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Bowering 1
Summative Math Assignment
Adam Bowering
March 4, 2016
1. Teaching Mathematics to Students withLearning Disai!ities " Annotated
Bi!iogra#h$%%%%%.Page 2
Instructional Design in Mathematics for Students with Learning
Disabilities………...2-
Math inter!entions for students with learning disabilities" m#ths and
realities…....2-$
Mathematics and learning
disabilities…………………………………………….........................$In!estigating elementar# mathematics curricula" focus on
students with learning
disabilities………………………………………………............................$-%
Preå teachers to teach mathematics to students with
learning disabilities…………………………………………………………………%-'
Mathematics instruction for elementar# students with learning
disabilities…………..'-(
2. A &ritica! 'eview o( )&TM Artic!es
*+ractions%%age -. 'eview o( Math
Mani#u!atives%%%%%%%%..Base-ten
Bloc)s…………………………………………………………………………………………
Pattern
Bloc)s…………………………………………………………………………………………..
Mone#……………………………………………………………………………………………
…………
*raction +iles…………………………………………………………………………………………….
+wo-Sided
,ounters…………………………………………………………………………………
4. Mathematics and
Techno!og$%%%%%%%%%.
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Smartboard lesson n Introduction to
*ractions………………………………………………
/ideo 0sing regrou&ing for addition and subtraction using base ten
bloc)s…….
Big boo) *all Into
,ounting………………………………………………………………………………
/. ther Areas o(
nterest%%%%%%%%%%%%%lementar# Mathematics course ournal
entries…………………………………………
Science obser!ation 3ournal gra&hs……………………………………………..
……………….
,ool 4ebsites and !ideo related to mathematics conce&ts………………………..
………………………………………………………………………………
1.+eaching Mathematics to ,hildren with
Learning Disabilities - nnotated Bibliogra&h#
Mathematics is regularl# seen as an area of struggle for man#
students. +his is es&eciall# true for students with learning disabilities.
Mathematics content can be o!erwhelming to these students since there are
man# strategies and s)ills the# must ac5uire in a relati!el# short amount of
time for their cogniti!e &rocesses. I belie!e it is u& to the educator to ta)e
note of these di6erences7 and ad3ust their instruction accordingl#.
+hroughout the following annotated bibliogra&h#7 I highlight some articles
that I ha!e found interesting that introduce a&&licable wa#s of teaching
mathematics content to children with learning disabilities.
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,arnine7 D. 188(. Instructional Design in Mathematics for Students with
Learning Disabilities. Journal of Learning Disabilities, 3092:7 1;-1$1.
+he author suggests that there are multi&le causes for low
achie!ement le!el for students with learning disabilities. cult# with memor#7 and strateg# ac5uisition. 4hen
conce&ts are introduced at a ra&id &ace that students are not
able to )ee& u& with7 these students often become o!erwhelmed
and disinterested in the material. Students with learning
disabilities bene?t more when the design is centered on big
ideas7 strategies7 e>cient use of time with mani&ulati!es7 clear
and e@&licit instruction7 and &ractice and re!iew. nother
strateg# for instruction is to use strand organiAation for lessons.
+he idea is to design lessons around strands to address di6erent
big ideas in %-1; minute &eriods. 4hen teachers a&&l# these
design &rinci&les7 it could ha!e a &ositi!e e6ect on achie!ement
for students with learning disabilities.
*leischner7 . .7 Manheimer7 M. . 188(. Math inter!entions for students
with learning disabilities" m#ths and realities. School Psychology
Review, 269:7 8(-$1.
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+wo t#&es of learning disabilities are discussed in this
&a&er. +he# include d#scalculia7 and !erbal learning disabilities7
which are referred to as Cgarden-!ariet# learning disabilities.
+he focus of this &a&er is to hel& educators understand e6ecti!e
instructional techni5ues that hel& students with disabilities.
Starting with children at the &recom&utation le!el7 methods for
whole number o&erations7 &lace !alue7 and fractional conce&ts
are discussed as well as algebra and &roblem sol!ing. n
e@am&le used for whole number com&utation is &ro!iding
students with wide-celled gra&h &a&er7 so that students who
ha!e di>cult# with &lace !alue can ha!e mar)ers that align
numerals.
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students with learning disabilities7 rather than a one-siAe ?ts all
class instruction.
Fear#7 D. 2;;$. Mathematics and learning disabilities. Journal of Learning
Disabilities, 3791:7 $-1%.
,hildren with Learning disabilities often ha!e trouble with
numbers7 counting7 and arithmetic. ,ounting &rinci&les and the
&rinci&les of &roblem sol!ing tended to be lac)ing in the children
used in this stud#. +he stud# also goes on to sa# that children
with learning disabilities continue to ha!e di>culties in retrie!ing
arithmetic facts from their long-term memor#7 which did not
im&ro!e o!er time. +he stud# found that children with LD had
interru&tions in attentional information &rocessing. +his resulted
in di>culties with information mani&ulation with math language.
Students with learning disabilities re5uire more time to &rocess
information7 which is 3ust as true for math content. Pro!iding
students with the time to &rocess new information allows these
students to ha!e a better understanding of conce&ts li)e
counting and basic arithmetic.
Gord7 ,.7 Eewton7 . . 2;1$. In!estigating elementar# mathematics
curricula" focus on students with learning disabilities. School Science
an !athe"atics, ##$9$:7 181-2;1.
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+he &ur&ose of this stud# was to e@amine curricula to see if
there was a wa# to &ro!ide more o&&ortunities for engagement
for students with LD. +he stud# focused on three mathematics
curricula that &ut a s&otlight on accessibilit# in regards to
challenges in wor)ing memor# for these students. Instruction
that was used in this stud# included the em&hasis of !isuals7
math con!ersations7 and de!elo&ing metacogniti!e s)ills. +he
?nding was that teachers must consider how their curriculum
outcomes &ro!ide e6ecti!e instruction for the storage and
organiAation of information for these students. +eachers must
also ta)e into account how the outcomes de!elo& students=
thin)ing &rocesses and their o!erall understanding of
mathematical conce&ts. +his s&eci?c stud# concluded that the
curriculum areas were e6ecti!e in teaching students with LD to
sol!e multi-ste& &roblems. Gowe!er7 su&&lemental materials
should be used in con3unction with curriculum to meet the needs
of students with LD.
Parmar7 H.7 ,awle#7 . 188(. Preå teachers to teach mathematics to
students with learning disabilities. Journal of Learning Disabilities, 3092:7
1-18(.
Se!eral standards are &ut forth in this stud# which reJect
the wa# in which teachers need to &re&are to teach children with
learning disabilities. Standard one is to model good mathematics
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teaching. +eachers need to ha!e )nowledge of the curriculum7
and the !ariet# of instructional techni5ues a!ailable to them.
+eachers shouldn=t 3ust focus on one area to strengthen
their teaching7 the# must focus on a well-rounded a&&roach
when teaching mathematics. collaborati!e a&&roach between
the teacher and s&ecial education teacher is suggested when
teaching children with LD. n e@&ectation of this a&&roach would
be enhanced 5ualit# of education7 allowing students to reach
higher le!els of &roblem sol!ing7 reasoning7 and connections with
the mathematics content.
Standard two is for students to ha!e the a&&ro&riate
)nowledge of mathematics. +e@tboo)s ha!e gotten longer7 and
therefore more com&le@. Brea)ing the information down into
?ner details will decrease the time for co!erage of the content.
+eachers need a balanced a&&roach to &ro!ide students with an
understanding of the meanings7 &rocesses7 and &rinci&les of
mathematics.
+he stud# stressed that acti!ities need to be &lanned
around real-life &roblem sol!ing and big ideas. Eew teachers
must constantl# )ee& in contact with other teachers and sta67
creating a collaborati!e a&&roach when focusing on a&&ro&riate
instruction. +his stud# also cited the E,+M as one of the
organiAations to use as a reference for teachers.
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Thornton, C., Langrall, C., & Jones, G. 1997. Mathematics instruction for elementary students
with learning disailities. Journal of Learning Disabilities, 30!"#, 1$"%1'.
+his stud# highlights the need for teachers to constantl# be
5uestioning their long standing beliefs when it comes to
teaching7 learning7 and using curriculum. ,ommon &ractice in
toda#=s classroom reJects a narrow focus on com&utation. +he
&a&er s&ea)s about how the E,+M &ro&osed ?!e goals to rethin)
math teaching and learning. +he# include instilling !alue of math
on students7 increasing con?dence in their abilit# to do math7
become mathematical &roblem-sol!ers7 to reason
mathematicall#7 and to communicate using math. +eachers
should do this b# decreasing their &encil-and-&a&er com&utation
and rote memoriAation of the content. Students should be gi!en
multi&le wa#s to re&resent the same answer. +his could be
through illustration or mani&ulati!e use.
In the end7 the article suggests that teachers need to ha!e
a balanced a&&roach when teaching mathematics7 engaging
students in meaningful tas)s7 accommodating for the di!erse
wa#s in which students learn7 and to allow students to discuss
and 3ustif# their &roblem-sol!ing strategies. +eachers need to
&ro!ide students time to engage in and share their own solutions
to meaningful &roblems.
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2. A &ritica! 'eview o( )&TM Artic!es "
+ractions
Artic!e 1 "
D=mbrasio7 B. Kastberg7 S.. 92;12:. Building understanding of Decimal
*ractions. %eaching &hilren !athe"atics' 198:7 %%8-%'%.
ntroduction
BeatriA D=mbrosio is a &rofessor at Miami 0ni!ersit# in
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+his article discusses an in!estigation loo)ing at decimal
understanding amongst &re-ser!ice teachers and students. +he
in!estigators &resented &re-ser!ice teachers and students with
decimal grids and were told to illustrate and e@&lain their
thin)ing when ordering ;.';'7 ;.;'''7 ;.'7 ;7 ;.'' and ;.;';.
Se!eral of the &re-ser!ice teachers struggled with using the grid
&a&er to re&resent the decimal numbers. +he# had ?!e students
com&lete this 5uestion and the# used their answers to draw their
conclusion. *our out of the ?!e students struggled li)e the &re-
ser!ice teachers. B# anal#Aing the students wor) it re!ealed to
them three distincti!e sources of di>cult# of understanding
decimal numerals and their re&resentations of 5uantities. +he
?rst source of di>cult# is that students didn=t understand the
relationshi& between the !arious subdi!isions of a whole. +he
second source of di>cult# man# students did not see the
additi!e nature of the decimal &laces. +he last7 and most
im&ortant7 di>cult# is that rel#ing on &rocedures for ordering
decimals can mas) a student=s true understanding of the relati!e
siAe of decimal numerals. D=mbrosio and Kastberg o6er
solutions for these three di>culties the# unco!ered. *or the ?rst
di>cult# teachers should focus on onl# one grid at a time and
instruct students to use that &articular grid to show di6erent
decimal 5uantities. +he second solution is for teachers to &lace a
larger em&hasis on additi!e nature of the &lace-!alue s#stem.
+he last strateg# is to &ro!ide follow u& tas)s for students that
continue to build the students understanding of the relationshi&s
between and among subdi!isions. +he article concludes with the
recommendation that &rocedural routines that mas) students
misunderstanding of how to re&resent decimals can be a!oided if
teachers begin with idea of what ma)es decimals di>cult for
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students to full# gras& and as the# teach to )ee& those
di>culties in mind.
resentation
ersona! 'es#onse
&onc!usion
Artic!e 2 *
Kent7 L.B.7 m&son7 S.B.7 Eielsen7 L. 92;1%:. +he richness of children=s
fraction strategies . %eaching chilren "athe"atics. 2292:7 %-8;.
ntroduction
Summar$
resentation
ersona! 'es#onse
&onc!usion
Artic!e "
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Mc,ormic)7 K.K. 92;1%:. Ma)ing *ractions Meaningful. %eaching chilren
"athe"atics. 229$:7 21-2.
ntroduction
Summar$
resentation
ersona! 'es#onse
&onc!usion
. 'eview o( Math Mani#u!atives
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(ase)#0 bloc*s
Image source" htt&"NNwww.amaAon.comNLearning-Hesources-Plastic-Base-
StarterNd&NB;;;*/BB<
Base ten bloc)s are !er# !ersatile mani&ulati!es that can be used to
re&resent man# abstract conce&ts in mathematics. +he# can be used to
illustrate addition7 subtraction7 multi&lication7 and di!ision &roblems. I would
use base ten in an# grade7 since the# ha!e such a range of use. +he# can be
used to introduce &lace !alue to lower grades7 and decimal &oints in u&&er
elementar#. During m# student internshi&7 I introduced base ten to m# grade
2 class. I started with the basics. 0nits were ?rst7 followed b# rods which I
associated with being 1; units. +his was a tough conce&t for students to
gras& at ?rst7 since the# couldn=t 5uite understand the !alue of each unit. I
would use base ten bloc)s for an# of the abo!e mentioned to&ics.
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Pattern (loc*s
Image Source" htt&"NNwww.mathed&age.orgNanglesN
s the &icture illustrates abo!e7 a set of &attern bloc)s consist of a blue
rhombus7 green triangle7 tan rhombus7 orange s5uare7 #ellow he@agon7 and
the red tra&eAoid. +hese mani&ulati!es can be used to meet a number of
curriculum outcomes. Students can use &attern bloc)s to e@&lore basic
geometric sha&es or &ut them together to build more intricate or com&le@
sha&es. +hese bloc)s allow students to see that larger sha&es are com&osed
of smaller ones7 which relates to fractions. +hese bloc)s would also be a
great tool for students to create and identif# &atterns. I would use them to
indicate conce&ts7 li)e teaching about the &attern core to lower lementar#
students. +hese mani&ulati!es also &ro!ide students with a wa# of
identif#ing and disco!ering 2D sha&es in a hands on wa#. Pattern bloc)s are
an educational tool that students ha!e fun using7 which ma)es their learning
more engaging. +he# are !er# !ersatile and are found in most schools.
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!oney as !ani+ulatives
Image source" htt&s"NNwww.rainbowresource.comN&rodlist.&h&O
sub3ectMathematicsN1;categor#Mone#Q-QMaterialsQR2'QMani&ulati!esN2%'8
In a t#&ical set of mone# mani&ulati!es7 #ou will ?nd &ennies7 nic)els7
dimes7 5uarters7 loonies7 toonies and bills. Mone# is a tangible mani&ulati!e
since it is something students will use in their e!er#da# li!es. +he
mathematic s)ills related to mone# are necessar# for all ages.
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dollar. +he fact that students use this mani&ulati!e in a real life content
strengthens its abilit# to connect students to the content.
raction %iles
Image source" htt&s"NNwww.enasco.comN&roductN+B1%11+
s seen in the &icture abo!e7 a full set of fraction tiles consists of one
whole7 two hal!es7 three thirds7 four 5uarters7 ?!e ?fths7 si@ si@ths7 eight
eighths7 ten tenths and twel!e twelfths. *raction tiles are an e@cellent wa#
for students to e@&lore and understand how &arts ma)e u& a whole. I would
use the sim&ler fractions at ?rst when I am introducing fractions7 and
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graduall# introduce fractions with a greater denominator as students become
more comfortable with this t#&e of mani&ulati!e. +his mani&ulati!e also
furthers students understanding of addition7 subtraction7 multi&lication and
di!ision of fractions. +hese mani&ulati!es gi!e students a concrete
re&resentation of an abstract conce&ts. Since a lot of students belie!e that
each number in a fraction is di6erent7 these &ro!ide clarit# on that issue b#
getting students to see the set of numbers as a whole. Being able to see the
&ro&ortions of these fractions ma)es it easier for students to com&are7 order7
and e@amine e5ui!alent fractions. Some fraction tiles also ha!e decimal
notations on the o&&osite side7 which is e@cellent for seeing the relationshi&
between decimals and fractions which is bene?cial in the u&&er &rimar#
grades. I chose this mani&ulati!e to re!iew because I feel as though it
&ro!ides a great !isual for students when the# are learning fractions. It=s
im&ortant that students see conce&ts and are able to e@&lore them in a
hands-on wa#. Students struggle with understanding fraction siAes. Man#
children might belie!e that 1N( is bigger than 1N sim&l# because the ( is
larger than the . +he mani&ulati!e &ro!ides students a wa# to see these
fractions for what the# trul# are.
%wo)Sie &ounters
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Image source" htt&"NNwww.amaAon.comNLearning-d!antage-(21;-Magnetic-
,ountersNd&NB;;10F;B
+wo sided counters are another !ersatile mani&ulati!e that can be used
in so man# di6erent wa#s. *irst o67 the# hel& students with counting7
di6erentiating between sets of ob3ects7 addition and subtraction7 or
multi&lication and di!ision. I used counters in m# &eer teaching module to
di6erentiate between di6erent sets of fractions using ten frames. I as)ed for
students to use one colour to re&resent N1;7 and another colour to re&resent
2N1;. +his allowed me to bridge into the addition of fractions as well7 since
the two sets could be combined. I also use them to teach about the multi&le
wa#s a number good be re&resented using addition. *or e@am&le7 if students
were loo)ing for the number C'7 I would as) students to dro& their '
counters on their des)s to see how man# di6erent wa#s this number could
be re&resented.
I reall# en3o#ed using counters. +he# &ro!ide students with so man#
eas# wa#s to re&resent abstract conce&ts. I would use these as a regular &art
of m# math teaching in m# classroom.
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4. Mathematics and Techno!og$S"artboar lesson - .n introuction to fractions
+he focus of the following lesson is how to introduce fractions at a
basic le!el. +his lesson is intended for a grade introduction. +he foundation
of fractional conce&ts is so im&ortant in math7 as these conce&ts e@tend into
other math areas. In this lesson7 I s&ea) about the a&&ro&riate !ocabular# to
use when tal)ing about fractions7 and the wa# in which students should be
associating fractions to real-life a&&lication.
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/sing !ani+ulatives for Regrou+ingBase-ten bloc)s are a fantastic wa# to illustrate addition and
subtraction &roblems using a techni5ue called regrou&ing. +his techni5ue
&ro!ides a sim&le and concrete wa# for students to ha!e a !isual
re&resentation of a &roblem. In the following !ideo7 I tal) about the
ad!antages of using this techni5ue7 and the ste&s to follow to com&lete this
techni5ue.
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all into &ounting - (ig boo*
+he aim of this big boo) was to &ro!ide students with multi&le
re&resentations of counting using familiar ob3ects. s the authors7 we wanted
to include an element of interacti!it# with the boo) so that students could
use a hands-on a&&roach when counting. Students will also be e@ercising
their literac# s)ills7 re5uiring them to listen to hear the &roblem that the#
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must sol!e on each &age. 4e also introduced a sim&le wa# to include
addition within some of the &roblems.
/. ther Areas o( nterest
Journal entries from my math methods course.
Discussion uestions
3amine $our own regions curricu!um document. 5hat are the
!earning e3#ectations (or numer (or chi!dren in indergarten
to grade 27 &om#are these e3#ectations with ideas #resented
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in this cha#ter. 5hat ideas are stressed7 Did an$thing sur#rise
$ou7
+he learning e@&ectations are similar to those in this cha&ter. ,hildren are
e@&ected to s)i& count b# 2s7 %7 and 1;s7 count on from a number7 count b#
ones from a gi!en number and count forwards and bac)wards. ,hildren are
also e@&ected to count using coins. +his last e@&ectation sur&rised me since
mone# is an abstract conce&t. Students must ?rst learn to associate the
corres&onding worth of each coin to the coins &resented in a math &roblem.
+his could ser!e to be &roblematic at ?rst since children at this stage ha!e
onl# learned to count ob3ects based on how man# ob3ects there are. Eow7
ob3ects ha!e di6erent 5uantities associated with them7 &enn# being one7
nic)el being ?!e7 and a dime being ten. ,hildren ha!e been learning using
singular mani&ulati!es to this &oint7 so adding this conce&t of coins being
worth more than others ma# be confusing to children in the beginning.
Gowe!er7 this is an im&ortant s)ill since children will e!entuall# be using
mone# in the real world.
8ouve noticed that a student $ou are woring with is counting
o9ects with an accurate se:uence o( numers words, ut is
not attaching a numer to each o9ect. There(ore, the
students ;na! count is inconsistent and inaccurate. 5hat
wou!d $ou #!an to he!# this student deve!o# a etter gras# o(
one*to*one corres#ondence7
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+here a cou&le of strategies I might use to hel& this child with one-to-one
corres&ondence. +he ?rst strateg# would be to ha!e a set of Jash cards with
the corres&onding number. s the student counts the ob3ect7 the Jash card
with the correct number could be located underneath that ob3ect so the
student could associate the number with the ob3ect. nother strateg# would
be to use a calculator or counter. s the student counts each ob3ect7 ha!e
them record the number on the calculator or counter so the# are associating
that number with the ob3ect in the se5uence.
3#!ore a we*ased #rogram (or dri!!ing asic (acts. 5hat
(eatures does the #rogram have that are good7 )ot so good7
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answer would be. +he teacher would use the answer on the website to see if
it would be correct or not.
5hat is the im#ortance o( menta! mathematics and
com#utationa! estimation (or dai!$ !iving7 5hat are the
advantages o( eing a!e to estimate in #u!ic situations and
settings7
Mental mathematics aids in the e!er# math conce&t that we use in a real life
conte@t. 4e use math to count mone# when we bu# groceries from the store7
or we use mental math to count our hours we=!e wor)ed during the wee).
4e use mental math while &la#ing s&orts to add u& scores to determine who
defeats who. +he ad!antage of being able to estimate in &ublic situation
shows #our com&etenc# with being able to &erform sim&le math &roblems
without the use of a calculator or other de!ice that &eo&le tend to de&end
u&on. +his is es&eciall# true with the use of cell &hones. Most of these de!ice
come e5ui&&ed with a read# to use calculator7 ma)ing them eas# and
con!enient7 which ham&ers our abilit# to use our mental math s)ills.
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Science Observation Journal
+he following images de&ict gra&hs that I created in a &ro3ect that we
com&leted as &art of m# lementar# science course. +he ?rst &hotogra&h
shows a line gra&h7 which com&ares two di6erent sets of tem&eratures o!er
a s&an of 1% da#s. I gathered the data from a thermometer outside of Brian
Mulrone# Gall on the S+0 cam&us7 as well as from theweathernetwor).com.
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In this second &hotogra&h7 I chose to use a bar gra&h to show the di6erent
5uantities and t#&es of garbage I found in the habitat we were in!estigating.
+his )ind of gra&h gi!es a good re&resentation of the t#&e of data I collected.
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*or this ?nal illustration7 we had to ma& the area we were stud#ing7 and
measure the total area using length width. I found this acti!it# eas#7 and
found that it would easil# be translated into a classroom acti!it#.
Cool websites and videos!
htt&"NNwww.a&lusmath.comN
+his website &ro!ides a ton of cool and interacti!e games to hel& students
with their math s)ills. +here are also wor)sheets7 digital Jashcards7 and
!ideosT
htt&"NNwww.coolmath-games.comN
http://www.aplusmath.com/http://www.coolmath-games.com/http://www.aplusmath.com/http://www.coolmath-games.com/
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8/20/2019 Math Summative Assignment.docx
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+his is another cool and fun interacti!e ?lled with information on !arious
to&ics in mathematics.
Uou+ube !ideo for decimals"