5 th grade Common Core Standards

Transcription

1 5 th grade Cmmn Cre Standards 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. Grade 5 Overview Operatins and Algebraic Thinking Write and interpret numerical expressins. Analyze patterns and relatinships. Number and Operatins in Base Ten Understand the place value system. hundredths. Perfrm peratins with multi-digit whle numbers and with decimals t Number and Operatins Fractins Use equivalent fractins as a strategy t add and subtract fractins.

2 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. 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. Write and interpret numerical expressins. 1. Use parentheses, brackets, r braces in numerical expressins, and evaluate expressins with these symbls. 2. 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 (8 + 7). Recgnize that 3 ( ) is three times as large as , withut having t calculate the indicated sum r prduct. Analyze patterns and relatinships. 3. 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 terms in the ther sequence. Explain infrmally why this is s.

3 Understand the place value system. 1. 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. 2. 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 Read, write, and cmpare decimals t thusandths. Read and write decimals t thusandths using base-ten numerals, number names, and expanded frm, e.g., = (1/10) + 9 (1/100) + 2 (1/1000). Cmpare tw decimals t thusandths based n meanings f the digits in each place, using >, =, and < symbls t recrd the results f cmparisns. 4. Use place value understanding t rund decimals t any place. Perfrm peratins with multi-digit whle numbers and with decimals t hundredths. 5. Fluently multiply multi-digit whle numbers using the standard algrithm. 6. Find whle-number qutients f whle numbers with up t fur-digit dividends and twdigit 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. 7. 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 Use equivalent fractins as a strategy t add and subtract fractins. 1. 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.) 2. 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. Apply and extend previus understandings f multiplicatin and divisin t multiply and divide fractins.

4 3. 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? 4. 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 visual fractin mdel t shw (2/3) 4 = 8/3, and create a stry cntext fr this equatin. D the same with (2/3) (4/5) = 8/15. (In general, (a/b) (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. 5. 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. 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 = (n a)/(n b) t the effect f multiplying a/b by Slve real wrld prblems invlving multiplicatin f fractins and mixed numbers, e.g., by using visual fractin mdels r equatins t represent the prblem. 7. Apply and extend previus understandings f divisin t divide unit fractins by whle numbers and whle numbers by unit fractins. 1 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) 4, and use a visual fractin mdel t shw the qutient. Use the relatinship between multiplicatin and divisin t explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. Interpret divisin f a whle number by a unit fractin, and cmpute such qutients. Fr example, create a stry cntext fr 4 (1/5), and use a visual fractin mdel t shw the qutient. Use the relatinship between multiplicatin and divisin t explain that 4 (1/5) = 20 because 20 (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? 1 Students able t multiply fractins in general can develp strategies t divide fractins in general, by reasning abut the relatinship between multiplicatin and divisin. But divisin f a fractin by a fractin is nt a requirement at this grade.

5 Cnvert like measurement units within a given measurement system. 1. 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 multistep, real wrld prblems. Represent and interpret data. 2. 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. Gemetric measurement: understand cncepts f vlume and relate vlume t multiplicatin and t additin. 3. Recgnize vlume as an attribute f slid figures and 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. 4. Measure vlumes by cunting unit cubes, using cubic cm, cubic in, cubic ft, and imprvised units. 5. 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 whle-number 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. Apply the frmulas V = l w h and V = b 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.

6 Mathematics» Grade 5» Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 1. 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). 2. 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. Classify tw-dimensinal figures int categries based n their prperties. 3. 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. 4. Classify tw-dimensinal figures in a hierarchy based n prperties.