Application of Percents

Transcription

1 Grade 7 Mathematics, Quarter 3, Unit 3.1 Applicatin f Percents Overview Number f instructinal days: 15 (1 day = minutes) Cntent t be learned Cnvert between different frms f ratinal numbers (psitive and negative decimals, fractins, and percents). Slve multistep rati and percent prblems invlving simple interest; tax, gratuities, and cmmissins; markups and markdwns; fees; percent increase and decrease; and percent errr. Slve prblems using numerical and algebraic expressins and equatins. Use mental cmputatin and estimatin strategies t assess the reasnableness f answers. Mathematical practices t be integrated Make sense f prblems and persevere in slving them. Explain the meaning f a prblem. Analyze relatinships between prprtinal quantities (part/whle relatinships). Check answers t prblems using a different methd, and cntinually ask, Des this make sense? Use apprpriate tls strategically. Cnsider the available tls when slving a mathematical prblem (including pencil and paper, cncrete mdels, calculatr, spreadsheet). Use estimatin and mental cmputatin t detect pssible errrs. Apply prprtinal reasning t make predictins. Attend t precisin. Cmmunicate precisely t thers. Calculate accurately and efficiently. Express numerical answers with a degree f precisin apprpriate fr the prblem cntext. 25

2 Grade 7 Mathematics, Quarter 3, Unit 3.1 Applicatin f Percents (15 days) Essential questins Hw can yu use a prprtin t slve prblems invlving percents? Hw can yu use an equatin t slve prblems invlving percents? What is the difference between finding the percent f a number and finding the percent f change (percent increase/decrease)? Hw d yu cnvert between fractins, decimals, and percents? Why wuld yu cnvert between different frms f ratinal numbers? Hw is estimatin with percents useful? Written Curriculum Cmmn Cre State Standards fr Mathematical Cntent Ratis and Prprtinal Relatinships 7.RP Analyze prprtinal relatinships and use them t slve real-wrld and mathematical prblems. 7.RP.3 Use prprtinal relatinships t slve multistep rati and percent prblems. Examples: simple interest, tax, markups and markdwns, gratuities and cmmissins, fees, percent increase and decrease, percent errr. Expressins and Equatins 7.EE Slve real-life and mathematical prblems using numerical and algebraic expressins and equatins. 7.EE.3 Slve multi-step real-life and mathematical prblems psed with psitive and negative ratinal numbers in any frm (whle numbers, fractins, and decimals), using tls strategically. Apply prperties f peratins t calculate with numbers in any frm; cnvert between frms as apprpriate; and assess the reasnableness f answers using mental cmputatin and estimatin strategies. Fr example: If a wman making $25 an hur gets a 10% raise, she will make an additinal 1/10 f her salary an hur, r $2.50, fr a new salary f $ If yu want t place a twel bar 9 3/4 inches lng in the center f a dr that is 27 1/2 inches wide, yu will need t place the bar abut 9 inches frm each edge; this estimate can be used as a check n the exact cmputatin. Cmmn Cre Standards fr Mathematical Practice 1 Make sense f prblems and persevere in slving them. Mathematically prficient students start by explaining t themselves the meaning f a prblem and lking fr entry pints t its slutin. They analyze givens, cnstraints, relatinships, and gals. They make cnjectures abut the frm and meaning f the slutin and plan a slutin pathway rather than simply jumping int a slutin attempt. They cnsider analgus prblems, and try special cases and simpler frms f the riginal prblem in rder t gain insight int its slutin. They mnitr and evaluate their prgress and change curse if necessary. Older students might, depending n the cntext f the 26

3 Grade 7 Mathematics, Quarter 3, Unit 3.1 Applicatin f Percents (15 days) prblem, transfrm algebraic expressins r change the viewing windw n their graphing calculatr t get the infrmatin they need. Mathematically prficient students can explain crrespndences between equatins, verbal descriptins, tables, and graphs r draw diagrams f imprtant features and relatinships, graph data, and search fr regularity r trends. Yunger students might rely n using cncrete bjects r pictures t help cnceptualize and slve a prblem. Mathematically prficient students check their answers t prblems using a different methd, and they cntinually ask themselves, Des this make sense? They can understand the appraches f thers t slving cmplex prblems and identify crrespndences between different appraches. 5 Use apprpriate tls strategically. Mathematically prficient students cnsider the available tls when slving a mathematical prblem. These tls might include pencil and paper, cncrete mdels, a ruler, a prtractr, a calculatr, a spreadsheet, a cmputer algebra system, a statistical package, r dynamic gemetry sftware. Prficient students are sufficiently familiar with tls apprpriate fr their grade r curse t make sund decisins abut when each f these tls might be helpful, recgnizing bth the insight t be gained and their limitatins. Fr example, mathematically prficient high schl students analyze graphs f functins and slutins generated using a graphing calculatr. They detect pssible errrs by strategically using estimatin and ther mathematical knwledge. When making mathematical mdels, they knw that technlgy can enable them t visualize the results f varying assumptins, explre cnsequences, and cmpare predictins with data. Mathematically prficient students at varius grade levels are able t identify relevant external mathematical resurces, such as digital cntent lcated n a website, and use them t pse r slve prblems. They are able t use technlgical tls t explre and deepen their understanding f cncepts. 6 Attend t precisin. Mathematically prficient students try t cmmunicate precisely t thers. They try t use clear definitins in discussin with thers and in their wn reasning. They state the meaning f the symbls they chse, including using the equal sign cnsistently and apprpriately. They are careful abut specifying units f measure, and labeling axes t clarify the crrespndence with quantities in a prblem. They calculate accurately and efficiently, express numerical answers with a degree f precisin apprpriate fr the prblem cntext. In the elementary grades, students give carefully frmulated explanatins t each ther. By the time they reach high schl they have learned t examine claims and make explicit use f definitins. Clarifying the Standards Prir Learning In grade 6, students used ratis, rates, and unit rates t understand and relate tw quantities; they als slved prblems using tables and percentages. Students applied and extended previus understandings f arithmetic t algebraic expressins in relatin t ratis. They used their prir knwledge f representing quantitative relatinships between dependent and independent variables with ratis and prprtinal relatinships. (6.EE.9) Current Learning Analyzing prprtinal relatinships and using them t slve real-wrld and mathematical prblems are critical areas and majr clusters. Students extend their knwledge f ratinal number applicatins t include the use f percents in slving prblems. They wrk with percents, including simple interest, 27

4 Grade 7 Mathematics, Quarter 3, Unit 3.1 Applicatin f Percents (15 days) percent increase and decrease, tax, markups and markdwns, gratuities and cmmissin, and percent errr. Future Learning Percents will nt be a fcus in grade 8; hwever, students will build n their knwledge f prprtinal relatinships and unit rates. They will use linear equatins and systems f linear equatins t represent, analyze, and slve a variety f prblems. Students will recgnize equatins fr prprtins (y/x = m r y = mx) as special linear equatins (y = mx + b), understanding that the cnstant f prprtinality (m) is the slpe and the graphs are lines thrugh the rigin. Additinal Findings Accrding t Elementary and Middle Schl Mathematics Teaching Develpmentally, Prprtinal reasning has been referred t as the capstne f elementary curriculum and the crnerstne f algebra and beynd. The ability t reasn prprtinally was a hallmark f Piaget s distinctin between cncrete levels f thught and frmal peratinal thught. It represents the ability t begin t understand multiplicative relatinships where mst arithmetic cncepts are additive in nature. (p. 298) Accrding t Principles and Standards fr Schl Mathematics, In grades 6 8, students shuld deepen their understanding f fractins, decimals, percents, and integers, and they shuld becme prficient in using them t slve prblems. (p. 215) Accrding t PARCC Framewrks, Meeting standard 7.EE.3 in its entirety will invlve using ratinal number arithmetic and percents. Wrk leading t meeting this standard culd be rganized as a recurring activity that tracks the students nging acquisitin f new skill sin ratinal number arithmetic and percents. (p. 32) 28

5 Grade 7 Mathematics, Quarter 3, Unit 3.2 Gemetric Drawing and Scaling Overview Number f instructinal days: 13 (1 day = minutes) Cntent t be learned Slve prblems invlving scale drawings f gemetric figures. Reprduce a scale drawing at a different scale. Cmpute actual lengths and areas frm a scale drawing. Draw (freehand, with ruler and prtractr, and with technlgy) gemetric shapes with given cnditins. Fcus n cnstructing triangles frm three measures f angles r sides, nticing when the cnditins determine a unique triangle, mre than ne triangle, r n triangle. Mathematical practices t be integrated Mdel with mathematics. Apply the mathematics they knw t slve prblems arising in everyday life, sciety, and the wrkplace. Identify imprtant quantities in a practical situatin. When cnstructing scale drawings and gemetric figures, rutinely interpret mathematical results in the cntext f the situatin and reflect n whether the results make sense, pssibly imprving the mdel if it has nt served its purpse. Use apprpriate tls strategically. Cnsider the available tls when slving a mathematical prblem (including pencil and paper, cncrete mdels, ruler, prtractr, calculatr, and dynamic gemetry sftware). Select apprpriate tls t use when cnstructing gemetric figures and scale drawings. Detect pssible errrs by using estimatin and ther mathematical knwledge. Essential questins Hw d yu use a scale drawing t find actual lengths? Hw d yu calculate actual areas frm a scale drawing? What steps d yu take t reduce r enlarge a scale drawing? What measures must be given when drawing gemetric shapes? Hw d yu cnstruct triangles frm three measures f angles r sides? What cnditins determine a unique triangle, mre than ne triangle, r n triangle? 29

6 Grade 7 Mathematics, Quarter 3, Unit 3.2 Gemetric Drawing and Scaling (13 days) Written Curriculum Cmmn Cre State Standards fr Mathematical Cntent Gemetry 7.G Draw, cnstruct, and describe gemetrical figures and describe the relatinships between them. 7.G.1 7.G.2 Slve prblems invlving scale drawings f gemetric figures, including cmputing actual lengths and areas frm a scale drawing and reprducing a scale drawing at a different scale. Draw (freehand, with ruler and prtractr, and with technlgy) gemetric shapes with given cnditins. Fcus n cnstructing triangles frm three measures f angles r sides, nticing when the cnditins determine a unique triangle, mre than ne triangle, r n triangle. Cmmn Cre Standards fr Mathematical Practice 4 Mdel with mathematics. Mathematically prficient students can apply the mathematics they knw t slve prblems arising in everyday life, sciety, and the wrkplace. In early grades, this might be as simple as writing an additin equatin t describe a situatin. In middle grades, a student might apply prprtinal reasning t plan a schl event r analyze a prblem in the cmmunity. By high schl, a student might use gemetry t slve a design prblem r use a functin t describe hw ne quantity f interest depends n anther. Mathematically prficient students wh can apply what they knw are cmfrtable making assumptins and apprximatins t simplify a cmplicated situatin, realizing that these may need revisin later. They are able t identify imprtant quantities in a practical situatin and map their relatinships using such tls as diagrams, tw-way tables, graphs, flwcharts and frmulas. They can analyze thse relatinships mathematically t draw cnclusins. They rutinely interpret their mathematical results in the cntext f the situatin and reflect n whether the results make sense, pssibly imprving the mdel if it has nt served its purpse. 5 Use apprpriate tls strategically. Mathematically prficient students cnsider the available tls when slving a mathematical prblem. These tls might include pencil and paper, cncrete mdels, a ruler, a prtractr, a calculatr, a spreadsheet, a cmputer algebra system, a statistical package, r dynamic gemetry sftware. Prficient students are sufficiently familiar with tls apprpriate fr their grade r curse t make sund decisins abut when each f these tls might be helpful, recgnizing bth the insight t be gained and their limitatins. Fr example, mathematically prficient high schl students analyze graphs f functins and slutins generated using a graphing calculatr. They detect pssible errrs by strategically using estimatin and ther mathematical knwledge. When making mathematical mdels, they knw that technlgy can enable them t visualize the results f varying assumptins, explre cnsequences, and cmpare predictins with data. Mathematically prficient students at varius grade levels are able t identify relevant external mathematical resurces, such as digital cntent lcated n a website, and use them t pse r slve prblems. They are able t use technlgical tls t explre and deepen their understanding f cncepts. 30

7 Grade 7 Mathematics, Quarter 3, Unit 3.2 Gemetric Drawing and Scaling (13 days) Clarifying the Standards Prir Learning In previus grades, students learned cncepts f angles, measured angles, classified plygns, and cnstructed plygns in the crdinate plane given crdinates fr the vertices. Students fund the lengths f a side using the crdinates and applied these techniques t slve real-wrld prblems. Current Learning This additinal cluster includes drawing, cnstructing, and describing gemetrical figures and describing their relatinships. Slving prblems invlving scale drawings and infrmal gemetric cnstructins are critical areas. Students reasn abut relatinships amng tw-dimensinal figures using scale drawings and infrmal gemetric cnstructins. In future units, students gain familiarity with the relatinships between angles frmed by intersecting lines and wrk with three-dimensinal figures, relating them t tw-dimensinal figures by examining crss sectins (slicing). Future Learning In grade 8, students will understand that translatins, rtatins, and reflectins are cngruent figures. They will als extend their understanding f scale drawings t similar figures. In high schl, students will make gemetric cnstructins with a variety f tls and methds. Additinal Findings Accrding t Principals and Standards fr Schl Mathematics, Students shuld cme t the study f gemetry in the middle grades with infrmal knwledge abut pints, lines, planes, and a variety f tw- and three-dimensinal shapes; with experience in visualizing and drawing lines, angles, triangles, and ther plygns; and with intuitive ntins abut shapes built frm years f interacting with bjects in their daily lives. In middle-grades gemetry prgrams based n these recmmendatins, students investigate relatinships by drawing, measuring, visualizing, cmparing, transfrming, and classifying gemetric bjects. (p. 233) All students in grades 6 8 shuld draw gemetric bjects with specified prperties, such as side lengths and angle measures. In additin, in grades 6 8, all students precisely describe, classify, and understand relatinships amng types f tw- and three-dimensinal bjects using their defining prperties. (p. 397) 31

8 Grade 7 Mathematics, Quarter 3, Unit 3.2 Gemetric Drawing and Scaling (13 days) 32

9 Grade 7 Mathematics, Quarter 3, Unit 3.3 Angle Relatinships Overview Number f instructinal days: 9 (1 day = minutes) Cntent t be learned Recgnize angle relatinships, specifically supplementary, cmplementary, vertical, and adjacent angles. Apply knwledge f angle relatinships in a multistep prblem t write and slve simple equatins fr an unknwn angle in a figure. Slve simple gemetric prblems using equatins. Mathematical practices t be integrated Mdel with mathematics. Apply the mathematics they knw t slve prblems arising in everyday life, sciety, and the wrkplace. Use diagrams t identify imprtant quantities in angle relatinships. Analyze angle relatinships t draw cnclusins. Interpret results in the cntext f the situatin and reflect n whether the results make sense. Attend t precisin. Use clear definitins in discussin with thers and in their wn reasning. State the meaning f the symbls chsen, including using the equal sign cnsistently and apprpriately. Use apprpriate units f measure. Calculate accurately and efficiently; express numerical answers with a degree f precisin apprpriate fr the prblem cntext. Essential questins Hw can yu tell the difference between cmplementary and supplementary angles? What is the difference between vertical and adjacent angles? Hw can yu use angle relatinships t find unknwn angles? What different methds can yu use t find the measure f an unknwn angle? 33

10 Grade 7 Mathematics, Quarter 3, Unit 3.3 Angle Relatinships (9 days) Written Curriculum Cmmn Cre State Standards fr Mathematical Cntent Gemetry 7.G Slve real-life and mathematical prblems invlving angle measure, area, surface area, and vlume. 7.G.5 Use facts abut supplementary, cmplementary, vertical, and adjacent angles in a multi-step prblem t write and slve simple equatins fr an unknwn angle in a figure. Expressins and Equatins 7.EE Slve real-life and mathematical prblems using numerical and algebraic expressins and equatins. 7.EE.4 Use variables t represent quantities in a real-wrld r mathematical prblem, and cnstruct simple equatins and inequalities t slve prblems by reasning abut the quantities. a. Slve wrd prblems leading t equatins f the frm px + q = r and p(x + q) = r, where p, q, and r are specific ratinal numbers. Slve equatins f these frms fluently. Cmpare an algebraic slutin t an arithmetic slutin, identifying the sequence f the peratins used in each apprach. Fr example, the perimeter f a rectangle is 54 cm. Its length is 6 cm. What is its width? Cmmn Cre Standards fr Mathematical Practice 4 Mdel with mathematics. Mathematically prficient students can apply the mathematics they knw t slve prblems arising in everyday life, sciety, and the wrkplace. In early grades, this might be as simple as writing an additin equatin t describe a situatin. In middle grades, a student might apply prprtinal reasning t plan a schl event r analyze a prblem in the cmmunity. By high schl, a student might use gemetry t slve a design prblem r use a functin t describe hw ne quantity f interest depends n anther. Mathematically prficient students wh can apply what they knw are cmfrtable making assumptins and apprximatins t simplify a cmplicated situatin, realizing that these may need revisin later. They are able t identify imprtant quantities in a practical situatin and map their relatinships using such tls as diagrams, tw-way tables, graphs, flwcharts and frmulas. They can analyze thse relatinships mathematically t draw cnclusins. They rutinely interpret their mathematical results in the cntext f the situatin and reflect n whether the results make sense, pssibly imprving the mdel if it has nt served its purpse. 6 Attend t precisin. Mathematically prficient students try t cmmunicate precisely t thers. They try t use clear definitins in discussin with thers and in their wn reasning. They state the meaning f the symbls they chse, including using the equal sign cnsistently and apprpriately. They are careful abut specifying units f measure, and labeling axes t clarify the crrespndence with quantities in a prblem. They calculate accurately and efficiently, express numerical answers with a degree f precisin apprpriate fr the prblem cntext. In the elementary grades, students give carefully frmulated 34

11 Grade 7 Mathematics, Quarter 3, Unit 3.3 Angle Relatinships (9 days) explanatins t each ther. By the time they reach high schl they have learned t examine claims and make explicit use f definitins. Clarifying the Standards Prir Learning In previus grades, students learned cncepts f angles, measured angles, classified plygns, and cnstructed plygns in the crdinate plane given crdinates fr the vertices. Students fund the lengths f a side using the crdinates and applied these techniques t slve real-wrld prblems. Current Learning Students gain familiarity with the relatinships between angles frmed by intersecting lines (supplementary, cmplementary, adjacent, vertical), which is part f a critical area. Althugh 7.EE.4 appeared in Quarter 2, students practice slving equatins when finding missing angles t achieve fluency. Slving real-wrld mathematical prblems invlving angle measures is an additinal cluster. Future Learning In grade 8, students will frm viable arguments based n the prperties f angles frmed when parallel lines are cut by a transversal. Additinal Findings Accrding t Principals and Standards fr Schl Mathematics, All students in grades 6 8 shuld draw gemetric bjects with specified prperties, such as side lengths and angle measures. In additin, in grades 6 8, all students precisely describe, classify, and understand relatinships amng types f twand three-dimensinal bjects using their defining prperties. (p. 397) In PARCC Framewrks, Students slve a variety f prblems invlving angle measure, area, surface area and vlume. (7.G.4-6) (p. 31) 35

12 Grade 7 Mathematics, Quarter 3, Unit 3.3 Angle Relatinships (9 days) 36