and how ins
truction will and does
impact students’ work within the mathematical practices.
Guide the discussion in such a way as to create a global list, but at the
same time make sure that participants generate ideas that will support
each of the eight Standards for
Mathematical Practice.
BACKGROUND INFORMATION:
Every grade level includes Standards for Mathematical Practice and
“describes ways in which developing student practitioners of the
discipline of mathematics ought to engage with the subject matter as
they g
row in mathematical maturity and expertise throughout the
elementary, middle and high school years” (Common Core State
Standards Initiative 2010a, 8).
At this point in the workshop, participants should be familiar with the
mathematical practices at a gener
al level. Section 2 focuses on how
instructional strategies can engage students in the mathematical
PW:
Page 18
Topic
Presentation Points
Presentation Preview
practices.
Display Slide 18.
Explain to participants that the video of Phil Daro they are about to watch
describes students’ required develo
pment of mathematic expertise and
character building in order to demonstrate the eight mathematical
practices.
Note:
Participants may have viewed this video during the Foundational
Overview of the Common Core State Standards for Mathematics
Professional De
velopment Workshop. If this is the case, you may want
to consider skipping the video and just continue working with the list
and instructional strategies as stated below with Slide 20.
As participants to watch the video clip titled, “Developing Mathematica
l
Expertise and Building Character,” ask them to think about the question,
“How is it that students build these described habits of minds?”
Debrief the video by asking participants if there is anything that they want
to add to the list that they generated.
Add any additional information to the
chart paper list that participants already started.
Display Slide 19.
Explain to participants that as they work through the next part of this
section, they will want to make note of various ideas for their
own
classroom practices.
o
Remind participants of the chart on pages 10
–
11 in the Participant
Workbook and the space provided there for these ideas.
PW:
Pages 10
–
11
Topic
Presentation Points
Presentation Preview
Bringing the Practice Standards to the
Classroom
Display Slide 20.
Use the list that par
ticipants created in the previous discussion as a
starting point for the next part of the section. Here, participants will use the
Instructional Carousel strategy in order to brainstorm ways that teachers
can develop the mathematical practices through vari
ous classroom
activities.
Note (Preview of Activity):
In the Instructional Carousel strategy, you will hang five pieces of chart
paper around the room with one instructional strategy (see Slide 20)
as a heading on each.
Then divide participants into five
groups and assign one teaching
practice and one corresponding brainstorming question from page 18
in the Participant Workbook to each group.
Participants generate answers and ideas to their questions using page
18 for note

taking purposes.
Then assign ea
ch group a different colored marker.
Groups use their colored markers to transfer their ideas to their
assigned chart papers.
Once groups finish transferring their ideas they will then “carousel”
around the room to other groups’ chart papers and, using th
eir
assigned colored markers, add additional ideas to what the original
group already started.
Briefly discuss five ways teachers can develop the mathematical practices
in the classroom. Use already

generated responses to begin the
conversation by connecti
ng those responses to the following five
instructional strategies:
o
Teacher modeling:
The teacher models the mathematical behavior
that he or she wants students to develop.
o
Task selection:
The teacher chooses tasks that allow students to
develop those behav
iors.
PW:
Page 19
Topic
Presentation Points
Presentation Preview
o
Relevant student discourse:
The teacher allows time for students to
discuss what they are thinking and doing when they develop solution
strategies. The teacher also has students share their strategies with
the class, requires a full explanation, and
allows other students to
assist and ask questions.
o
Questioning strategies:
The teacher poses questions to students
that assist them in delving deeper into the concept. Questions are
used in place of providing specific strategies for solving a task.
o
Classr
oom routines:
The teacher develops classroom routines that
create a safe environment for students to explore strategies, develop
their conceptual understanding, and work on becoming proficient in the
mathematical practices.
Explain that you want participan
ts to add to the ideas that they already
generated by digging deeper into the specifics of each as they relate to the
five instructional strategies listed above and on Slide 20. For example, if
one of the previously charted responses was “provide real

worl
d context,”
ask participants under which instructional strategy this would fall. It should
fall under task selection. Then, have participants expand on this by
thinking about what it is about the task and context they would be looking
for; for example, som
ething relevant to students’ lives, grade

level
appropriate situations, and so forth.
Divide participants into five groups.
Assign each group to one of the five brainstorming questions listed on
page 19 of the Participant Workbook. These questions relate
to the ways
that teachers can support the development of the mathematical practices:
o
Teacher modeling
: What are your current and future approaches to
modeling the Standards for Mathematical Practice for students?
o
Task selection
: What are your current and f
uture criteria for selecting
tasks that encourage the development of the Standards for
Mathematical Practice?
o
Relevant student discourse
: What are your current and future methods
for supporting student discourse that encourage the development of
the Standa
rds for Mathematical Practice?
o
Questioning strategies
: What are your current and future questioning
Topic
Presentation Points
Presentation Preview
strategies that encourage the development of the Standards for
Mathematical Practice?
o
Classroom routines
: What are your current and future classroom
routine
s that encourage the development of the Standards for
Mathematical Practice?
Have groups spend five minutes brainstorming ideas for their questions
while they use page 19 of the Participant Workbook to take notes. If they
have trouble, refer them to the St
andards for Mathematical Practice on
pages 6
–
8 of the CCSSM document for ideas.
Display Slide 21.
Assign one colored marker to each group. Explain to participants that once
they finish with their notes, they should transfer their ideas to
their
assigned piece of chart paper.
Let participants know that they will revisit their posters at the end of this
section of the workshop, so they should not worry if they feel they need
more time to come up with ideas.
Once groups finish with their ch
art papers, have them carousel around to
the other posters and add additional ideas. They should spend two to three
minutes reading each poster and making additions. Each group should
continue to use its assigned colored marker to distinguish the additions
that participants make.
Continue to have groups move around the room until they have visited all
of the posters. As a whole group, debrief the activity and ask participants
for highlights and thoughts on each instructional strategy.
Allow participants ti
me to go back and write down the ideas they want to
take away on the chart on pages 10
–
11 in the Participant Workbook.
Topic
Presentation Points
Presentation Preview
Display Slide 22.
Ask participants to reflect on the following questions:
o
At what point in time do teachers select a task?
o
When do
teachers model the Standards for Mathematical Practice?
o
When do teachers support student discourse?
o
When do teachers use questioning strategies?
o
What does it look like when teachers integrate the Standards for
Mathematical Practice into classroom instru
ctional processes?
Explain to participants that most of the various ways for developing the
mathematical practices happen within the routines and rituals of the daily
classroom. So, suppose you and participants are in a seventh grade
classroom at the begin
ning of a classroom lesson. Model the three

phase
lesson structure for the problem sets that are provided in the Participant
Workbook on pages 20
–
22.
BACKGROUND INFORMATION:
The three

phase lesson structure is a problem

based teaching method
described in
E
lementary and Middle School Mathematics: Teaching
Developmentally
by John Van de Walle, Karen Karp, and Jennifer
Bay

Williams (2010, 48). The lesson structure is divided into three
phases: Before, During, and After.
Before:
During the Before phase the tea
cher sets

up the problem,
activates student prior knowledge without giving away a solution strategy
to the current problem, makes sure the problem is understood, and
establish clear expectations for how the students will work the problem (i.e.
in groups, o
n their own, verbal responses, written responses, etc).
During:
In the During phase of the lesson students work the assigned
problem. While students are working the teacher circulates throughout the
room and allows the students to purposefully struggle wit
h the problem in
order to create their own strategy and understand the mathematics, listens
actively to determine which students need additional support and provides
those students with appropriate hints, and provides worthwhile extensions
Topic
Presentation Points
Presentation Preview
for those studen
ts who are ready to move on.
After:
In the After phase of the lesson the teacher brings the students
back together and holds a public discussion of the student work in which
students publically explain their strategies, other students and the teacher
asks
questions and listen actively without evaluation. This public
discussion is used to promote a mathematical community of learners. At
the end of the public discussion the teacher summarized the main (big)
ideas and identifies future problems.
Display Sl
ide 23.
Set up the Before phase of the model lesson by asking the following
questions displayed on the slide:
o
What is a comparison?
o
Why do we compare?
Chart participants’ responses so that you and participants can refer to
them throughout the lesson as nec
essary and use the responses to gauge
and clear up any content misconceptions.
Display Slide 24.
Together with participants, examine two types of comparisons by
completing Problems 1a and 1b on page 20 of the Participant Workbook.
PW:
Page 20
Topic
Presentation Points
Presentation Preview
Displ
ay Slide 25.
Begin the During phase by asking participants to continue to work through
the remaining questions. Tell them that you will be walking around to pose
questions and gauge pacing. Let them know in advance that not all groups
will finish, but that
you will ask various pairs of people to present different
questions.
While participants work on the task, be sure you adequately prepare for
the After phase, or public discussion, of their work.
BACKGROUND INFORMATION:
Before the workshop, carefully revie
w the companion document in the
Additional Resources folder. The companion document provides the
answers to the tasks, as well as sample solution strategies and
content considerations.
Display Slide 26.
Begin the After phase.
Have participants follow a
long in the Participant Workbook on pages 20
–
22 as various pairs or groups explain each set of problems.
After the presentations, ask participants what they noticed about the
structure of this problem set. Listen for specific ideas related to the
concept o
f scaffolding as noted in the background information below.
Bring the discussion to a close by facilitating a conversation about the big
ideas of this set of problems. Be sure you know the background
information well enough to pull out the big ideas
—
either
from the
participants’ presentations or by charting the ideas yourself. It is very
important not to read directly from your Facilitator Handbook during the
workshop.
BACKGROUND INFORMATION:
Topic
Presentation Points
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The big ideas include the following:
Students must understand
the concept of absolute comparison (using
subtraction to find a difference) versus the concept of multiplicative
comparison (using division to find a ratio).
When using subtraction, if two numbers are each multiplied by the same
factor, then the differenc
e also changes by that factor.
When using division, if two numbers are each multiplied by the same
factor, then the ratio stays the same.
Problems 1
–
3 give students an opportunity to explore the different
comparisons via several different examples created
by them. They also
ask students to defend their answers.
Problem 4 gives students an opportunity to contrast the comparisons and
demonstrate an understanding of when each comparison is
mathematically appropriate.
Problem 5, at a higher level of cognitive
demand, gives students the
opportunity to extend their knowledge by developing an applicable rule.
Note that if teachers use Problem 5 as an extension task, the final two
bullets above may not be big ideas for this lesson.
While the usefulness of scaffold
ing for increasing all students’ capacities
for engaging in mathematical practices is not highlighted until Day 2,
Section 6, it might be useful to mention to participants that this problem set
is also scaffolded.
Instead of asking students complicated qu
estions about the comparisons,
the problem set begins by giving students specific tasks in regard to the
number of stars, the number of triangles, and how the two quantities could
be compared.
Next, students extend this knowledge to create different sets
of stars and
triangles via the specific questions in the problem set. This gives them the
opportunity to begin making connections about the two different types of
comparisons.
Then, problem sets give students the opportunity to engage with specific
problem
situations that use relative and absolute comparisons.
Finally, the problem sets asks students (if they are able) to develop
Topic
Presentation Points
Presentation Preview
mathematical rules based on their experiences with the previous problems.
Display Slide 27.
Ask participants to consider the
following question: Describe how these
mathematical practices would emerge if students worked on this same
problem set.
Discuss the Standards for Mathematical Practice that relate to this set of
comparing quantities mathematical tasks. Possible responses t
o this
prompt may include the following (Common Core State Standards Initiative
2010a, 10):
o
Make sense of problems and persevere in solving them.
Students
must understand the difference between comparison by division and
comparison by subtraction.
o
Reason
abstractly and quantitatively.
Students are asked to
decontextualize from the situations of stars and triangles, different
people’s salaries, and tables and chairs to ratios.
o
Construct viable arguments and critique the reasoning of others.
Students must b
e able to explain why one comparison gives better
information for the problem than another comparison.
o
Attend to precision.
Students need to take note of the differences
between subtraction and division.
o
Look for and make use of structure.
After several p
roblems, students
need to write a rule that explains two different situations.
Topic
Presentation Points
Presentation Preview
Display Slide 28.
Use page 22 in the Participant Workbook to review with participants the
three

phase lesson structure they just worked through.
o
Describe what happened duri
ng each phase of the lesson and provide
background on the intent of each section. The background information
can be found above in the text that corresponds to Slide 22.
Explain to participants that the three

phase lesson structure is just one
way for teac
hers to deliver lessons that naturally allow them to implement
the five instructional strategies; this would allow teachers to integrate more
work with Standards for Mathematical Practice.
PW:
Page 23
Display Slide 29.
Have participants reconvene in t
heir original groups from the carousel
activity at the beginning of this section to make revisions to their posters
based on their experiences with the previous set of mathematical tasks.
As time permits, allow groups to select a spokesperson to share any
revisions that they made.
Reflection: 3

2

1 Activity
Display Slide 30.
Have participants complete a 3

2

1 reflection by following the instructions
on the slide and writing their responses on page 24 of the Participant
Workbook:
o
List
3
new ideas relate
d to teaching the Standards for Mathematical
Practice that you are going to implement in the classroom.
o
List
2
strategies you have previously used that support the Standards
for Mathematical Practice.
o
List
1
question you still have about the Standards for
Mathematical
PW:
Page 24
Topic
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Practice.
Essential Question Review
Display Slide 31.
Have participants turn to page 25 in the Participant Workbook and read,
discuss, and answer the Essential Question for Section 2.
o
How can strategies and classroom routines he
lp students develop the
mathematical practices of mature mathematical thinkers and learners?
PW:
Page 25
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