OwningYourMathCurriculum:
HowtoAdaptfortheChildreninYour
Care
LeadershipII-GradesK–5-Day2
Curriculum Review Sample Packet
Please return to UnboundEd.
Page 2
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NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
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Lesson&1&Problem&Set& 3•1&
Lesson&1&&
Objective:!!Understand!equal&groups!of!as!multiplication.&&
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Suggested&Lesson&Structure&
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Fluency!Practice!
Application!Problem!
Concept!Development!
Student!Debrief!
(5!minutes)!!
(10!minutes)!!
(35!minutes)!!
(10!minutes)!!
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Total&Time&
(60&minutes)&
Concept&Development&&(35&minutes) &
Materials:!!(S)!12!counters,!personal!white!board!
Problem&1:&&SkipLcount&to&find&the&total&number&of&objects.&&
T:! (Select!10!students!to!come!to!the!front.)!!At!the!signal,!say!how!many!arms!you!each!have.!!(Signal.)!
S:! 2!arms!!
T:! Since!we!each!represent!a!group!of!2!arms,!let’s!skipPcount!our!volunteers!by!twos!to!find!how!many!
arms!they!have!all!together.!!To!keep!track!of!our!count,!the!students!will!raise!up!their!arms!when!
we!count!them.!!!
S:! (Count!2,!4,!6,!…20.)!
T:! How!many!raised!arms!do!we!have!in!all?!
S:! 20.!
T:! Arms!down.!!How!many!twos!did!we!count!to!find!the!total?!!Turn!and!whisper!to!your!partner.!!
S:! 10!twos.!
T:! What!did!you!count!to!find!the!number!of!twos?!
S:! I!counted!the!number!of!volunteers!because!each!person!has!a!group!of!two!arms.!!
T:! SkipPcount!to!find!the!total!number!of!arms.!!
Sample&teacher&board:&
S:! (Say!2,!4,!6,!….)!!!
T:! (As!they!count,!write!2!+!2!+!2….)!
T:! Look!at!our!addition!sentence.!!Show!thumbs!
up!if!you!see!the!correct!number!of!twos.!!
S:! (Show!thumbs!up.)!!
T:! (Under!the!addition!sentence,!write!10!twos.)!!
Clap!3!times!if!you!agree!that!10!groups!of!two!
is!20.!!
S:! (Clap!3!times.)!
T:! (Write!10&groups&of&two&is&20!under!the!other!number!sentences.)!
Page 3
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NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
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Lesson&1&Problem&Set& 3•1&
Problem&2:&&Understand&the&relationship&between&repeated&addition,&counting&groups&in&unit&form,&and&
multiplication&sentences.&&&
Seat!students!at!tables!with!personal!white!boards!and!12!counters!each.!
T:! You!have!12!counters.!!Use!your!counters!to!make!equal&groups&of!two.!!How!many!counters!will!
you!put!in!each!group?!!Show!with!your!fingers.!
S:! (Hold!up!2!fingers!and!make!groups!of!two.)!
T:! How!many!equal!groups!of!two!did!you!make?!!Tell!at!the!signal.!!(Signal.)!
S:! 6!groups.!
T:! 6!equal!groups!of!how!many!counters?!!
S:! 6!equal!groups!of!2!counters.!
T:! 6!equal!groups!of!2!counters!equal!how!many!counters!all!together?!
S:! 12!counters.!
T:! Write!an!addition!sentence!to!show!your!groups!on!your!personal!white!board.!!
S:! (Write!2!+!2!+!2!+!2!+!2!+!2!=!12.)!
Sample&teacher&board:&
T:! (Record!the!addition!sentence!on!the!board.)!!In!unit!
form,!how!many!twos!did!we!add!to!make!12?!
S:! 6!twos.!
T:! (Record!6!twos!=!12!under!the!addition!sentence.)!!!
6!×!2!is!another!way!to!write!2!+!2!+!2!+!2!+!2!+!2!or!6!
twos.!!(Record!6!×!2!=!12!under!6!twos!=!12!on!the!
board.)!!These!number!sentences!are!all!saying!the!
same!thing.!!!
T:! Turn!and!talk!to!your!partner.!!How!do!you!think!
6!×!2!=!12!relates!to!the!other!number!sentences?!
S:! They!all!have!twos!in!them,!and!the!answer!is!12.!!#!!!
I!think!the!6!shows!how!many!twos!there!are.!#!You!
have!to!count!two!6!times!because!there!are!6!groups!
of!them.!!That’s!how!you!get!6!times!2.!!#!!6!×!2!might!
be!an!easier!way!to!write!a!long!addition!sentence.!
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It!may!be!necessary!to!explicitly!connect!
times!and!the!symbol!×.!!Have!students!
analyze!the!model.!!“How!many!times!do!
you!see!a!group!of!two?”!!Have!them!count!
the!groups,!write the number sentence, and
say the words together.
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NOTES&ON&MULTIPLE MEANS
OF ACTION AN E P ESSION
T:! Ways!that!are!easier!and!faster!are!efficient.!!When!we!
have!equal!groups,!multiplication!is!a!more!efficient!
way!to!find!the!total!than!repeated!addition.!
Repeat!the!process!with!4!threes,!3!fours,!and!2!sixes!to!get!
students!comfortable!with!the!relationship!between!repeated!
addition,!counting!groups!in!unit!form,!and!multiplication!
sentences.!
NOTES&ON&MULTIPLE&MEANS
OF&REPRESENTATION:&
" 6!groups!of!two!equal!12.!
§ 6!times!2!equals!12.!
Some!students!may!need!more!
scaffolding!to!realize!that!multiplication!
cannot!be!used!to!find!totals!with!
groups!that!are!not!equal.!!Use!the!
following!questions!to!scaffold.!!
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Problem&3:&&Write&multiplication&sentences&from&equal&groups.&
Draw!or!project!the!picture!to!the!right.!
T:! These!are!equal!groups.!!Turn!and!tell!your!partner!
why!they!are!equal.!
S:! There!are!the!same!number!of!grey!circles!in!each!
group.!!#!All!of!the!grey!circles!are!the!same!size!and!
" Does!the!drawing!show!3!fours?!!
" Does!3!times!4!represent!this!
drawing?!
" How!might!we!redraw!the!picture!
to!make!it!show!3!×!4?!
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Page 4
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NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
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Lesson&1&Problem&Set& 3•1&
shape,!and!there!are!4!in!each!group.!
T:! Work!with!your!partner!to!write!a!repeated!addition!
and!a!multiplication!sentence!for!this!picture.!
S:! (Write!4!+!4!=!8!and!either!2!×!4!=!8!or!4!×!2!=!8.)!
T:! (Project!or!draw!the!following!image.)!!Look!at!my!new!
drawing!and!the!multiplication!sentence!I!wrote!to!
represent!it.!!Check!my!work!by!writing!an!addition!
sentence!and!counting!to!find!the!total!number!of!
objects.!!!
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3&×&4&=&12&
MP.3& !
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S:! (Write!4!+!4!+!3!=!11.)!
T:! Use!your!addition!sentence!as!you!talk!to!your!partner!about!why!you!agree!or!disagree!with!my!
work.!
S:! I!disagree!because!my!addition!sentence!equals!11,!not!12.!!#!!It’s!because!that!last!group!doesn’t!
have!4!circles.!!#!!You!can!do!multiplication!when!the!groups!are!equal.!!#!!Here,!the!groups!aren’t!
equal,!so!the!drawing!doesn’t!show!3!×!4.!
T:! I!hear!most!students!disagreeing!because!my!groups!are!not!equal.!!True,!to!multiply!you!must!have!
equal!groups.!!&
Problem&Set&&(10&minutes)&&
Students!should!do!their!personal!best!to!complete!the!
Problem!Set!within!the!allotted!10!minutes.!!Some!problems!
do!not!specify!a!method!for!solving.!!This!is!an!intentional!
reduction!of!scaffolding!that!invokes!MP.5,!Use!Appropriate!
Tools!Strategically.!!Students!should!solve!these!problems!
using!the!RDW!approach!used!for!Application!Problems.!!
For!some!classes,!it!may!be!appropriate!to!modify!the!
assignment!by!specifying!which!problems!students!should!
work!on!first.!!With!this!option,!let!the!careful!sequencing!of!
the!Problem!Set!guide!your!selections!so!that!problems!
continue!to!be!scaffolded.!!Balance!word!problems!with!
other!problem!types!to!ensure!a!range!of!practice.!!Assign!
incomplete!problems!for!homework!or!at!another!time!
during!the!day.!
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Lesson&1& 4 3&
NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
Lesson&1!
Objective:!!Investigate!and!use!the!formulas!for!area!and!perimeter!of!
rectangles.!
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Suggested&Lesson&Structure&
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!
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Fluency!Practice!
Concept!Development!
Student!Debrief!
(15!minutes)!
(35!minutes)!
(10!minutes)!
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Total&Time&
(60&minutes)&
Concept&Development&&(35&minutes) &
Materials:! (T)!Grid!paper!(with!ability!to!project!or!enlarge!grid!paper),!chart!paper!!(S)!Grid!paper,!personal!
white!board!
Problem&1:&&Review&and&compare&perimeter&and&area&of&a&rectangle.&
T:! Draw!a!rectangle!on!your!grid!paper!that!is!four!units!wide!
and!seven!units!long.!!!
S:! (Draw!rectangle!on!grid!paper.)!!!
T:! (Monitor!to!see!that!the!students!have!drawn!the!rectangle!
correctly.)!!Tell!your!partner!what!you!notice!about!your!
rectangle.!
S:! The!opposite!sides!are!the!same!length.!!"!It!has!four!right!
angles.!!"!The!area!of!the!rectangle!is!28!square!units.!!!
"!The!perimeter!of!the!rectangle!is!22!units.!
T:! Place!the!point!of!your!pencil!on!one!of!the!corners!of!the!
rectangle.!!Now,!trace!around!the!outside!of!the!rectangle!
until!you!get!back!to!where!you!started.!!What!do!we!call!the!measurement!of!the!distance!around!a!
rectangle?!
S:! The!perimeter.!
T:! Trace!the!perimeter!again.!!This!time,!count!the!units!as!you!trace!them.!!What!is!the!perimeter!of!
the!rectangle?!
S:! 22!units.!
T:! When!we!know!the!measurements!of!the!length!and!width!of!a!rectangle,!is!there!a!quicker!way!to!
determine!the!perimeter!than!to!count!the!units!while!tracing?!
S:! We!could!add!the!measurements!of!all!four!sides!of!the!rectangle.!
T:! Take!your!pencil!and!count!all!of!the!squares!within!your!rectangle.!!These!squares!represent!the!
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Lesson&1:&
Date:&
Investigate!and!use!the!formulas!for!area!and!perimeter!of!rectangles.!
1/7/17!
3.A.3&
Page 6
©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org!
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This!work!is!licensed!under!a!!
Creative!Commons!AttributionENonCommercialEShareAlike!3.0!Unported!License.!!
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Lesson&1& 4 3&
NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
area!of!the!rectangle.!!How!do!I!find!the!area!of!the!rectangle?!
S:! You!count!the!squares.!!"!You!can!multiply!the!length!times!the!width!of!the!rectangle.!!"!Four!
units!times!7!units!is!28!square!units.
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Lesson&1:&
Date:&
Investigate!and!use!the!formulas!for!area!and!perimeter!of!rectangles.!
1/7/17!
3.A.4&
Page 7
©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org!
!
This!work!is!licensed!under!a!!
Creative!Commons!AttributionENonCommercialEShareAlike!3.0!Unported!License.!!
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!
Lesson&1& 4 3&
NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
Problem&2:&&Use&the&formula&2&×&(l&+&w)&to&solve&for&perimeter&and&to&find&an&unknown&side&length&of&a&
rectangle.&
T:! Draw!a!rectangle!on!your!graph!paper!that!is!3!units!wide!and!9!units!
long.!!(Draw!and!display!the!rectangle.)!!Watch!as!I!label!the!length!and!
width!of!the!rectangle.!!Now,!label!the!length!and!width!of!your!
rectangle.!!How!can!I!find!the!perimeter?!
S:! Add!up!the!lengths!of!all!of!the!sides.!!3!+!9!+!3!+!9!=!24.!!The!perimeter!is!
24!units.!!"!You!could!also!add!3!+!3!+!9!+!9.!!The!answer!is!still!24!units.!!
The!order!doesn’t!matter!when!you!are!adding.!
T:! Use!your!pencil!to!trace!along!one!width!and!one!length.!!Along!how!
many!units!did!you!trace?!
S:! 12!units.!
T:! How!does!12!relate!to!the!length!and!width!of!the!rectangle?!
S:! It’s!the!sum!of!the!length!and!width.!
T:! How!does!the!sum!of!the!length!and!width!relate!to!finding!the!perimeter!of!the!rectangle?!
S:! It’s!halfway!around.!!"!I!can!double!the!length!and!double!the!width!to!find!the!perimeter!instead!of!
adding!all!the!sides!(2l"+!2w).!!"!I!could!also!add!the!length!and!the!width!and!double!that!sum,!!
2!×!(l!+!w).!!"!Both!of!those!work!since!the!opposite!sides!are!equal.!
T:! You!have!just!mentioned!many!formulas,!like!counting!along!the!sides!of!
the!rectangle!or!adding!sides!or!doubling,!to!find!the!perimeter.!!Let’s!
create!a!chart!to!keep!track!of!the!formulas!for!finding!the!perimeter!of!a!
rectangle.!!Talk!to!your!partner!about!the!most!efficient!way!to!find!the!
perimeter.!
S:! If!I!draw!the!shape!on!grid!paper,!I!can!just!count!along!the!edge.!!"!I!am!
good!at!adding,!so!I!will!add!all!four!sides.!!"!It!is!faster!to!double!the!
sum!of!the!length!and!width.!!It’s!only!two!steps.!
T:! We!can!write!the!formula!as!P!=!2!×!(l!+!w)!on!our!chart,!meaning!we!add!the!length!and!width!first!
and!then!multiply!that!sum!by!2.!!What!is!the!length!plus!width!of!this!rectangle?!
S:! 3!plus!9!equals!12.!!12!units.!
T:! 12!units!doubled,!or!12!units!times!2,!equals?!
S:! 24!units.!
T:! Now,!draw!a!rectangle!that!is!2!units!wide!and!4!units!long.!!Find!the!perimeter!by!using!the!formula!
I!just!mentioned.!!Then,!solve!for!the!perimeter!using!a!different!formula!to!check!your!work.!
S:! 2!+!4!=!6!and!6!×!2!=!12.!!The!perimeter!is!12!units.!!"!Another!way!is!to!double!2,!double!4,!and!
then!add!the!doubles!together.!!4!plus!8!is!12!units.!!Both!formulas!give!us!the!same!answer.!
Repeat!with!a!rectangle!that!is!5!units!wide!and!6!units!long.!
Instruct!students!to!sketch!a!rectangle!with!a!width!of!5!units!and!a!perimeter!
of!26!units!on!their!personal!white!boards,!not!using!graph!paper.!
T:! Label!the!width!as!5!units.!!Label!the!length!as!an!unknown!of!x!units.!!
How!can!we!determine!the!length?!!Discuss!your!ideas!with!a!partner.!
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Lesson&1:&
Date:&
Investigate!and!use!the!formulas!for!area!and!perimeter!of!rectangles.!
1/7/17!
3.A.5&
Page 8
©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org!
!
This!work!is!licensed!under!a!!
Creative!Commons!AttributionENonCommercialEShareAlike!3.0!Unported!License.!!
!
!
Lesson&1& 4 3&
NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
S:! If!I!know!that!the!width!is!5,!I!can!label!the!opposite!side!as!5!units!since!they!are!the!same.!!If!the!
perimeter!is!26,!I!can!take!away!the!widths!to!find!the!sum!of!the!other!two!sides.!!26!–!10!=!16.!!If!
the!sum!of!the!remaining!two!sides!is!16,!I!know!that!each!side!must!be!8!since!I!know!that!they!are!
equal!and!that!8!+!8!=!16,!so!x!=!8!(shows!sketch!to!demonstrate!her!thinking).!
S:! We!could!also!find!the!length!another!way.!!I!know!that!if!I!add!the!length!and!the!width!of!the!
rectangle!together!that!I!will!get!half!of!the!perimeter.!!In!this!rectangle,!because!the!perimeter!is!26!
units,!the!length!plus!the!width!equals!13!units.!!If!the!width!is!5,!that!means!that!the!length!has!to!
be!8!units!because!5!+!8!=!13.!!"!26!÷!2!=!13,!x!+!5!=!13!or!13!–!5!=!x,!so!x!=!8.!
Repeat!for!P!=!28!cm,!l!=!8!cm.!
Problem&3:&&Use&the&area&formula&(l&×&w)&to&solve&for&area&and&to&solve&for&the&unknown&side&length&of&a&
rectangle.&
T:! Look!back!at!the!rectangle!with!the!width!of!3!units!and!the!length!of!9!units.!!How!can!we!find!the!
area!of!the!rectangle?!
S:! We!can!count!all!of!the!squares.!!"!We!could!also!count!the!number!of!squares!in!one!row!and!then!
skipEcount!that!number!for!all!of!the!rows.!!"!That’s!just!multiplying!the!number!of!rows!by!the!
number!in!each!row.!!"!A!quicker!way!is!to!multiply!the!length!times!the!width.!!Nine!rows!of!3!
units!each!is!like!an!array.!!We!can!just!multiply!9!×!3.!
T:! Talk!to!your!partner!about!the!most!efficient!way!to!find!the!area!of!a!rectangle.!
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Discuss!how!to!find!the!area!for!the!2!×!4!rectangle!and!the!5!×!6!rectangle!drawn!earlier!in!the!lesson.!!
Encourage!students!to!multiply!length!times!width!to!solve.!!Ask!students!to!tell!how!the!area!of!each!
rectangle!needs!to!be!labeled!and!why.!
T:! We!discussed!a!formula!for!finding!the!perimeter!of!a!rectangle.!!We!just!discovered!a!formula!for!
finding!the!area!of!a!rectangle.!!If!we!use!A!for!area,!l!for!length,!and!w!for!width,!how!could!we!
write!the!formula?!
S:! A"="l"×"w.!
T:! (Sketch!a!rectangle!on!the!board,!and!label!the!area!as!50!square!
centimeters.)!!If!we!know!that!the!area!of!a!rectangle!is!50!
square!centimeters!and!that!the!length!of!the!rectangle!is!10!
centimeters,!how!can!we!determine!the!measurement!of!the!
width!of!the!rectangle?!
!
Lesson&1:&
Date:&
Investigate!and!use!the!formulas!for!area!and!perimeter!of!rectangles.!
1/7/17!
3.A.6&
Page 9
©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org!
!
This!work!is!licensed!under!a!!
Creative!Commons!AttributionENonCommercialEShareAlike!3.0!Unported!License.!!
!
!
Lesson&1& 4 3&
NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
S:! I!can!use!the!area!formula.!!50!square!centimeters!is!equal!to!10!centimeters!times!the!width.!!10!
times!5!equals!50,!so!the!width!is!5!centimeters.!!"!The!area!formula!says!50!=!10!×!!!!!!!!!.!!I!can!solve!
that!with!division!!!So,!50!square!centimeters!divided!by!10!centimeters!is!5!centimeters.!
Repeat!for!A!=!32!square!m,!l!=!8!m!and!for!A!=!63!square!cm,!w!=!7!cm.!
!
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Problem&4:&&Given&the&area&of&a&rectangle,&find&all&possible&whole&number&combinations&of&the&length&and&
width,&and&then&calculate&the&perimeter.&
T:! If!a!rectangle!has!an!area!of!24!square!units,!what!whole!numbers!could!be!the!length!and!width!of!
the!rectangle?!!Discuss!with!your!partner.!
S:! The!length!is!3!units,!and!the!width!is!8!units.!!"!Yes,!but!the!length!could!also!be!4!units!and!the!
width!6!units.!!Or,!the!other!way!around:!!length!of!6!units!and!width!of!4!units.!!"!There!are!many!
combinations!of!length!and!width!to!make!a!rectangle!with!an!area!of!24!square!units.!
T:! With!your!partner,!draw!and!complete!a!table!similar!to!mine!until!you!have!found!all!possible!whole!
number!combinations!for!the!length!and!width.!
Circulate,!checking!to!see!that!students!are!using!the!length!times!width!formula!to!find!the!dimensions.!!
Complete!the!table!with!all!combinations!as!a!class.!
T:! Now,!sketch!each!rectangle,!and!solve!for!the!perimeter!using!the!perimeter!formula.!
Circulate,!checking!to!see!that!students!draw!rectangles!to!scale!and!solve!for!perimeter!using!the!formula.!!
Check!answers!as!a!class.!
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Lesson&1:&
Date:&
Investigate!and!use!the!formulas!for!area!and!perimeter!of!rectangles.!
1/7/17!
3.A.7&
Page 10
©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org!
!
This!work!is!licensed!under!a!!
Creative!Commons!AttributionENonCommercialEShareAlike!3.0!Unported!License.!!
!
!
Lesson&1& 4 3&
NYS&COMMON&CORE&MATHEMATICS&CURRICULUM&
Problem&Set&&(10&minutes)&
Students!should!do!their!personal!best!to!complete!the!
Problem!Set!within!the!allotted!10!minutes.!!Some!
problems!do!not!specify!a!method!for!solving.!!This!is!an!
intentional!reduction!of!scaffolding!that!invokes!MP.5,!
Use!Appropriate!Tools!Strategically.!!Students!should!solve!
these!problems!using!the!RDW!approach!used!for!
Application!Problems.!
For!some!classes,!it!may!be!appropriate!to!modify!the!
assignment!by!specifying!which!problems!students!should!
work!on!first.!!With!this!option,!let!the!purposeful!
sequencing!of!the!Problem!Set!guide!your!selections!so!
that!problems!continue!to!be!scaffolded.!!Balance!word!
problems!with!other!problem!types!to!ensure!a!range!of!
practice.!!Consider!assigning!incomplete!problems!for!
homework!or!at!another!time!during!the!day.!
Student&Debrief&&(10&minutes) &
Lesson&Objective:!!Investigate!and!use!the!formulas!for!
area!and!perimeter!of!rectangles.!
The!Student!Debrief!is!intended!to!invite!reflection!and!
active!processing!of!the!total!lesson!experience.!
Invite!students!to!review!their!solutions!for!the!Problem!
Set.!!They!should!check!work!by!comparing!answers!with!a!
partner!before!going!over!answers!as!a!class.!!Look!for!
misconceptions!or!misunderstandings!that!can!be!
addressed!in!the!Debrief.!!Guide!students!in!a!
conversation!to!debrief!the!Problem!Set!and!process!the!
lesson.!!!
You!may!choose!to!use!any!combination!of!the!questions!
below!to!lead!the!discussion.!
#
What!is!a!formula!for!solving!for!perimeter?!!
What!formula!is!most!efficient?!
#
Compare!the!units!used!to!measure!perimeter!
and!the!units!used!to!measure!area!(length!units!
and!square!units).!
#
What!was!challenging!about!solving!Problems!
6(a)!and!6(b)?!!How!did!the!process!of!solving!
Problems!4!and!5!help!you!to!figure!out!how!to!
solve!Problems!6(a)!and!6(b)?!
!
Lesson&1:&
Date:&
Investigate!and!use!the!formulas!for!area!and!perimeter!of!rectangles.!
1/7/17!
3.A.8&
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