Triangle Congruency. Overview. Geometry Mathematics, Quarter 2, Unit 2.1. Number of Instructional Days: 15 (1 day = 45 minutes)

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1 Gemetry Mathematics, Quarter 2, Unit 2.1 Triangle Cngruency Overview Number f Instructinal Days: 15 (1 day = 45 minutes) Cntent t Be Learned Apply and describe the effects f rigid mtins (translatin, reflectin, rtatin) n a given figure. Cmpare crrespnding parts f triangles t determine cngruency. Explain and utilize the criteria fr triangle cngruency. ASA, SAS, SSS AAS, HL CPCTC culd be taught here. Prve therems abut lines and angles using the triangle cngruency criteria, fr example: Pints n a perpendicular bisectr f a line segment are exactly thse equidistant frm the segment s endpints. Pints n an angle bisectr are equidistant frm sides f the angle. Prve therems abut triangles using the triangle cngruency criteria. Examples include: Base angles f issceles triangles are cngruent. The segment jining midpints f tw sides f a triangle is parallel t the third side and half the length. The medians f a triangle meet at a pint (centrid). Mathematical Practices t Be Integrated Reasn abstractly and quantitatively. Interpret a given situatin abstractly, representing it symblically and cnversely. Manipulate symblic representatins t visualize situatins in a diagram in rder t slve quantitative prblems with gemetric figures. Knw and flexibly use prperties f rigid mtin t make sense f cngruent relatinships. Cnstruct viable arguments and critique the reasning f thers. Make cnjectures and cnstruct a lgical argument t prve cngruent triangles. Analyze a given situatin and justify cnclusins using triangle cngruence criteria. Listen r read the arguments f thers and decide whether they make sense; ask useful questins t clarify r imprve the arguments. Lk fr and make use f structure. Recgnize and explain the sequence f rigid mtins t shw triangle cngruence. Predict the effects f certain rigid mtins t transfrm a given figure. Recgnize the significance f a perpendicular segment and angle bisectr as well as their prperties in relatin t segments and angles. 9

2 Gemetry, Quarter 2, Unit 2.1 Triangle Cngruency (16 days) Essential Questins Hw d yu identify transfrmatins that are rigid mtins? Hw d yu use cngruence criteria in prfs and t slve prblems? What are the different methds that may be used t create a frmal argument? Describe special segments and angles in triangles and explain hw can they be used t slve prblems? Given a prf, hw can yu use lgical reasning t critique, analyze, and imprve the argument? Why are AAA and SSA invalid criteria fr prving triangle cngruence? Written Curriculum Cmmn Cre State Standards fr Mathematical Cntent Cngruence G-CO Understand cngruence in terms f rigid mtins [Build n rigid mtins as a familiar starting pint fr develpment f cncept f gemetric prf] G-CO.7 G-CO.8 G-CO.6 Use the definitin f cngruence in terms f rigid mtins t shw that tw triangles are cngruent if and nly if crrespnding pairs f sides and crrespnding pairs f angles are cngruent. Explain hw the criteria fr triangle cngruence (ASA, SAS, and SSS) fllw frm the definitin f cngruence in terms f rigid mtins. Use gemetric descriptins f rigid mtins t transfrm figures and t predict the effect f a given rigid mtin n a given figure; given tw figures, use the definitin f cngruence in terms f rigid mtins t decide if they are cngruent. Prve gemetric therems [Fcus n validity f underlying reasning while using variety f ways f writing prfs] G-CO.9 Prve therems abut lines and angles. Therems include: vertical angles are cngruent; when a transversal crsses parallel lines, alternate interir angles are cngruent and crrespnding angles are cngruent; pints n a perpendicular bisectr f a line segment are exactly thse equidistant frm the segment s endpints G-CO.10 Prve therems abut triangles. Therems include: measures f interir angles f a triangle sum t 180 ; base angles f issceles triangles are cngruent; the segment jining midpints f tw sides f a triangle is parallel t the third side and half the length; the medians f a triangle meet at a pint. 10

3 Gemetry, Quarter 2, Unit 2.1 Triangle Cngruency (16 days) Cmmn Cre Standards fr Mathematical Practice 2 Reasn abstractly and quantitatively. Mathematically prficient students make sense f quantities and their relatinships in prblem situatins. They bring tw cmplementary abilities t bear n prblems invlving quantitative relatinships: the ability t decntextualize t abstract a given situatin and represent it symblically and manipulate the representing symbls as if they have a life f their wn, withut necessarily attending t their referents and the ability t cntextualize, t pause as needed during the manipulatin prcess in rder t prbe int the referents fr the symbls invlved. Quantitative reasning entails habits f creating a cherent representatin f the prblem at hand; cnsidering the units invlved; attending t the meaning f quantities, nt just hw t cmpute them; and knwing and flexibly using different prperties f peratins and bjects. 3 Cnstruct viable arguments and critique the reasning f thers. Mathematically prficient students understand and use stated assumptins, definitins, and previusly established results in cnstructing arguments. They make cnjectures and build a lgical prgressin f statements t explre the truth f their cnjectures. They are able t analyze situatins by breaking them int cases, and can recgnize and use cunterexamples. They justify their cnclusins, cmmunicate them t thers, and respnd t the arguments f thers. They reasn inductively abut data, making plausible arguments that take int accunt the cntext frm which the data arse. Mathematically prficient students are als able t cmpare the effectiveness f tw plausible arguments, distinguish crrect lgic r reasning frm that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can cnstruct arguments using cncrete referents such as bjects, drawings, diagrams, and actins. Such arguments can make sense and be crrect, even thugh they are nt generalized r made frmal until later grades. Later, students learn t determine dmains t which an argument applies. Students at all grades can listen r read the arguments f thers, decide whether they make sense, and ask useful questins t clarify r imprve the arguments. 7 Lk fr and make use f structure. Mathematically prficient students lk clsely t discern a pattern r structure. Yung students, fr example, might ntice that three and seven mre is the same amunt as seven and three mre, r they may srt a cllectin f shapes accrding t hw many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparatin fr learning abut the distributive prperty. In the expressin x 2 + 9x + 14, lder students can see the 14 as 2 7 and the 9 as They recgnize the significance f an existing line in a gemetric figure and can use the strategy f drawing an auxiliary line fr slving prblems. They als can step back fr an verview and shift perspective. They can see cmplicated things, such as sme algebraic expressins, as single bjects r as being cmpsed f several bjects. Fr example, they can see 5 3(x y) 2 as 5 minus a psitive number times a square and use that t realize that its value cannt be mre than 5 fr any real numbers x and y. 11

4 Gemetry, Quarter 2, Unit 2.1 Triangle Cngruency (16 days) Clarifying the Standards Prir Learning In grade 7, students created triangles with rulers and prtractrs given the measures f all three angles and sides, and they recgnized the cnditins that create a unique triangle, mre than ne triangle, r n triangle (7.G.2). In grade 8, students understd that a tw-dimensinal figure is cngruent t anther if the secnd can be btained frm the first by a sequence f rtatins, reflectins, and translatins (8.G.2). Students encuntered cngruent figures in the unit n transfrmatins [unit 1.2]. Current Learning Students use the definitin f cngruence in terms f rigid mtins t shw that tw triangles are cngruent nly if crrespnding pairs f sides and crrespnding pairs f angles are cngruent (CPCTC). Students explain hw the criteria fr triangle cngruence (ASA, SAS, SSS, AAS and HL) fllw frm the definitin f cngruence in terms f rigid mtins. They use gemetric descriptins f rigid mtins t transfrm figures and t predict the effect f a given rigid mtin n a given figure. Given tw figures, they use the definitin f cngruence in terms f rigid mtins t decide if they are cngruent. Students prve therems abut lines and angles (e.g., pints n a perpendicular bisectr f a line segment are exactly thse equidistant frm the segment s endpints). Students prve therems abut triangles (e.g., base angles f issceles triangles are cngruent; the segment jining midpints f tw sides f a triangle is parallel t the third side and half the length; the medians f a triangle meet at a pint). Future Learning Students will encunter cngruent figures in future units (e.g., similarity [unit 2.3], plygns [unit 3.1] and circles [unit 3.3]). The cncept f cngruency is utilized in fields such as arts, architecture, interir design, engineering, industrial design, and cnstructin-related wrk. Additinal Findings Benchmarks fr Science Literacy by the American Assciatin fr the Advancement f Science states, Things can change in detail but remain the same in general. Smetimes cunterbalancing changes are necessary fr a thing t retain its essential cnstancy in the presence f changing cnditins (p. 275). Teacher Ntes: Students have truble reading a diagram fr the rder f sides and angles; emphasize the meaning f included sides/angles. Students struggle with lgical thinking and cmmunicating their ideas precisely in written r tw-clumn prfs. Incrprate multiple methds f prving therems frmally. A hands-n apprach might be an apprpriate intrductin t the triangle cngruence criteria. 12

5 Gemetry Mathematics, Quarter 2, Unit 2.2 Triangle Similarity Overview Number f instructinal days: 12 (1 day = 45 minutes) Cntent t be learned Establish the AA, SSS, and SAS triangle similarity criteria using similarity transfrmatins. Determine if tw triangles are similar by using the definitin f similarity. Explain the meaning f similarity fr triangles using similarity transfrmatins. Prve therems abut triangles using triangle similarity; examples include: A line parallel t ne side f a triangle divides the ther tw prprtinally. The Pythagrean Therem. Prve therems and slve prblems invlving similarity using cngruence and similarity criteria. Mathematical practices t be integrated Make sense f prblems and persevere in slving them. Analyze givens, cnstraints, relatinships, and gals. Translate verbal descriptins and prblems int diagrams, including the imprtant features and relatinships. Cnstruct viable arguments and critique the reasning f thers. Make cnjectures and cnstruct lgical arguments t prve triangles are similar. Analyze a given situatin and justify cnclusins using triangle similarity criterin. Listen r read the arguments f thers and decide whether they make sense; ask useful questins t clarify r imprve the arguments. Essential questins What can yu cnclude abut similar triangles, and hw can yu prve tw triangles are similar? Hw can similar triangles be used t measure bjects, and what are the benefits f using indirect measurement? What relatinships exist within a triangle when a line is drawn parallel t ne f the sides? Hw can yu use triangle similarity t prve the Pythagrean Therem? What are the similarities and differences between triangle similarity and triangle cngruence criteria? 13

6 Gemetry, Quarter 2, Unit 2.2 Triangle Similarity (12 days) Written Curriculum Cmmn Cre State Standards fr Mathematical Cntent Similarity, Right Triangles, and Trignmetry G-SRT Understand similarity in terms f similarity transfrmatins G-SRT.3 Use the prperties f similarity transfrmatins t establish the AA criterin fr tw triangles t be similar. G-SRT.2 Given tw figures, use the definitin f similarity in terms f similarity transfrmatins t decide if they are similar; explain using similarity transfrmatins the meaning f similarity fr triangles as the equality f all crrespnding pairs f angles and the prprtinality f all crrespnding pairs f sides. Prve therems invlving similarity G-SRT.4 Prve therems abut triangles. Therems include: a line parallel t ne side f a triangle divides the ther tw prprtinally, and cnversely; the Pythagrean Therem prved using triangle similarity. G-SRT.5 Use cngruence and similarity criteria fr triangles t slve prblems and t prve relatinships in gemetric figures. 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 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. 14

7 Gemetry, Quarter 2, Unit 2.2 Triangle Similarity (12 days) 3 Cnstruct viable arguments and critique the reasning f thers. Mathematically prficient students understand and use stated assumptins, definitins, and previusly established results in cnstructing arguments. They make cnjectures and build a lgical prgressin f statements t explre the truth f their cnjectures. They are able t analyze situatins by breaking them int cases, and can recgnize and use cunterexamples. They justify their cnclusins, cmmunicate them t thers, and respnd t the arguments f thers. They reasn inductively abut data, making plausible arguments that take int accunt the cntext frm which the data arse. Mathematically prficient students are als able t cmpare the effectiveness f tw plausible arguments, distinguish crrect lgic r reasning frm that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can cnstruct arguments using cncrete referents such as bjects, drawings, diagrams, and actins. Such arguments can make sense and be crrect, even thugh they are nt generalized r made frmal until later grades. Later, students learn t determine dmains t which an argument applies. Students at all grades can listen r read the arguments f thers, decide whether they make sense, and ask useful questins t clarify r imprve the arguments. Clarifying the Standards Prir Learning In grade 7, students slved prblems invlving scale drawings f gemetric figures. They calculated actual lengths and areas frm a scale drawing and reprduced scale drawings frm a different scale (7.G.1). In grade 8, students used a sequence f rtatins, reflectins, transfrmatins, and dilatins t understand similarity between tw-dimensinal figures. Students als made infrmal arguments t establish the AA criterin fr similar triangles (8.G.4). In additin t using the Pythagrean Therem, students explained a prf f the therem and its cnverse (8.G.6 and 7). Als, students used similar triangles t derive the equatin y = mx and y = mx + b (8.EE.6). In unit 1.1, students learned abut angle relatinships frmed by parallel lines. In unit 2.1, students learned abut rigid mtins, cngruency in bjects, and mre specifically, triangles. In unit 1.2, students studies similarity in terms f transfrmatins. Current Learning Students use the prperties f similarity transfrmatins t establish the AA criterin fr tw triangles t be similar. Given tw figures, students use the definitin f similarity in terms f similarity transfrmatins t decide if they are similar. Students explain the meaning f similarity fr triangles as the equality f all crrespnding pairs f angles and the prprtinality f all crrespnding pairs f sides. Students prve therems abut triangles. Therems include: A line parallel t ne side f a triangle divides the ther tw prprtinally. The Pythagrean Therem (prved using triangle similarity). 15

8 Gemetry, Quarter 2, Unit 2.2 Triangle Similarity (12 days) Students use the cngruence and similarity criteria fr triangles in rder t slve prblems and t prve relatinships in gemetric figures. Future Learning Students will use their knwledge f triangle similarity when studying trignmetric ratis (unit 2.3) and circles (unit 3.3). The cncept f similarity is useful in fields such as arts, architecture, interir design, engineering, surveying and cnstructin related wrk. Additinal Findings Accrding t Principles and Standards fr Schl Mathematics, the cncept f similarity is ne f the fundamental tpics in gemetry, Students are becming increasingly independent explrers, which will allw them t develp deeper understanding f imprtant gemetric ideas such as transfrmatin and symmetry. These understandings will help students address questins that have always been central t the study f Euclidean gemetry Are tw gemetric figures similar, if s, why? (p. 309) In additin, study f similarity will allw students t analyze characteristics and prperties f tw-and three-dimensinal gemetric shapes and develp mathematical arguments abut gemetric relatinships. (p. 310) Fr a detailed example f hw cncepts f similarity and cngruence can be used alng with tasks that require deductin and prfs, refer t pages in Principles and Standards fr Schl Mathematics. Cmputer transfrmatins, especially thse in carefully designed micrwrlds, can help students learn similarity and rati cncepts (A Research Cmpanin t Principles and Standards fr Schl Mathematics, p.161). 16

9 Gemetry, Quarter 2, Unit 2.3 Right Triangle Trignmetry Overview Number f instructinal days: 13 (1 day = 45 minutes) Cntent t be learned Define trignmetric ratis (sine, csine, tangents) in right triangles by understanding that by similarity, side ratis in right triangles are prperties f the angles in the triangle, leading t definitins f trignmetric ratis fr acute angles. Calculate the side lengths and the trignmetric ratis assciated with special right triangles. Explain and use the relatinship between the sine and csine f cmplementary angles. Use trignmetric ratis and the Pythagrean Therem t slve right triangles in applied prblems. + Apply trignmetry t general triangles: Understand and apply the Law f Sines and the Law f Csines t find unknwn measurements in right and nn-right triangles (e.g., surveying prblems, resultant frces). Prve the Laws f Sines and Csines and use them t slve prblems. Derive the frmula A = 1/2 ab sin(c) fr the area f a triangle by drawing an auxiliary line frm a vertex perpendicular t the ppsite side. Mathematical practices t be integrated Make sense f prblems and persevere in slving them. Explain t self the meaning f a prblem and lk fr entry pints t its slutin. Make cnjectures abut the frm and meaning f the slutin and plan a slutin pathway rather than simply jumping int a slutin attempt. Cnsider analgus prblems and try special cases and simpler frms f the riginal prblem in rder t gain insight int its slutin. Mdel with mathematics. Apply the mathematics knwn t slve prblems invlving triangles arising in everyday life, sciety, and the wrkplace. Interpret mathematical results in the cntext f the situatin and reflect n whether the results make sense. Fr example, ask, Des my answer fr the height f a building r an angle I have fund make sense? Attend t precisin. State the meaning f the symbls chsen, including using the crrect trignmetric ratis cnsistently and apprpriately. Use care when specifying units f measure. Calculate accurately and efficiently and express numerical answers with a degree f precisin apprpriate fr the prblem cntext, especially when runding their results. Lk fr and express regularity in repeated reasning. Maintain versight f the prcess, while attending t the details. Cntinually evaluate the reasnableness f intermediate results. 17

10 Gemetry, Quarter 2, Unit 2.3 Right Triangle Trignmetry (13 days) Essential questins Hw can yu find a side length r angle measure in a right triangle? Hw wuld yu explain a slutin t a realwrld scenari where yu wuld apply yur knwledge f right triangle relatinships t find a height, distance, r an angle measure? What are the imprtant ratis between sides f a right triangle, and hw des changing the sides affect these ratis? Written Curriculum Cmmn Cre State Standards fr Mathematical Cntent Similarity, Right Triangles, and Trignmetry G-SRT Define trignmetric ratis and slve prblems invlving right triangles G-SRT.6 Understand that by similarity, side ratis in right triangles are prperties f the angles in the triangle, leading t definitins f trignmetric ratis fr acute angles. G-SRT.7 Explain and use the relatinship between the sine and csine f cmplementary angles. G-SRT.8 Use trignmetric ratis and the Pythagrean Therem t slve right triangles in applied prblems. Apply trignmetry t general triangles G-SRT.11 (+) Understand and apply the Law f Sines and the Law f Csines t find unknwn measurements in right and nn-right triangles (e.g., surveying prblems, resultant frces). G-SRT.10 (+) Prve the Laws f Sines and Csines and use them t slve prblems. G-SRT.9 (+) Derive the frmula A = 1/2 ab sin(c) fr the area f a triangle by drawing an auxiliary line frm a vertex perpendicular t the ppsite side. ***Teacher Nte: (as per the PARCC Mdel Cntent Framewrks, mathematics grades 3 11, G-SRT.9, G-SRT.10, and G-SRT.11 may be an extensin t right triangle trignmetry (p. 53).*** 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 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 18

11 Gemetry, Quarter 2, Unit 2.3 Right Triangle Trignmetry (13 days) 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. 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 explanatins t each ther. By the time they reach high schl they have learned t examine claims and make explicit use f definitins. 8 Lk fr and express regularity in repeated reasning. Mathematically prficient students ntice if calculatins are repeated, and lk bth fr general methds and fr shrtcuts. Upper elementary students might ntice when dividing 25 by 11 that they are repeating the same calculatins ver and ver again, and cnclude they have a repeating decimal. By paying attentin t the calculatin f slpe as they repeatedly check whether pints are n the line thrugh (1, 2) with slpe 3, middle schl students might abstract the equatin (y 2)/(x 1) = 3. Nticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them t the general frmula fr the sum f a gemetric series. As they wrk t slve a prblem, mathematically prficient students maintain versight f the prcess, while attending t the details. They cntinually evaluate the reasnableness f their intermediate results. Clarifying the Standards Prir Learning In grade 6, students fund the area f right triangles by cmpsing int rectangles. (6.G.1) 19

12 Gemetry, Quarter 2, Unit 2.3 Right Triangle Trignmetry (13 days) In grade 7, students used facts abut supplementary, cmplementary, vertical, and adjacent angles in a multistep prblem t slve fr an unknwn angle in a figure (7.G.5). Students als recgnized and represented prprtinal relatinships between quantities. (7.RP.2a-c). In grade 8, students applied the Pythagrean Therem t determine unknwn side lengths in right triangles in real-wrld and mathematical prblems in tw and three dimensins. (8.G.7) In algebra 1, students simplified radicals. (Refer t the algebra 1 scpe and sequence, quarter 4.) In unit 2.2 f this curse, students used the definitin f similarity in terms f transfrmatins t decide whether tw triangles are similar. They explained similarity fr triangles as the equality f all crrespnding pairs f angles and the prprtinality f all crrespnding pairs f sides. (G.SRT.2) Current Learning Students understand that by similarity, side ratis in right triangles are prperties f the angles in the triangle, leading t definitins f trignmetric ratis (sine, csine, and tangent) fr acute angles. Students can explain and use the relatinship between the sine and csine f cmplementary angles. Students utilize trignmetric ratis and the Pythagrean Therem t slve applied prblems invlving right triangles t determine heights, distances, and angle measure. + Students understand and apply the Law f Sines and the Law f Csines t find unknwn measurements in right and nn-right triangles in real-wrld prblems such as surveying prblems and resultant frces. Students prve the Laws f Sines and Csines and use them t slve prblems. Students derive the frmula A = 1/2 ab sin(c) fr the area f a triangle by drawing an auxiliary line frm a vertex perpendicular t the ppsite side. Future Learning Students will build upn their knwledge f trignmetry in Algebra 2 and furth-year mathematics curses, as well as cllege curses. Students will als use this knwledge in engineering, cnstructin, and cmputer science prfessins. Additinal Findings A Research Cmpanin t Principles and Standards fr Schl Mathematics states that intermediategrade students ften pssess ne f tw schemes fr measuring angles. In the schema, slanted lines are assciated with 45 degree turns, and hrizntal and vertical lines, with 90 degree turns. (p. 163) Writing Team Ntes Reinfrce simplifying radicals cvered in Quarter 4 f Algebra 1. In additin, it is necessary t teach students hw t divide radicals and ratinalize the denminatr. These skills are nt cvered in the Algebra I curriculum. 20