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Unless footers note otherwise, all pages are copyrighted to © Mona Toncheff 2013 and are REPRODUCIBLE. • solution-tree.com
How Do You Collaboratively Plan for CCSS-M (K–5)?
Mona Toncheff [email protected] http://puhsdmath.blogspot.com
WE
Paradigm Shifts• Professional development
– Ongoing collaborative team learning• Instruction
– Teaching for conceptual understanding as well as procedural fluency
• Content– Focus, coherence, rigor; conceptual
understanding and procedural fluency• Assessment
– Multifaceted process; emphasis on formative assessment
• Intervention– Required, not invitational
Today’s Learning Targets • I can examine criteria and effective
lesson planning and design on a unit-by-unit basis.
• I can create great tasks that develop student access to CCSS-M.
• I can define the learning progressions of the K–12 CCSS-M.
Three Big Ideas
1. Focus on student learning
2. Focus on collaboration
3. Focus on results
--DuFour, DuFour, Eaker, & Many, Learning by Doing (2010)
Four PLC Questions1. What do we expect students to learn?2. How will we know students learned it?
3. What will we do when students do not learn?
4. What will we do when students do learn?
--DuFour, DuFour, Eaker, & Many, Learning by Doing (2010)
Seven Stages of Teacher CollaborationStage 1: Filling the timeWhat exactly are we supposed to do?
Stage 2: Sharing personal practiceWhat is everyone doing in their classroom?
Stage 3: Planning, planning, planningWhat should we be teaching and how do we lighten the load for each other?
Stage 4: Developing common assessmentsHow do you know students learned?What does mastery look like?
Stage 5: Analyzing student learningAre students learning what they are supposed to be learning?
Seven Stages of Teacher Collaboration
Stage 6: Adapting instruction to student needsHow can we adjust instruction to help struggling students and those who exceed expectations?
Stage 7: Reflecting on instructionWhich Mathematical Practices are most effective with our students for this lesson or unit?
Seven Stages of Teacher Collaboration
Where Are We Now?
Think about your current course based or grade level collaborative teams.
Which Stage? Scan and decide.
How Do We Get to Stage 7?
• Norms for collaboration • Shared vision of mathematics
Action orientation: Teams do and produce stuff on a unit-by-unitbasis.
Dr. Timothy Kanold tkanold.blogspot.com
Brief Excerpt from Common Core Mathematics in a PLC at Work (Dr. Timothy D. Kanold (tkanold.blogspot.com)
Seeking Stage Seven as a Team
Graham and Ferriter (2008) offer a useful framework that details seven stages of collaborative team development.
Adapted for our purposes, the stage at which teams fall is directly correlated to each team’s level of effective
collaboration. Table 1.1 highlights these seven stages.
Table 1.1: The Seven Stages of Teacher Collaboration Diagnostic Tool
Stage Questions That Define This Stage
Stage 1: Filling the time What exactly are we supposed to do? Why are we meeting?
Is this going to be worth my time?
Stage 2: Sharing personal practice What is everyone doing in his or her classroom? What are
some of your favorite problems you use for this unit?
Stage 3: Planning, planning, planning What content should we be teaching, and how should we
pace this unit? How do we lighten the load for each other?
Stage 4: Developing common assessments How do you know students learned? What does mastery
look like? What does student proficiency look like?
Stage 5: Analyzing student learning
Are students learning what they are supposed to be learning?
What does it mean for students to demonstrate understanding
of the learning targets?
Stage 6: Adapting instruction to student needs How can we adjust instruction to help those students
struggling and those exceeding expectations?
Stage 7: Reflecting on instruction Which of our instructional and assessment practices are most
effective with our students?
Visit go.solution-tree.com/commoncore for a reproducible version of this table.
Common Core State Standards for MathematicsTwo type of standards:• Standards for Mathematical
Practice
• Standards for Mathematical Content
How Familiar Are You With Standards for Mathematical Practice?
Rate your knowledgeon a scale of
5 (high) to 1 (low)
Students Standards for Mathematical Practice
“The Standards for Mathematical Practice describe 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 education.”
--Common Core State Standards Initiative (2010)
How Do We Develop Student’s Mathematical Practice?“Shift to include within daily lesson plans intentional strategies to teach mathematics in different ways – in ways that focus on the process of learning and developing deep student understanding of the content.”
—Kanold (Ed.), Common Core Mathematics in a PLC at Work™, K– 2 (2012)
Collaborative Team Work
Develop a common understanding of the Standards for Mathematical Practice:1. What is the intent of this
Mathematical Practice?
2. What teacher actions facilitate this Mathematical Practice?
3. What evidence is there that students are demonstrating this Mathematical Practice?
How Do We Develop Student’s Mathematical Practice?Intentional strategies:• Purposeful planning
• Rich mathematical tasks that develop each practice
What is your experience with collaborative lesson design?
Strengths Weaknesses
How Do We Develop Student’s Mathematical Practice?
What Questions Do You and Your Team Ask Each Other When Planning?
Collaborative Lesson Design Activity
1.OA.3: Apply properties of operations as strategies to add and subtract.
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (commutative property).
To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (associative property).
4.OA.2: Multiply or divide to solve word problems involving multiplicative comparison.
Examples: Using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
Grade-1 Lesson Grade-4 Lesson
R E PRO DUCI B LE
Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.
Table 2.1: Elements of an Effective Mathematics Classroom Lesson Design
Probing Questions for Effective Lesson Design Reflection
1. Lesson Context: Learning Targets
Procedural Fluency and Conceptual Understanding Balancing
What is the learning target for the lesson? How does it connect to the bigger focus of the unit?
What evidence will be used to determine the level of student learning of the target?
Are conceptual understanding and procedural fluency appropriately balanced?
How will you formatively assess student conceptual understanding of the mathematics concepts and of the procedural skill?
What meaningful application or model can you use?
Which CCSS Mathematical Practices will be emphasized during this lesson?
2. Lesson Process: High-Cognitive-Demand Tasks
Planning Student Discourse and Engagement
What tasks will be used that create an a-ha student moment and leave “mathematical residue” (insights into the mathematical structure of concepts) regardless of content type at a high-cognitive-demand level?
How will you ensure the task is accessible to all students while still maintaining a high cognitive demand for students?
What strategic mathematical tools will be used during the lesson?
page 1 of 2
R E PRO DUCI B LE
Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.
Probing Questions for Effective Lesson Design Reflection
2. Lesson Process: High-Cognitive-Demand Tasks
(continued)
How will each lesson example be presented and sequenced to build mathematical reasoning connected to prior student knowledge?
What are the assessing and advancing questions you might ask during guided, independent, or group practice? What are anticipated student responses to the examples or tasks?
How might technology and student attention to precision play a role in the student lesson experience?
3. Introduction, Daily Review, and Closure
What activity will be used to immediately engage students at the beginning of the class period?
How can the daily review be used to provide brief, meaningful feedback on homework? (Five minutes maximum)
How will the students summarize the lesson learning targets and the key vocabulary words?
4. Homework How does the homework assignment provide variety and meaning to the students—including long-term review and questions—that balance procedural fluency with conceptual understanding?
page 2 of 2
CCSS Mathematical Practices LP Tool
—Kanold & Larson, Common Core Mathematics in a PLC at Work™, Leader’s Guide (2012)
Collaboration Begins With the End in Mind …
• Grade 1: OA. 3• Grade 4: 4.OA.3• Review your standards and complete
the table.
Lesson Design Questions for Exploring Standards
Grade LevelΥ
StandardsΥ
What content needs to beunpacked for LessonDesign around this
cluster?
Which topics need to beemphasized as parts of the
lesson?
How will students beengaged in the
Mathematical Practices asthey learn the content?
What resources will beneeded?
How will studentsdemonstrate learning of
this content standardcluster?
1.OA.3
Domino Fact Families
Draw a picture of the domino and then write a fact family for the
domino you chose.
Domino (picture) Fact family
Domino (picture) Fact family
Domino (picture) Fact family
Domino (picture) Fact family
Formative Assessment Task Grade 4: Operations and Algebraic Thinking
(Source: Howard County Public School System, Ellicott City, Maryland,
grade4commoncoremath.wikispaces.hcpss.org)
4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by
using drawings and equations with a symbol for the unknown number to represent the
problem, distinguishing multiplicative comparison from additive comparison.
Materials and Directions
1. Put the following story problem up for students to see (chart paper, document camera).
Then read it aloud.
The bakery stocks 4 times as many donuts as muffins. If they stock 144 donuts, how many
muffins are there?
2. Distribute grid paper (attached) to students. Ask students to use the paper as a model to
solve the problem.
3. Give time for the students to solve the problem.
4. Have the students come up to show their thinking while they explain their strategy.
5. To help all students learn to think aloud and solve problems through mental math or
estimation, have another student repeat the strategy given in their own words.
6. Repeat these steps using other word problems similar to this one.
Considerations
Ask students if they could estimate an answer before trying to solve it.
Observe students thinking while solving the problem. Do they use the grid paper or just use
the numbers and divide?
Some students may need paper and pencil, but others may not. However, it is important for
students to show how they solved a problem.
Explanations can be oral or written.
R E PRO DUCI B LE
Common Core Mathematics in a PLC at WorkTM, Grades K–2 © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.
Unit: Date: Lesson:
Learning target: As a result of today’s class, students will be able to
Formative assessment: How will students be expected to demonstrate mastery of the learning target during in-class checks for understanding?
Probing Questions for Differentiation on Mathematical Tasks
Assessing Questions
(Create questions to scaffold instruction for students who are “stuck” during the lesson or the lesson tasks.)
Advancing Questions
(Create questions to further learning for students who are ready to advance beyond the learning target.)
Targeted Standard for Mathematical Practice:
(Describe the intent of this Mathematical Practice and how it relates to the learning target.)
Tasks
(The number of tasks may vary from lesson to lesson.) What Will the Teacher Be Doing?
What Will the Students Be Doing?
(How will students be actively engaged in each part of the lesson? )
Beginning-of-Class Routines
How does the warm-up activity connect to students’ prior knowledge?
Figure 2.11: CCSS Mathematical Practices Lesson-Planning Tool
page 1 of 2
R E PRO DUCI B LE
Common Core Mathematics in a PLC at WorkTM, Grades K–2 © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.
Tasks
(The number of tasks may vary from lesson to lesson.) What Will the Teacher Be Doing?
What Will the Students Be Doing?
(How will students be actively engaged in each part of the lesson? )
Task 1
How will the learning target be introduced?
Task 2
How will the task develop student sense making and reasoning?
Task 3
How will the task require student conjectures and communication?
Closure
How will student questions and reflections be elicited in the summary of the lesson? How will students’ understanding of the learning target be determined?
page 2 of 2
• What are learning targets for the lesson?
• How would you assess the targets?
Collaboration Begins With the End in Mind …
What Learning Experiences Will Develop These Targets?
How Will We Create Access to the Tasks?
What MPs Will Be Targeted?
Handouts for Teachers Improving Learning Through Questioning
Handout 3: Five principles for effective questioning
1. Plan to use questions that encourage thinking and reasoning
Really effective questions are planned beforehand. It is helpful to plan sequences of questions that build on and extend students’ thinking. A good questioner, of course, remains flexible and allows time to follow up responses.
Beginning an inquiry
• What do you already know that might be useful here? • What sort of diagram might be helpful? • Can you invent a simple notation for this? • How can you simplify this problem? • What is known and what is unknown? • What assumptions might we make?
Progressing with an inquiry
• Where have you seen something like this before? • What is fixed here, and what can we change? • What is the same and what is different here? • What would happen if I changed this ... to this ... ? • Is this approach going anywhere? • What will you do when you get that answer? • This is just a special case of ... what? • Can you form any hypotheses? • Can you think of any counterexamples? • What mistakes have we made? • Can you suggest a different way of doing this? • What conclusions can you make from this data? • How can we check this calculation without doing it all again? • What is a sensible way to record this?
Interpreting and evaluating the results of an inquiry
• How can you best display your data? • Is it better to use this type of chart or that one? Why? • What patterns can you see in this data? • What reasons might there be for these patterns? • Can you give me a convincing argument for that statement? • Do you think that answer is reasonable? Why? • How can you be 100% sure that is true? Convince me! • What do you think of Anne’s argument? Why? • Which method might be best to use here? Why?
Communicating conclusions and reflecting
• What method did you use? • What other methods have you considered? • Which of your methods was the best? Why? • Which method was the quickest? • Where have you seen a problem like this before? • What methods did you use last time? Would they have worked here? • What helpful strategies have you learned for next time?
http://map.mathshell.org/static/draft/pd/modules/4_Questioning/html/index.htm
Mathematical Practices “Look Fors”
Goals of Assessment“We must ensure that tests measure what is of value, not just what is easy to test.
“If we want students to investigate, explore, and discover, assessment must not measure just mimicry mathematics.”
—National Research Council, Everybody Counts (2000)
Eng
agin
g in
the
Mat
hem
atic
al P
ract
ices
(Loo
k-Fo
rs)
All
indi
cato
rs a
re n
ot n
eces
sary
for p
rovi
ding
full
evid
ence
of p
ract
ices
. Eac
h pr
actic
e m
ay n
ot b
e ev
iden
t dur
ing
ever
y le
sson
. M
athe
mat
ics P
ract
ices
St
uden
ts
Tea
cher
s
Overarching habits of mind of a productive math thinker
1. M
ake
sens
e of
pr
oble
ms a
nd
pers
ever
e in
so
lvin
g th
em.
U
nder
stan
d th
e m
eani
ng o
f the
pro
blem
and
look
for e
ntry
poi
nts t
o its
solu
tion.
Ana
lyze
info
rmat
ion
(giv
ens,
cons
train
s, re
latio
nshi
ps, g
oals
).
Mak
e co
njec
ture
s and
pla
n a
solu
tion
path
way
.
Mon
itor a
nd e
valu
ate
the
prog
ress
and
cha
nge
cour
se a
s nec
essa
ry.
C
heck
ans
wer
s to
prob
lem
s and
ask
, “D
oes t
his m
ake
sens
e?”
Com
men
ts:
I
nvol
ve st
uden
ts in
rich
pro
blem‐b
ased
task
s tha
t enc
oura
ge th
em to
per
seve
re
to re
ach
a so
lutio
n.
P
rovi
de o
ppor
tuni
ties f
or st
uden
ts to
solv
e pr
oble
ms t
hat h
ave
mul
tiple
so
lutio
ns.
E
ncou
rage
stud
ents
to re
pres
ent t
heir
thin
king
whi
le p
robl
em so
lvin
g.
Com
men
ts:
6. A
tten
d to
pr
ecis
ion.
C
omm
unic
ate
prec
isel
y us
ing
clea
r def
initi
ons.
S
tate
the
mea
ning
of s
ymbo
ls, c
aref
ully
spec
ify u
nits
of m
easu
re, a
nd p
rovi
de
accu
rate
labe
ls.
C
alcu
late
acc
urat
ely
and
effic
ient
ly, e
xpre
ssin
g nu
mer
ical
ans
wer
s with
a d
egre
e of
pre
cisi
on.
P
rovi
de c
aref
ully
form
ulat
ed e
xpla
natio
ns.
L
abel
acc
urat
ely
whe
n m
easu
ring
and
grap
hing
. C
omm
ents
:
E
mph
asiz
e th
e im
porta
nce
of p
reci
se c
omm
unic
atio
n by
enc
oura
ging
stud
ents
to
focu
s on
clar
ity o
f the
def
initi
ons,
nota
tion,
and
voc
abul
ary
to c
onve
y th
eir
reas
onin
g.
E
ncou
rage
acc
urac
y an
d ef
ficie
ncy
in c
ompu
tatio
n an
d pr
oble
m-b
ased
so
lutio
ns, e
xpre
ssin
g nu
mer
ical
ans
wer
s, da
ta, a
nd/o
r mea
sure
men
ts w
ith a
de
gree
of p
reci
sion
app
ropr
iate
for t
he c
onte
xt o
f the
pro
blem
. C
omm
ents
:
Reasoning and explaining
2.
Rea
son
abst
ract
ly a
nd
quan
titat
ivel
y.
M
ake
sens
e of
qua
ntiti
es a
nd re
latio
nshi
ps in
pro
blem
situ
atio
ns.
R
epre
sent
abs
tract
situ
atio
ns sy
mbo
lical
ly a
nd u
nder
stan
d th
e m
eani
ng
of q
uant
ities
.
Cre
ate
a co
here
nt re
pres
enta
tion
of th
e pr
oble
m a
t han
d.
C
onsi
der t
he u
nits
invo
lved
.
Fle
xibl
y us
e pr
oper
ties o
f ope
ratio
ns.
Com
men
ts:
F
acili
tate
opp
ortu
nitie
s for
stud
ents
to d
iscu
ss o
r use
repr
esen
tatio
ns to
mak
e se
nse
of q
uant
ities
and
thei
r rel
atio
nshi
ps.
E
ncou
rage
the
flexi
ble
use
of p
rope
rties
of o
pera
tions
, obj
ects
, and
solu
tion
stra
tegi
es w
hen
solv
ing
prob
lem
s.
Pro
vide
opp
ortu
nitie
s for
stud
ents
to d
econ
text
ualiz
e (a
bstra
ct a
situ
atio
n)
and/
or c
onte
xtua
lize
(iden
tify
refe
rent
s for
sym
bols
invo
lved
) the
mat
hem
atic
s th
ey a
re le
arni
ng.
Com
men
ts:
3.
Con
stru
ct
viab
le
argu
men
ts a
nd
criti
que
the
reas
onin
g of
ot
hers
.
U
se d
efin
ition
s and
pre
viou
sly
esta
blis
hed
caus
es a
nd e
ffec
ts (r
esul
ts) i
n co
nstru
ctin
g ar
gum
ents
.
Mak
e co
njec
ture
s and
use
cou
nter
exam
ples
to b
uild
a lo
gica
l pro
gres
sion
of
stat
emen
ts to
exp
lore
and
supp
ort i
deas
.
Com
mun
icat
e an
d de
fend
mat
hem
atic
al re
ason
ing
usin
g ob
ject
s, dr
awin
gs,
diag
ram
s, an
d/or
act
ions
.
Lis
ten
to o
r rea
d th
e ar
gum
ents
of o
ther
s.
Dec
ide
if th
e ar
gum
ents
of o
ther
s mak
e se
nse
and
ask
prob
ing
ques
tions
to
clar
ify o
r im
prov
e th
e ar
gum
ents
. C
omm
ents
:
P
rovi
de a
nd o
rche
stra
te o
ppor
tuni
ties f
or st
uden
ts to
list
en to
the
solu
tion
stra
tegi
es o
f oth
ers,
disc
uss a
ltern
ativ
e so
lutio
ns, a
nd d
efen
d th
eir i
deas
.
Ask
hig
her-
orde
r que
stio
ns th
at e
ncou
rage
stud
ents
to d
efen
d th
eir i
deas
.
Pro
vide
pro
mpt
s tha
t enc
oura
ge st
uden
ts to
thin
k cr
itica
lly a
bout
the
mat
hem
atic
s the
y ar
e le
arni
ng.
Com
men
ts:
© Fennell 2012. solution-tree.comReproducible.
REPRODUCIBLE
Mat
hem
atic
s Pra
ctic
es
Stud
ents
T
each
ers
Modeling and using tools 4.
Mod
el w
ith
mat
hem
atic
s.
A
pply
prio
r kno
wle
dge
to so
lve
real
-wor
ld p
robl
ems.
I
dent
ify im
porta
nt q
uant
ities
and
map
thei
r rel
atio
nshi
ps u
sing
such
tool
s as
diag
ram
s, tw
o-w
ay ta
bles
, gra
phs,
flow
cha
rts, a
nd/o
r for
mul
as.
U
se a
ssum
ptio
ns a
nd a
ppro
xim
atio
ns to
mak
e a
prob
lem
sim
pler
.
Che
ck to
see
if an
ans
wer
mak
es se
nse
with
in th
e co
ntex
t of a
situ
atio
n an
d ch
ange
a m
odel
whe
n ne
cess
ary.
C
omm
ents
:
U
se m
athe
mat
ical
mod
els a
ppro
pria
te fo
r the
focu
s of t
he le
sson
.
Enc
oura
ge st
uden
t use
of d
evel
opm
enta
lly a
nd c
onte
nt-a
ppro
pria
te
mat
hem
atic
al m
odel
s (e.
g., v
aria
bles
, equ
atio
ns, c
oord
inat
e gr
ids)
.
Rem
ind
stud
ents
that
a m
athe
mat
ical
mod
el u
sed
to re
pres
ent a
pro
blem
’s
solu
tion
is a
wor
k in
pro
gres
s, an
d m
ay b
e re
vise
d as
nee
ded.
C
omm
ents
:
5. U
se
appr
opri
ate
tool
s st
rate
gica
lly.
M
ake
soun
d de
cisi
ons a
bout
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© Fennell 2012. solution-tree.comReproducible.
REPRODUCIBLE
Rich Learning Experiences
What decides the cognitive demand of a task?
It is decided not by whether it is a hard problem, but rather by the complexity of reasoning required by the student.
(Kanold, Briars, & Fennel, What Principals Need to Know About Teaching and Learning Mathematics (2011)
Hess’ Cognitive Rigor Matrix & Curricular Examples: Applying Webb’s Depth-of-Knowledge Levels to Bloom’s Cognitive Process Dimensions – Math/Science
© 2009 Karin Hess permission to reproduce is given when authorship is fully cited [email protected]
Revised Bloom’sTaxonomy
Webb’s DOK Level 1Recall & Reproduction
Webb’s DOK Level 2Skills & Concepts
Webb’s DOK Level 3Strategic Thinking/ Reasoning
Webb’s DOK Level 4Extended Thinking
RememberRetrieve knowledge fromlong-term memory,recognize, recall, locate,identify
o Recall, observe, & recognizefacts, principles, properties
o Recall/ identify conversionsamong representations ornumbers (e.g., customary andmetric measures)
UnderstandConstruct meaning, clarify,paraphrase, represent,translate, illustrate, giveexamples, classify,categorize, summarize,generalize, infer a logicalconclusion (such as fromexamples given), predict,compare/contrast, match likeideas, explain, constructmodels
o Evaluate an expressiono Locate points on a grid or
number on number lineo Solve a one-step problemo Represent math relationships in
words, pictures, or symbolso Read, write, compare decimals
in scientific notation
o Specify and explain relationships(e.g., non-examples/examples;cause-effect)
o Make and record observationso Explain steps followedo Summarize results or conceptso Make basic inferences or logical
predictions from data/observationso Use models /diagrams to represent
or explain mathematical conceptso Make and explain estimates
o Use concepts to solve non-routineproblems
o Explain, generalize, or connect ideasusing supporting evidence
o Make and justify conjectureso Explain thinking when more than
one response is possibleo Explain phenomena in terms of
concepts
o Relate mathematical orscientific concepts to othercontent areas, other domains,or other concepts
o Develop generalizations of theresults obtained and thestrategies used (frominvestigation or readings) andapply them to new problemsituations
ApplyCarry out or use a procedurein a given situation; carry out(apply to a familiar task), oruse (apply) to an unfamiliartask
o Follow simple procedures(recipe-type directions)
o Calculate, measure, apply a rule(e.g., rounding)
o Apply algorithm or formula (e.g.,area, perimeter)
o Solve linear equationso Make conversions among
representations or numbers, orwithin and between customaryand metric measures
o Select a procedure according tocriteria and perform it
o Solve routine problem applyingmultiple concepts or decision points
o Retrieve information from a table,graph, or figure and use it solve aproblem requiring multiple steps
o Translate between tables, graphs,words, and symbolic notations (e.g.,graph data from a table)
o Construct models given criteria
o Design investigation for a specificpurpose or research question
o Conduct a designed investigationo Use concepts to solve non-routine
problemso Use & show reasoning, planning,
and evidenceo Translate between problem &
symbolic notation when not a directtranslation
o Select or devise approachamong many alternatives tosolve a problem
o Conduct a project that specifiesa problem, identifies solutionpaths, solves the problem, andreports results
AnalyzeBreak into constituent parts,determine how parts relate,differentiate betweenrelevant-irrelevant,distinguish, focus, select,organize, outline, findcoherence, deconstruct
o Retrieve information from a tableor graph to answer a question
o Identify whether specificinformation is contained ingraphic representations (e.g.,table, graph, T-chart, diagram)
o Identify a pattern/trend
o Categorize, classify materials, data,figures based on characteristics
o Organize or order datao Compare/ contrast figures or datao Select appropriate graph and
organize & display datao Interpret data from a simple grapho Extend a pattern
o Compare information within oracross data sets or texts
o Analyze and draw conclusions fromdata, citing evidence
o Generalize a patterno Interpret data from complex grapho Analyze similarities/differences
between procedures or solutions
o Analyze multiple sources ofevidence
o analyze complex/abstractthemes
o Gather, analyze, and evaluateinformation
EvaluateMake judgments based oncriteria, check, detectinconsistencies or fallacies,judge, critique
o Cite evidence and develop a logicalargument for concepts or solutions
o Describe, compare, and contrastsolution methods
o Verify reasonableness of results
o Gather, analyze, & evaluateinformation to draw conclusions
o Apply understanding in a novelway, provide argument orjustification for the application
CreateReorganize elements intonew patterns/structures,generate, hypothesize,design, plan, construct,produce
o Brainstorm ideas, concepts, orperspectives related to a topic
o Generate conjectures or hypothesesbased on observations or priorknowledge and experience
o Synthesize information within onedata set, source, or text
o Formulate an original problem givena situation
o Develop a scientific/mathematicalmodel for a complex situation
o Synthesize information acrossmultiple sources or texts
o Design a mathematical modelto inform and solve a practicalor abstract situation
Task Sort In a group of 34, sort the questions by DOK Level (1, 2, or 3).
There are three of each.
DOK ?
1
2
3
4
Instructional Alignment What percent of your instructional materials develop each level of Depth of Knowledge?
DOK ?
1 ?
2 ?
3 ?
4 ?
What Are Great Mathematical Tasks?
• Center on an interesting problem, offering several methods of solution
• Are directed at essential mathematical content as specified in the standards
• Require examination and perseverance (challenging to students)
• Beg for discussion, offering rich discourse on mathematics involved
• Build student understanding, following a clear set of learning expectations
• Warrant a summary look back with reflection and extension opportunities(www.mathedleadership.org/ccss/greattasks.html)
Five-Finger Rule
Sustained Implementation of CCSS requires four pursuits:
1. A thorough review of your current local assessments on a unit-by-unit basis
2. High-quality common assessments and the accurate scoring of those assessments
3. A robust formative assessment process for students and adults, using each assessment Instrument
4. Instruction that provides evidence of student understanding via the mathematical practices
K–8 Content Standards Connected to High School
Review Key Features of the Standards for Mathematical Content
Focus questions:Considering the new CCSS-M features that you have been discussing, what implications do these features have for your curriculum, instruction and assessment?
For your collaborative team work?
Where Do I Find Resources to Create a Coherent Learning Progression?
• PARCC model content frameworks
• Sample scope and sequences from the CCSS toolbox
• Learning progressions
• Evidence tables
Grades K–5 CCSS Progressions Table
STANDARDS PROGRESSION
KINDERGARTEN GRADE 1 GRADE 2 Count to 100. Rote count by 1s and 10s.
Count to 120. Establish understanding of place value into the hundreds; rote counting by 1s, 2s, 5s, and 10s.
Count to 1,000. Establish understanding of place value into the thousandths; count by 5s, 10s and 100s.
Fluently add and subtract within 5. (Fluency at this level does not mean can compute quickly. It means a student can determine an answer using mental strategies or physical tools without hesitation)
Fluently add and subtract within 10.
Fluently add and subtract within 100.
Solve word problems using addition and subtraction within 10 using models.
Solve word problems using addition and subtraction within 20 using models.
Solve word problems using addition and subtraction within 100 using models.
Add and subtract within 10 using a variety of strategies and models. Compose and decompose numbers to 19.
Add within 100 and subtract within 20. Subtract to 100 with multiples of 10 using concrete models or drawings and strategies based on place value, including composing or decomposing into tens and ones.
Add and subtract within 1,000 using concrete models or drawings and strategies based on place value, including composing or decomposing into tens or hundreds.
Describe and compare measurable attributes of objects and shapes.
Measure the lengths of up to three objects indirectly and by iterating length units. Tell and write time in hours and half‐hours using analog and digital clocks.
Measure and estimate lengths in standard units.Relate addition and subtraction to length. Tell and write time to the nearest five minutes. Identify, count, recognize, and use coins and bills in and out of context.
Identify and describe 2D and 3D shapes. Classify and analyze shapes using informal language.
Distinguish between defining attributes of shapes. Compose and decompose two‐dimensional shapes (rectangles, squares, trapezoids, triangles, half circles, and quarter circles) or three‐dimensional shapes.
Recognize and draw shapes having specified attributes. Partition a rectangle into rows and columns of same‐size squares and count to find the total number of them. Partition circles and rectangles into two, three, or four equal shares.
Understand the meaning of the equal sign.Determine an unknown number in an addition or subtraction sentence. Write addition and subtraction sentences to represent a given situation.
Solve word problems with multiple operations.
STANDARDS PROGRESSION
GRADE 3 GRADE 4 GRADE 5 Multiply within 100. Include multiplying one‐digit whole numbers by multiples of 10.
Solve problems based on the multiplication counting principle. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Multiply a whole number of up to four digits by a one‐digit whole number, and multiply two two‐digit numbers, using strategies based on place value and the properties of operations. (Algorithm is introduced in grade 5.)
Explain why problems with multiplication and division by powers of 10 make sense. Fluently multiply multidigit whole numbers using the standard algorithm. Simplify expressions using the Order of Operations, including all 4 operations and some simple fractions and decimals.
Divide within 100. Find whole‐number quotients and remainders with up to four‐digit dividends and one‐digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. (Algorithm is introduced in grade 5.)
Find whole‐number quotients of whole numbers with up to four‐digit dividends and two‐digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Have students estimate first to determine a reasonable range of answers.
Use place value understanding to round multidigit whole numbers to the nearest 10 or 100.
Use place value understanding to roundmultidigit whole numbers to any place.
Use place value understanding to round decimals to any place.
Add and subtract within 1,000 using strategies based on place value. (Algorithm is introduced in grade 4.)
Fluently add and subtract multidigit whole numbers using the standard algorithm and explain why the algorithm works.
Add, subtract, multiply, and divide decimals to hundredths. Have students estimate first to determine a reasonable range of answers.
STANDARDS PROGRESSION
GRADE 3 GRADE 4 GRADE 5 Develop an understanding of fractions and generate simple equivalent fractions. (Grade 3 is limited to fractions with denominators 2, 3, 4, 6, and 8.) Students should connect their understanding of fractions to measuring to one‐half and one‐quarter inch using a ruler. Third graders need many opportunities measuring the length of various objects in their environment.
Generate equivalent fractions and compare. (Grade 4 is limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12 and 100.)
Decompose fractions, add and subtract fractions and mixed numbers with like denominators, and multiply a fraction by whole number by using unit fractions, not a standard algorithm.
Represent and compare fractions using part–part–whole models, number lines and line plots.
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with unlike denominators. (Do not use standard algorithm.)
Use decimal notation for fractions with denominators 10 or 100.
Add and subtract fractions with unlike denominators. Multiply fractions, including mixed numbers. Divide fractions with whole number divisors and unit fraction dividends. (Division of a fraction by a fraction is not a requirement at this grade.)
Identify arithmetic patterns. Write a rule in the form of an expression to represent a number pattern.
Write and interpret simple numerical expressions from a context. Identify apparent relationships between corresponding terms of two numerical patterns. Create ordered pairs from these and graph in first quadrant of a coordinate plane.
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit within one system of units. In addition, solve problems with money. At this point, students should have significant opportunities to solve problems that include fractions and decimals.
Convert among different‐sized standard measurement units within a given measurement system.
Make a line plot to display a data set of
measurements in fractions of a unit (1/2, 1/4, 1/8).
Solve problems for such data.
STANDARDS PROGRESSION
GRADE 3 GRADE 4 GRADE 5 Measure areas by counting unit squares. (Make sure students have lots of opportunities to develop an understanding of area before connecting to multiplication. Formula should not be formally introduced until grade 4.) Solve real‐world and mathematical problems involving perimeter. Find unknown side lengths given a perimeter and explore polygons with the same area but different perimeters.
Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
Measure volumes by counting unit cubes, using cubic centimeters, cubic inches, cubic feet, and improvised units. Have students derive the volume formula.
Students recognize shapes that are and are not quadrilaterals by examining the properties of the geometric figures. Students should be encouraged to provide details and use proper vocabulary when describing the properties of quadrilaterals. They sort geometric figures (see examples below) and identify squares, rectangles, and rhombuses as quadrilaterals. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
Understand concepts of angles, measure, and sketch angles. Develop an understanding of benchmark angle measures including 360, 180, 90, 45 and 30 degrees. Decompose an angle into smaller angles and find an unknown angle measure. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two‐dimensional figures. Identify right triangles. Identify and draw lines of symmetry.
Use a pair of perpendicular number lines, called axes, to define a coordinate system. Represent data from real world problems on a coordinate plane. Classify two‐dimensional figures in a hierarchy based on properties.
Continue to develop an understanding of the meaning of the equal sign. Determine an unknown number in an addition, subtraction, multiplication and division sentence, including fractions and decimals in grades 4–5. Write addition, subtraction, multiplication and division sentences to represent a given situation, including fractions and decimals in grades 4–5.
Solve word problems with multiple operations, including fractions and decimals in grades 4–5.
Grades 6–8 CCSS Progressions Table
GRADE 6 GRADE 7 GRADE 8
RATIOS AND PROPORTIONAL RELATIONSHIPS Ratios, rates, simple proportions: “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non‐complex fractions.)
Make tables of equivalent ratios relating quantities with whole‐number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Complex unit rate, proportional relationships, constant of proportionality with a focus on representations (tables and graphs): Example: If a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per
hour.
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Graph proportional relationships, interpreting the unit rate as the slope of the graph: Compare two different proportional relationships represented in different ways. For example, compare a distance‐time graph to a distance‐time equation to determine which of two moving objects has greater speed.
Use similar triangles to explain why the slope m is the same between any two distinct points on a non‐vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
**These standards are in the Expressions and Equations Domain but progresses from grades 6 to 7 standards in the Ratios and Proportional Relationships Domain.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error
GRADE 6 GRADE 7 GRADE 8
THE NUMBER SYSTEM Fluently divide multidigit whole numbers using standard algorithm. (Grades 3–5: Divide using place value models.)
Interpret and compute quotients of fractions. (Grade 5: Divide unit fractions and whole number.)
Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation. (Grade 5: Same but using place value)
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).
Convert between expressions for positive rational numbers, including fractions, decimals, and percents.
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
GRADE 6 GRADE 7 GRADE 8 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Understand ordering, absolute value, comparing, and representing using number lines and a coordinate plane. (Grade 5: first quadrant only)
Apply and extend previous understandings of addition, subtraction, multiplication and division to rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram (positive/negative numbers, including fractions and decimals). Solve real‐world and mathematical problems involving the four operations with rational numbers. Example: Division of positive/negative number: It took a submarine 20 seconds to drop to 100 feet below sea level from the surface. What was the rate of the descent?
ft/sec -5second1
feet 5seconds20
feet 100
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of
expressions (e.g., 2). Example: By truncating the decimal expansion of √2, show that √2 is between 1and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
EXPRESSIONS AND EQUATIONS
GRADE 6 GRADE 7 GRADE 8 Simplify expressions using order of operations focusing on formulas and expressions with exponents, as well as fraction and decimal operations. (Grade 5: order of operations with fractions and decimals, but no exponents)
Know and apply the properties of integer exponents to generate equivalent numerical expressions.
Example: 323–5 = 3–3 = 1/33 = 1/27
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. Example: Estimate the population of the United
States as 3108 and the population of the world as 7109, and determine that the world population is more than 20 times larger. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
GRADE 6 GRADE 7 GRADE 8 Apply the properties of operations to generate equivalent expressions. Examples: Apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients (extension of grade‐6 standard).
Suzanne thinks the two expressions
aa 4232 and 210 a are equivalent?
Is she correct? Explain why or why not?
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.
Example: a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Solve linear equations in one variable.a. Give examples of linear equations in one
variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true (no algorithm at this point; include fractions and decimals).
Solve multistep real‐life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals). Estimation strategies for calculations with fractions and decimals extend from students’ work with whole‐number operations (front‐end estimation; friendly and compatible numbers, benchmark numbers).
Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irra onal.
Use variables to represent numbers and write expressions when solving a real‐world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
GRADE 6 GRADE 7 GRADE 8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real‐world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Use variables to represent two quantities in a real‐world problem that change in relationship to one another. Write a one‐step equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
Use variables to represent quantities in a real‐world or mathematical problem, and construct simple 2‐step equations and inequalities to solve problems by reasoning about the quantities.
Analyze and solve pairs of simultaneous linear equations. (Solve a system of Equations.)
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
FUNCTIONS
GRADE 6 GRADE 7 GRADE 8 Interpret the equation y = mx + b as defining a
linear function, whose graph is a straight line; give examples of functions that are not linear. Example: The function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
GRADE 6 GRADE 7 GRADE 8 GEOMETRY
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes
Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real‐world and mathematical problems.
Solve real‐world and mathematical problems involving area, volume and surface area of two‐ and three‐dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real‐world and mathematical problems.
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real‐world and mathematical problems.
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Understand that a two‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that exhibits the congruence between them.
Represent three‐dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real‐world and mathematical problems.
Describe the two‐dimensional figures that result from slicing three‐dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
GRADE 6 GRADE 7 GRADE 8 Use facts about supplementary, complementary,
vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.
Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to
line segments of the same length. b. Angles are taken to angles of the same
measure. c. Parallel lines are taken to parallel lines. Describe the effect of dilations, translations, rotations, and reflections on two‐dimensional figures using coordinates.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle‐angle criterion for similarity of triangles. Example: Arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Explain a proof of the Pythagorean theorem and its converse. Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real‐world and mathematical problems in two and three dimensions. Apply the Pythagorean theorem to find the distance between two points in a coordinate system.
GRADE 6 GRADE 7 GRADE 8
STATISTICS AND PROBABILITY Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
Understand that statistics can be used to gain information about a population by examining a sample of the population. Generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. Example: Estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. Example: The mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team. On a dot plot, the separation between the two distributions of heights is noticeable.
GRADE 6 GRADE 7 GRADE 8 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
Example: In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
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Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two‐way table. Construct and interpret a two‐way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
Example: Collect data from students in your class on whether they have a curfew on school nights or assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
GRADE 6 GRADE 7 GRADE 8 Summarize numerical data sets in relation to their context using measures of center and describing patterns of the data.
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. Example: Decide whether the words in a chapter of a grade‐7 science book are generally longer than the words in a chapter of a grade‐4 science book.
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long‐run relative frequency, and predict the approximate relative frequency given the probability. Example: When rolling a number cube 600 times, predict whether a 3 or 6 would be rolled roughly 200 times.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies. If the agreement is not good, explain possible sources of the discrepancy.
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Today’s Learning Targets • I can examine criteria and effective
lesson planning and design on a unit-by-unit basis.
• I can create great tasks that develops student access to CCSS-M.
• I can define the learning progressions of the K–12 CCSS-M.
End-of-Day Note Card Reflection
1. Has our work today caused you to consider or reconsider any aspects of your own thinking and/or practice? Explain.
2. Has our work today caused you to reconsider any aspects of your students’ mathematical learning? Explain.
3. What would you like more information about?
