Mendham Township School District Mathematics Curriculum Grade 5 General & Advanced

Transcription

1 Mendham Twnship Schl District Mathematics Curriculum Grade 5 General & Advanced In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin f fractins and f divisin f fractins in limited cases (unit fractins divided by whle numbers and whle numbers divided by unit fractins); (2) extending divisin t 2-digit divisrs, integrating decimal fractins int the place value system and develping understanding f peratins with decimals t hundredths, and develping fluency with whle number and decimal peratins; and (3) develping understanding f vlume. 1. Students apply their understanding f fractins and fractin mdels t represent the additin and subtractin f fractins with unlike denminatrs as equivalent calculatins with like denminatrs. They develp fluency in calculating sums and differences f fractins, and make reasnable estimates f them. Students als use the meaning f fractins, f multiplicatin and divisin, and the relatinship between multiplicatin and divisin t understand and explain why the prcedures fr multiplying and dividing fractins make sense. (Nte: this is limited t the case f dividing unit fractins by whle numbers and whle numbers by unit fractins.) 2. Students develp understanding f why divisin prcedures wrk based n the meaning f base-ten numerals and prperties f peratins. They finalize fluency with multi-digit additin, subtractin, multiplicatin, and divisin. They apply their understandings f mdels fr decimals, decimal ntatin, and prperties f peratins t add and subtract decimals t hundredths. They develp fluency in these cmputatins, and make reasnable estimates f their results. Students use the relatinship between decimals and fractins, as well as the relatinship between finite decimals and whle numbers (i.e., a finite decimal multiplied by an apprpriate pwer f 10 is a whle number), t understand and explain why the prcedures fr multiplying and dividing finite decimals make sense. They cmpute prducts and qutients f decimals t hundredths efficiently and accurately. 3. Students recgnize vlume as an attribute f three-dimensinal space. They understand that vlume can be measured by finding the ttal number f same-size units f vlume required t fill the space withut gaps r verlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit fr measuring vlume. They select apprpriate units, strategies, and tls fr slving prblems that invlve estimating and measuring vlume. They decmpse three-dimensinal shapes and find vlumes f right rectangular prisms by viewing them as decmpsed int layers f arrays f cubes. They measure necessary attributes f shapes in rder t determine vlumes t slve real wrld and mathematical prblems. Based upn NJSLS 2016

2 Grade 5 Overview (Mad Minute multiplicatin / divisin / fractins decimal cnversins) Operatins and Algebraic Thinking Write and interpret numerical expressins. Analyze patterns and relatinships. Number and Operatins in Base Ten Understand the place value system. Perfrm peratins with multi-digit whle numbers and with decimals t hundredths. Number and Operatins Fractins Use equivalent fractins as a strategy t add and subtract fractins. Apply and extend previus understandings f multiplicatin and divisin t multiply and divide fractins. Measurement and Data Cnvert like measurement units within a given measurement system. Represent and interpret data. Gemetric measurement: understand cncepts f vlume and relate vlume t multiplicatin and t additin. Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. Classify tw-dimensinal figures int categries based n their prperties. Based upn NJSLS 2016

3 Mathematical Practices Make sense f prblems and persevere in slving them Reasn abstractly and quantitatively Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls strategically Attend t precisin Lk fr and make use f structure Lk fr and express regularity in repeated reasning. CCCS CCCS# Cmp & Cntent Skills Cncepts Operatins and Algebraic Thinking 5OA Write and interpret numerical expressins. Use parentheses, brackets, r braces in numerical expressins, and evaluate expressins with these symbls Write simple expressins that recrd calculatins with numbers, and interpret numerical expressins withut evaluating them. Fr example, express the calculatin add 8 and 7, then multiply by 2 as 2 X (8 + 7). Recgnize that 3 X ( ) is three times as large as , withut having t calculate the indicated sum r prduct. Analyze patterns and relatinships Generate tw numerical patterns using tw given rules. Identify apparent relatinships between crrespnding terms. Frm rdered pairs cnsisting f crrespnding terms frm the tw patterns, and graph the rdered pairs n a crdinate plane. Fr example, given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences, and bserve that the terms in ne sequence are twice the crrespnding 5OA1 Test2#8-11 5OA2 Test2#12 5OA3 Test1#15-17 Math Symbls ( parentheses, brackets, and braces) Duble means twice as much Triple means 3X as much Based upn NJSLS 2016 Evaluating expressins Write simple expressins t represent calculatins f numbers Evaluate the value f expressin thrugh inspectin Graphing pints in crdinate plane Create patterns using tw rules Identify the rule assciated with a specific pattern Using rder f peratins t simplify expressins

4 terms in the ther sequence. Explain infrmally why this is s. Number and Operatins in Base Ten 5NBT Understand the place value system Recgnize that in a multi-digit number, a digit in ne place represents 10 times as much as it represents in the place t its right and 1/10 f what it represents in the place t its left. Explain patterns in the number f zers f the prduct when multiplying a number by pwers f 10, and explain patterns in the placement f the decimal pint when a decimal is multiplied r divided by a pwer f 10. Use whle-number expnents t dente pwers f 10. Read, write, and cmpare decimals t thusandths Read and write decimals t thusandths using base-ten numerals, number names, and expanded frm, e.g., = 3 X X X X (1/10) + 9 X (1/100) + 2 X (1/1000). Cmpare tw decimals t thusandths based n meanings f the digits in each place, using >, =, and < symbls t recrd the resultsf cmparisns 5NBT1 Test 1#4-9 5NBT2 Test1#9, 10, 11 5NBT3 Test 1 #12 5NBT3A Test 1 #13 5NBT3B Test 1# 14 Names f each place in a number Identifying numbers are pwers f 10 Definitin f expnents Reading decimals t thusandths Writing decimals t thusandths Symbls: >, =, < Cmparing decimals t thusandths Represent numbers in expanded frm using base-ten numerals and number names Cmpare 2 decimals t thusandths using symbls Each place is 10x greater than each place t its right + 1/10 th f each place t its left Explain patterns when multiplying r dividing by pwers f 10 as: 1) mvement f decimal pint 2) additin f zers Use place value understanding t rund decimals t any place 5NBT4 Test2#18-20 Runding decimals t any place Perfrm peratins with multi-digit whle numbers and with decimals t hundredths. 5NBT5 Test 1 #2 Fluently multiply multi-digit whle numbers using the standard algrithm. Find whle-number qutients f whle numbers with up t fur-digit dividends and tw-digit divisrs, using strategies based n place value, the prperties f peratins and/r the relatinship between multiplicatin and divisin. Illustrate and explain the calculatin by using equatins, rectangular arrays, and/r area mdels. Add, subtract, multiply, and divide decimals t hundredths, using cncrete mdels r drawings and strategies based n place value, prperties f peratins, and/r the relatinship between additin and subtractin; relate the strategy t a written methd and explain the reasning used. 5NBT6 Test 1#1,3 5NBT7 Test2#13-17 Number and Operatins Fractins 5NF Use equivalent fractins as a strategy t add and subtract fractins Multiply multi-digit whle numbers using the standard algrithm Divide whle numbers up t 4- digit dividends & 2- digit divisrs Illustrate & explain the calculatin by using equatins, rectangular arrays and/r area mdels Add, subtract, multiply and divide decimals using cncrete mdels and standard algrithms Explain reasning used Illustrate & explain the calculatin by using equatins, rectangular arrays and/r area mdels Based upn NJSLS 2016

5 Add and subtract fractins with unlike denminatrs (including mixed numbers) by replacing given fractins with equivalent fractins in such a way as t prduce an equivalent sum r difference f fractins with like denminatrs. Fr example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) Slve wrd prblems invlving additin and subtractin f fractins referring t the same whle, including cases f unlike denminatrs, e.g., by using visual fractin mdels r equatins t represent the prblem. Use benchmark fractins and number sense f fractins t estimate mentally and assess the reasnableness f answers. Fr example, recgnize an incrrect result 2/5 + 1/2 = 3/7, by bserving that 3/7 < 1/2. 5NF1 Test 3#1-4 5NF2 Test3# 5-7 Knwledge f equivalent fractins Cmmn denminatr = LCM f denminatrs Benchmark fractins 0, ½, 1 Add and subtract fractins & mixed numbers with unlike denminatrs by writing equivalent fractins w/a cmmn denminatr Slve wrd prblems by additin and subtractin f fractins w/unlike denminatrs by using mdels Eestimate sums & differences f fractins by using benchmark fractins Apply and extend previus understandings f multiplicatin and divisin t multiply and divide fractins. Interpret a fractin as divisin f the numeratr by the denminatr (a/b = a b). Slve wrd prblems invlving divisin f whle numbers leading t answers in the frm f fractins r mixed numbers, e.g., by using visual fractin mdels r equatins t represent the prblem. Fr example, interpret 3/4 as the result f dividing 3 by 4, nting that 3/4 multiplied by 4 equals 3, and that when 3 whles are shared equally amng 4 peple each persn has a share f size 3/4. If 9 peple want t share a 50-pund sack f rice equally by weight, hw many punds f rice shuld each persn get? Between what tw whle numbers des yur answer lie? Apply and extend previus understandings f multiplicatin t multiply a fractin r whle number by a fractin. Interpret the prduct (a/b) q as a parts f a partitin f q int b equal parts; equivalently, as the result f a sequence f peratins a q b. Fr example, use a 5NF3 Test3 #8-10 5NF4 Test 3 #28 5NF4A Test 3 #12 A fractin is same as the qutient f numeratr & denminatr Interpret a fractin as divisin Slve wrd prblems in which divisin f whle number leads t fractin r mixed number qutients Multiply a fractin by a fractin Multiply a fractin by a whle number Create a stry cntext fr multiplicatin f: 1) fractin with fractin and Based upn NJSLS 2016

6 visual fractin mdel t shw (2/3) X 4 = 8/3, and create a stry cntext fr this equatin. D the same with (2/3) X (4/5) = 8/15. (In general, (a/b) X (c/d) = ac/bd.) Find the area f a rectangle with fractinal side lengths by tiling it with unit squares f the apprpriate unit fractin side lengths, and shw that the area is the same as wuld be fund by multiplying the side lengths. Multiply fractinal side lengths t find areas f rectangles, and represent fractin prducts as rectangular areas. Interpret multiplicatin as scaling (resizing), by: Cmparing the size f a prduct t the size f ne factr n the basis f the size f the ther factr, withut perfrming the indicated multiplicatin. 5NF4B Test #13-14, NF5A Test3 #8 Definitin f area f a rectangle Finding area using fractin files Definitin f: 1) scale factr 2) dubling 3) tripling Finding area by multiplying side lengths f sides Cmparing size f prduct t size f the scale factr used withut multiplying 2)fractin with a whle number Area is equal t number f square units enclsed in a 2-D figure Derive area frmula f rectangle A = l x w Understanding the impact n prduct (enlarge/reduce by size f sale factr) i.e.: if the number is greater than ne it will enlarge the number, (>1 =enlarge) if the number is greater than zer and less than 1 it will reduce the number ( 0<x <1= reduce ) based n the scale factr used Explaining why multiplying a given number by a fractin greater than 1 results in a prduct greater than the given number (recgnizing multiplicatin by whle numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fractin less than 1 results in a prduct smaller than the given number; and relating the principle f fractin equivalence a/b = (nxa)/(nxb) t the effect f multiplying a/b by 1. Slve real wrld prblems invlving multiplicatin f fractins and mixed numbers, e.g., by using visual fractin mdels r equatins t represent the prblem. Apply and extend previus understandings f divisin t divide unit fractins by whle numbers and whle numbers by unit fractins Interpret divisin f a unit fractin by a nn-zer whle number, and cmpute such qutients. Fr example, create a stry cntext fr (1/3) x 4, and use a visual fractin mdel t shw the qutient Use the relatinship between multiplicatin and divisin t explain that (1/3) x 4 = 1/12 because (1/12) x 4 = 1/3. 5NF5B Test 3 #15 5NF6 Test3 #11 5NF7 Test #19,20 5NF7A Test 3#31 Explain why the prduct is greater when scale factr >1 and smaller when scale factr between 0 and 1 Definitin f mixed # Slve real wrld prblems invlving multiplicatin f fractins and mixed numbers by using mdels r equatins Definitin f 1) unit fractins 2) whle numbers Based upn NJSLS 2016 Divide unit fractins by whle numbers and whle numbers by unit fractins Create a stry cntext fr divisin f unit fractin by a nn-zer whle number and Calculate divisin f a nn-zer whle number by a unit fractin Understand that the prcess f writing equivalent fractins (multiplicatin & divisin f numeratr & denminatrs by same numbers) is same as being multiplied r divided by 1

7 Interpret divisin f a whle number by a unit fractin, and cmpute such qutients. Fr example, create a stry cntext fr 4 x (1/5), and use a visual fractin mdel t shw the qutient. Use the relatinship between multiplicatin and divisin t explain that 4 x (1/5) = 20 because 20 x (1/5) = 4. Slve real wrld prblems invlving divisin f unit fractins by nn-zer whle numbers and divisin f whle numbers by unit fractins, e.g., by using visual fractin mdels and equatins t represent the prblem. Fr example, hw much chclate will each persn get if 3 peple share 1/2 lb f chclate equally? Hw many 1/3-cup servings are in 2 cups f raisins? 5NF7B Test 3 #21 5NF7C Test 3#30 Create a stry cntext fr: divisin f a whle number by a unit fractin Calculate qutient f whle number by unit fractin Slve real wrld prblems invlving divisin f unit fractins & whle numbers & whle numbers by unit fractins by using fractin mdels and equatins Measurement and Data 5MD Cnvert like measurement units within a given measurement system Cnvert amng different-sized standard measurement units within a given measurement system (e.g., cnvert 5 cm t 0.05 m), and use these cnversins in slving multi-step, real wrld prblems. Represent and interpret data Make a line plt t display a data set f measurements in fractins f a unit (1/2, 1/4, 1/8). Use peratins n fractins fr this grade t slve prblems invlving infrmatin presented in line plts. Fr example, given different measurements f liquid in identical beakers, find the amunt f liquid each beaker wuld cntain if the ttal amunt in all the beakers were redistributed equally. 5MD1 Assessed in Science Benchmark 5MD2 Test3#22-24 Standard units f measure including metric & custmary Line plt Cnvert frm ne unit f measure t anther within a given measurement system Use cnversins t slve multistep real wrld prblems Making a line plt using fractins f a unit Using peratins with fractins t slve prblems presented in line plts Gemetric measurement: understand cncepts f vlume and relate vlume t multiplicatin and t additin Recgnize vlume as an attribute f slid figures and 5MD3 Vlume measurement tls Understand vlume as the # f Based upn NJSLS 2016

8 understand cncepts f vlume measurement A cube with side length 1 unit, called a unit cube, is said t have ne cubic unit f vlume, and can be used t measure vlume. A slid figure which can be packed withut gaps r verlaps using n unit cubes is said t have a vlume f n cubic units. Measure vlumes by cunting unit cubes, using cubic cm, cubic in, cubic ft, and imprvised units. Relate vlume t the peratins f multiplicatin and additin and slve real wrld and mathematical prblems invlving vlume. Find the vlume f a right rectangular prism with whlenumber side lengths by packing it with unit cubes, and shw that the vlume is the same as wuld be fund by multiplying the edge lengths, equivalently by multiplying the height by the area f the base. Represent threefld whle-number prducts as vlumes, e.g., t represent the assciative prperty f multiplicatin Test3#26 5MD3A Test3#26 5MD3B Test3#26 5MD4 Test3#26 5MD5 Test 2 #7 5MD5A 1x1x1 cube is a unit cube and is said t have ne cubic unit f vlume and is used t measure vlume A slid figure which can be packed w/ gaps r verlays using n unit cubes is said t have a vl. f n cubic units Units f vlume: Cubic centimeter Cubic inch Cubit feet Assciate prperty Cumulative prperty Measure vlume by cunting unit cubes then assigning the apprpriate unit f vlume (fr ex 3 unit cubes = 3 cm cubed) Slve real wrld & mathematical prblems invlving vlume Calculate vlume f a right rectangular prism by using unit cubes & then by multiplying edge lengths cubic units cntained in a 3-D figure Relate vlume t peratins f x & / & derive vl. frmula fr right rect. prism Vlume is calculated by multiplying the area f the base times the height Apply the frmulas V = l x w x h and V = b x h fr rectangular prisms t find vlumes f right rectangular prisms with whle number edge lengths in the cntext f slving real wrld and mathematical prblems. Recgnize vlume as additive. Find vlumes f slid figures cmpsed f tw nn-verlapping right rectangular prisms by adding the vlumes f the nn-verlapping parts, applying this technique t slve real wrld prblems. Gemetry 5G 5MD5B Test1#21-24 Test2#2,3,5,6 5MD5C Test2, #4,6 Test3#25,27-29 Graph pints n the crdinate plane t slve real-wrld and mathematical prblems Use a pair f perpendicular number lines, called axes, t define a crdinate system, with the intersectin f the lines (the rigin) arranged t cincide with the 0 n each line and a given pint in the plane lcated by using an rdered pair f numbers, called its crdinates. Understand that the first number indicates hw far t travel frm the rigin in the directin f ne axis, and the secnd number indicates hw far t travel in the directin f the secnd axis, with the cnventin that the names f the tw axes and the crdinates crrespnd (e.g., x- axis and x-crdinate, y-axis and y-crdinate). 5G1 Test 1 # 18 Test 1 # 19 Test 1 # 20 Define crdinate plane parts (axes, rigin, rdered pair, crdinates, x,y) Based upn NJSLS 2016 Apply vlume frmulas l x w x h and B x h fr rectangular prisms w/whle # edge lengths in cntext f real wrld & mathematical prblems Understand the meaning f x,y crdinates as far as mvement n the axis

9 Represent real wrld and mathematical prblems by graphing pints in the first quadrant f the crdinate plane, and interpret crdinate values f pints in the cntext f the situatin 5G2 Test 1 # 18 Test 1 # 19 Test 1 # 20 Graphing pints in the first quadrant f the crdinate plane Interpret crdinate values f pints f the cntext f the situatin Classify tw-dimensinal figures int categries based n their prperties Understand that attributes belnging t a categry f tw dimensinal figures als belng t all subcategries f that categry. Fr example, all rectangles have fur right angles and squares are rectangles, s all squares have fur right angles Classify tw-dimensinal figures in a hierarchy based n prperties 5G3 Test2# G4 Test2#21-25 Classify tw-dimensinal figures in a hierarchy based n prperties Understand that attributes belnging t a categry f tw dimensinal figures als belng t all subcategries f that categry. Fr example, all rectangles have fur right angles and squares are rectangles, s all squares have fur right angles Based upn NJSLS 2016