Helping ng All Children en Become e Part of f the Whole
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Tr Trish sh Al Alex exander nder: : ◦ 19 years s experi erience ence teachin hing g middl dle e school ool math ◦ Curri urricul culum um team for r 15 years rs ◦ Ta Table ble Le Leader er for state e assessme ssment nt trainin aining g Joann nn Ba Barnett: ett: ◦ 28 years s experi erience ence teachin hing g middl dle e school ool math ◦ Help lped ed deve velo lop p EMS S course rse work rk and presently ently facilit cilitating ting Elementary entary Math Sp Specialist ialist program ogram at Misso sour uri i St State e Univer versi sity. y. ◦ Instruc tructor or of Deve velo lopmental pmental Math classes sses at Ozark k Comm mmuni unity ty Coll llege ege
Our challeng nge e is to design our lessons s so studen ents ts can articulat ate e important nt ideas. Tomlinson, C. A. (2014). Differentiated classroom: Responding to the needs of all learners . Ascd.
Stu tudents ents sh shou ould ld not ot si simply ly be e hea eard rd as t s the e teacher’s ec echo. o.
"St Stud udents ents joini ning ng your ur classroom ssroom have ve va vari ried d conception nceptions s of what it means s to do mathematic ematics. s. It is imp mporta ortant nt to establi blish sh imm mmediately diately that mathemat hematic ics s is about ut ideas, s, that all the students ents in the class s are capable ble of havi ving ng ideas s in mathemati matics, cs, and that all are expected cted to contrib tribute ute to the s. St Students udents must st deve velopmen lopment t of the class' ss' ideas. le learn rn that id ideas s need not be fu full lly y fo form rmed ed to be expres ressed; sed; tentative ative id ideas as provide vide im importa rtant nt begi ginnings nings." ." (P (Page e 10 10) Russell, Susan Jo, Deborah Schifter, and Virginia Bastable. Connecting arithmetic to algebra: Strategies for building algebraic thinking in the elementary grades. Heinemann, 2011.
The The big bigger ger the the num number er is is in in the the bot otto tom, m, the the sma smalle ller r it i it is. s.
An abil ilit ity to p perceiv ive the ordered pair ir in in a a fra ract ction ion symbo bol l as a c conce ceptual tual unit it (ra rather er than n two in indiv ividua idual l number bers) s) has been found nd to b be an in indic icato tor for successful ssful perf rform rmance ance wit ith ra ratio ional l numbe bers. rs. Behr, , Wachsmu muth, th, Post, and Lesh (1984) ask, “what meaning, for example, do 2/3 × 5/6 or or 2/ 2/3 + + 5/ 5/8 h have for chil ildren n who la lack a well ll-in inte ternal rnaliz ized ed concept of the big igness ss of rational numbers?” Chil ildren ren have dif ifficul iculty ty in internal aliz izing ing that the symbol ol for r a fr fract ctio ion n re repre resents ents a si singl gle entit ity. y. (RNP)
Kinder dergarten garten Whe hen n cou ount ntin ing g ob obje jects, ts, say the th e nu numb mber er na name mes s in n th the e sta tandard ndard or order, r, pairing ring each h ob obje ject ct wi with th on one an and on only on one nu number er na name and nd each h nu number er na name wi with th on one an and on only on one ob e obje ject. t.
Transition from Whole Number to Fraction Notation • Gr Grade 1 Partit ition ion and describ ibe e two a and four equal al shares es of cir ircle les and rectangl gles es …recognize that decomposing into more equal l shares s created d sim imil ilar fig igures es Gr Grade 2 •Partition and describe two, three or four equal l shares s of cir ircle les and rectangles, …recognize that equal shares es of id identica ical l whole les s need not have e the same shape
Wh Why ar y are we we able e to ma make two wo wh whole ci circles cles wi with h 4 2 yet we couldn’t even make 1 who ma whole ci circle cle wi with h ? 6 12
Look at all the 6’s in the two numbers belo low: 6 6 6 12 What does each 6 m mean, , and why do th the ci circ rcle le pie iece ces lo look so di differ ferent ent for r the two numb mber ers? s?
5 How can be equal l to mo more re 3 than 1 c cir ircle le whil ile is is le less than 1 w whole le 5 cir ircle le? ? 8
What do you know about the fractions below?
Ba Basi sically: cally: ◦ the he top nu number r is is the he co coun unt ◦ the he bottom nu number r is is wha hat you u ar are co e coun unti ting ng Activity adapted from John Van de Walle’s “Teaching Student -Centered Mathematics, Grade 3- 5.” Pearson Education, 2006. Pages 138 -140.
The connecti ection on betwe ween en the comparison rison of fracti tions ons and dev evel elopm opmen ent of numb mber er sen ense se is clear. . Comparing aring fractions ions is necessa ssary ry for obtaining ining an intuiti tive ve feel of the size of fracti tions. ons. If a fracti tional onal number ber is recogn gnized zed to be e close e to 1/ 1/3 3 or 1/ 1/2, 2, for ex examp mple, e, one e has s a better er feel for its magn gnitude. tude. This fracti tional onal number ber sense se is parti ticul cular arly y imp mportant rtant wh when en es estimati mating ng wi with fractions. ons. (Sowd wder er)
In support of this, Post et al. (1986) noted that “children who do not have a workable concept of rational number size cannot be expected to exhibit satisfactory performance across a set of tasks which varies the context in which the number concept of fraction is involved” (p. 2). Further, Behr, Wachsmuth, Post, and Lesh (1984) ask, “what meaning, for example, do 2/3 × 5/6 or 2/3 + 5/8 have for children who lack a well-internalized concept of the bigness of rational numbers?”
Wr Write e the followi wing g fractions ons in order r from lea east st to gr grea eates est. 5 9 3 12 12 12
3 5 9 12 12 12
Wr Write e the followi wing g fractions ons in order r from lea east st to gr grea eates est. 4 4 4 3 6 9
4 4 4 3 9 6
Wr Write e the followi wing g fractions ons in order r from lea east st to gr grea eates est. 3 5 2 6 7 5
1 2 2 3 5 5 6 7
Wr Write e the fo follow lowing ing fr frac actio ions ns in ord rder er from fr m le leas ast to gr great atest. st. 6 2 13 15 12 18
Wh Whic ich h Str Strate tegy gy? Hand nds s Up Up/Pair ir Up Up Rol olling ling Som Somethi thing ng Cl Clos ose
Whic ich Fra ractio ion n is is Gre reater ter Pa Pair ir up wit ith someone one near r you Game on…. Quiz each other Tra rade e papers rs Pa Pair ir up wit ith someone one new
11 4 12 5
Each ch time e a digi git t is rolle lled, d, place ce it in one of th the boxes. es. A digi git t that t is rol olle led d may y be placed ced in the SAVE E Stars ars at any point int durin ring g the ga game. e. After er we have ve rolled led all 14 digi gits, ts, the numb umbers ers in the SAVE E Stars rs may replace lace any two digi gits ts on the e number mber line ne IF doing ing so will ll help lp you create ate a more re accurate rate fraction. tion. Afterw erwards, ards, check k fractions tions with h partner tner and circle rcle any y fracti tion on yo you created ated on on the number mber line ine that has the appr propriate opriate va value. ue.
Ple lease se email il Jo Joann n pers rsonally ally at the email il above e if if you would ld li like a co copy of the comparing ing fractio ions ns activ ivit ity y usin ing the Hands Up Up/Pair ir Up Up activ ivity ity .
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