85-294

Implementing Interventions ti 85

As childr~n with disabilities solve problems, explicit strategy instruction can help guide
them m carrymg out tasks. First ask them to read and restate the problem, draw a picture or
make a model with_materials, develop a plan by linking this problem to previous problems,
,,Tite the p_roblem m a mathematical sentence, break the problem into smaller pieces, carry
out ope.raoons, and check answers using a calculator, hundreds c.hart or other appropriate
tools. Th~se ~elf-instructive prompts, or self-questions, structure the entire learning process
from begmrung to end. Unlike more inquiry-based instruction, the teacher models these
steps :md exp_lains components using tenninology that is easily understood by children who
struggle-children who did not discover them independentlv through initial Tier I or 2
activities. Yet, consistent with what we know about how all children learn, children are still
doing problem solving (not just skill development).

Concre~e models can support explicit strategy instruction. For example, a teacher
demonstratmg the properties of two-dimensional shapes, might say, "Watch me. Now
make a rectangle with these four straws that looks just like mine." In contrast, a teacher
with a more inquiry-oriented approach might say, "Using these different sized pieces of
straws, how can you show me a rectangle?" Although initially more structured, the use of
concrete models in this fashion will provide children with disabilities with greater access
to abstract concepts.

There are a number of possible advantages to using explicit strategy instruction for
children with disabilities. This approach helps you make more explicit for these children
the covert thinking strategies that others use in mathematical problem solving. Although
children with disabilities hear other children's thinking strategies in the A.ftei- phase of each
lesson, they frequently cannot keep up with the rapid pace of the sharing. \Vithout extra
time to reprocess the conversation, children with disabilities may not have access to these
strategies. More explicit approaches are also less dependent on the child's ability to draw ideas
from past experience or to operate in a self-directed manner.

There are some aspects of e:xl)licit strategy instruction that have distinct disadvantages for
children with special needs, particularly the times children must rely on memory-often one
of their weakest skills. There is also the concern that highly teacher-controlled approaches
promote prolonged student dependency on teacher assistance. This is of particular concern
for children with disabilities because many are described as passive learners .

Children learn what they have the opportunity to practice. Children who are never given
opportunities to engage in self-directed learning (based on the assumption that this is not
an area of strength) will be deprived of the opportunity to develop skills in this area. In fact,
the best explicit instrnction is scaffolded, meaning it moves from a highly sm1ctured, single-
strategy approach to multiple models, including examples, and nonexamples. It also includes
immediate error correction followed by the fading of prompts to help children move to inde-
pendence. To be effective, explicit instruction must include making mathematical relationships
explicit (so that children, rather than only learning how to do that day's mathematics, make
connections to other mathematical ideas). Because making c01mections is a major component
in how children learn, it must be central to learning strategies for children with disabilities.

Concrete, Semi-Concrete, Abstract (CSA)

The CSA (concrete, semi-concrete, abstract) intervention has been used in mathematics
education for a variety of topics for years (Dobbins, Gagnon, & Ulrich, 2014; Griffin, Jossi,
& van Garderen, 2012; Heddens, 1964; Hunter, Bush, & Karp, 2014). Based on Bruner
and Kenned 's stages of representation (1965), this model reflects concrete representa-
tions such a/manipulative materials that encourage learning through movement or action
(enactive stage) to semi-concrete representations of dr~w!ngs or_pictures (ic~nic stage) and
learning through abstract symbols (symbolic stage). Bmlt mto this approach 1s the return to

86 - CH hild wi'th Exceptionalities
.. APTER 6 Planning, Teaching, and Assessing C ren

.· ncrete representations as children _need or as children b .

vmotttn\iooi1.,ef.ssoaswuamudteba:m1esulclslhntoiimnron-In1ecJwogepccosdertr.nt1e.heptcsoshlet'vrnses1ee1mannt;a1oentbferaa,tdroenrte1eieI.ctspo.,xaorbnttltehehescsne1gcot·s1.nonOO1.ntuv0tnaitgne1,o-h.ehnrs.oserognoautcttsfJt.iO.ohocTtarnhourssinehnesissdcts.hp.eaeiaJoppsrn_stnptnrssrgdnotooein.ltaneadtcoagdohnrt,nhsn,.aI.teeutloyldm1m•_ytseheopxaearr_putseathsr:vtI,eehs1cmnmi•0.iatsmt•ulIan·yai.tss.ddiI.lcIay.taeae.hrsl1qiaAagycuttsieooadtuntchtia_.chhoe~peeirnleppedlsrctp.or,oiesaCtcnnhchpcSheiarslArerthdeehatgarealacelirItaeenrsnnOnJ0ttaontobnel~·dayenoees_cdl~laollsP\elnSlOIJad1·io·V~nnnered•o~g,tc,i

ItthJh·11e·..msi.rtaoopnwp,rn&oathcShIt'lrlokhizan1.seg.1m., U2e0t•s1ew4d1'.,tihMJaarh'tniigccuhl 'la'lM_relv·yielllmesr,oa&fcsoKumcecbnemns.esadt1fyoo,rn2c0wh1ii.2lth,drMeexni.plllweirci.ti&ht sdtKri.asaatfefbag1'rIy,1•t.21i.ne0ssItrI(ut)F1.ccluot11.r0aentse,,

Peer-Assisted Learning

Children with special needs also benefit from other children's modeling and support (M

treorle&oFf uacnhsa'p2p0re1n6)ti.cTehweobraksiincgnowti.itohn aismthoartecshkilildlreednplee.aerrn.obre"sfet xwhpheernt. "t1h·Aely.thaoreugphlacthede~ipMneteahrse..
assisted learning approach shares some of the charactensucs o t e exp 1c1t strategy instru

tion pmreoddeelte' rimt isinde-idstisnecqtubeneccaeu. sTehkenocw~1.lelddrgeen icsapnrebseenpatei.rde oclnwa.inth"oa_sId-neerecdhei.dld"rebna sis as opposecd-
to a or peers who

have more sophisticated understandmgs of a concept. At other times, tutors and tutee

reverse roles during the tasks. Having children with disabilities "teach" others is an impos can

part of the learning process, so giving children with special needs a chance to explain :~;

thinking to a peer or younger child is a valuable learning tool.

Think-Alouds

When you use a "think-aloud" as an instructional strategy you demonstrate the steps to

accomplish a task while verbalizing the thinking process and reasoning that accompanythe
actions. Remember, don't start with where your thinking is; assess and start where thechild's

thinking is. Let's look at an example. Consider a problem in which first graders are giventhe

task of determining which of two children is taller and by how much by having the students
10ng on the floor. Rather than merely demonstrating, for example, how to use connecting
cubes to measure the length of the children, the think-aloud strategy would involve talking

through the steps and identifying the reasons for each step while measuring each childand

then comparing the difference between the children's heights. As you place a tick markonthe

floor to indicate where the cubes should start, you might state, "I used this line to markoff
where Rebecca's foot ends. How should I use this line as I measure Rebecca? I know I have to
add on cubes, but how do I know when to stop adding cubes?" "How can I use both lengthsof
cubes to find out if Rebecca is taller than Emma and by how much?" "Do I need to countall
the cubes?" All of this dialogue occurs prior to counting the difference in the amount ofcubes.

Often teachers share alternatives about how else they could have carried out the iask.
When usm· g th'1s metacogm·tt·ve strategy, try to talk about and model possible approaches(and
the reasons behind these approaches) in an effort to make your invisible thinking processes

visible to children.

Although you will choose any of these strategies as needed for intenrentions, your go~i~

always to work toward high stude.nt responsibility for learning. Movement to hsitgaih.~esrtelepvsce\ol\s1u_torb·
understanding of cont~nt can be likened to moving up a hill. For some, fonnal

support along the way is n~ces~ary (explicit strategy instruction); for others, ramps W1th e:t:h up

agement at the top ofthe hill will work (peer-assisted rleeaprrneisnegn)t.aOtiothnesr(CchSiAldraepnprcoaan':1f1in)·dAallrPpl)el0ldPeJfe
the hill on thei.r own WI'th some ~1·dance from visual
can relate to the need to have different support dun·ng diffimnt u.mes of their lives 0

150 • CHAPTER 9 Developing Meanings for the Operations

ther1: arc three quJnn.n.e n<d1. 1rwhoehr.ce.,wI··Iram•n gin'-,i,t,1ih0a1l111o1hr(srthae,retrob1tmeai1Inota1g1m1"1ao,duadncethdaa"fnttgeer a 1c1h1.11a11(tghectPka
being ·adckd or joined), a 1110

thhe
erp lace ). In Table 9. 1(a ), thi , is illustrated by t e c ang h

smPorolovdvienildgethacehcpihlradonrbeglneemtuhnoeknJnotohiwennsSraotdor,yrI'\,.s'nr.•:ot\uncamn·p•n.r·rtno~I.)g,1Pnc·ampgh·erY.cwoohureg•rwear.n1t1r..z,e.ce.ycr..
an swe r using doubl es facts sI1c aIrea d.v know s· c\,ha,',o!nawtwe·.Iso1hrtekh, 'Orrwsetrvati.shdeoeSnc.,to,sau0rtinofJt3e!0o;1rccs•t1a~aI00t1n,1
th,

Separate /Take From Problems

Sep111-n1r problems are common I}' k'11"".n .as 1t1ke au·n·y o r rak.ef.rom p.rob lem s in wh·reh Pan
3thqeusatnartit•taymr.soPuhnyt srr.scath11eYw1Jheom· leg removed goerstta.akmenouanwra•v·w. :h"Je'ore.nacsei nthtahte lm·_oit11hPe rsoebplaermates, pthrOebrieenioi.f
or the lar
r·s the largest amount (the whoIe). Ch'1l 'lre,n '""'11 use the Se11arate Stor·y Acti vr·ty 1,age a1su1at
graphic organizer.

Part- Part- Whole Problems

Pan-parr-u·hole problems, al so known as p,u t ogether and tak e apart problems in the
Common Core State Standards (CCSSO , 20 l 0), 111vokc two parts ch ar,are conceptuallya,
mentalh· combined into one collection or whole , as Ill Tabl e 9.1 (c) . f hese probl ems
diffc re;t from change problems in that there is no action of phys icall y joinin g or separ::t
ing the quantities . In these situations, either the missing whole (total unknown), one of
the missing parts (one addend unknown) , or both parts (two add en ds unknown ) rnust he
found. T here is no meaningful distinction between the two pa rts in a part-part-whole
situarj on, so there is no need to have a different problem for ea ch part as th e unknown.
Th e third situation in which the whole or total is known and the two parts arc unknown
c r eate s opportunities to think about all the pos sible decompos it io n s of the whole. This
structure links directly to the idea that numbers arc embedded in o t her numbers (pan
of the "Jumber Core). For example, children can break apart 7 into 5 and 2, where each
of the addend s (or parts) is embedded in the 7 (whole). T hese kin ds of probl ems can
conce ptuall y combine diffe rent kinds of objects, such as 5 re d balls a nd 6 blu e balls or
3 cars and 4 trucks , into one collection. Combining different kin ds of objects into one
set can he challen g ing for y,rnng children to understand because they have to gcneraliz,
what th e different obj ect s have in common . Sec the Pa rt- part- whole S tot') Actil'it,
Page for the corresponding graphic organi zer.

Compare Problems

Compare problems invol ve the comparison uf two quantitie s. The t hird amount in the;e
problems docs nut actuall y exi st hut is the dif]i-n·11ce between t he two quantities (sec
'fable 9 .1(d)). Like part- part- whole problems, comparison situations do not t ypicallv in rohr
a physical action . The corresponding Compare Story Acti,ity Page can help children model
rhe situation . The unknown quantity in compare problems can be o ne of rhree quantities:
the smaller amount, the larger amount, ur the difference. For each of t hese situations, rwo
examples arc provided: one problem in which the difference is stated in te rms of"how man,
more?" and the other in t<:rms of "how many less:" :'\Jore that the language of "more'' a_nJ
"less" can present a challenge to many children as they attempt to interpret the relationships
between the quantities using these idea s.

You can find more examples of compare problems as well a~ the o ther problem t:'Pc'

cincthsesoC,o2m0m1o0n, Care Stall' Standards (sec 'lable ] in the CCSSI Math Standards GlossJrr:
p. 88).

J

242 • CHAPTER 12
8uiiding Stratfgies f o r ~ Computation

prem:irureh· abandon concrete approaches usm· _g marerials-f-riondud.ing- thei·r 6n ~ ,

e,.--er_som e cliildren rruy ne,.,"'<i encouragement ro n_10,·e :iwar m d trect m odeling_I-{ l~ -

some id eas to promote the fading of direct modehng:

I' • Record children's ,·er bal e.x:plan atio ns on the board .

tha t thev. and others can follow. Ill ~ .

Accepting direct modeling ;as a necessary developmen- Ask children who have just solved a problem With
taJ J>hase provides children who are not ready for more
"·materials if they can do a similar pro blem rnentl.l~'
efficient methods a way to explore the same problems Ask childre n to pmroakbeleamv_.-wT..i t1ht.tenconnucmreetericmraetceoriradls·of~•..-,_.
as d~srnates Who have progressed beyond this stage. they soh·ed the

have them tn- the same wntten m ethod on a new ·
• Prob!cr.i_

Invented Strategies

An im:ented strategy is any strategy, o~er than the standard algori th ~ , that does nae in, ~
the use of phnicaJ materials or counong by ones (Carpenter, Franke, Jacobs, Fennema
Empson, 1998). For first and second &rraders, the Common Core State Standards (CCsst
2010) describe these as "strategies based on place value, properues of operations, and/ ·
the relationship between additi on and subtra~on" (pp. 16, 19). More specifically, chi!~
are expected to "develop, discuss, and use efficient: accurate, and ge_neraliz_able methods to
compute sums and differences of whole numbers m base-ten notation, using their Uod .
standing of place value and the properties of operations" (p. I 7). At times, invented stracegier
become mental methods after ideas have been ell.-plored, used, discussed, and undersc~

For example, after some experience, children may be able to do 75+ 19 mentally (i5 .._20

is 95, less I is 94). For 648+ 25 7, children may need to write down intermediate steps
(i.e., add 2 to 648) to support their memory as they work through the problem. ( Dy that onr
yourself.) In the classroom, \\Tirten records of thinking are often encouraged as Strategies
develop because they are more easily shared and help children focu s on the reasoning use.d_
Distinctions among wrirten, partially wrircen, and mental computation are not important
especially in the development period.

A number of research studies have focu sed artention on how children handle computa-
tional siruations when they have been given options for multiple strategies (see, for example.
Kei ser, 20 IO; Lynch & Star, 2014; Rittle-Johnson, Star, & Durkin, 20 IO; Verschaffel, Greer.
& De Corre, 2007). "There is mounting e,;dence that children bo th in and out of school
can construct methods for adding and subtracting multi-digit numbers without explicit
instruction" (Carpenter et al., I998, p. 4). One of the best wa ys fo r children to grow their
repertoire is to ljsten to the strategies invented by classmates as they are shared, explored.
and tried out by others. However, children should not be pennirted to use any strategy
without understanding it.

Contrasts with Standard Algorithms

Consider the following significant differences between invented strategies and scandanl
algorithms.

I. In v en~ed rrrategi:s are number orienud rather than digit oriented. U sing the srnnd_!UO

,-al:;algonthm for 4) + 32 , children think of 4 + 3 instead of 40 and 30. Kamii, loognrlle

advoc~te for in~ented strategies, clainis that _standard algorithms " unteach" place

(Kamu & Donuruck, 1998). By contrast, an mvented strategy works with the co~P1- j,

numbers. For example, a~ invented strategy for 618-2 54 might begin .,.,;th 600-.00 ·
400. Another approach might begin with 254, and add 46 which is 300. Then 300 mart
to 600 and so on. In either case, the computation is number-oriented.

r 248 .,

CHAPTER 12 Building Strategies for Whole -Number Computation

f 'ig u re J 2 ,~ ,( n.x-, 1rJin~ l.h dJr1•11\ 1h1>t1,t: ht 1'111,:cv,,.., 1'll I 1H. 1,11·,rtJ ,ci l h.,11 hc1.. l,, .... l.mf,,Hc1wtheq 1.u I.:\'

r.
\\ ll ntcrh,-..f.,

(a) How much Is 86 and 47? (b) What Is 84 minus 68?
S : I know lhat 80 and 20 more Is 00.
5 1started at 84 . First. I jumped ba ck 4 to 98110 80
T: Where do the 80 and the 20 come from ?
'I - -- ------,,-8:C0s4;\f-
S : I splrt the 4 7 into 20 and 20 and 7 and !he 86 into
80 and 6. T: Why did you subtract 4 first ? Why not 8 ?

T: (illustrates the splitting \\ti th hnes) S: 11 was easier to think abou t 80 than 84 . 1will s
So then you added one of the 20s to B07

S : Yes, 80 and 20 is 100. Then I added the other !he other part of 8 until la ter. T hen I Jumped ba~~e
20 and got 120. 50
to gel to 20 .

T: (writes the equations on the board) 60 4

S : Then I added lhe 6 and lhe 7 and gol 13. 20 8C0 8\4

T : (writes this equation)

S : Then I added the 120 lo the 13 and gol 133. S: Then I jumped back 4.

T: Indicates with jo,nlng lines. n ~4 60 n4

/ 4,7-........ 80/ 86"-6 16 20 80
=20 20 7
100 + 20 = 120
80 + 20 100
T: Why 4?
6+7=1~ / S: That was how much I still had left over from 68

133

Standards for The empty number line (also known ,1s the open 1111111be1-/i11e) shown in Figure 12.6(b) is a
Mathematica Prac:-tice number line with no prewrittcn numbers or tick marks. Children can use it to suppon their u;e
of a sequential jump mmegy char is ,-cry cffccti,·c for thinking about addition and subrraction
D use appropriate siruations (Ca ldwell, Kobett. & K:irp. 201-!; Gm-emcijer & van Galen. 2003; \ 'crschaftd er JI.
2007). The empty number line is much more flexible than the usual number line l~c.1u<c it
tools strategically. works with any numbers, eliminates the confusion with tick marks and the spaces between them.
and ir can be jotted do,rn anywhere. Also, children arc less prone to making comput1tion.1! errors
when using it (Gravemeijer & va n Galen, 2003; Klein, Beishu izen, & lrefte rs. 2002; \ 'm chaffd
er al. , 2007).

Y<JU can introduce the empty number line by using it to model a child's thinking for che
class. Make sure ro emphasize that the number line, like a ruler, is a length marked off into
particular units-and in the case with the open numher line, children create their own unm
with jumps. Children have ro detennine the range of numbers to use o n the empty numb..·r
line based on the problems they are solving. Ther often use " liien<lly numbers" for the jumpi

or jump to benchmark numbers on rhe number line and then calculate the total of the jump,

(Barker, 2009). The jumps o n the number line can be recorded as chi ldren share or expl.un

each step of their solution. counting up or down from an initi ,1 1number. \ \'ith time and prJc·

tice, children find the empty number line to be an elfoctive tool to support and explain their

reasoning.
Bar diagrams can also be used to support chi ldren's thinking and help them exp_lJin

their ideas to others_ Bar diagrams work particularly well for contexts that fir a con1pawon
situation and a part-part-whole model. See Figure 12 .i for a sa mple of eKh-

254 "' CHAPTER 12 Building Strategies for Whole-Number Computation

& Ro bitzsch. 2012). For exom1,k, for -!; - 29. the idea is tc~ th -i½n,"l,:-, ("}iHi >oecwa um"icuocfh do I a,td t<,
29 tog-ct t o 45~.. :'\:oricc th,lt this str3 tL' ~7)- 1s not 3s et.t•ici.cnt tor <listan
th e
1u.c.".twc•cn t he r\HJ numbers. 4 ,· a1ll16. rv•sm· g1·01·11 ·..--,rb change 1.111/mrr.:._·11 probl.e m. s o r "1J.Ssm_'<.-p.r,,e.,
pro blem s (di, cu,scd in Chapter 9) will cncuungc tl1c use nt the th111 k-add1t1 o n strategy. 11.,,

is an example of each .

Sam had 46 baseball cards. He bought some more cards for his collection . Now he has
73 cards. How many cards did Sam buy' (join w ith change un known)

Standard s for Juanita has 73 crayons in all. Some are broken and some ar e not . 46 cr ayons are not
Mathematical Practice broken . How man y are broken ? (missing-part)

D Reason abstractly Fig ure l . 1O show~ invcn ccd -:tratL" gies that U'-C ad d ition t o soh-c ··s ubtrau ion~
sto r y problem s. .-\ s you c::rn '-Ct' . u, in g ten s is :il;;o an import an t pJ.rt of th e-;c ,trJtt--
and quantitatively. gi cs. Simpl~· askl ng for the difference between two numbers m ar also prompt the,l
strat egic s.

f 1~ure 12 . l O
fhrt.'c d:ikn:nt ur.-t. ntl:d ...tratc~1e-.. fnr -.ulit rJ c..tto n b\ "' 1h111I-. ..1JJ11 1o n

Add Tens to Get Close , Then Ones 46 + 20 = 66 Add Ones to Make a Ten, Then Tens and Ones

73 -46 66 + 4 == 70 73-46 46 + 4 - , 50
70 + 3 73 50 + 20 - , 70
46 a nd 20 is 66 . (30 more is too much) 20 + 4 + 3 = 27 46 and 4 1s 50. 50 and 20 is 70 and 3 70 + 3 - > 73
The n 4 more 1s 70 and 3 1s 73 . That s 20 more 1s 73. The 4 and 3 15 7 and 20 1s 27 . 4 + 20 + 3 = 27
and 7 or 27.

27 4 10 10 3

2~ \ 46 50 60 70 7 3
~ 0403

46 66 70 73

Add Tens to Overshoot, Then Come Back

73- 46 46 + 30 - , 76 - 3 _ , 73 S1m1larly. 46 and 4 15 SO 46 + 4 - , 50
46 and 30 ,s 76 T hat"s 3 too 30 - 3 = 2 7 50 and 23 1s 73 50 + 2 3 _, 7 3
23 and 4 is 27 23 + 4 = 2 7
much. so it"s 27 .

4 23
46 50
73

°Development of InvcnIed Strategies fur Addition 11tl Subtraction "' 255

How Far to My Number? CCSS M· 2 NBT.B !>. 2 NBT B 9

1111 Chttdren work In pelrs with o single sot of littl e Ton-Frame, One child usos tho

- cards to meke a number less then 60 Whilo the other child wrltoa o number greater
than 50 on o piece of paper, as shown In Figure 12.11), For children with dlsnbilltlcs or chi l-

dren who need a challenge, you m ay choose to suggest th e slzo of th o soconcl numbor
(e.g.. less than 100; less than 600), The chttdron work togo th or to find out how mu ct,

more must be added to the number shown with th o ton-fromo to got to th o Iorgo, number
written on paper and to write o corresponding eq uation. Onco on answer Is determined, thoy

should demonstrate how their answer combined with th o smaller number motchos th e larger
number. Over time, you can fade th e use of the littl e-ten frames ,

You can repeat " I low Far to My :'-lu 111l,er' " with three-di git I isutc• I.' 11 p1nhll 111 ,
numbns with or without t he use of models, As :i lways, it i, i111por- 1·.,,r tl1111k ,1\.ld1t1011111 'tih\. 'I l,1\\ ! .11
1.1nt to Juve children share t heir strategics durin~ a disn, ,sion ,

Take-Away Subtraction How In, lrom 48 lo 73? IJ

1:ikc-away subtraction is co nsid c r:tbl v 111 ore dil'l, cult to do lll cn-
t,111
I . J·lowc ve r rake -aw av strat egies arc co111111j.,11 , prol',a·lil v(
.• ·
,
hcc~ use manv textbooks c111ph asi2.c takc-:1\\':l)' .'" I ,c inc:1111'.·1f o

subtracti on (~ve n rhou g h th ere arc oth er 111 ca111 ng,). "'''."' d1tlcr-
cnr strategi.es are shown in Fi gure 12.1 2 fu r th e fo ll " wlll g st m)'
rrohlern .

There_ were 73 children on the playgroun d · Th e 46 second-graders came inside first.
How many children were st ill on th e playgrourid?

lat\1ti1d\ori'adthnhiteneTcnblgahhant,eshJefbeom-l>erts•aeeuac8ntnkb·h3-t.poc-ri'aSade2uscsc'1·1tme·et.shmdc!.Ja1e-nctIt'huphII1>r:m•colcidrgdbonri..ctncct·,Irountebil·,dlsawY,.ur,·1,',·tnImaImtekI·,tcu~eihn•ncylnt•ct•grn-.dotpllstmolle)eln,e(.n'os:.otclfv·h1to.cr!·on(t11)O1811,1J'u1·3rt"I1sc.e,_,ek,cro,.,2·~IlIflO119'wt.1.i,.,,hc1inh1,ne'rue,tIlon)b1,I'1oIt1ra1rthuc1lai1.mifr'1ccl.c'th,u·vrcceolIrttahuisidpfl,'r·i"aluen1Iecr'1t.i)ad1o',afc-lf•ricItIirIl1t5tel:h0io1ts;,aHastdt1·1an.1k1o1ks"k•.11lw·1ct1l1·lIIc»1-eg·ca1r1Hdw.:s1c'c1.l\au1'hs'yIc••sti'1,)l1Irs·d1,rr.1u·:ha131r1Irr0cJ1ks·nl1lr.,t.,t1Inolr1>dm.1cctnt.1)r--.I',
ca use rr:-,uu added I to 29 t,>makc r.t- )10' the n :r·rI.wIuauoyu"tJiO1o1·'uirthh( l:1tc n.:1 l, ( b)Iack. l1' 11c \I :i, 1.e1.
usmecbtehrasta"n·htIc··J1e1, t ht·hec,·rntak:,ect B.1, c-T c 11 \ tl)' .11·11 I 1I1·.11 11·I1)·
3p2ro)bsole·mthawt itt:hhe t he n t l')' too k I too 111 :1 11)' :lll' !.
v can
"·

lhcr need to ajd I back .

256 • CHAPTER 12 Building Strategies for Whole-Number Computation

- - - - __Hg u re 12. 12 ---------

!·our diffrrc-111 1m·c111ed '>tr.1 tcgin frn 1akt"-aw.a:.r_<i uhrractHin.

Take Tens from the Tens , Then Subtract Ones Take Away Tens, Then Ones

73-46 73 - 46 73 - 40 - > 33 _3

70 minus 40 is 30. 73 minus 40 is 33. Th en take away 6: 30 - 3 -> 27
Take away 6 more
is 24. 70 - 40 - > 30 - 6 A;-\,,/:: ==:-----,,_,3 makes 30 and 3 more is 27. -4 0 n-
2730 33
Now add in the 24 + 3-, 27
3 ones - · 27.

- 6 -40 Take Extra Tens, Then Add Back

oh~ 73-46 73 - 5 0 -> 23 + 4

2 4 2 7 3 o fo 73 take away so is 23. That's 4 too many.

23 and 4 is 27. SO 27

Or 70 - 40 = 30 tggM;-qC 73
(73 - 3 = 70)
70 m inus 4 0 is 3 0. t can 30 - 3 = 27 - -2~3---=2"-7,,- - - - -
lake those 3 away, but I
need to take away 3 more Add to the Whole If Necessary
from th e 30 to make 2 7. 73-46

- 40 Give 3 to 73 to make 76 . 76 take away +3
7 6 _7436-_4,63-0,)

2 7 3 0 o 73 46 is 30. Now give 3 back - 27.

46 - 3 - , 27

A~
27 30 7 3 76

Stop and Reflect 500 ')_.- I

Try computing 82 - 57. Use both take-away and "think addition • methods. Can you use all of
the strategies in Figures 12.1Oand 12.12?

Remcml,cr the "equal Jdditions" or "same difforence" str:lteb'Y(1-lumphreys & Parker, 201 5)
that was described earlier in this chapter? There .1r~ some children who use this strategy as an
inn~ntcd strategy wi th take-away subtr,1ction. For example, for 32 - I 7, children might think that
I 7 is 3 awa_,· from 20, so if ther add .l to 17 to equal 20, they need to add 3 to 32 to get 35 {to
maintain the difference between the two nwnbcrs). :\low the problem is 35 -20 or 15. Encourage
c hildren to u"e an c 1npty nwnhcr line to sec why 32 - 17 :ind 35 - 20 ha ve equivaJcnt answe rs.

Keep in mind that for man y subtrJction problems, especially those with three digits,'
" think additio n" approach is significrntly easier thJn a take-away approach. For children w_ho
could benefit from a "think addition" approach but arc not using it, yo u 1nay want to re"1sir
some sim ple mi ssi ng- part acti1·itics to encourage that type of th inking .

Extensions and Challenges

Each of the cxJmples in the preceding sections im·oh·ed smns less than J00 and ail involred
crossing 11 ren: that is. if done with J standard algorithm, they required regroupi ng or rracling.
As yo u choose problems whe n planning instruction, you sho uld consider whether the prob·
!ems require c rossing a ten (hundred, or thousand), the size of the nu mbers in the problems,
and the potential fo r doing- t he probl ems menta ll y.

Standard Algori thms for Addition and Subtraction 'ii 259

Iigurr 1US
• '-~ 111\Jt'C't1 n-c1ird ,m J"-l!'lCrl' Jl 1' "h'Plill\ <l, ' on t Ii c1 r plai..r \'Jlu e Ill.II'

Ill 8 Tens Ones How much is in the ones column? (14)

Ill/ /D Will you need to make a trade? (yes )

36 How many tens will you make? (1)
+4 8 How many ones will be left? (4)

Ill Tens Ones Goodl Make th e trade now.
Ill/
/ D Let's stop now and record exactly what
Trade for a ten . l we have done.You had 14 ones. and you
3 6 made 1 ten and 4. Write a "1" in the tens
( /// 8 column to show th e ten you put there
+4 4 and a "4" in the answe r space ol th e
/ //II Ones ones column for the 4 ones left.
Tens
Group tens. D Look at the tens column on your mat.
/ You have 1 ten on top, 3 from the 36,
( I I I , 1\ I 6 and 4 more from the 48. See how you r
I// I 1 I j J j 3 8 paper shows the same thing?
////I 1- 1-l•\•1 \ 4
+4 Now add all th e tens together. Write how
many tens that is in lhe answe r space tor
8 the tens column.

I I I \3\

Standard Algorithm for Subtraction H gure 12.16

The general approach to developing the subtraction ,1lgorithm is the same as for adJitinn . 1 hl· p .1111-1 1 '-ll!Jl', :l!'\,1 11:lt h (.I ll hl
\\'hen th e procedure is completely un derstood with models, a do-and-rcrn rd approac h lhlJ fin11 1 !dt t11 l l}!h l ,I\ \\di,\\
connects it with a written form. 11n111 1t:~ht t•, idL

Begin with Models Only 358
Starr by having children treat the subu,1ction problem as a "rn kc-aw ,1)'" situation. \ \C, th
this meaning of subtracti on, they model onl y rhe Virsr (rop) nt11nucr in a subtracti on + 276
problem (minuend) on the top half uf rheir placc -\'a luc 111,1ts. 1:or the amount robe sul,-
tracred, have chilJrcn write each di git on a snull piece of paper and place th ese pieces 500
near the bottom of their mats in rJ1c respective co lumn s, as in Fib'11rC 12.1 7. ·1u avoid 120
errors, suggest making all necessary tr,1dcs iirsr. Th,1t way, rJ1e full amount on the paper 14
slip can be taken off at once. Also cxp"1 in to chi ldren that they arc to beg·in workin g with
634
thc ones column fir st, as they did with additio n.

260 • CHAPTER 12 Building Strategies fo r Whole•Number Computation

t ,i;un• 12 , ~ L/// '- i- •\•• lld0os
"•' J,~ 11 ,ul,r1 ••~ 111 m \\ 1th 1hr ,11 11 d_11 d ,1\~11111h,11 ,,ml 11H •drl , not me.11or
L-------i& - W~lch Oollt
-:4151 comoo11
Pu1lho
Not onough onos to tako off 7 lollovers
Trada a ton for 1Oonos
toge111or.
And now I can take elf 2 Ions

Now there ore 15 ones
I can take 7 off easily.

Virtual base•ten blocks for addition and subtracti on are availabl e at the National Library of V1nual
Manipulatives. These two similar applets use base -ten blocks on a place•value chart. You can form any

problem yo uwish up to four digits. The subtraction model shows the bottom number in red instead of blue

When the top blocks are dragged onto the red blocks, they disappear.Although you can begin in any column.
the model fo rces a regrouping strategy as well as a take-away model for subtraction.These digital versions

of physical base ten materials are use ful for reinforcing the standard algorithm.

Ant icipate Difficulties with Zeros

Proli lcm ':i that incl ude numbers in wh ich zeros ;1rc in volved tend to cause .., peL i.d dirtiu 1ltil',.
' J'hc comm on errors thar emerge when "regroup ing :.tcross 1.cro·• ;H e hc~t :1ddn.: ,::, t.'d .H thl'
model ing stage . Fo r cxo 1n plc, in 403- 138, child re n must m ake a d o ubl e trad e: tr.ul ing 3
h un dreds pi ece fo r IO tens and the n " ten, piece fur IO ones. Afte r c h ild re n ha\'c experience
with making tr.i des using basl.!-tcn matt' rials, use the ti ,llowing nctl vity hcfo n.: givln ~childrt'n
any hints abo ut how they mi g ht deal wirh regro uping :i cross a zcru.

Activity 12.8 CCSS- M 2. NBT.B.7 2.NBTB9

Tricky Trading

Pose a problem to the class th at requires regroupln& across zero, such as 103 - 78. Children work In pairs using Base-Ten
M aterials (Blackllne Master 32) and Place-Valuo Mats (Blackllne Master 17). Once they have Identified an answer, they now
check their answer using an Invented strategy. II they did not get the same answer with the base-ten materi als and their invented
strategy, encourage th em to try to determine why. Follow up with a discussion that starts with children sharing their Ideas.

rigu1e12 . I 9 Computational Estimation Strategies • 265

nHflJ \,,111n.1tton in ,11,.I J11ion (b) Front-end addition, numbers not In columns

(•) Front-<>nd addition, column form

Adjust~ Adjust
about 150 (ten place)
9+5+2 = 16
$160 to 170 more

Gi:E'

to place va lu e and o nl y co ns id er digits Fig u re- 12. 20 Rounding u,111~ pbu.: \ ,1!u1..·.
in the largest pl ace, especia l! )· when the What is the approximate total of these Items?
numbers vary in the nurnber of digits.
The front-e nd strategy can b e casv to u se
because it does not require rounding or
ch:rnf_ring nun1 be r s.

Rounding Methods

Children are not requi red to use rounding in
the Co71nno11 Core Stt1te St(//lilm-ds until third
grade, but rounding ca n help children build
number sense an d use invented strategi cs
efficiently to find exact sums and differences.
If rour children are ready, you ca n use tl1e
following activity to introd uce th em to the conce pt o f rou nding. \ Vhen several numbers arc
to be added. it is usually a good idea to round t hem to the sa me place va lue. Keep a rnnnin g
sum as ~-ou rou nd each nurn ber. figu re J2.20 shows an exa mpl e of ro unding.

Activity 12.11 CCSS-M : 2.NBT.8 .5

Round Up?

Create a number line on the floor with painter's tape, a rope, or cash register tape. Use sticky notes or pieces of paper lo label

the benchmark numbers of tens (10. 20, 30, • - • ), hundreds (l00, 2 00. 3 oo. · · · ) or whatever range of numbers you are

considering as a class. Have the numbers far apart so that 3 to 4 children can stand facing forward between each number.

Then distribute numbers for children to round. For example , give a child the number 53. The child stands on the line where

53 should be. then rounds to the nearest ten (50). Talk about the case of a 5 In the ones position (or In other halfway positions

for larger numbers) as a convention-we all agree that we round up when we are midway between numbers.

For addition and subtroction probl ems invo lving on ly two numbers, one strategy is to
round onJv one o f the two numbers. For examp le, yo u con row1d on ly the subtracted num ber
such as 6i4-83 becomes 624- 100, resul ting in 524. You ca n stop here, or you can adjust .
Adjusting might go like th is: Beca use rou subtracted a bigger number, the_result must be too
1111all-by about 20, t he difference berween I00 and 80. Ad)lJSt to abo ut ,40.

276 "' CHAPTER 13 Promoting Algebraic Reasoning

and their commonalitics, chi.ldren wi•111.,e 3 1)1e· ro general ize a nd recogni ze whcn Parr1cu1a,
stra tSe,egtiscsofcopnrobbeleumsesfaurlc. good ways ,,or c·h'1I'jren .to loo k for an.d <Iescn·he patte rns aero,, the
problems, panern s th at Iiu1.1<1 an unders· t·anrlin•g /or the opcrat1 o n and related algurith 1111:

]+8 5+8 8+9 8+i 6+8

Once children ha\'e solved sets of related problen:s, Y_0." can _start w it h gen e ral <]Ucsti,Jns:
\Vhat do yot1 notice? I-low are the problems alike, Ditfc~entc But yot1. m ay need to fo(us
their attention o n what 1·ot1 want them to gene ralize by ,isk:ing m ore specific question s: Jlo,
can using 10 help you s~lve each problem: \\lhy: In this set of proble m s, such a discussi"n
can help ch il dren appreciate how the relationship hct:wccn 8 ~nd IO ca n make_ th e problcrn,
ea sier to solve. Children can also develop a better understandmg o f additton situations-for
example, how if you add an ,1mount to one addend (to make I0) yo u must subtract that sarne
amount from the other addend. This strategy of Making IO can be applied to anr of the basic
addition facts for 8 (and i and 9 for that m,mer), hclping ch ildren with what is often some
of the more cha llenging facts to learn .

Meaningful Use of Symbols

Algebraic notation in the fon11 of equations and YJriables arc powerful tools for representing
mathematica l ideas and the primarr grades are not too early to begi n using them. >lute rhJ1
algebraic notation needs to be built from a firmly established understand ing o f the arithmetic
symbols+ ,- . X , -e-, and= . Many misconceptions often develop in th ese earls- vcJ rs thJt
unfortunately stay with children into later rears. So before we talk further abo ut algebraic think-
ing and generali1.1tion, in particular, we first emphasize how preK-2 teachers can help children
develop a strong understanding of the equal sign and ,·ariables through meani ngful insm1ction.

The Meaning of the Equal Sign

T he equa l sign is one of the most important symbols in mathemati cs, especia l!)· in clcmcntm
arithmetic and in early algebra . The Co111111011 Corl' State Standards expli ci tl y address dewlup-
ing an understanding of the equal sign as ea rl y as the first grade . Sometimes children bc~~n
to write equations using the equal sign as early as kinderga rten , and they co ntinue to work
with equations in every subsequent grnde . Unfornm,1te ly. resc,m.:h dating fro m i 9i5 to the
present indicate s that chi ldren ha,·e a ,·cry poor understandin g of the equal sign (Kicr.111.
2007; .\lc_'\/eil. 2014; R:\_'JJ) Mathematics Stud)' Panel, 200J). Jn additio n, the equal sig11 is
rarely represented in l;.S . tcxthooks in ways that facilitate children's understanding ofrhc
equivalence relationship (Powell , 2012)- an understanding that is critica l to undcrstsnding
algebra (Knuth, Aliba li , Mc'-/eil, \Veinherg, & Stephens, 20 I I; .Vlc:--Jcil c t al. , 2006).

Stop and Reflect

In the following equation, what number do you think belongs in the box?
8+4= 0 +5

How do you think children in the early grades or even in middle school typlcally answer this
question?

Meaningful Use of Symbols ti 271

,,•t;rh0r1incnnd1cmI,tn·h1llo~nonIsb.n.\orl.re.ocex~.sr6(p.sr·l.u'o''p):dn\ ul."syk'·<t,c'nOsnn~nIowre,c.eL1ocr11eefoovtrr1ihrc,2ee&ctashti.nxC'nndtnauhrm1p1"b7'l0erJ·n'ne(,rpt1rIee-c1,(rror,7cswe)1(9no1.l·19Itu1'I9dtot)o1f,1cfchcih1.bl4diol5rxd)er·npe•uIn.gtheie·.nt1 Do not connect multiple expressions using equal
. \\.hc1c du su.ch 1111scunceptioiis co111c from? /\. large signs unless they are equal. For example, when add ·
=,nn1onrv of cqu nt.1 ons .that child·ren e.,nco unter .,n cl crncn- Ing 6 and 6 and then adding 3 more to that sum, do
1,m·· school look •hkc.thi s: 5+7 - = 0r 4 X 3 -_ . 1\s n not record those computations as (J l 6 IZ+.l - I , .
sec ""S s1·gn1·fyi·ng-;;a-n- d thc Doing this incorrectly reinforces the idea that the equal
consequ.e,n, ce, ch1ldrcn come to sign means "and the answer is" rather than indicating
answer 1s rather than as :1s1~11bol thar 1· ,1c1,·c:1 tcs cqw·v:1 1cnn·
,. • :· • equivalence.

(C:ll'pc_nt~r, Franke'. & Lev,, 200 3; McNeil & Alibali, 200 5;

1\k;\ic1I, _014; i\1\01111:1 & /\rnbrosc, 2006).
. Suhrle s. hifts, in how )'OU 'n1·>1Jr''"·1c·I1 tc,ac J11·11g computan·on ca n all cvr·nte tlm· s1·gm'fi,ca nt

,rnswnccpt,on. l•or exa mpl e, '' si111plc change such as writin g l,asic fac ts as 7 = 2+ 5 ca n

W, :rn se children to st0P and qu estion why thi s is tl1 e s:irne as 2+5 = 7. Also, rath er than

:1lways asking children solve a prnl,lcm like I9+ 23, ask them to find an equi valent cxprcs-
s1t1n a11<I use both to wnte :i n equ ation (Blanton, 2008). For 19+ 23, chil dren might wri te

l'i+ 2.l = 20+ 22 · ~ct1 v1ty 13.3 is a w:iy to work on equi va lent expressions while supporting

1hc dcvclop111cnt ot the Ma ki ng IO strategy for lea rning basic fa cts.

Activity 13.3 CCSS-M: 1 .0A.C.6 ; 1 .0A.D.7: 2.0A.B.2

Ten and Then Some

i W In this activity (based on Fosnot & Jacobs, 2010), each pair of children will need eight note cards labeled with the

- equations 10 + 1 to 10 +8 (see the Equation Cards). They lay the cards out on their desks face up.
Each pair will also need a deck of playing cards with all the face cards. aces, and tens removed . Each child draws one play-

Ing card from the deck. Together the partners decide which note card ts equivalent to the sum of their playing cards and they
place the playing cards behind the Identified note card (I.e.. If they drew 8 and 5, they place these cards behind the 10 + 3 note
card). If the sum of the playing cards ts less than 10, they slide the cards back In the deck In random places . Have children
write the two expressions as an equation (e.g., 5 + 8 ; 10 + 3) to reinforce the Idea that these quantities are equal. Children

can also play this game Independently.
As an alternative to using note cards, you can prepare a game board whose spaces are all the expressions 10 + 1 to 10 +8.

The children then place their two playing cards below th e appropriate space on the board .
For struggling learners. have ten-frames or arithmetic racks available so th ey can model the sums. tnttlally have them

move counters to make 10 to reinforce the Making 10 stra tegy. Eventually have them only describe how they would move
the counters to make 10 and then fade the use of manlpulatlves altogether.

Jniti all v, children will struggle to understa nd how amounts th at look different ca n

Jllu all y be· equi val ent. I11 oth er wo rd s, they wonder wh )' the numl,crs do not have_rn Standards for
be identical. Consequentl y, at fi rst children will ne ed to add th e numhers on eac h side
Mathematical Practice
tu rcrifv for th emse lves th at they are th e sam e qu ant1 ry. As the y come to undersrnnd
l-0111 pcn·sat1·011 (e.g. , I10w part of one num b.er. ca n be mo,·cd to ano.ther .numb.er), th e)· D Look for and make

use of structure.

begin to understand how equi val ent q11ann t1es do not have to he 1dc nt1 cal. Once yo u
knoWlIlJ-1C1rcncanu sec.o mpe-n.s·•,,ti on, ch·all c1wc t hem find th e cqu1 va lcnt ex press ion
c to

without co mputi ng fir st.

278 1 CHAPTER 13 Promoting Algebraic Reasoning

The next acti ,·iry conn.n ues to chall enge chi ldren to work on equiva lent exp,c,. 11,n

.· Activity 13.4 ccss-M 1.0A C.6: 1 OA D 7 2 0A B,

Different but the Same

ChaIIenge children to find different ways to express a particular number. say, 6· Encourage children to create both addition
and subtraction expressions. Give a few examples, such as 3+3 or 12- 6. Ask questions such as, "How many ways can You

make 6 using at least one number treater than 107" "Wh at patterns do you notice?" Have children write equa tions using
their expressions (e.g., 12 - 6 = 0+ 6; 15- 9 = 2 +4). For equations that use the same operation, challenge the Children to
explain why the quantities on each side are equivalent without doing the compu tation. Adding a context (e.g., Legos, trading
cards) can support children 's reasoning.

Why is it so important that children in grades preK- 2 correc~y under_,tand the equl)
sign? First, it is impona nt for children to understand and symbolize rela aonship, in r,ur
number system and the equal sign is a principal method of represen a ng these relacionsfup,

For example, S+; = 8+2+ J sho11·s th e basic fact Strategy ofMaking I0. L'nderstancLng dk,
eq ual sign as equiva lence supports children as they explore the bcha1ior ofoperations beat1;e

they can focu s on the relationship between the num bers and not on doi ng the rnmput.rar,n.
A second reason, although removed from the prcK-2 classroom, is that when older stuckll!s
hal'e a poor understanding of the equal sign. they typically have difficuln·\I orking 11ith alg,.
braic expressions (Knuth , 1Vi bali, .\'lc.\Teil, \\'einbcrg, & Stephens, 20 JI ; Knuth, Stephen.,

,\1c:\'ei l, & Alibal i, 2006). For exam ple. the eq uation fr+ 24 = H regwres srudenn to see

both sides as eq uivalent. \Vhen srudents interpret the equal sign as "do a
they think it is impossible to "du" the left -hand side of the cquation.1!011e1er. if both 11de-

are understood to be equivalent expressions, students can reason that fr must be 1-! Jes,
than 54 or fr = JO. Therefore, x must equal 6. Helping preK- 2 children develop a,ohd

understanding of the equal sign ca n in turn help them avoid such difficu lties m later grad,s.

Conceptualizing the Equal Sign as a Balance

Children's un dcrsranding of the idea of equi ralence can and must be de,·eloped through
meaningful contexts and concrete methods. Chi ldren 's literature offers some 11·ars to intro-
duce eq uivalence in tern1s of a bala nce. For exa mple, Equal Shmeq11el (Kroll, 2005 ) isastol\

about a mouse and her frie nds who want to plar rug-of-war. lo do so, they must detenrune

how to make both sides equ al so that the ga me is fa ir. In the end, ther use a teeter-toner to
bala nce the weight of the friends. This focu, on equal sides and bal~nce make this a grtJt
book for focu sing on the mconing of the equal sign as a balance.

T he next two activities use kinesthetic approaches, tactile objects. and ,isualiz.irions to
reinforce the "bala nci ng" not inn of the c11ual sign (ideas based on .\lann, 200-I).

., Activity 13.5 CCSS-M 1 .0 A D.7 2 ,BT A 4

Seesaw Comparisons

&) Ask children to raise their arms to look like a seesaw. Explain that you have softballs, all weighing the same. and ter>
nls balls, all weighing the same. but the softballs are heavier than the tennis balls. (Have some softballs and tennis
fl
balls available In case children, especially Ells. are not famlllar with these Items.) Ask children to Imagine that you
have placed a softball In each of their left hands. Ask them whar would happen to their seesaw (children should bend

to the left side). Ask children to imagine that you place another softball In their right hands (children should /eve/

Meaningful Use of Symbols fl 279

off). Next, with the softballs Still th ask them to Imagine a tennis ball added to the left. finally. say you are adding another
tennis ball In the left hand ag ere,
important activity tor children aIInth. Then ask the m tO Imagine a tennis ball moving over to the right hand. This Is a particularly

After acting out several w disabilities who may be challenged with the abstract Idea of balancing values of expressions.
seesaw exam PI85• 5hare , "If
you have a balanced seese ask thlldren to share their observations. For example, one child may

away the same object f w and add something to one side, It will tilt to that side : Another child may explain. "If you take
rom bolh sides of th8 seesaw, the seesaw will still be balanced:

·· Activity 13.6 CCSS-M : K.MD.A.2: 1 .0 A.D.7: 2. NBT.A.4

What Do You Know about the Shapes?

Show children a balance scale with objects on both sides. Here Is an example:

lliii tilooo

Tell children that the cubes weigh the same and the balls weigh the same. Then ask the
children , "What do you know about how the weights of the balls and the cubes comparer
Have children explain their thinking first with a partner and then with the class.

Figure 13. I shows a series of other examples for the pan balance. "1-vo or more balances for a Standards for
single problem provide different information about the shapes or variables. Problems of this type Math ematica l Practice
can be adjusted in difficulty fo r children in grades K- 2. \-Vhen no numbers arc given, as in the
top three examples in Figure I 3.1 , children look for combinations of numbers for the shapes that fJ Reason abstractly
make the pans balance. T he different sha pes represent different amo,mts (variables) ancl so would
have ,lifferent values. There arc often different paths to finding a solution so discussion ofsolution and quantitatively.

strategies is a must. To create your own pan balance problems, start by assigning va lues to two

or three shapes. Place shapes in groups and add the values. Ile sure yo ur problems can he solved'

Stop and Reflect

How would you solve the last problem in Figure 13.1? Can you solve it in two ways?

Th ere are several excellent online pan balance explorations to continue work on equivalence:

, NCTM's Illuminations "Pan BalancE>-Shapes" and '"Pan BalancE>-Numbers." where children enter

what they believe to be equivalent expressions.
• PBS Cyberchase Poddle We,·gh-In, where shapes are balanced with numbers between 1 and 4.
• Agame Monkey Math BaIance. w here children select numbers for each side of a balance to make th e

two sides balance. The level of difficulty can be adapted.

28 0 ., CHAPTfR 1,> prun,ot1no Alyeh,air Hed,onIfl \l I\

l lj( lllt' J I I fJ(I
l \ 11111•1,,, ,t 111,11 t il l 11 , 111 1~ 1• 111hil .111< 1"

i\ CJfl

W h1c t\ Ml,aµn wnlglu1 !1 10 rl1 0HI ? I .1'pln lll

fJ

Wtilc h shupo WOl(l h S II Hl mor. t? r Mµlnin
Whic h shnpo WOIQllS tho Ion s!? I xrt, 111 1

g (J

Wh nt wlll lm lr1 nc o l wo sphoro s? Cx plw n

@El

/ 12

How much doos onch shnpe wolgh? Explain

Aft e r c hild rl' ll h:1vl' l''l: pt.: rie n1.·l.·tl pan hal:1 11 n : 1:1 -. k11 in,·, ,lvi ng ._ hapl".;, they <::111 c ,pl Pn· pJn
hala ncl'" u-; ing 1u 1ml,l' r, and tl u: n ,·,iri ,d ,k·....

Activity 13.7 CC !\ lNHTc,4 2NBTA4 2 NBTB7

Tilt or Balance?

li'1iil Draw or proj ect a sim ple two-pan balance. In each pan, wru e a numeri c expression and ask w hich pnn w ill go down

- or w hether the two will balance (see Figure 13.2(0)). Selec t expres sions that are appropriate fo r the grade level ot
children (e .g ., sum s withi n 1 00 for gra de 1 and within 1000 grod e 21. Challenge children to writ e expressions for each side

of th e scale 10 make It balance and th en to wrlt o tho corr esponding equation to Illustr ate t he meaning of =. Note th ~t when

Meaningful Use of Symbols fl/ 281

(itOnhceSluosdpceaploeesxetaidlmtstp.oleedistohsienurgcahth•8ge1retchaoetmetrph-tlh,~uanan• dorfo•ulersths·tbh8aInan• cseysmfobrowl(h>ichorc<hil)d1r1enusaerde' and u It Is balanced , an equal slen (• ) 11 used·
asked to analyze the relationships on bOth sides

thRn have them write express! u18fotlon) to determine whether the pan tilts or balances. For children with disabilities. rather

Identify the ones that will maokns r each slde 01 the scale , share a small collectlon of cards with expressions and have them

e the scale balance.

l·i)lure 11 .2

' t ' ,ilnh,t•1q11u~:.d( 'i\1J1)tl· l'Il l
')1\ 1101,1\ J \ 111 l'l!UJ t1011 ') ,ind
Ill o1r1,1
• ' · h ! \\ n •p,rn h.il,11 1t t lwlp-. dl'vdnp 1111·
111 <.111111~ 1)f · , · • ,ind

(a) 3+4
12 - 7

12- 7@3+4

4+4+4 6+6

4+4+4(§)6+6
455 + 197

455 + 197 Q 460 + 192

Can you determine whether the expressions
balance without doing the addition?

(b) 63 - 27
64 - 26

64-26 Q 63 - 27

Can you determine whether the expressions
balance without doing the subtraction?

1 I+ 3 8 +2

Can you determine how to make the expressions
balance without doing the addition?

True/Fa lse and Open Sentences
Carpenter and coUc,1gues (20U.l) suggest that a guorl starting point for helping children make
sense ofthe equal sign is to explore cquatiom as bei ng either tru(' or fa lse. Clnri~1ng the menning
ofthe equal sign is jusronc of the outcomes of thi srypc ofexploration, as seen in the next acri,ity.

282 "' CHAPTER13

Promoting Algebraic Reasoning

Activity 13.8 CCSS-M 1 .0 A.B 3 1.0 A D 7 l NBT B 4 2 l¼BT BS

True or False Equations

li! lntrOduce true/ false sentences or equations with simple examples to explain what is m eant by a tr ue equation and
=a false equation (e .g., 2 + 3 5 Is a true equation : 4 + 3 = 2 Is a false equation). Then put s everal simple equations
on the board, some true and some false. Here are som e e,amples:
5+2 7 4+1 =6

8 = 10- 1 7 12 - 5

To begin with, keep the computations sim ple. Ask c hildren to talk with th eir partners and decide w hich of the equations are
true (and why) and which are fa lse (and why).

After this Initial exploration , hasc children explore equations that are in a less familiar form :

3 + 7 = 7+ 3 10 - 3 ° 11 ·- 4 9 + 6 0 + 14

8 = 8 15 + 7 + 3 •• 16 +10

listen to the types o f reason s children use to Justify their answers and plan additional eq uat ions accord ingly, Ells and
children with dlsabllltles will benefit fro m first e,pla lnlng (or showing) t he ir reasoning to a pa rtner as a low-risk speaking
opportunity and then sharing with t he w hole group. Children can use "Pan Balance-Numbers· o n NCTM 's Illum inations website
to explore and/ or verify equivalence .

Children will t,vica lly agree :1hout cquJti o ns when there i, :1 11 expression on one - hk

and J si ngle number o n the other. although in iriJlh- cqu:1rio ns ,uch :1 , i = 12 - 5 ma) !!tner-

atc di scussion. for equatio ns with no opcr'Jt:ion (e.g.. 8 = 8). the discu"ion nu, be li,el:

C hild ren o ften Lcl ien , there m ust he Jn opcrnion o n o ne side and an "answer" on the othtr
Reinforce that the equa l si!-'11 m ean s "is the same a, " bi· u~ing th :1t langu age a, ) Ou rl'Jd the
S)m l x , I.

.\ftcr ch ildren h:1 ,·e expe rienced true/ fa lse sentences, introd uce o pen sen tcnce,-thJt
is, eq uJ tiuns with a hox or bl ank (or varia hl cJ tu he replaced by J n um be r. T,, de,clop Jn
understan d ing of o pen ,e nrcncc~/alsu t:a lled missing-n lue cquatjon, ). enn,urJgc children to
look at the num ber sente nce hulist icalk ,llld dc, crihe in "ords what th <' equa tion rcpre,onr,

Activity 13.9 CCSS M 1 .0A D 7 1 OA D 8 2 QA .A 1

What's Missing?

Write several open sentences on the board . Ask children to figure out wh at number Is m issing
and how they know. Notice that most of the equations are set up so t hat children do not have
to do the computation to figure out what Is missing. Encourage them to first tr y to fig ure out
t he missing number without doing the computation . Challenge children to fi nd more than one
way to figure out the missing number. Here are some e, amples:

5+2 0 4 + 0 - 6 4 .,. 5 = 0 - 1

6- 7- 4 +5 5+8 15 + 27 = + 28

r
Meaningful Use of Symbols • 283

Relational Thinking

•aCd5honilosdotremedneptuhrsem'uvagw·ll.y"uSsthIeyic,notkhneadyb, ocmuhtialyt.dhrheeanevqmeuaaan1shoi.gpner1•a11tz.oonnae/0_vfu.t'hWr,emc ewaanyi.sn(gSttehpaht etnhse et al., 2013) . F'1rst,
equal si.gn means
ign symbolizes a relation between y O1da relatwnal--computational view in which the equal
sthe only way to dcterminc itfhetheeqtuwaa~ns~w1.deerss to two ca Icu Iatt.ons, I)ut they see computat1·0n as Standards for
,elationa/-str11ct11ral view of are e_qual or not. Finally, children can develop a Mathematical Practice

cional conception of the equal sign sih~l (we will refer to this as relational). With the rela- D Reason abstractly
da'iccutcucarcI'InaYtcceoxImpdlpauunstaem.tsigonnutshmeicoanrm·cPo.IrauecnIma.tts1.g·.onasnh18·psinbtehtewebeonx s1'dcs
of thCe oenqus.iadlesrigtwn oradt1h. set.rtntchtalyn the two open and quantitatively.
sentence 7+0 = 6+9. for the

8E,xspolathneatiboonx Iis: B8.ecause 6+9 is 15' 1 need to figure out 7 plus what equals 15 . It is

sEhxopulladnabteioonn2e:lSesesvethnanis o<,'n, es.om1•tormeutsiitalnJet8h.e 6 on the other side. That means that the box
Th.e fitrhst chi.ld. .computes the sum on one s•i·de Of the equati·on and then uses the sum to deter-
mine e m.JSSthn'g pbart on the other side (reJati·ona I--<0111pmatio11al view). The second child
uses a re1attons 1p etween the expressi·ons o.n c1'ther si·de of the equal sign. This child docs
not need tIo c.ompIute the value.s on. each si'de, (re Iat1·onal-J1111ct11ra/ view). vVhen numbers are
large, a re attona -structural Y!Cw 1s much more efficient and useful.

Stop and Reflect

If thechildren used the same reasoning they used to solve 7+0 = 6+9, how would they each

solve the following open sentence?

534+175 = 174+0

The first child would likely do the computation and Children sharing their reasoning promotes relational
might have difficulty finding the correct addend because of thinking and can help other children improve their
the size of the numbers. The second child would reason that noticing skills and analysis of relationships in problems.
174 is one less than I75, so the number in the box must be

one more than 534.

D Formative Assessment Note

As children work on these types of tasks, you can interview them one on one (though you may not get to
everyone). Listen for whether they use relational-structural thinking. If they do not, ask, ·can you find the
answer without actually doing any computation?" This questioning helps nudge children toward relational
thinking and helps you decide the next instructional steps.

Children need many and ongoing opportunities to explore problems that encourage
relational thinking (Stephens ct al ., 2013). Explore increasingly complex true/false and open
sentences with your class, perhaps as daily problems, wann-ups, or at centers/stations. Posing
problems with larger numbers that make computation difficult (not impossible) can prompt
children to try a relational-strucrural approach. Herc are some examples to consider.

284 • CHAPTER 13 Promoting Algebraic Reasoning

TRUE/FALSE :

-_- - - -- - - -- - - - - -66~74~~- 3~6~9~=~6~6~4--=~3~79=-~3~7+~54'.'._.'.'.=:_3~8~+~5::3:__3::.7:_6::_-__3_29_= _76_- 2_9 _________

OPEN SENTENCES:

- - - - - - - - - - - -73 +56 = 71+0 __6=8::_+__5:._8_ =_5_7_+_6_9_+_ _ _ __ _ _ _ __
126 - 37 =

Stop and Reflect

Before readi•ng on, try us.ing re1a11-onaI-structural thinking to reason about these true-false and
open sentences.

U sin g such equ ation s ca n pro mpt chil dren to look at eq uatio ns in th eir entirety

rather than just jumping ri g ht into a series of comput atw ns, an impo rtant as pect of al ge-
braic thinlcing (Blanto n, Levi, C rites, & Do ug hertv, 20 I I ) . Mo lm a and A mbrose (2006)

found that aslcin g children to write the ir own o pen sentences wa s parti cul arly effecti,·e
in h elping ch em so lidify their und ersta ndin g of th e eq ua l sign. T his is th e focus on the
next acti vity.

-4 Activity 13 .10 CCSS-M: 1 .0A .D.7 : 1 .0A.0 .8 ; 2.0A.A.1

Make a Statement!

Ask children to make up their own true/ false and open sentences that they can use to challenge their classmates.

a . ' To support student thinking, provide dice with numerals on them. They can turn the d ice to differe nt faces to try

different possibilities.

Each child should write three equations with at least one true and at least one false sentence. For children who need add;.
tlon al structure, In particular children with disabilities, provid e th e Make a Statement Recording Sheet. Children can trade

their equations with other children to find each others false statem ents. interesting equations can be the focus of a foilow-<JP,
whole-class discussion.

. \Vhe'.1 children write th eir own true/fa lse Sentences, thev o ften are intri ,rned with the
,dea of .usmg large numbe_rs and lo t s omfonvuemcbheilrdsriennthtoew,·rasrednrtee·lnacteios.nSalu-pstpronrctruthronctl•rtheiftn·okrirnsg· .as
these kmds of pro bl ems tend to help

Meaningful Use of Symbols -, 285

The Meaning of Variables

ps,qt1..e1.t,ul0a1r•,1,·r1rciriuncan'h.tbnoh·ei.golnrc.~i,elsmn.r.scltacuoerorcmcrs·rphpftrq_ioce1,ruearstxatre•sn1dmmp1.cte•i.e1artc1int1.•si.he.fotimscic_otntahnrhn.nnaeua•vtodt.mcswivatnbu·'1laedrytsy.rhei•sst·en•u(UtTaCts1t11Cih1.,1Ioi1·rcSO.nqey,rSu·s·mtc.c•ui.,naMonnuramnest.bictkada·c1cnn)cI'Ueo,udcwpascethsrndcdai.lhtsqdtouou.1r1osea1irlnngne(trgBpoi.atlrlidv-.ateeacenssren)ti6oan,atbnJthnrlnuced.u1sttnkniarwoeqlu.hsut,efe·e2ranac0rlhpctIIuhrhIvcet)ea.syruncVs·nenaasnkr1ur1nc1nigtaoc.oi.gbsouwelnneasnss-st
hfourilndiuml oeapnl~inogr feovrebnomtht.mitc va lues (qua ntities that vary). Children need ex periences t hat

variables Used as Unknown Values Standards for
Mathematical Practice
In the open scntenc1: ex plorations, the O and the __ arc precursors of a va riable used to
D Reason abstractly
represent a umquc, unknown valu e. You ca n also use a letter, such as an 11, instead of a box
or a bl ank in your open sentences, to sta nd for the missing number. and quantitatively.

Consider the fo ll owing open sente nce:

0+0+7 = 0+ 17

This equation could have also been writte n as 11 +11 + 7 = 11 + 17. When rhc same symbol or

letter is used in multiple places in an expression or equa tion, the conve ntion is th at it stands
for the same number eve ry place it occurs. (Point this our to childre n.) In th.is exa mple, the

Oor II must be 10.

Ma ny sto ry problems describe a situation in volvi ng a specifi c un known. Herc is an
example where the change is unlmown:

Rebekah had 5 apples in her basket. She picked some more after lunch. Then she had 13.
How many apples did she pick after lunch?

Although chil d ren ca n so lve t hi s prob lem with out using alge bra , they ca n represe nt

•u'1c equ ati·on as ,· + _ - 13 o r eve n •as 5+,, = 13 as a wa y to bcgm to lea rn about

variab les .
\,vhen wn·o·ng eq uati·ons ,r,0 r sto•n,, JJroblcms, rliffcrent eq uations may occur. For examp le,

consider thi s story problem:

1-f·Gabbi·e h-as -12 -card-s a-nd K. arI. has 5 cards• write an equation for how many more cards
Gabb ie has than Karl.

sseT·,,t·'oh.qahiOounneutaeci•cIltJdiaeatosth.ntbtlaeGe1·tr,eSa·.abeoi·bbqo11mbltuse'1Ite1aertuu'seco.··ocqhItnw1u'l.-.,lal'r)clit.ncra-it·eaosnenrnddba,i..nsned.u5d".hingnojoFhtuwatetscratwpdt.si·hf1rkry.ir.eentct·e/·.,etshoeadi'rImlb.a7da_erscce1q'••ir·nsue,"Lc'.Iatuaaon.tsr•1.·osIs0'oi·sn·onls'vn•sehhcw•iv·arphirstc<ehIonspnncprondholtciu.Jbl1bcsdleroeir,srtmenoshn""mmbatetrau·oheytmehcrwoaeeorrqtlri.ehpurteeeactrchat,•ittre+o.odmCnOsw,shaurtir.in=luctdedrttc1hhea2rnene-. Standards for
Mathematical Practice
1tnry.
DModel with

mathematics.

286 • CHAPTER 13 Promoting Algebraic Reasoning

\,\'jth a conre.~l, tchheill.dorlleonwcm.ang aecvtei"nvt te)x. p(uldonrept th rec ,•a r i a b l c ~. each o ne standing or in
1mknown value as in ed fro VI a,'d3 7004 )
m ,
I •·

Activity 13.11 ccss-M 1 OA A 1 1 OA C.6 2 OA,l\ 1

Toys, Toys, Toys

Chlldren are challented to 1111ure out the coll of three toys. given th e following three facts:

1. C . • • 53

,..,2. e. t!, - S4

3, _, • ss

Ask child ren to look at each fact and meke observations that ca n help th em figure ou t
the cost of each toy. For example. they may notice that the soccer ball costs $1 more than
the teddy bear (I.e.. compare the first two equations). Help chlldren writ e this observa-
tion In the same format as the other statements . Continue until these discoveries lead
to flndln& the cost of each toy. Encourage children to use manlpulatlves to represent and
explore the problem .

Stop and Reflecr ~,oo q)

Work on the problem In Activity 13.11 before reading further. Using the observation that
the soccer ball cost S1 more than the bear, that means that the third fact can be thought

of as a bear+ S1 + t>eer = $5. So, how much Is a teddy bear? Notice the work done in

building the concept of the equal sign is now applied to understanding ;1nd solving for
variab les.

Variables Used as Quantities That Vary

The s hi fr fro m the va riable reprcsc ntill!a( a <pccifi c quanti ty to 3 ,·ariablc reprc,cnring mulupk
possibiliti es c3 n be diffi cull fo r childre n and i, nor as explici t in tlw nirrirnlum J, it ,houhl
be. Children n eed e.xpe riences "'ith v:iriahlcs that vu rv car1 1, in the ele 111 ,,nrJr\ currietJunt
For exa m p le , second g raders c,rn hcgin tn d cscrihe pat~crns ~, ing , ariJhle,, J, ,~hen ,k<cnh-

ing ho w m uch many leg, fo r :rn r number uf dog s: L = I) x 4 (th e numhcr nf le!{, i, th<'

number o f dogs times four).

'.\fake sure to emphasize th at d tc variable <tn nds fo r the ll1tmbcr o/ hcc.JU, C ,·hildren on

J, ,,co nfuse the va rnble w1th a lahd (lllantnn ct al. , 101 I; Russell , Sc h ifrer, & Ba, tablc . 201 ll-

Fo r example. in so lving t he problem , There r1r,• I3 dogy: one/ ems flt 11.,1, kmm1. H,r.,· 111nm
and bozl' 111a11_y t fl ts an•ill the kennel?, yo u may find a va rien • of so luti o ns tlia r use t he t'ollv" 1" ~

n o t1ti o n :

6D +7 C 5D +8C 10O +.!C

H e rc c,hi.ldre n are usi.ng letters as la. be ls, not va ri ables• D does n ot represent a ,,u,,,brr, but,
rJtl1cr ,t ts s ho rthand tor dog. \,\' he n 1mroducing the concept o f variable mu nceJ to be owan:

of this to::ndenc·y to use lette rs as label s. '-

287

Structure in the Number System: Properties •

- - - -I,.c,;irsSictlhiti1hrlh<eltrpcsrhneifv<tIi.seoviunesIohspotowtrhvai.srpitsrhoemcbolcenrm"dic.mpbreoaIn>l.mcmgsoafrvca rpi·rac1,>s1.cc,ns t(c1,l11acmanoanl,so20o0p8e)n. up an exa .. Ictro's
r, . ' ut remow the result· As mple,

.

Rebekah had 5 apples in her b;;ket · She pi.cked som - - - - - -.
descri be the total number of apples Rebe . e more after lunch. How might you

-- _ kah had 111 her basket after lunch?

\\i th the tota l re moved , the gua l be c o m e s· wn.n.ng th c ex p ressi·o n : 5+11 . · t-·h · - a Iso I,c
il lustrated on a number line: is
c.111

5?

modAifsieadnforothmerthexeafmirpstle: , consider the fr,ll owi ng' tas•ks.. ·1·1· k a I,o ut huw the sccon,I usk ·
1111 •
IS

NUMBER TASK:

Sandra has 10 pennies . George has 4 more pennies than Sandra . How many penn ies does
George have?

ALGEBRA TASK.

Sandra has some pennies . George has 4 more pennies th an Sandra . How many pennies
does George have?

,'soticc in the modified ve rsion there is no way to do a computation because you arc not given

specific values . C hildren can li st possi ble ways in a rnhlc (sec Fi gure I.l.3 ) and c,·cntu ,111)·

represent the answer as Gcorw = S11111/m +4, or more briefly, G = S+4.

Another o•t JJJortunity to use va riabl es as qu anti ti es that vary is l·igurc I J. J
when children make conj ectures about the nu111hcr S)'Stcm (e .g ., p, .,,1lil l' W,l\' \ .ir1.1hJ4,., , l ,111 \JI\" Ill ,,, Ill\.'

when r ou add ze ro to an y number, you get th at nu111hcr ha ck).

These statements arc the properties of our nu111hcr systc lll an d arc Number of Number of

tn,c for all numbe rs, m aking them appro priate places to use vari - Pennies for Sandra Pennies for George

ables that vary. The n ext section looks at ways to help childre n rec - 6
ognize, understa nd , and describe these im portant general iza tions

about o ur number sys te m . 48

Structure in the Number System: 10 14
Properties 23 27

The importance o f properties ca nnot he ovcrstJtcd .. la hie 1.1 .I providcs a li st of the ones

altlhpl1.apgtlyhcitnhgdJ'ledths.rceenn·pb1•r1eo1 tl1c pn•ma ry gra des. rnust kno.w and Ch,eC,.S1h.Sl.c-/\t1o.· utshee,, acn,Imd p1h1a1ds.hu.Je1.ss. how c.htldn: n
the t_.he nn 11st11'1; an d
property• Importantly' 111 tor exa111 pl~, 111 g ra, e ~. one standard state,,
pert ies (not identifying them);

"&plain \1·hy .t t· . anJ s.u b~·i cti on stratc11ics work, usin g pbcc va lue and the properties

:H l mon u, ""

of operatio ns" (CCSSO, 20 I0, P· 19).

Patterns and functions "' 293

TheY must decide If It Is posslbi e and If so' Share an example of how to do It Finally they are to prepare a Justification or ill us-
tratlon to describe why It does
or doesn't w .
For children with disabilities ork (In general).
, Provide linking b k' Invite
°early finishers to take on the oth cu es so they can concretely model numbers to support their thin mg.
er problem or t k hlld en
for other patterns or generallzat1 Write their Justifications using variables. In the follow-up discussion, as c r

ons they notice about odd and even numbers.

Cthl,'isssMhroaaopkpmienn(gR., utsessetliln, gS,cahnifdterp,r&ovBinagstacbolncj.'e2c0tu1r1e)s· Tshho·tsunIdexbt eaccotimv ietyasuroggueti.sntse ac a•v1· ty i·n yo ur
one way to make

4 Actiy_ity 13.16 CCSS- M: 1.0A.B.3; 2.NBT.8.9

Convince Me Conjectures

To begin, offer children a conj ec1ure lo test (see Conjecture Cards for K-2 for Ideas).
For example, "If you add one to one addend and take one away from the other

i i addend, the answer will be the same .• Ask children to (1) test the conjecture and
(2) prove It Is true for any number. Point out the difference between testing and
proving. Te sllng Is seeing If the conjecture Is true for specific examples. Proving is
providing a convincing argument (visual or explanation) that the conjecture will work for all
numbers. Then, Invite children to create (In words) their own conjecture that they believe
Is always true. Then they must prepare a visual or explanation to convince others that It
Is always true.

All children, particularly Ells, may struggle with correct and precise terms. When
needed , ' revolce " their Ideas using appropriate phrases to help them learn to communicate

mathematically, but be careful to not make this the focus-the focus should be on the
Ideas presented . Children with disabilities will benefit from the presentation and discussion

of counterexamples.

Using and applying the properties is central to mathematica l proficiency-it is nor only
emphasized in the CCSS-M content, but al so in the Mathematical Practi ces (CCSSO, 20 10).
An explicit focus on seeki ng general izations and looking for structure is also important in
supporting the range of learners in the classroom from those who struggle to those who excel
(Schifter, Russell , & Bastable, 2009). Doing so requires planning- deciding what problems
to pose and what questions to ask to help children think about genera li zed ideas-across the

mathematica l strands, not just during the "algebra" unit.

Patterns and Functions

Patterns are found in all areas of mathematics. Learning to look for, describe, and extend

patterns are important processes in thinki~g algebraical!::, T,~o of,,rhe eight mathen:atica l

practices (CCSSO, 20 10) actua lly begin with the phrase look for, 1mply111g that children
who are mathematically proficient pay attention to patterns as they do mathemati cs. Func-

tional th' ki b • • K-7 when chil dren make obscrvanons hke, "Each person we
grnogup egms 111 pre 2- fe.e t" lanton et al 20 11 )
add to thien mea ns we add more (!3 ·• ·

294 • CHAPTER 13 Promoting Algebraic Reasoning

Standards for ttnIt·u·oooocrvt.luocsdmIo,ncckehsb,gnmi•fulro1dta·rbrttdehsmh·ent3ersgt,•ud·,ppcoaarnarnepet1Kdtraecnte-rotanxe2tn5trc,ndcanxsrerdepex1.pm·lupiensa_rgtao.etttltsntio.rsnautcgr1sIt1ceo1eega.cnr:nupn1ntI,I<11,•,..1·mtottItco·1n•ht1t'e).eI"·n.fldn.aC1·s.•1.,cI11·i1gt1•r1,s.•.-r1ittodn.hh,nwi1eea·ctmittrn1.(e.'•0'gr.cocnnmhkpsst.\.ml·acpd•tollrrt.onocehpnrveCni.t.dpsushheuu.ehom.oaus.\pk.tvtl1dpne.1.lo~ltbre5otteuhSft1to.aete1ntnn1l1ot.et=11h-odl<.-iceno>cks,·ce·g'ff,csoo1IQ•ar1.3r1111<cit.phslrhoiruiso0ile0kd0dr1JIu<ranelrrtene,g\,,
Mathemat,cal Pract1Ce and express rcgu laritr in all nrn thc111,t0 cal s1n1.1t1ons. .

D look for and make

use of structure.

Ill look for and

~xpress regularity
1n reasoning.

\\ lien po;siblc. ·patterning acti l'itics should in n,lvc some to rm nf physica l lttJterial
ks a' Iran's ts howhncnadps,•ci1n.·
fc,,,r',Turaenv.ofk·a'mpdactrtgeranrtscunchanads bc
f-t rst-gra<Ie tex.t I ,ilir·•b1,,.c··I-ph••1c<rr-cr:asswk· .htse re chtldrcn
a string of co lo to coIo r tIie
10 0

n.,d

the string accordi ng tu the pattern . T here ,ire ,1 tcw differe nces between thi s and t he <nine

activity done with physic tl nrntt· rials. Firsr, by colormg 0 11 the pa ge without an ini tia l in"c,.

t.ig:irion with p hysica l mate ri als, the acti,~ty ta kes o n a sense of ri g ht versu s _wrong rarhcr

than n sense-m aking acti,~ty. If a mi stake is made, correction o n the page 15 diffi cult and tan

ca use fru strn tio n. P hysica l materi als. on the other hand, ,1Uow a tnal -a nd -en-or approach thar

can allow for learning from mistakes or mi ssteps. Second , pattern acti,~ties o n work;hect;

p revent childre n from extending pattern s beyond t he few spaces provided o n t he page. U,

u si n g m aterials such as colo red bl oc ks, buttons. and co nnecti ng cubes, children gain Illar~

experience thi nking about patterns beca use t.hc patterns can he exte nded well hc,·ond th ose

few sp,1 ces. Plus, by using ,1 variety o f materi als tu create patterns, children can begin to

genernlize patterns ,1cross the diffe rent materi als.

Repeating Patterns

Repeating patterns arc those patterns that have a co re that repeats. For exampk. if rcd-hluc

is the co re, a stri ng o f beds could be used to continu e to repea t thi s pattern: rcd-bluc-rcd-

blu c-rcd- blue . . . . :-Joticc in Fi gure 13.7 th ,u the core is always fu lly repeated and ne,er

o nl y p,1rtially sho wn.
An important goa l when worki ng wi th repeating patterns is to hd p c hildren identify the

con' o f the pattern (\Varren & C ooper, c008). O ne possible w,1y to cm phasi Le the core is to

pbcc shape pattern s un de r a docu me nt camer:t , sta te ,1lo ud what is there and :tsk wh:tt come,
next . After adding on a few shapes, ask chil dren to describe the pattern. To help chi ldren.

especia lly those with disabili ties, track on the core of the pattern , highli ght the cure bvdm11 •

ing a circl e aro und t he shapes tha t fo rm the core, repeatin g the circle each time the core i,

repeated. You ca n also hi ghl ight the core by la belin g the p:1 ttern with letters as sho11 n in

Figu re 13 .7 . For exa m ple, th e fi rst pattern in Figu re 13.7 sho ws an ,\B pattern because the

core has two di fferent elements, A and B. T he red-blue pattern desct; bed earlier is also an ,\ ll

pa ttern. An !\B BB patte rn (the last pattern shown in Figu re 13.7) uses 2 different element;

but because B is repeated 3 times, th e co re has -l elements.
Repea ting patterns are c,·erywhcre ! T he seasons, days oi

the week, and month s o f the yea r are just a beginning. ChJI·

lengc children to find rea l-li fe AB patte rn s-for e.xarnpk,

When first starting to work with patterns, ask children "dar, ni ght," "open a door, close the door,·• "to school, hotne

to describe three things they notice. Once they share from school. " or "se t the table before eati ng , cbtr ruble after

C hildre n's books o fte n have pa tte rn s in 2r0e0p0eu) a3nngd
words, o r phrases. Pattern Fish (Harri s,
their ideas. then point out the pattern if no one else rwoeati ng.''
does, and ask them. •what comes next?"
r h y m e s,
.. . MgryeaMt cohm~t·aenesd. Dari Ninh M e Lnu.,rrh (Sharra tt, 1996),ure d ,n
. A very long re peating pattern can be ,oun, h
ff You Gn·e fl Mmise a Cookie (Numcroff, 1985) (o r anv books in this series). in which , nc

event even tually leads back to givi ng a mo use a cookie, ~vit h the implication that the sequl!t1Ct'

would be repeated. In addi tion, you ca n recite om! patte rns. Fo r example, "do, m i, tni, do,