295-379

Patternsand Functions fl 295

,midi'cwmaiy.s.,.a.n" disthaesnimdoplwensicnanoni· ng upsaettde tron.mAalkseo'pbaott,eIYmms:ouvpe,msi.ednet,sdsouwchn as moving the arm up,
be , up, side, d own (ABC).

Figure 13. 7
[xa mplc!I- of repealing' I, att rrn., using manipulative,.

Multlllnk Cubes

.... .. .i~[i~~~1~"

.... .,.,.Pattern Blocks

,,.,. , ,..A B C O
Two-Color Counters
B
•• •• a•A A B
Color TIies

ABBB

There are numerous websites that offer children opportunities to explore repeating patterns. For example,
PBSKids offers several interactive pattern activities for young children (search for PBS Kids patterns). An
Activity called "Patch Tool" on NCTM's Ill uminations website allows children to flip and rotate shapes to
createvarious pattern s, which informally introduces them to geometric transfo rmations while they explore
patterns.The National Library of Virtual Manipulatives also has several applets that support the exploration
of repea ted (and growing) patterns, incl uding AttributeTrains, Color Patterns, and Pattern Blocks.

The followi n g activity helps child ren work on identif)~ng the core of a pattern .

CCSS-M : K.MD 8 .3; K.G .A.1: K G.8 .4 1 .G.A.2

Can You Match It?

Show six or seven patterns with different m aterials or pictures (see Figure 13.8 for examples). In pairs, one child closes his
eyes while the other child uses the A, B, C method to read a pattern tha t you point to. After hearing the pattern , the child who
had his eyes closed examines the patterns and Identifies which pattern was read. If two of the patterns In the list have the
same structure. the discussion can be very lntereSling.

296 CHAPTER 13
Promoting Algebraic Reasoning

Fig ure I l .8
Lxarnpk, of pattern ca,d s on c:1rJ ,toek. Each pci ttc-rn co m plcrely
, cpcat s ,t, u,rc ,11 lc,l\t l w1ee_ _ _ _ _ _ _ _ _ _ _ _ _1

,~,~~Q-Q-:--~ ~G] G~J Il

Patte rn blocks

Standards for A sign ificant step fo rward m athem atica ll y is to see that two p attern s con structed
Mathematical Practice with different m ateri als are actually the sa me pattern. For exa mpl e, the second pattern in
F igur e 13 .7 and the second pattern in Figure 13.8 can both be "read" A-B-C -D-A-B-C -D, and
D Look for and make the third pattern in both fi gures is A-A-B-B-A-A-B-B . Tra nslati ng two o r m o re patterns ,,; th
the same structure into some form of symbolism (i n thi s case, the alphabet) m m·es beyond doing
use of structure. pattern s for the sa ke o f doing patterns and helps children gai n expe ri ence with generalization.

Activity 13.18 CCSS-M: K.MD.8 .3 ; K.G.A.1 ; K.G .8.4 : 1 .G.A.2

Same Patte rn, Different Stuff

Give each child a pattern strip like those shown In Figure 13.8 and a set of materials that Is different from that on the pattern
strip. Have children make a pattern with the set of materials that has the same structure as the pattern on the strip. You can
also mix up the pattern strips and have children find strips that have the same pattern structure . To test if two patterns are the
same, children can translate each of the strips Into a third set of materials or they can write down the A, B. C pattern for each.

Afte r chilclrcn hnvc c " . 297
trrn s, you can step up thllePc· rh11.:ncc' ~ identify in, I . Pattern! and Functions •

bn,le'Xa1sa1o.uc: lm1.~·1.trIyi.em to predi ct wch 1·aclhie n•I.,,,c, 1)y .int rllnrgIti,ct1c1:.>1·Ut1niiagd13)11 gfu,n.acntIci.1oCnKatle nd1.n g rep eat.ing pal-
c nt Wi nspcct to th eir work
c ein

I articu ar positio n, as done in the

Predict Down the Line CCSS-M: K.MD.B.3: K.G.A.1;
K.G.8.4; 1.G.A .2; 2.0A .C.3
Provid e children with a Patt

Befor e children beg1 t err, to extend (e,e.. ABC

n o ext end the Pattern made with colored links)
(colored links) WIii be In, say, th e twelf~:ttern, have them predict exactly what element~

sixth , ninth, and twelfth term , aro th Position. (Notice tha t In an ABC Pattern the thi rd,

children predict, have th em com I e C elemen t becauae they are multiples of 3.) After
d ff P ete the Pa tt ern tO
can I erentlate this lesson b I t th a mo, chec k. Ask th em ho w they knew You
Y art1n11 Wi ·
eroups of Children work on dlfferent types of Patte basic AB Pattern or by having dllferent
should be required to provid e 8 reason for their erns based on thei r readlneu. Children
After the dlscu u lon extend the h di
pre ctlon In Writing su pported with vlauals.
1a1nllee,nsguechbyaBs itkhineg2c0hthlldorren24tothppreodsiictitown.hAatgaeilnem, cehniltdwreinll
be In a position furt'h er down thC
should be required to provl d e

e a reason fo r thei r prediction.

In "Predict Down th e Linc " o •h'l I · I
to bui.l d onto the patterns th cv h,adsalm,e cI I crcn w.il dchOec k their pred ictio•n by c·onti nuing

. , , rcacy constru cte . thcrs may comh111 c their 12 -cube
pattern WI~ a partn_e r s an<l use the new 24-cube pattern to check their prediction. Still others
may use skip countin g by 3s to help th em check their prc<liction .

Growing Patterns

Growing patterns, which involve a progression fro m step to step, are precu rsors to fu ncti ona l

relationships and shou ld he explored in the primary graclcs. In techn ical tcnns, these patterns

are called sequences; we will simply call th em growing p1111er11s. With the~e patterns, children

ide nti fy the cure, but they also look fo r a generalization or a relation~hip th at will te ll them

how the core is changi ng- and , ultimately, in late r E',Tacles, what the pattern will be at anr

point along the way (e.g. , the nth term). .. .

Chi ldre n's eKperi enccs wi th gro wi ng patt erns shou ld start with fa irl y ~tra1ghtforw~ rd

patterns using visua ls, such as those in FiE,•tm.: 13.9. Geumcm c pamrns made_with_phys ical
1natcn·aIs (e.g., t1·1es, counter~, cul)cs) arc guod cx' cm.1Jlars hcca u~e.the partc: rn 1s easier to sec

(th an with numbers) and beca use chilclren ca n ma111pulatc the oL1ccts. .

h.. gure 13 9(d) 1·~ a grow·ing patt crn in which a S(]uare lfladc o.f 4 cu bes II added to the

mi·dd le as you •go fro m one ; tep to 1I,e 11ext· Describing how a grow ing patte rn chan!:(cs from

ste · k · rerun'/·Ve th1'11 k'mg (r[3eeazs,uosnzkaaI,u&utKthccn•n' cr,ki2n0d0,8o, fU'11iaantttcornn,s.20T0h8e)raenfot1re1s·
howPtmoossttepyo1sun gneorwcnhialsdren w1·11Iic aIJIe, to
Teach.t.n Tt.cwohnesn.iddeirschuuswsineagcah gstrcnpw.1m11 gtI1pcatptaetrtner,nendcifo'feurrsagferom( h1tlhdcr'epnret·-o
',/

11

ceding step . When a chilJ observe'! t hat c·a, h .new ste, p ctahne
be built hy add ing on to or changi·ng, th e prevf ious stthqjis, ca n
d1,scuss1on should 111durk a cjemonij'trltl OflI () Il,OthWe same in Two important questions help children analyze growing
1u,c_done. Circling ur color cu,1m· g, the 11.a rt t mt" , patterns in order to determine the general relationship:
What ls staying the same7 What is changing?

ad1.accnt tem1S can help I·c1entl' fYwhat 1s changrn g.

IIIIIIIIIIL...

ha • CHA!tlU n
~Mg A!gN),A,c Rl't\Ontog

1 111 0 11" 1 111

<••" ·m,1,,- qr,,-. '"II p.,tt, n1, '"1111{ 111 m,J'ul~111 r•

(1) Mulllllnk cubet

S11111 I SIPP 2 Slop 'l

II fiJ

{bl PIH11n blockt

Step 1 Step 2 Stop :1

(cl Color lilH Step 2 Slap 3

Step I

X

(di Multllink or c1nllmtt1r cubea

Step 1 S1ep2 Step 3

#1 I Step 3
(•l Color IIIH or peper 1quar"
[J
Step 1 Step2 [0J
0
B0

• CHAPTER 13 Promoting Algebraic Reasoning

. ntinues. Th . gtyapceos.v0t.tfyri.enlca1o des some repeatin g pattern s and so rn e .
stnng co as well as a e fol.lowm o n shi.p s. growing
patterns few other

Activity 13.21 ccss-M : 1 .oA.C.6 ; 1 .NBT.c-4.
2 .0A.8 .2 ; 2 .NBT.A.2 ; 2.NBT.B.s

What's Next and Why?

be done In pairs at a station. Provide children five or SI
This activity can as a whole class or pattern for several more nu X
number pattern. Chlldren are to extend the rn-
numbers from a the rule the
bers and explaln for generating the pattern. The dlfflc ulty of the task depe nd s on

number pattern and how famlllar the children are with searching for patterns. Here are sorne

recommended patterns to try.
1, 2, 1, 2 , 1, 2, . . . a simple alternating pattern

1, 2, 2 , 3 , 3, 3, . . . each digit repeats according to Its value

2, 4 , 6 , 8, 10, . . . even numbers-skip counting by 2
1, 2, 4, 8 , 16, . . . double the previous number
2, 5, 11, 23, .. . double the previous number and add 1

1, 2, 4, 7, 11, 16, . .. successively Increase the skip count

2, 2 , 4 , 6, 10, 16, ... add the preceding two numbers
Have children make up their own number pattern rules to challenge their classmates.

You can use a calculator to skip-count by any amount beginning anywhere. For example,

to count by fives, enter O+ 5 = . (fhe Oisn't necessary but is a good idea with young chil-
dren.) Successive presses of the = key will count on from 5 by fi ves . To skip count by fives
from a different number, say, 16, simply press 16+5 and continue pressing the = key. The
calculator "remembers" the last operation, in this case " +5," and adds that to whatever is

currently in the window whenever the = key is pressed. The = will continue to have this
effect until an operation key is pressed. The next activity uses the calculator as a tool to

explore patterns.

-:,I Activity 13.22 CCSS-M : 1 .NBT.C.5 ; 2 .NBT.A.2

Calculator Skip Counting

i'iil Teach children how to make their calculators skip count by different numbers. As they

. . work In pairs, challenge them to say the next number before pressing the = key. Have

y;;:, them hold their finger over the = button and say the next skip count, press the = to

confirm or correct, and then continue, always trying to say the next count before press-

Ing the key. Record the numbers on the board or chart paper as they are displayed on

the calculator so children can refer back to them to describe the patterns they see. Providing

a Hundreds Chart (Blackllne Master 3) for children to record theIr number sequences Is also
helpful, especially for children with dlsabllltles· For exam PIe, when counting by twos starting

with o, they see numbers ending In 0, 2, 4' 6' and 8 , and then that pattern repeats.

When children are comfortable starting their skip counting from zero, teach them how to
use the calculator to start at a different number' and note any patterns. For example, what do
they notice about the number sequence when they begl n with any even number such as 34,
and skip count by twos? What happens If they count bY twos and begin with 1? '

Common Misconcepti.ons w1'th Algebraic Reasoning ti 301

•v••/hen children. have become comfortable· sk. 1' p counti.ng by smaII num hers , suggest that Standards for .
tJicYtr)' skip cou~ong by a "big" number such as 20 or 50. Children will be surpri sed to see Mathematical Practice

(:irniliar patterns m the tens and hundreds places. Continue as appropriate with challenges DMake sense
cocount b~ oth er number~ sue~ as 15, 25, or JO. . of proble~s and
persevere in
focusing on patterns m skip counting will support children's use of invented strategies solving them.
for addition as well _as mul~plicati on. In fact, a key reason for exploring number patterns_ in
thee)cment~ry curriculum is to strengthen children's understanding of number relationships

and properties. T he _m o re often you ask children, "Did you notice a pattern?," and then
e~pect them t~ exp lam the pattern , the more often they are making sense of the math emat-
ics they are domg.

Common Misconceptions with Algebraic Reasoning

Table I3.2 provides a summary of the most common errors and misconceptions children
will demon strate with algebraic reasoning, along with suggestions about how to help them
work through these issues. Although all of these mi sconceptions are ones that you sho~ld
help children clear up, the misconceptions related to children's understa~ding ~f equality
are critical to address. Children who hold misconcepti ons of the equal sign will struggle
when solving more sophisticated problems in the elementary curriculum. Moreover, a robuSt
understanding of equality is crucial to understanding and doing well in algebra (Knuth ,
Alibali, McNeil, Weinberg, & Stephens, 2011; McNeil et al., 2006).

Table 13.2. Comm o n errors and m isconcepti ons in algebraic reasoning and how to help.

Misconception/Error What It Looks Like How to Help

1. Child thinks of the Child identifies 12 as th e missing • Read and emphasize the equal sign as "is," "1s the same as, or
equal sign as a signal
to "do something ." number for problems like: "is equal to ." For example, read 4+5 = 9 as "four plus five is the

5+7 = _ + 8. same as 9."
• Restate children's remarks that imply the equal sign "makes"
Child resists w riting equations in the
another number or signals an action to emphasize = means "is"
form :
or "is the same as ."
8 = 5+3
• Relate the concept of equality to a balance (see Section called
because he or she thi nks the answer "Conceptualizing the Equal Sign as a Balance").

should alw ays be w ritten on the right • Use true/false and open sentences to emphasize equality .
• Record children's equivalent expressions when
side after doing the computation.
they solve problems in different ways (e.g .,
=Equations such as 6 6 does not 8+7 = 8+2+5 = 5+3+7 = 10+5 = 5+10 = 15).

make sense to the child because no

computation is invo lved.

2. When check ing Child explains that two numbers can • Although the child may not be ready to generalize using variables,
to see " does this be added in any order because you can encourage them to check a wider range of numbers (e. g .,
always wo rk7" child 3+5 = 5+3 and 1+3 = 3+1. two rea lly large numbers, one small number and one rea lly large
only checks a few Child is satisfied with one or two number, two even numbers, two odd numbers, one even and one
examples. examp les. odd number, and so on).

• Ask the ch ild to describe in general terms what is happening,
without referring to specific numbers .

• En courage t hem to try to use physical materials or
representations in a genera l w ay to justify the ir cla im
(see Figure 13.4 as an exam ple).

(continued )

l06 • CHAPTER 14

Exploring Early Fraction Concepts

Introducing Fraction Language

Fraction symboli sm represents a fairly complex conveno·~n that can be mi.s1ead'mg-_to-ch-ild-re-n

T h at is why it is important in grades preK-2 to use fraco~n words and postpone tntroducin ·
fraction symbolism (e.g., Empson & Levi, 201 L). Allow children to first focus o n _making sens!
of fractions without the complication of also trying to make sense of tbe symbolism.

In the Common Core State Standard for Mathematics, children are expected to learn the
fraction vocabulary of halves, fourths, and quarters in first grade a~d the_n thirds in second
grade. A good time to introduce the vocabulary of fractional parts is d~nng the discussions
of children's solutions to story problems and not before. When a browrue or other whole has
been partitioned into equal shares, simply say, "We call these fourths. The whole is cut into
four equal-sized parts-fourths."

Initially, children need to be aware of two aspects of fractional parts: (1) the number of
parts and (2) the equality of the parts (in size, not necessarily in shape). Emphasize that the
number of equal parts or fair shares (i.e., unit fractions) that make up a whole determines
the name of the fractional parts or shares. For example, it takes four one-fourths to create
the whole. Children will Likely be familiar wi th halves but should quickly learn to describe
thirds and fourths. 1n time, children should begin to consider the relationship between the
size of the part and the whole.

D Formative Assessment Note

Some children think that all fractional parts are called halves. Once a child has completed a partitioning
story pro~lem, ask the child to tell you how much each person gets. The word the child uses to describe the
fractional amount will tell you if the child is overgeneralizing the word half.

Standards for Notice that in the CCSM-M, halves and fourths are developed in first grade prior to
Mathematical Practice thirds in second grade. T his is done because successive halvi ng of parts is a natural process
for young children. Once children have been successfu l dealing with and explaining halves
El Attend to predsion. and fourths, pose sharing tasks that involve eighths. They wilJ likely rely on their hahing
strategy to find a soluti on. Likewise, once children have demonstrated an understanding of
thirds, pose sharing tasks that involve sixths. Using this progression from parts that can be
found through halving to parts that require a different strategy is a great way to differenti-
ate as some children will require more time than others to demonstrate understanding of
partitioning into different amounts.

In addition to helping children use the words halves, thirds,fourths, and quarters, be sure
to make regular comparisons of fractional parts to the whole. Make it a point to use the rem1
whole, one whole, or simply one so that children have a language that they can use regardless
of the model involved.

/Models for Fractions

Substantial evidence suggests that the effective use of visuals in fraction tasks is impcrtanr
in building children's understanding of fractions (Cramer & Henry, 2002; Empson & Le\~,
2011; Petit, Laird, Marsden, & Ebby, 2016; Siebert & Gaskin, 2006). Unfortunately, when
textbooks incorporate visuals or manipulatives, they tend to use only area models (Hodges.
Cady, & Collins, 2008; Watanabe, 2007). Using a variety of physical tools that repre~ent
both continuous and discrete quantities is critical for children to make sense of frac n°0 5

Models for fractions • 307

£as1tiheert)o' ne,x2p0loIrSe) fractions in a varie ty of s1.tuat1.o ns (S'ieg1er et al. , 2010; Zhang, Clements, &
•.

Table 14. 1 proVJdes tahqeuiricrkeloavteedrvi.ew of three types of models-area, length, an d set-
defining the wholes and

ow]ptb1·.haoneaerYt0etwsssa5aretocmoemafnentaohtadwehm.cenethlikhmvocaialatnecbyncaoawensrueitchwtihhnfeerilfltalpolwdcracrtocsieloeandtnrhi'tisfhfefiy:deuernerienedcallsoemae.attrii·s.Frsoo.etonnapprsnarohedefI'xrsm~paeamngbffotrePaa'arttI5cw,e·eto,tiaehnocae.sehnnnammaatnhsrdooeeadadaseemsmi.llkzs.ee.ionaUodosgfesuffcliretnhehcrgi.aeldndWdip.rifbeaffhfenereetrhrntetoeeonnaltmpttphofapmuepkrlopoewm.dopcherrotiolV·uamsl•tJmefepsou,.twa1r.ardnehi·•fossI·rzJ·oam·meocngg-ras

Table 14.1. Model s for frac tion co ncepts. and '.c,J atcd v.isuals and co nt ex ts.

Model Description Sample Contexts Sample Manipulatives
Type and Visuals
Area Fractions are determined based Ouesadillas (circular food) Fraction Circles/ Rectangles
on how a part of a region or Pan of Brownies Pattern Blocks
Length area relates to the w hole area Garden Plot or Playground Tangrams
or region. Geo boards
Set Grid paper regions
Fractions are represented as Walking/Distance travelled Cuisenaire Rods
a subdivision of a length of String lengths Paper strips
a paper strip (representing a Music measures Number lines
whole), or as a length/distance Measuring with inches,
between O and a point on a fractions of miles, etc. Objects (e.g., pencils, toys)
number line, subdivided in Counters (e .g., two color
relation to a given whole unit. counters, colored cubes,
teddy bear, sea shells)
Fractions are determined based Children in the class,
on how many discrete items school, stadium
are in the whole set, and how Type of item in a bag of
many items are in the part . items

It is important for children to experience fractions through contexts that are meaningful
to them (Cramer & Whimey, 2010). These contexts may align well with one representation
and not as well with another. For example, if children are being asked who walked the far-
thest, a linear model will be better than an area model in supporting their thinking.

Area Models

Circular fraction p ieces are by far the most commonly used area model. One advantage of

the circular model is that it emphasizes the part- whole concept of fractions and so are good

for introductory activities but they also can be used to emphasize the relative size of a part

to the whole (Cramer, Wyberg, & Leavitt, 2008). What is being compared is the area of the

pan to the area of the whole. Because we can cut area into as many equal-sized pieces as we

want and because area is measured, it falls into the category of a continuous quantity. Notice

~at when drawing circles, children (and adults) can find it difficult to partition the circle

into reasonably equal-sized parts. Regions, such as recta ngles, can b~ dra~n on blank or grid

paper and are easier to partition. There a.re many o~er area mod~ ls ,~cludmg pattern blocks,

hgoewobdoi.afrfdesr,ecnot lsohraptielessc,aannrdepfrraecsteionnt bars. The physical models m Figure 14.1 demonstrate
the whole.

308 ., CHAPTER 14 Exploring Early Fraction Concepts .·t..• • • • •

figure 14.J C•- • • •~-:,
A r<'.! models t.,r fracti,ins. •••••
•••••
Circular "pie" pieces Rectangular regions
Any piece can be selected as the whole. Fourths on a geoboard
One- third . . . . .One-half or two-fourths
Pattern blocks Paper folding
I -~......a...?.......

I

II

I

II

II

III

Drawings on grids or dot paper

Too often we o nly provi de area models for children that are alrea dy p artitioned into
equ al-sized parts so they do not have to attend to thi s critical fea ture of fracti onal parts
(\Vatanabe, 2007). It is important to provide children opportunities to explicitly confront this
issue o f equal-sized parts. Activities like Activi ty 14.1 provide such oppo rtunities.

Activity 14.1 CCSS-M : 1 .G.A .3 ; 2 .G.A.3

Halves or Not Halves?

liii'I Use the Halves or Not Halves Activity Page showing examples and nonexamples of halves. (Note that examples and

- nonexamples are very Important to use with children with dlsabilltles.) Have children Identify the wholes that are

correctly divided Into halves (equal shares) and those that are not. For each response , have children explain their reasoning.
Repeat with other fractional parts, such as fourths (see Figure 14.2) or thirds. (See Fourths or Not Fourths Activity Page

and Thirds or Not Thirds Activity Page.)

Figure 14. 2
Ch ildren need tn recogni ze when shares arc nn t cqu:d.

(e) (I) (g)

In the "H. akes o r ~ot H a1ves" .. . .. . Models for Fractions fl 309
correctlr o r mcorrec tl r; the ch . ac ti vity, the wholes .
Standards for
ost important part o f th' . . ddren are not • are al ready part1t1oned eith er Mathematical Practice
sr1h11e
Dconstruct
th viable arguments
'dea \stCh\hu'aahmtr1'1ica1n1hllg0p1ta1hi.eCfeocop0erie·d\e\'SctohctuasalttdneitsehYSeaetodcyunta1odvi.lkl1b.e•l._e.1i·aks1n·eshdaatnhwredehdw•vda~·ihns"ecdanun5sthidshn·a1."evro\reonVedl·viaieu.eriqedesue,u·a1entnlrelhyyq1e.o1uilndeaec1'lsip!tpciauca1r1·hrstntsi1s·kl1.,tdo•itprnhoeomentsy•oenthoqgraeu.pvm.eerseoftaotieoc,tranhnci·deers and critique the
•I are reasoning of others.
sueeh\. a,.. "

,,harTe .:h»e h " equal-sizeda11p aanrdss" sde t
",II. -. t at as .
recognize an expectation for second graders that they

tursJitnegthan1.5. acroenacmepotd. el'lh. Seonewxitthtwsoeco d oes not necessa rily mean the same shape whe~
n• g. r.aders' make sure to m. cl ude examples that illus-
acttVJnes address thi s idea.

Activity 14.2

Size and Shape Check CCSS-M: 2.G.A.3

II Challenge Children by askl g th
listed below for any gradenappreompr1taotedrfarawcstihoanpael sptahratst.fl(tNeoaticche of the four categories
that creating fourths

and eighths tends lo be easier than creating thirds and sixths because children can use a

halving strategy to create fourths and eighths.)

1. Same shape, same size (equivalent)
2. Different shape, same size (equivalent)
3. Different shape, different size (not equivalent)
4. Same shape, different size (not equivalent)

Provide children with recording sheets such as the Eighths or Not Eighths Activity Page and
the Sixths or Not Sixths Activity Page. Many children, In particular children with dlsabilltles,
wlll need to cut and move pieces around to check to make sure that parts that are different
shapes are Indeed equal In size.

· Activity 14.3 CCSS-M: 1.G.A.3; 2.G.A.3

Different Shapes for Fair Shares

0 Give children Dot Paper (Blackllne Master 8) and ask them to enclose a region that
d It If to partitioning wIth a Particular fractional part (e.g., halves, fourths,
len s se I mfoiugrhtthsasokr t ahe3m-b Yto-6 enclose a 4-by-2 rectangle If they are going
thirds). For You rectangle If they are going to partition Into
examp e,

to partition Into halves or t ' they are untlng the space In between the dots, not the
It that co
thirds. (Make sure to po ngloeus ) If using, say, the 4-by-2 rectangle, ask children to find a way
dots, to create the rectan · have them draw another rectangle that Is the
to partition their rectangIe In to halves. Then
t Yto show halves. Encourage children to find
III It a dlfferen wa
same size (4-by-2) and part on different shapes. See how many ways they can
a way to show halves where the halves are

(continued)

310 • CHAPTER 14 Explon·ng Early Fraction Concepts

find . Note that the areas do not need to be adjacent. See Figure 14·3 for some P0SS1biiities,
Emphasize "ths" as you say the fractional parts, particularly for Ells who may not hear th e
dltterence between fractional parts and wholes (e.g., fourths sou nds like "fours "), Explicltly
discuss the ditterence between four areas and a fourth of an area. Also discuss the meaning

of the words whole and hole.

[J Formative Assessment

The previous three activities offer good diagnostic interviews to assess whether children understand that
it is the size that matters, not the shape. When working with fourths as shown in Figure 14.2, if children
get all correct except (e) and (g), they hold the misconception that parts should be the same shape. Future
tasks are needed that focus on equivalence.

Figure 14.3

Gi ven a whole, find fractional parts that arc
different shapes but the same size.

DJ B i Cyberchase, a popular PBS television series, offers fraction activities on their website
such as "Thirteen Ways of Looking at a Half.· In th isactivity, children are challenged
to find all the ways they can shade half of a geometric shape that is cut into eighths.

Length Models

\½th length model s, lengths or linear measurements are compared inst ead o f area s. In this
model , a unit of length is compared to the whole length , \\'e can cue a length into as many
equal-sized pieces (units) as we want; therefore, length m odels represent continuous quanti-
ties. Length model s appropriate for preK-2 include fraction strips , paper strips (e.g., adding-

machine tape), Cui sen ai re rods, and line segments (see Figure 14.-+)..-\II of these models

provide flexibility because an y length can represent the who le .

Virtual Cuisenaire rods and accompanying activities can be found online at various websites including the
University of Cambridge's NRICH website.

Researchers have identified number line models as useful in fra cti o n instruct.ion because
they help children underscanrl that a fraction is a quantity or 3 number (Pe tit, Laird, &
Marsden , 20 JO; Siegler et al. , 20 IO; \Varanabe, 200 7). Locatin g fraction s on a number line
also highlights their relationship to other numbers, inclurling o ther fractions . The num ber

line has been shown as e~trcmel y effc~tive with _young child;cn when working "irh whole

numbers (e.g. , Booth & Siegler, 2008; f·osnot & Dolk, 200 I ; Si egle r & Ramani , ~009). Using

Mode ls for fractions fl 311

I igur,, 14.4

rIe11~ t h l'r lHt",hu rl~nwnt n1n dc-h1I \ ) 1·
-
t. .

l ,H. l lllJl\

l

Fraction strlps or Cuisenal re rods

1- n -TII I I I I I I
t 6 7
0 412 43 1 45 4 2
4
I I i0
2 345678 10

Measurement toots

Folded paper strips

mthi~ : ode! wit~ who le numbers ca n help prepare these children to be ready to use thi s mode l

WI . a~tJ o n s later grades. Usi ng li_n~ segments, a modified version of a num ber line, for
fraction 1~struct1 0 11 '.n grades pre_K-2 1s ideal. C hildren can develop an understanding o f the
number lme by fo lding paper stn ps as described in Activity 14.4 (adapted from Z hang, C le -

ments, & Ell erton , 2015).

Activity 14.4 CCSS-M: 1 .G.A.3 ; 2 .G.A.3

Pap er Strip to Number Line

The Paper Strip to Number Line I Activity Page, wh ich focuses on ha lves, fourths, and eighths,

can be used for this activity. On this activity page, you will see several line segm ents t hat are
all the same length. Give pairs of children three different colored strips of paper (e.g., blue ,
green , red) that are each the same length as one of the line segments on th e Activity Page .
Their task Is to fold, say, t he blue strip Into two eq ual parts, the green strip Into four equal parts,
and the red strip Into eight equal parts. Then they use their unfolded st rips to locate various
fractions (e.g., one-half, two-halves, one-fourth , two-fourths, and so on) on the nu m ber lines on
the Activity Page . To rein force the notion that three-fourths, for example, is 3 one-fourths , have

(continued)

31 2 "' CHAPTER 14 Explori·ng Early Fraction Concepts

children cut each strip into Its respective unit fraction parts. They can use the unit fractions to

count the fractional amounts.
When your children are ready, use the Paper Strip to Number Line II Activity Page, Which

Involves thirds and sixths.

Set Models

The whole in a set model is understood to be a group of objects, and subsets of the whole
make up fractional parts. For example, 3 red counters arc one-fourth of a group or set of I7
counters. T he set of 12, in this example, represents the whole or_ I. The idea of referring ~
a collection of counters as a single entity can make set models difficult for young children.
Another challenge with set models is that children may focus on the size of the subset rather
than the number of equal-sized subsets in the whole. For example, if 12 counters make a
whole, then a subset of 4 counters is one-third, not one-fourth, because 3 equal-sized subsets
make the whole. To help children with these challenges, put a loop of yam around the set to
help them "sec" the whole. Then use additional pieces of yam in a different color or sticks
to group the subsets within (see Figure I4j),

Figure 14.5
Set models for fractions.

Two-color counters show Two-color counters in a loop
two-thirds. of yarn show one-fourth.

Discrete objects, like two-color counters, are effecti\'C for set models. They can easily
be flipped to change their color to model nrious fractio nal parts of a whole set. .-\111· count-
able objects, such as a box of crayons or a tin of muffins, can work as a set model (one box
or tin could be the unit or whole). The next activity engages children in fairly sharing with

a set model.

Activity 14.5 CCSS-M 1.G.A.3 : 2.G.A.3

Sets in Equal Shares

Give pairs of children 16 crayons In a box or clear plastic bag. Explain to the children that
although there are 16 Individual crayons, the ·whole" consists of alf 16 crayons In the box
(or bag). Their task Is to determine and be ready to explain how the 16 crayons can be
equally shared with two, four, and then eight friends. They can use the Sets in Equal Shares
Activity Page to record their ideas. When children demonstrate confidence and accuracy
with halves, fourths, and eighths. they can be challenged to share , say, 12 crayons with
three and then six friends.

Building Fractional Parts through Part·Irioni·ng and Iterating • 31l

V1u111ia1r1tnu1.eiapr,lu&lamtA1a.vnnedispe,urvsla.iorttnuiv,ae2ls0m1aa2ren) .i.pYauvolaautriliavcebhslieldhrafeovre ab11 three mod eIs Of fracti.ons. When paired with using actual
een found to ·improve student achievement (Moyer-Packenham,

applet called "Fract.ions Model" at the NnCcTaMn explo.re !e. ngth, area, and set models of fractions us.ing the
llluminat1ons website.

........,J

0 Formative Assessment Note

Children should have opportun 1·ries to explore fractions across the three models for fractions.
A good way to assess h"ld ,
f a Fract ion Act· •t pc I ren s knowledge Of a fract1•onal amount is to give them the Meani•ng
oOf the fractional 1v1 y a.ge along w1·th a fracti.onal value (e.g., three-fourths) . Their task is to think

) ·t t amdount in terms of area, length, and set and for each model (1) draw a picture and

(2 wn e a sen ence escribing a context or example for the selected fraction.

Building Fractional Parts

through Partitioning and Iterating

The first goal in the development of fractions is to help children construct the idea offrac-
tional parts of the whole-the parts that result when the whole or unit has been partitioned
into equal-sized portions or fair shares. Recall that Table 14.1 describes the meanings of parts
and wholes for each type of fraction model. Vle can build on children's experiences of fair
sharing to begin to establish this idea of fractional parts.

One of the most significant ideas for children to develop about fractions is the sense that
fractions are numbers-quantities that have values. You may not be familiar with the terms
partitioning (splitting equally) and iterating (counting a repeated amount) but, as you will see,
they connect to whole-number concepts you will recognize. Researchers have acknowledged
for some time how important these two actions are to meaningfully working ,,~th fractions
(e.g., Olive, 2002 ; Pothier & Sawada, 1990). These actions, in particular, emphasize the
numerical nature of fractions . Look to create and use tasks embedded in contexts that explic-
icly require children to engage in these actions. When children explain how they thought

about fractional situations, listen for these ideas.

Partitioning

Partitioning can be thought of as splitting or cutting a quantity equally. Young children are

engaged in the act of partitioning from an early a~e when ~ey spli_t a gr~up of~ into 3 a~d
3on h th h • ng family and friends . Given children s expenences with
fcdtutfauuaeensirlpvrdkeletsiesl··1·roomvossphtffpameaI"nfondrnsed·eanennicngdttt1ig.e1.oooci•1yt.·fnno1ef.s1msfrntaphfgsracaeorhnamre.maotm-.lnivwotaIseennnhsm(,gdoStashtI1'cfLheea1n"asIg'm•fvnfIrad11eia·"pt0slrcyy,poe·frtatooohnnaEuaadlscrtq·t•hehfum2rmsaic0ae'lheknIaSiOredlngdsh)sde.ra'seSresasnihonshnasaduaoernoinS.rnnte.notggso1outtr•ttahilhaetsebesksmkoessesgffu,am.tafrlhrrlraoaeeEWtwci·harna.ie.trochfgrlaynhot·ttai.rohrFlladdasrdrphnaetpa.catn1rlhtn.aito·asotncnovengaam.sdlnt1·goe1dfp1voatbecmrhrelhteoaeg-iplawml.sdentcrhsri·eosnutonhongcl'enes---
contexts. See Expande esson: :,

designed for grades I or 2.

320 fll CHAPTER 14 Explori·ng Early Fraction Concepts

II

How much Holly shared

II I'

1

How much Holly shared

II I I
I I
' 'I
2

How much Holly shared

In the story problems you pose, make sure to include a variety of situations that can
be represented with these different models. For example, the following story problems fit
a length model, a set model, and an area model, respectively:

• Marley, Zack, Rita, and Hannah want to share 5 pretzel sticks equally. How much will
each of them get? (length)

• Marley and Zack want to share 11 grapes equally. How much will each of them get?
(set)

• Marley, Zack, and Mia want to share 7 pieces of construction paper equally. How much
will each of them get ? (area)

Make sure to use fractions greater than 1 (ten-fourths) The objects to be shared can be drawn on paper as rect-
and mixed numbers {two and two-fourths) along with angles, circles, or line segments along with a statement of the
fractions between O and 1. This helps children see problem. Some children may need to cut and physically dis-
that fractions can be any size and that they often fall tribute the pieces, so another possibility is to cut out construc-
between whole-number values. tion paper circles or rectangles to represent the objects to be
shared. You could use thin rectangles to represent objects of
length. Children can use connecting cubes to make bars that
they can separate into pieces. Or they can use more traditional
fraction models such as circular "pie" pieces (area), Cuisenaire
rods (length), or counters of different colors (sets).

Iterating

In whole-number learning, counting precedes learning to add and subtract. This is also true
of fractions. Children should come to think of counting fractional parts in much the same way
they might count apples or any other objects. Counting fractional parts to see how multiple
parts compare to the whole helps children to understand the relationship between the partS
and the whole. Children should be able to answer the question, "How many fourths are in
one whole?" just as they know how many ones are in ten. Counting a repeated amount (e.g.,
unit fraction) is called iterating. Understanding that three-fourths can be thought of as a count
of three parts called fourths is an important idea for children to develop (Post, Wachsmuth,
Lesh, & Behr, I 985; Siebert & Gaskin, 2006; T zur, 1999).

Children should engage in counting by fractional amounts to reinforce that fractions
are numbers. With young children, iterating or counting should be done with length or
area models. (Set models are too difficult for young children to use for iteration.) f or

326 fll CHAPTER 14 Exploring Early Fraction Concepts

choose to do so, you will need to spend time helping ch'ld
develop a strong understanding of what the nurner I ten

The phrase "improper fraction" is a misleading phrase denominator of a fraction tell us. ator and
that implies something is wrong or unacceptable about
the fraction, when it is simply an equivalent representa- (tBioyntTh-aehnewwaayra,byiattlrhwaoartyywsaewgrwne.tteiemtefrefanrcattcfi.tooiornnshsowsw1'ythmtoabrohelopicrraieJz.sloeynnitstaa&l cbaoCnt1.voen1s1•·
tion. Instead use the phrase "fraction greater than 1." a slanted one:¾, not ¾ . It is easier for children to tell w~~•

Standards for m. g o f ti1e conventi.on can be devtheelonpuedmtehrraotourghanadttwenhtii.cohn itsothheowdewneomusienathtoer.n)uUmnedraertostracnrji.s
Mathematical Practice denomm· ator, m· partt·cu Jar, m• 1•teratm• g actt•vi•tt•es. 311 d

ta sks to children that include a range of fracti ons less than I, equal to
r than I (e.g., After the children have counted the fracti onal pans in th
El Reason abstractly f).Pose iterating 1

and quantitatively. (e.g., 44), and greate

iterating task, write the corresponding fraction symbol. T hen pose questions to help childre~
make sense of the symbols, such as:

What does the numerator in a fraction tell us?
What does the denominator in a fraction tell us?

How do you know if a fraction is greater than or less than l? Greater than or less than 2?

Stop and Reflect 1500 C,~250

Before reading further, answer these three questions in terms of what we have been talking
about-namely, counting, or iterating, fractional parts.

Here are some reasonable explanations for the numerator and denominator.

• Numerator. This is the counting number. It tells how many equal shares or parts we have.
It is the number of repetitions of the unit fraction.

• Denominator. This tells what size piece, or fractional part, is being counted. It is the
number of repetitions of the unit fraction needed to create the whole. For example, it
takes four one-fourths to create the whole.

Making sense of symbols requires making connections to visuals. Illustrating what¾
looks like in tenns of the amount of pizza (area), the distance on a number line (length), or
the number of objects in a bag (set) will help children make sense of this value. One of the
best tilings tllat we can do for children is to emphasize equivalence and different ways to
write fractional amounts.

Teaching Considerations for Fraction Concepts

Because tile teaching of fractions is so important, and because fractions are often not well
under stood even by adults, we revisit some of tile more significant ideas offered in this chap-
ter h ere. Hopefully you have recognized tllat one reason fractions are not well understood is
tllat tllere is a lot to know about them, from part-whole relationships to divi sion (fair shar-
ing), and understanding includes being able to represent fractions across area, lengtli, and ser
models and includes conte.xts that fit these models. Many of these ideas may not have been
part of your own learning experience, but tlley must be part of your teaching experienc~ so
tllat your children can fully understand fractions and in tile future be successful witll fracoon
computation, algebra, and matllematics beyond.

r;

Building Measurement
• Concepts

\ 1 Measurement involves a comparison of an attribute of an item or situation with a unit that

330 ., has the same attribute. Lengths are compared to units of length, areas to units of area,
time to units of time, and so on.

2 Before anything can be measured meaningfully, it is necessary to understand the attribute

to be measured.

3 Estimation of measures and the development of benchmarks for frequently used units of

measure help children increase their familiarity with units, preventing errors and aiding in
the meaningful use of measurement.

4 Measurement instruments(e.g., rulers) are tools that group multiple units so that you do

not have to iterate a single unit multiple times .

Measurement can be thought of as the "assignment of a numerical value j'

to an attribute or characteristic ofan object" (National Counlil ofTeachers I.
of Mathematics, 2003 , p. I). It is one of the most useful mathematics
content strands because it is an important component in everything from

~ccupa~onal tasks tu life skill s for the mathematically literate citizen.
From gigabytes that measure amounts of infonnation, fo nt size on
computers, recipes for a meal, to deciding whether vou need the 6-foot
or I2-foot power cord, we are surrounded with me;surcment concepts
that apply to many real-world contexts and applications.

334 • CHAPTER 1S Building Measurement Concepts

that t h ey Wl' ll tmdersta nd ho.w an in. str um .ent. m. eas· ures. A
ru Ier 1•s a good exa mple children line u.p md1VJdual phys 1.-
· lf
ca l units, such as I-inch-long paper cl ips o r I -i nch-Ion

If your children have been using individual paper clips tiles, along a strip of card stock and mark them _off, the~

as units of measurement, have them create a chain of can see that it is the spaces on ruler~ and n o_t the tick marks

paper clips to use as a ruler for an initial transition or numbers that are important. It 1s essential that children

from single units to instruments. discuss how measurement with iterating indivi dual units

compares with measurement using an instrument. Wi thout

this comparison and discussion, children may not underSta~d"that :~es_e two methods are

essentiall y the same. T hen they are ready to compare their ruler with standard rulers

and can compare their use .

Introducing Nonstandard Units

A common approach in primary grades is to begin measurement of any attribute with non-
standard units. Although the primary emphasis at the preK-2 level is on the development of
linear measurements, establi shing a conceptual foundation for measuring other attributes,
such as area, volume, capacity, and weight, is common. The use of no nstandard units for
beginning measurement activities is beneficial for the following reasons:

• Nonstandard units can emphasize the attribute being measured. For example, when
discussing how to measure the length of a bulletin board, units such as toothpicks,
straws, or paper clips may be suggested. Each of these units covers length-and actually
accentuates length because each unit is thin and long. With the visual suppon of such
nonstandard units, the discussion can foc us on what it means to measure length.

• The use of nonstandard units avoids conflicting objectives in introductory lessons.
ls your lesson about what it means to measure length or about understanding

inches?
• By carefully selecting nonstandard units, the size of the numbers in early measurements

can be kept reasonable. Length measures for first graders can be kept at fewer than
20 units even when measuring long di stances simply by using longer units.

• Nonstandard units provide a good rationale for standard units. T he need for a standard
unit has more meaning when your children measure the same objects with their o,m
collection of nonstandard units and arrive at different answers.

~ hough nonstandard units_ offer these benefits, ba sed on several research studies,
C lements and Sarama (2009) cauoon that early measuring experiences with several different
units can confuse chi ldren. While children are grappling with the concept of measurement,
it is important to use a few nonstandard units that clearly demonstrate the attribute being
measured (e.g., for length, toothpicks as opposed to sq~are tiles or linking cubes). Early
on, children need to first understand the attribute being measured, and then the notion of
matching and the use of units of equa l size. Once they demonstrate understanding of these
concepts, you can introduce nonstandard units of different sizes to provide the rationale
for standard units. The move to standard units should be guided by how well your children
understand measurement of the target attribute.

How Big ls a Foot? (Myller, 1991) tells the story of a king who measures the queen using
his feet and orders a bed to be made that is 6 king's feet long and 3 king's feet wide. T he
carpenter's apprentice, who is very small, makes the bed according to hi s own feet, demon-
strating the need for standard units. Another tale about nonstandard units that children enjoy
is Twelve Snails to One Lizard (Hightower, 1997) .

111troducing Standard u The Meaning and Process of Measuring fl 335
t,.0u-P..h...1".•eeee,erashfs:saatui•p(1ansIu.nn)rdtgehuaentraoddbperi.auergrscngttoite.atcgsnsuntdli.ecaiznrorergroanatrntrthnrdI."eotnbsneUrrlnnpteeeaaanrsanaiun1t.terndsegtm(w2aeo)nndltteyaptt.enrecnsshtirnonuficqotui.boe·Jneoei-fs
artrt·bute. Y Used to measugreabthout As you teach standard units, make sure children, espe-
Teaching standard units f at cially Ells, understand the terms (e.g., feet, yard) and
around three broad goals·· O tn.easure can be organi.zed abbreviations (e.g., in, ft, cm). Include these on your
math word wall.

J. AauFsrbaubi.mltsietiiylaninatgordiastwybellwehecaittttohatnhtmheaeeypaupmsnruoietrpa.ersCiutahrhteeei.1·udlBenrneeitgm.ntChsghaoob.fulteIhdteoha5ehvseetiI·afmabacatcesuicara1s"tdheeleyal.foafst5hecresei.zte1oonfgc1o.smasmi.omnplyorutsaendt
2.

Yoof.umdrelaaswunretboinpaugrcivheanses1·gtruaasus•osneead·ndwhitthihledtrphereencs1sa.shm1.oouenldthkanto_1·w~ rebqomth•.rwedh.a(tW1.soaulrdeaysoounambleeasuurru.et
uwrmu.tsoawndtoJ.Udugym. agpathnee loefvgellaossf?p)rCechiislidorne.n need prapcrteicceisi.inonseyIeocutiwngoualpdpruospertioatemsetaasnudraerda

3. Knowledge ofrelationships between u . Ch.
commonly used, such as those b ;:,ts. . tldren should know the relationships that are
e een inches, feet, and yards or minutes and hours.

Children who exhibit measurement sense ar f: T . i:11easurement units,
are able to make estimates in terms of theese am1_iar with the s~andard interpret measures
dep1.cted w1"th standard uru.ts. umts, and meaningfully

Developing Unit Familiarity

Two types of activities can develop familiarity with standard units: (I) comparisons that focus
on a single unit and (2) activities that develop personal referents or benchmarks for single
units or easy multiples of units.

Activity 15. 1 CCSS-M: 1 .MD.A.2 ; 2 .MD.A.1 ; 2 .MD.A.3

About One Unit

Give children a phys1ca I model of a standard un It tahnedmheatveer,thgeivme search for objects that have about the same measure as that
one unit. For example, to deve1op f mlliarlty with d und the children a piece of rope 1. meter long and challenge them find
a school building, or at home that are about 1. meter In length
the playgroun , aro
objects In the classroom, outside on arate lists of things that are about 1. meter long, things that are a
H e them make sep
(see About aM et er Activity Page). av towbicjeectas1sI01onngg(eor,r half as long). Be sure to Include curved or circular lengths.
little shor te r (or a little longe r), or things that are shorter, or close to 1 meter. (Notice the use of terms such
given
Later, children can try to predict whether a t use precise language!)
as longer rather than more th8n. Remember o

"About O ne U ru.t" can be dobne wt ilthmoitlheeorru1n1.kti!leonmgethted.sr..ESnultgihsgetefssactmhinoiloiealstoatrokgeh-reohlcopemcryehis1ledtotrtreeenr,

find familiar distances that are a ou d the neighborhoo ' to ( ruse in class) a I-meter or

that families check the di'stances aroauthns. If possibIe, se'.1dvhidoemoe(hottp:// www.youtube.coml
oralong other frequently traveled
1-rard trundle wheel to measure dPistan·mceast·inWgawtchhi.ltehiuss1·ng a tru ndle wheel to measure the

11·atch?v:6iK.5b2uA2Ac) of a class est!

length of the school hallway.

Money • 353

,iow the children's task Is to
add the nu b

tllflre Is almost always more than one COOd rn ers rnen1a11y. Do not suggest how they add the numbers or In what order oocause

1he iyplcal way coins are taugi,t In boo way lo do this. For example, rather than add from the largest values to the smallest-

note that It Is easy to add 5 and 25, th:-i1ot Is also reasonable to use the 5s to make tens or other methods. For this collection,
, then 7 (the laSt 5 and two ls). Discuss with children how they add each collection.

10 25
5
5

You can also pose story problem b
ind subtraco.on d1' scus sed Chaptesr a9•oCuotnms1o. dneerytuhse·m1c0gIItohwe i•pnrgo blem strucrures add'm.an
in examples: for

, Alexis has some coins. John gives h .
bHeg1.nerW1.itdhi?m· (eJoainnding2..psetnanrti·eUsn.
What coi.ns di.d Alexis have to Now Alexi•s has 36¢ .
Jay has 2 quarters and a dim known)
money does Jay need? (Part-eP. e needs 95¢ to buy a notebook. How much more
, Andy has $11 .65. He gave Kevianrts-oWmhole: Part Unknown)

di.d Andy gi.ve Kevi.n? (Separate: Chanegme oUnnekyn. oNwonw) Andy has $8.15. How much money

, Wendy has $2.67. That is $1.25 more than what K ·th h H h d
Keith have? (Comparison: Smaller Unknown) e1 as. ow muc money oes

mAsoddiisfycuthsseeldevienl Cofhdaipfftiecrul9ty, .changing the numbers and the location of the unknown wi ll

Making Change

When you pay for something at a store using cash, you may give the clerk more money
than the purchase price and expect to receive change back. Knowi ng how to make change
is important for the clerk but also for the customers-so they can check to make sure they
received th e correct change1

Stop and Reflect 500 ')j250

Before reading further, think about how you would make change from a hundred dollar bill for
apurchase that cost $82.

You may have thought, I can start at I00, go back IO to 90,

then back 8 more to 82-so the change would be IO + 8 or 18

dollars. You could also start at 100, go back 20 to 80, up 2 to When introducing children to the process of making

82, so that's 20 - 2 or 18. Or you may have th0ught to start at change, start with smaller values and connect the dime
82, go 8 more to 90 and then IO more to IDO-that's 8 + I0 to the ten-frame.

or 18 dollars. .

¼sp,\ec1dM'fii c3akl.liny,g . h me as finding a d1fference- pri ce .
those
change 1s t be sa n the amount gw• en to the clerk and the purchase
a diffe.rencef erwee_es to fin d a dw'«erenee in Chapters 9 and 12. One of
eBs~terceaeatreui.gessein·cecumse,ssasaekwddi.ndiatgihnvcgtahhnaeoensntegyeosotr1•rsa"srttteehrgIama·it.teeeksgd-iabtedorcdosm.trre.oattnhe'ge,,iy.ewsaarbesasauessdkeedodnitnonutmhmeabkteehrcisrhedannssgetr,eactwheii.gtlyhdrmjeunosntshedyo·e.uslcdrihbaevde.

354 • CHAPTER 15 Building Measurement Concepts

T he followi ng at1:ivity builds fromchI.1u1 cn,5 experiences using various strateg'l·es to fi nd
a difference.

".' Activity 15.19 CCSS-M· 2,MD.c.a

How Much Is the Change?

Write a target number on the board that Is an amount of money that might be given to a store
clerk for a purchase, most likely 25, 50, 75, or 100. To the left of this target, write a smaller
starting number (the purchase price) and an arrow. Here are some examples:

13 -+ 25 56-+ 75 29-+50

In creating the amounts for this activity, think In terms of purchases. If the target Is 75, that
means you gave the clerk 75¢. You would only do this for Items costing more than 50¢.
Similarly, for a target of 50, use numbers greater than 25. For a target of 100, any smaller
number would be appropriate because you may have given the clerk a dollar bill.

Embed the numbers in story problems that describe making a purchase. Explain to the
children that the first number written represents the amount of the purchase and the second
number represents the amount of money given to the store clerk. The children 's task Is to
find the difference (I.e., the change) using a strategy that they can explain. They may need
to write down Intermediate results. Discuss the solution methods used by different children.

When your children are ready, extend to target values greater than a dollar.

T he next activity extends "How Much Is the Change?" and attempts to drawchildren's
attention to the notion of using the fewest possible coins.

Activity 15.20 CCSS...M: 2.MD.C.8

The Fewest Coins

On the board, write start and target numbers as In ' How Much Is the Change? ' Then write
on the board the values of the coins: 25, 10, 5, 1. Children must use only the numbers (i.e.,
coins) In the list to create the difference. As they use a number, they should write It down.
Challenge children to try to use as few "coins• as possible or, In other words, as many of the
larger numbers as possible. For example, If the target Is 75 with a start of 58,they would write
1, 1, 10, 5. Have children discuss their solutions.

The Coin Box applet found at Illuminations on NCTM's website poses tasks that allow
children to count coins, collect a given amount of money, exchange a given collection
of coins for the fewest coins, and make change. There are two options to display
coins-one that uses only the pictures of coins and another that places the co in onto
a grid that indicates its value (e.g., a nickel is displayed on a 1 by 5 grid; a dime is
displayed on a 2 by 5 grid). The National Library of Virtual Manipulatives applet called
"Money" uses coins and bills. Children can count the money displayed, pay an exact
amount, and make a dollar by dragging coins into a box. These applets are good
tasks for math centers where children can practice skills related to counting money.

figore 16.9 Shapes and Properties ,_ 379

four types of tangram puzzles ill ustratc Fit all seven
, range uf difficulty level tangram pieces
in this shape.
S.

[SJ

To make Full -sized
outlines

Full-sized
outlines

Fit in the
tangram pieces.

Each of these Outlines are

Dog shapes can be made to scale but

using all seven pieces. much smaller.

fi v6eTdi'heuereIrveanlut eanogfleths e(wmhoischaiccopuulzdzlleeaids tthoatd1'tshceusssew.t ncsonatbaom.ust
·
m~nna _angle comparisons and groupings of angle measures
Many pattern block designs and artwork created by
sue as right, acute, and obtuse). children will have an element of symmetry in them.
Although symmetry is not a core idea in the Common
Core State Standards for grades K-i, you can use
children's creations to introduce this idea informally.

"Patch Tool" at NCTM's Illuminations site is an interactive environment where children
c~n compose shapes using different pattern blocks. It also provides five outlines of
different designs where children have to decide which shapes are used to complete
th_e designs. Also, the National Library of Virtual Manipulatives has a tangram applet
With a set of fourteen puzzle figures that can be made using all seven of the pieces.
Thee-version of tangrams has the benefit of requiring the child to be much more

.___ - - --deliberate in arranging the shapes because they have to anticipate how they will
transform (e .g ., flip, rotate) a given shape to make it fit.

The geoboard is one of the best devices for "constructing" two-dimensional shapes. Here
adrejust a few of the many possible activi ti es appropriate fo r thinking about composing and
eco111posi.ng shapes using a geoboard•