Figuring Out Fluency - Multiplication and Division With Fractions and Decimals: A Classroom Companion

225 Pages·2022·28.821 MB·English

by  Jennifer M. Bay-Williams, John J. SanGiovanni, Sherri L. Martinie, Jennifer Suh

WHAT YOUR COLLEAGUES ARE SAYING . . .
“This book places mathematical thinking front and center! The authors make
the compelling case using myriad examples that fluency requires reason-
ing and relies on conceptual understanding. They provide a treasure trove of
engaging activities teachers can use to help their students develop fluency with
fractions and decimals. If you want your students to confidently choose and
flexibly use a range of strategies when working with fractions and decimals,
you must read this book!”
Grace Kelemanik
Co-Founder, Fostering Math Practices
“As a high school mathematics teacher, I listened year after year to students
who said they lost their passion for mathematics right around the time they
began to learn fractions. Thankfully, Figuring Out Fluency—Multiplication and
Division With Fractions and Decimals is a book designed to make fractions more
accessible and understood by all students. The book is a must-read for mathe-
matics educators. This book clearly defines fluency and also shares action steps
and strategies teachers can employ to help students develop a deeper under-
standing of, and fluency with, fractions and decimals.”
Mona Toncheff
President, National Council of Supervisors of Mathematics (2019–2021)
Preservice Teacher Supervisor, University of Arizona
Author and Educational Consultant
Former District Mathematics Content Specialist and High School
Mathematics Teacher
“This book is such an eye-opener for providing interventions and strategies for
students and supporting teachers that I coach. Fluency isn’t about timed tests
or ‘drill and kill’ but providing students with rich discourse, games, tasks, rou-
tines, and more. I’m so excited to delve into this much-needed resource more.”
Tomika R. Altman
Math/Science Specialist, Seawell Elementary School
Chapel Hill-Carrboro City Schools
“Figuring Out Fluency—Multiplication and Division With Fractions and Decimals is an
essential resource for math educators who teach fraction and decimal opera-
tions. This companion book connects the strategies introduced in the anchor
book to multiplying and dividing with decimals and fractions. The explicit
strategy instruction provided in the book allows access for ALL students in a
practical, hands-on approach. This is a go-to for all math educators, coaches,
specialists, and interventionists!”
Nichole DeMotte
K–5 Mathematics Coach, Atkinson Academy
Jefferson County Public Schools, Louisville, KY
“A needed addition to mathematics education! This book establishes a
broader definition of fluency and emphasizes the importance of helping
students develop and use relationships and strategies to solve problems.
Bay-Williams, SanGiovanni, Martinie, and Suh outline which major rela-
tionships and strategies students need to develop stronger mathematical
reasoning!”
Pam Harris
Texas State University
Owner of Math is Figureoutable
“Figuring Out Fluency: Multiplication and Division With Fractions and Decimals is the
resource that the educational world has been wishing and waiting for as it
works to tackle mathematical concepts that traditionally leave students con-
fused and teachers frustrated. This carefully crafted guide will inspire and
support educators in their journey to integrate quality and rigorous fluency
practice. It provides a plethora of practical, well-designed activities and games
rooted in research that will actively engage students in rich discussion, criti-
cal thinking, and the strengthening of their conceptual understanding needed
to select appropriate strategies to multiply and divide fractions and decimals
accurately and efficiently. As an instructional leader, I view this text as the cat-
alyst for refocusing the goals of math instruction so both teachers and students
can confidently engage in practices that build mathematical proficiency.”
Allyson Lyman
Elementary Principal, Emporia Public Schools
“Figuring Out Fluency—Multiplication and Division With Fractions and Decimals is a
must-have for any classroom teacher, mathematics supervisor, or anyone that
wants to think about advancing their students’ fluency with fractions and deci-
mals. The ideas in this book are supported through the key ideas of figuring out
fluency and the detailed ways of its strategy model of looking at fractions and
decimals. The numerous resources that are highlighted in this book support
anyone from a new teacher all the way to a 25-year veteran. The ease of use is
what will help anyone improve their practice.”
Spencer Jamieson
Past President, Virginia Council of Mathematics Supervision
Mathematics Specialist, Virginia Council of Teachers of Mathematics
Mathematics Educational Specialist, Fairfax County Public Schools
Figuring Out Fluency—Multiplication
and Division With Fractions and
Decimals: A Classroom Companion
The Book at a Glance
Building off of Figuring Out Fluency, this classroom
companion dives deep into five of the Seven
Significant Strategies that relate to procedural
fluency when multiplying and dividing fractions
and decimals, beyond basic facts.
FIGURE 10 Reasoning Strategies for Multiplying and Dividing Fractions and Decimals
REASONING STRATEGIES RELEVANT OPERATIONS
1. Break Apart to Multiply (Modules 2 and 4) Multiplication
2. Compensation (Module 3) Multiplication
3. Halve and Double (Module 4) Multiplication
4. Partial Products (Module 2) and Partial Quotients Multiplication and Division
(Module 6)
5. Think Multiplication (Module 5) Division
TEACHING TAKEAWAY
Sometimes representations can be confused with strategies. For example, if
a student multiplies using an area model, they may be implementing a Break Representations are
not strategies. When
Strategy overAvpiaerwt sst raatnegdy foar ma Ciloym bpreniesfast iocno smtrmateugny,i caamtoen gh ootwhe resa. Wchh esnt raa stteugdyen ht sealypss,
a student says, “I used
students deve“lIo ups flede xainb ialrietya, mefofidceile,”n acsyk, haocwcu thraecyy u, saeudt oitm—tahteicni tyyo,u a wndil lr leeaasronn wahbalet nstersast-. an area model,” ask
egy they used. Using base-10 pieces, fraction circles, pattern blocks, Cuisenaire H ow did you use it?
rods, or number lines is a way to represent one’s thinking. The thinking is to learn what strategy
the strategy. they selected.
Compensation
MODULE HOW DO I TEACH, PRACTICE, AND ASSESS
3
SCTomRApTSeEnGsTaYt iORonV EARVTIEWE: GIES? trehlea tperdo bstleamnd maredn atlaglolyr, iothr mit .m ay lend to a written method but one that is less messy than the MODULE 1Foundations for Reasoning Strategies
Wnw2ins3uu h4h.mmo 3aClbb7teh .ee ia nsWrrn,. uC gtIimtotre halmb oct EwMfeohkprraikiensscsxn ,gt ttlsho iiehhkoa 2eeontp 0 iwstsroe hta× nmrilna rs?2nd:tu i T3e cdih gc(hei4ya sct6 ihsiiis0mta m)trctt.aao Ttlc m sehh,i gsimatsyhn-s ao egitnnnW es svltdywronr aultevhatsree eeeisogsd a cty1 nwh b itsgeanhwrnveo.oegsnugr li v,npa Laegt nonsy o fudec enm2 hce3 aobr notmtieogfr,p noi ’tni esh smg neec sa lusno a cnustxuheuimtn. m tSgborpu nbef baoeru strrbl r tpttaeohoicn ca tacact a h 2m ncm3tehio caaaartirnernbkd ogk cy, eotl mwih.nk nWe ehv e ota1itnln9eh iehs e ×wan eta ert nt peshn “rdfauss lteloy e rwxephvlieecniatl seatd rs a CS Ibtthtoe iet stOpm rirarme raiMatnp ng otpeertaertPsyrae exgniEnnt t tyt Nghiopher na BstSpyc rfuhsarAromes iuotieiTillt isnre sfoesIfs uwusO f nsl uladfeecNeeotntrt csedr:tyrgia s. n ToFo dhrow e atsnshhemeao sr . setit r rtdaitaAhtlteeeihgm geyci oe absostr caip penaufdrsoe aktnrtntero ic awood vtn hofooeniawrl efn otnhg rc eedagysou. wiTnn ohgrete tkyh d osaao-rt e.t Yhaoavuta itclahabenly ei nc fcaolnur de MODULE 2Break Apart to Multiply (by Addends)
Example nm eastsh” e(m “ COEFMRaPxA 5EC×NtTp3SIi78OAclT NIiSOa cN:l i tr,”e l2a0t2i CoO1MDn )PE 7 8.EC.9 ×NI s ×M 3S I3.AAh5.n5 TL SIOi Npm: sa vthiseimblea.t iAcs s tteraatcehgyin igs,d oawbnl oeflad i seon thxagt yi obu e cxalne apdj ulsmit cthiemet atsm nheedeoed.a dn tso msoalkvien ag MODULE 3Compensation
Change problem. E 5 ×x 4 plicit s (Htalrvea antd eDogublye st riantegsy utserd uherce, tion, then, is engaging students in ways to clearly
M Coumltpipelnys asste terea the Tohge wainsew 2 e0sar− is58 2 5 n0= e× 1189md t83o o pmwuochh.w ye T rhaes a n sssw 2e8ttotr – it rs0uh 0.i3an 2.15k8d ×= 4t 32×ee.7 57.6 )t 5nog o mtyusc h.w, doervkesl.o Lpeianrgn ain pgo hsiotiwve t om uasteh eamnda twichse inde tnot cithyo aonsde ETxEpAliCciHtlIyN tGMODULE e 4aTcAhBreak Apart to KiMultiply nE(by Factors)gA aW AY
a sense of agency. This is a major shift from traditional teaching and assess- strategy does not mean
H TsuhbiOst rsaWtcrattioe Dng isyisO u nustsEeesrdgS t.h aeC, d tOiswteMribugPhtiEvyeNi pcr SopsAhereTty ,I jlOupsetN lrikc eWi btvreOiakioRinlgKne a?pga .rt einEtos aad dcsendhts, ab utsn int tdhris caaasetr edg ayl gmoroidthumle si na nPda rat c2c uorfafecrys otveaerc hfl ienxgi baiclittiyv iatineds tinutron ianng atlhgeoMODULE r5sittrhThink amtMultiplicatione.g y
EXAMPLE STUDENT THINKING WRITTEN RECORD
12×tIi3nmh78 tapot o aac T I S couhrambannrtttp rciaeseha cn 1 atsa2n a 1×gn8t2dee18 ,l .w3 tgSte78oho oi,ilc no 1hmsty2 oiu s× ric4 ,1 h4 ga 21i. n = .e td T4n ht8hxha. tee’nspm 4d621l . i .tc Soitt hlryae ttlepega sictehusid nreegnq atus si rutern afldt eeexgrsiybt daleno tedhs hi nnookrewsotoiunrcl ni nemesagn.eTmh duealstei p rnlweysdoi uvtrihdceeusfy rcaarcn t ibanoen dd ioeswcninmtloagarl .dea dt atth e reseogu rcyses.t cworrwaino.ctorem/kFgOsFy/. MODULE 6Partial Quotients
4.9 × SQ 0.2t5u uadl Iih 1e siw.a 21vti5.nle2l0 t5ty h–o. ti T0nsh u.k0sab o2pttf5 ri s at= hnc0r 1it.s1 .0 a2a ×.e2s0 0525c. 5 e2q.5 ut atdrotioe cm rsau, wechhc,i csohci I se sfso ctou sqeud a lointy ap rsatrcatitceeg yth, avta irsie NndOoT EtiSn a twhoer tkysphee eotf! See Chapter 6 (pp. 130–1MODULE 573) oStandard fAlgorithms for Fraction
a W Cbwoerma, yHpp a(eerEnt.gisNc.a,u t4li eda.o9Drn liy snmO i w1ad h tkgYeeenns,Ot siaht e Uianaswgs ae a n Cuywen fhHirteod mfnmOr ao c 5nOt)ce.i oe CSonrofe E antawh tneCian ytfgO r(a ne,tch.Mg tei.po, e w3 nPhs78r Eco oislreNo dt 18neSu ecacmiwAmbadeaTeyl rsf I rmsi osOt macsylN oo 4lse)ee? ao dt row dta ot th ebe neh bntinheg xoy atar b whlt uhen ott dloethr senodhultvmehe- e syt uardee nleta tron minagk. e sense of what they Fq iugualriitnyg p Orauctt Ficleu.e ncy foMODULE r8 moStandard rAlgorithms for eDecimal about
Each practice section begins with worked examples. Worked
6644 examples are opportunities for students to attend to the thinking Part 2 • Module 3 • Compensation 65
involved with a strategy, without solving the problem. We feature three types
to get at all components of fl uency:
1. Correctly worked example: effi ciency (selects an appropriate strategy) and
fl exibility (applies strategy to a new problem type)
Part 1 • Figuring Out Fluency 19
Each strategy module starts with
teaching activities that help you
explicitly teach the strategy.
TEACHING ACTIVITIES for Compensation
A key to fluently knowing when to choose a strategy is being able to fully use and understand it.
Compensation is not as widely used with fractions and decimals as it is with whole numbers. But
there are circumstances in which compensation leads to a very effi cient method to solve a problem.
Thus, the activities in this section focus fi rst on why the Compensation strategy works and then
on fi nding opportunities to use it. Being able to use the strategy and noticing when it is a good fi t
helps students to develop flexibility and effi ciency.
ACTIVITY 3.1
HOW MUCH TOO MUCH? FRACTION
CIRCLES (FRACTIONS)
In this activity, students compare the area of a changed, simpler problem from the original problem to see how much
too much there is. Pose the two similar problems and ask, “How much too much?” Here is an example:
Original equation: 4×27 Changed equation: 4 × 3
8
Using physical or virtual fraction circles, students build the original equation:

Ask, “How much too much in the changed expression?” Students can reason about this question or actually build the
changed problem and then compare visually:

Connect the visuals to language and to symbols: “We had 4 eighths too much, or 4 groups of one-eighth too much. To
compensate, we subtract 4 eighths, which is one-half.”
4×27=4×3−4×1
8 8
=12−1
2
=111
2
Repeat with other examples where there is a whole number and a fraction close to the next whole number. Adapt and
use this idea with other fraction manipulatives (physical or virtual).
66 Figuring Out Fluency—Multiplication and Division With Fractions and Decimals
Each strategy shares worked examples for
ACTIVITY 5.6 you to work through with your students
THINK MULTIPLICATION WORKED EXAMPLES
as they develop their procedural fluency.
Worked examples can be used as a warmup, as a focus of a lesson, at a learning center, or on an assessment. Here we
share examples and ideas for preparing worked examples to support student understanding of Think Multiplication.
There are diff erent ways to pose worked examples, and they each serve a diff erent fluency purpose.
TYPE OF WORKED EXAMPLE PURPOSES: COMPONENT QUESTIONS FOR DISCUSSIONS
(FLUENCY ACTIONS) OR FOR WRITING RESPONSE
Correctly Worked Example Efficiency (selects an appropriate What did do?
ssttrraatteeggyy) t aon ad n felewx ipbriolibtyle (map tpylpiees) a Why does it work?
Is this a good method for this
problem?
Partially Worked Example Efficiency (selects an appropriate Why did start the problem
strategy; solves in a reasonable this way?
a(ccomormoreupcnltet atoenfss t wsimteerep)) sa ancdc aucracuteralyc;y g ets Wfinhiasht dthoee sp roblem n? eed to do to
Incorrectly Worked Example Accuracy (completes steps What did do?
accurately; gets correct answer) What mistake does make?
How can this mistake be fixed?
The following are some challenges students may have with the Think Multiplication strategy, which can inform the
creation of worked examples:
1. After they restate the expression as a missing factor equation, students don’t know the answer and are “stuck.”
• 7 kn13o÷w23 w ish tarta tnos ldaote nde txot .“ How many 23 are in 7 13? ” But since the divisor is not a unit fraction, the student doesn’t
• 10.4 ÷ 0.8 is interpreted as “How many 8 tenths in 104 tenths?” Since that is not a basic fact, the strategy (or
problem) is abandoned.
2. a Ins s5o, ltvhineg s t 1u78d÷en83t , trheeco srtdusd e58n . t translates it to 185÷83=? and notices that 15 ÷ 3 = 5. Rather than record the answer
While there are worked examples throughout this module and you can use student responses from the prompts in
Activity 5.5, we off er a few ready-to-go ones here.
104 Figuring Out Fluency—Multiplication and Division With Fractions and Decimals
CORRECTLY WORKED EXAMPLES
Compare Susie and Tad’s methods for solving 3÷23 .
Susie’s work for 3÷23 : Tad’s work for 6 ÷ 0.4:

PARTIALLY WORKED EXAMPLES
How might you finish Henley’s start for 16 ÷21=? Sarah started 443÷83=? by finding out how many
eighths are in 443(468) :

Explain what Sarah’s equations mean.
Explain how to finish this problem.
Compare Lily and Peyton’s methods for solving 1.44 ÷ 0.4.
Lily’s start: Peyton’s start:

INCORRECTLY WORKED EXAMPLES
Explain how Kayla is thinking about the problem, find Tucker’s work for 6 ÷ 0.4:
the mistake that she needs to fix, and explain why it is
incorrect.

Part 2 • Module 5 • Think Multiplication 105
ACTIVITY 7.8
“Stand Up, Hand Up, Pair Up” Routine
Name: Type:
About the Routine: “Stand Up, Hand Up, Pair Up” is a physically active routine in which students stand up and look
for their match. This routine can be used for many purposes; in this case, the focus is on multiplication and division
as inverse operations.
Materials: set of “Stand Up, Hand Up, Pair Up” cards and recording sheet
Directions: 1. Give each student a card.
2. Students stand up, put their hands up, and start to mix and mingle, sharing their card and looking
for the inverse operation that matches their expression like this:
3÷31 3 × 3
q Thueosteie tnwt oa ncda rtdhse a prero mduatccth aered tbheec asuamsee d. ividing by 13 is the same as multiplying by 3; therefore, the
3. Once students fi nd a match, they stand together. The routine ends when all have found their match.
4. Ask pairs to talk about whether they need the standard algorithm to solve, or if they can use a
reasoning strate Agy.C TIVITY 7.10
5. Students record the pairs on their recording sheets along with their solutions. Routines, Games, and Centers
Roll to Win Game
Name: Type: for each strategy offer extensive
RESOURCE(S) FOR ATboHuItS th Ae GCaTmIeV: RIoTllY to Win helps students reason through how the placement of digits in the numerator or
denominator can yield the greatest or least product or quotient. The activity reinforces and uses estimation and opportunity for student practice.
knowing the relative size of fractions. This game can be played with two to four players.
Materials: four 10-sided dice (0 on the dice changed to 10) or digit cards; Roll to Win recording sheet
Directions: 1. On a player’s turn, they roll the four dice and construct an equation of two fractions, trying to win
each round.
2. Round 1: The greatest product.
3. Round 2: The least product.
4. Round 3: The greatest quotient.
5. Round 4: The least quotient.
6. The best out of four rounds wins the game. If it is a tie, Round 5 is choice operation—the greatest
result wins.
resoounrclinees These resources can be dRoEwSnlOoaUdeRd CatE re(Sso)u rFceOs.Rco rTwHinI.cSom A/FCOTF/ImVuIltTipYlydividefractiondecimal .
Part 2 • Module 7 • Standard Algorithms for Fraction Multiplication and Division 141
ACTIVITY 8.11
Partials to Algorithms Center
Name: Type:
About the Center: Connecting algorithms to other strategies, especially Partial Products and Partial Quotients
with and without the area model, helps students make sense of the algorithms. In this center, students are given a
problem completed with Partial Products with and without area models. Students then have to rewrite the problem
resoounrclinees This resource can be downloaded at resources.corwin.com/FOF/multiplydividefractiondeucsiimnga lt .he standard algorithm. An extension for this center is to task students to come up with cards.
Materials: Partials to Algorithms cards and recording sheet
Directions: 1. Students select a card that shows a problem completed with Partial Products with or without an
area model.
144 Figuring Out Fluency—Multiplication and Division With Fractions and Decimals
2. Then students solve the problem again with the standard algorithm.
3. Students tell which approach they thought was the better choice for the problem (and why).

Download the resources you RESOURCE(S) FOR THIS ACTIVITY
need for each activity at this
book’s companion website.
resoounrclinees These resources can be downloaded at resources.corwin.com/FOF/multiplydividefractiondecimal .
160 Figuring Out Fluency—Multiplication and Division With Fractions and Decimals
FIGURING OUT
Fluency
MULTIPLICATION & DIVISION
With Fractions and Decimals
Grades
4–8
A Classroom Companion
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FIGURING OUT
Fluency
MULTIPLICATION & DIVISION
With Fractions and Decimals
Grades
4–8
A Classroom Companion
Jennifer M. Bay-Williams
John J. SanGiovanni
Sherri Martinie
Jennifer Suh