Curriculum Development Overview Unit Planning for 8 th Grade Mathematics MA10-GR.8-S.1-GLE.1 MA10-GR.8-S.4-GLE.2

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1 Unit Title It s All Greek t Me Length f Unit 5 weeks Fcusing Lens(es) Cnnectins Standards and Grade Level Expectatins Addressed in this Unit MA10-GR.8-S.1-GLE.1 MA10-GR.8-S.4-GLE.2 Inquiry Questins (Engaging- Debatable): Hw has technlgy changed ur cncept f number? (MA10-GR.8-S.1-GLE.1) Unit Strands Cncepts The Number System, Expressins and Equatins, Gemetry Pythagrean Therem, right triangle, indirect measurement, ratinal numbers, irratinal numbers, divisin, integers, terminate, repeat, decimals, apprximatin, representatin, number line Generalizatins My students will Understand that Factual Guiding Questins Cnceptual The Pythagrean Therem cmmunicates the relatinship amng the sides f a right triangle. (MA10-GR.8-S.4- GLE.2-EO.a) The Pythagrean Therem facilitates indirectly finding the lengths f measurements f right triangles. (MA10-GR.8-S.4-GLE.2-EO.b, c) and (MA10-GR.8-S.1- GLE.1-EO.e, f) Ratinal numbers, written as the divisin f tw integers, result in a terminating r repeating decimal (MA10-GR.8- S.1-GLE.1-EO.b.i, b.ii)) What is the Pythagrean Therem? Hw can yu use the Pythagrean Therem t prve that a triangle is a right triangle? Hw can yu use the Pythagrean Therem t find unknwn side lengths in a right triangle? Hw can the Pythagrean Therem be used fr indirect measurement? (MA10-GR.8-S.4-GLE.2-IQ.2) Hw can yu apply the Pythagrean Therem t determine the distance between tw pints n the crdinate system? What are examples f tw integers whse qutient results in a terminating decimal? What are examples f tw integers whse qutient results in a repeating decimal? Hw can yu determine if the decimal expansin f a ratinal number will terminate r repeat? Why des the Pythagrean Therem nly apply t right triangles? (MA10-GR.8-S.4-GLE.2-IQ.1) Hw can yu prve the Pythagrean Therem and its cnverse? Why are the Pythagrean Therem and the distance frmula equivalent equatins f the same relatinship? (MA10-GR.8-S.4-GLE.2-IQ.3) Why is slving square rt and cube rt equatins similar and different t slving linear equatins? Why d ratinal numbers create terminating r repeating decimals? Why des 1/3 repeat in a base ten decimal system but nt in a base 60? 8 th Grade, Mathematics 8 f 79

2 Irratinal numbers, which cannt be written as the divisin f tw integers, require a decimal apprximatin fr representatin n a number line (MA10-GR.8-S.1- GLE.1-EO.a, c) What are sme examples f irratinal numbers? What is the difference between ratinal and irratinal numbers? (MA10-GR.8-S.1-GLE.1-IQ.3) Why are irratinal numbers unable t be written as a divisin f tw integers? Why is the square rt f tw an irratinal number? Key Knwledge and Skills: My students will What students will knw and be able t d are s clsely linked in the cncept-based discipline f mathematics. Therefre, in the mathematics samples what students shuld knw and d are cmbined. Knw that numbers that are nt ratinal are called irratinal (MA10-GR.8-S.1-GLE.1-EO.a) Students encunter irratinal numbers in elementary schl when intrduced t the number π. Hwever, students first encunter with irratinal numbers as numbers that can be expressed as a decimal but cannt be expressed as the divisin f tw integers, cmes in middle schl frm their experiences with right triangles and the Pythagrean Therem [17]. While they are nt expected t prve it, using a calculatr, they learn that all square rts that are nt perfect squares are irratinal numbers. Since students are able t cnvert ratinal numbers t decimals, they may first make the assumptin that all decimals can be cnverted t ratinal numbers. This nly hlds true when the decimal terminates r repeats. Using a calculatr t find the decimal equivalent f an irratinal number, they may be cnvinced that the decimal des nt appear t repeat r terminate. (CC.8.NS.1) Understand infrmally that every number has a decimal expansin (MA10-GR.8-S.1-GLE.1-EO.b) Students are able t assciate ratinal numbers with fractins and the divisin f integers. They als assciate decimals with ratinal numbers. Students knw that all ratinal numbers can be expressed in the frm a/b, where a and b are integers such that b. This develps as they encunter three cases (integers, terminating decimals and repeating decimals) (CC.8.NS.1) Students are able t assciate ratinal numbers with fractins and the divisin f integers. They als assciate decimals with ratinal numbers [17]. Students knw that all ratinal numbers can be expressed in the frm a/b, where a and b are integers such that. This develps as they encunter three cases (integers, terminating decimals and repeating decimals): Case One: Integers. Students knw that integers can be written in the frm f a fractin by writing them ver 1. Fr example, Case Tw: Terminating Decimals. Students can change a terminating decimal int a fractinal equivalent. Fr example, Fr example, students are asked t cnvert 0.65 int a fractin. Students identify 65 as the string f digits t the right f the decimal and cunt 2 digits in this string. They first write the fractin as and then simplify t find the equivalent fractin. They als knw that sme fractins create repeating decimals, such as,, and therefre recgnize that ratinal numbers can have decimal equivalents that repeat ne r mre numerals withut terminating. Students have nticed that every ratinal number has a decimal equivalent in which the decimal either terminates r. Students justify this utcme by examining the prcess f lng divisin p q fr the ratinal number p/q, recgnizing that each time they divide the divisr, q (denminatr), int the dividend (riginally the numeratr), they either have n remainder and the divisin terminates r they have a remainder which is smaller than the divisr, q. After at mst q 1 times, the remainder must repeat after which the pattern in the qutient repeats. Due t the fact that adding any number f decimal places f 0 s t the right f a number des nt change its value, every frm can be interpreted as a repeating decimal. Case Three: Repeating Decimals. Students can als reverse the prcess and change repeating decimals int fractinal frm. Students learn t cnvert 8 th Grade, Mathematics 9 f 79

3 repeating decimals t fractins using their knwledge f wrking with equatins and simultaneus equatins, as described belw (see als Standard 8.EE.8 in the Linear Equatins, Inequalities, and Functins LT). Students first write an algebraic equatin setting the repeating decimal equal t a variable. They then identify the string f repeating digits and cunt the number f decimal places ccupied by each iteratin f the repeat). T cnstruct the equivalent fractin fr the repeating decimal, students multiply bth sides f the equatin by the pwer f 10 equal t the number f repeating digits (a 1 fllwed by ne 0 fr each repeating digit in the string). They then subtract the riginal equatin frm the new equatin, and slve fr the value f the variable. The resulting fractin is nt necessarily in simplest frm; it shuld be reduced if it is pssible t d s. If students can imagine repeating patterns, they can imagine nn-repeating patterns. Fr example, There is a pattern, but that pattern des nt repeat. These numbers are defined as irratinal numbers. Students encunter irratinal numbers in elementary schl when intrduced t the number π. Hwever, students first encunter with irratinal numbers as numbers that can be expressed as a decimal but cannt be expressed as the divisin f tw integers, cmes in middle schl frm their experiences with right triangles and the Pythagrean Therem. While they are nt expected t prve it, using a calculatr, they learn that all square rts that are nt perfect squares are irratinal numbers. Since students are able t cnvert ratinal numbers t decimals, they may first make the assumptin that all decimals can be cnverted t ratinal numbers. This nly hlds true when the decimal terminates r repeats. Using a calculatr t find the decimal equivalent f an irratinal number, they may be cnvinced that the decimal des nt appear t repeat r terminate. Fr example, find the length f the missing side in the fllwing right triangle: Students use the Pythagrean Therem t determine that Students are als intrduced t the existence f ther irratinal numbers that d nt result frm nn-perfect squares, such as thrugh the ntin f generating a decimal by infinitely adding digits t the end f the decimal that d nt create a repeating pattern. In particular, students shuld be aware f the existence f knwn useful irratinals that appear frequently in mathematics, such as [17]. Nte t Teachers: It is imprtant fr students t understand that the values given by a calculatr are limited by what can fit and be seen n the screen. Therefre, while students cnjecture that decimals that d nt appear t repeat r terminate (n the calculatr) are irratinal number is a reasnable ne, they shuld als understand that this cnjecture cannt be assumed t be true in all instances, because a decimal des nt repeat r terminate within this fixed number f digits n the calculatr may nnetheless eventually repeat r terminate. Shw that fr ratinal numbers the decimal expansin repeats eventually, and cnvert a decimal expansin which repeats eventually int a ratinal number (MA10-GR.8- S.1-GLE.1-EO.b.i, b.ii) Repeating Decimals. Students can als reverse the prcess and change repeating decimals int fractinal frm. Students learn t cnvert repeating decimals t fractins using their knwledge f wrking with equatins and simultaneus equatins, as described belw (see als Standard 8.EE.8 in the Linear Equatins, 8 th Grade, Mathematics 10 f 79

4 Inequalities, and Functins LT). (CC.8.NS.1) Students first write an algebraic equatin setting the repeating decimal equal t a variable. They then identify the string f repeating digits and cunt the number f decimal places ccupied by each iteratin f the repeat). T cnstruct the equivalent fractin fr the repeating decimal, students multiply bth sides f the equatin by the pwer f 10 equal t the number f repeating digits (a 1 fllwed by ne 0 fr each repeating digit in the string). They then subtract the riginal equatin frm the new equatin, and slve fr the value f the variable. The resulting fractin is nt necessarily in simplest frm; it shuld be reduced if it is pssible t d s. The examples belw shw hw students cnvert and justify that 1/3 =.3333 and 115/333 = (CC.8.NS.1) Example One: x = (Given) 10x = (Multiplicative axim f equality by 10 n bth sides) 9x = 3 (Frm subtracting the first frm the secnd equatin) x = 1 / 3 (Multiplicative axim f equality by 1 / 9 n bth sides) (CC.8.NS.1) Example Tw: x = (Given) 1000x = (Multiplicative axim f equality by 1000 n bth sides) 999x = 345 (Subtract the first tw equatins) x = 345 / 999 (Multiplicative axim f equality by 1 / 999 n bth sides) x = 115 / 333 (Multiplicative identity; divide by 3 / 3 ) (CC.8.NS.1) If students can imagine repeating patterns, they can imagine nn-repeating patterns. Fr example, There is a pattern, but that pattern des nt repeat. These numbers are defined as irratinal numbers. (CC.8.NS.1) Use ratinal apprximatins f irratinal numbers t cmpare the size f irratinal numbers, lcate them apprximately n a number line diagram, and estimate the value f expressins (MA10-GR.8-S.1-GLE.1-EO.c) Students knw hw t represent and lcate whle numbers and ratinal numbers (fractins and decimals) as pints n a number line frm Standard 6.NS.6.c earlier in this LT. They are challenged t apprximately lcate irratinal numbers n a number line and further characterize the relatinship between ratinal and irratinal numbers. (CC.8.NS.2) As a first rugh apprximatin f irratinal numbers that are nn-perfect square rts, students determine which tw integers the rt wuld lie between n a number line using cnsecutive perfect squares. (CC.8.NS.2) Fr example, students are asked t apprximate and lcate it n a number line. (CC.8.NS.2) They determine that 45 lies between the cnsecutive perfect squares 36 and 49, being clser t 49. Therefre, lies between 6 and 7, being clser t 7. S, the psitin f a nn-perfect square rt (in this example,) will be clser t the integer (in this example, 7) that is the rt f the perfect square (in this example, 49) t which the nn-perfect square (in this example, 45) is clser. (CC.8.NS.2) This apprximatin can be refined by finding the distance between 45 and 36 (the clsest perfect square less than the radicand), and then dividing that by the distance between 36 and 49 (the tw clsest surrunding perfect squares). This yields, and therefre (using a calculatr and runding t tw decimal places, ) (CC.8.NS.2) Hwever, students are nw able t cnvert ratinal numbers t precise decimals and vice versa. Using their knwledge that every number has a decimal expansin (see Standard 8.NS.1 earlier) students use algrithms r technlgy t find the decimal expansin f any irratinal number. (CC.8.NS.2) 8 th Grade, Mathematics 11 f 79

5 Fr example, using a calculatr, students find that (CC.8.NS.2) Students recgnize that by runding the decimal expansin f any number, including irratinals, a ratinal apprximatin, fr which the decimal terminates, is btained. T cmpare the relative sizes f irratinal numbers and ratinal numbers, students lcate the numbers r their apprximatins n a number line. (CC.8.NS.2) Fr example, t cmpare the values f and, students first use a calculatr t find the decimal expansin f each f these numbers, resulting in and They then determine ratinal apprximatins f the numbers by truncating (runding) the decimal expansins t an apprpriate number f digits (here tw decimal places), giving and. Finally, they graph the ratinal apprximatins n a number line, as shwn belw, and determine which number lies t the right, and is therefre greater. (CC.8.NS.2) (CC.8.NS.2) Frm their wrk with the Pythagrean Therem, and, in particular, drawing n the prf f its cnverse, students are equipped t lcate irratinal numbers mre precisely n a number line. (CC.8.NS.2) Use square rt and cube rt symbls t represent slutins t equatins f the frm x2 = p and x3 = p, where p is a psitive ratinal number (MA10-GR.8-S.1-GLE.1- EO.e) Tasks might fr example take the frm f algebraic wrd prblems leading t equatins r, r gemetric prblems such as finding the edge length f a cubical bject with a given vlume. (PARCC) In prblems where and are bth relevant as slutins t, bth f these slutins shuld be given. Nte that is nnnegative by definitin. (PARCC) Slutins t equatins r are represented as and r, respectively. (PARCC) Manipulatins such as are beynd the scpe f grade 8. Student need nt simplify a slutin such as. But students shuld ultimately express the fllwing cases in the frm f whle numbers: (a) the square rts f 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100; (b) the cube rts f 1, 8, 27, and 64. (PARCC) Evaluate square rts f small perfect squares and cube rts f small perfect cubes (MA10-GR.8-S.1-GLE.1-EO.f) Tasks might fr example take the frm f algebraic wrd prblems leading t equatins r, r gemetric prblems such as finding the edge length f a cubical bject with a given vlume. (PARCC) In prblems where and are bth relevant as slutins t, bth f these slutins shuld be given. Nte that is nnnegative by definitin. (PARCC) Slutins t equatins r are represented as and r, respectively. (PARCC) Manipulatins such as are beynd the scpe f grade 8. Student need nt simplify a slutin such as. But students shuld ultimately express the fllwing cases in the frm f whle numbers: (a) the square rts f 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100; (b) the cube rts f 1, 8, 27, and 64. (PARCC) Explain a prf f the Pythagrean Therem and its cnverse (MA10-GR.8-S.4-GLE.2-EO.a) In 50% f tasks, the answer is a whle number and is t be given as a whle number. (PARCC) In 50% f tasks, the answer is irratinal and is t be given apprximately t three decimal places. (PARCC) The testing interface can prvide students with a calculatin aid f the specified kind fr these tasks. (PARCC) 8 th Grade, Mathematics 12 f 79

6 Students are first able t justify and explain the Pythagrean Therem using cncrete examples. The applicatin f the Pythagrean therem is dne prir t the prfs, because it is nt a prf that can easily invented by students. Hwever, they shuld knw there are numerus prfs and encuraged t find thers than presented here. Fr example, wrking with the right triangle with respective side lengths f 3, 4 and 5 as shwn belw: By tiling unit squares alng the lengths f each f the legs and extending then cnstructing areas equivalent t the square f each f thse sides, students see that the same number f unit squares builds an area equivalent t the square f the hyptenuse. This is demnstrated belw: 8 th Grade, Mathematics 13 f 79

7 Students are nt expected t be able t develp a frmal prf f the Pythagrean Therem n their wn, but they can justify the reasning used in a presented prf using explanatins related t cncepts they have previusly mastered. Numerus prfs fr the Pythagrean Therem exist, but nt all are expected t be understd by students at this level. Tw examples are prvided belw. Given is a right triangle. Squares are cnstructed ff f each side with resulting areas a 2, b 2, and c 2 respectively, as shwn belw: Using the riginal triangle and shared side lengths with the respective squares t create cpies f the riginal triangle, tw new squares can be cnstructed frm the diagram, each with side length (a + b). These are shwn belw: 8 th Grade, Mathematics 14 f 79

8 Students apply the area frmula fr squares and triangles t reasn and explain that the newly frmed larger squares each have areas f. They then justify the equality f a 2 + b 2 and c 2 using algebra as well as gemetric areas and deductin. Algebraically, students determine the area f the newly frmed squares t be equivalent t a 2 + 2ab + b 2 and the area f every cpy f the riginal triangle t be. Since the tw newly frmed squares have equal area, subtracting the same area frm each wuld leave equal remaining areas. The area f fur cpies f the riginal triangle is equivalent t. Therefre it is cncluded that a2 + b 2 = c 2. A secnd prf f the Pythagrean Therem that students justify and explain is based n similarity applied t right triangles. In the figure belw, students determine that using the AA similarity pstulate (earlier Bridging Standard 6.TNT.B). They then setup prprtinal relatinships fr crrespnding pairs f sides using rati tables r rati bxes and determine that the prf shws that a 2 + b 2 = c 2. The prf states that frm the figure abve it can be determined that and, which imply that a2 = ce and b 2 = cf. Therefre, a 2 +b 2 = ce + cf = c(e + f) = c 2. 8 th Grade, Mathematics 15 f 79

9 Students are als presented with the cnverse f the Pythagrean Therem, any triangle with side lengths a, b, and c such that c is the lngest leg (where and ) and is a right triangle. They are nt expected t prvide a frmal prf f the cnverse either, but shuld be able t explain a given prf, within reasn. Students understand that the claim behind a triangle being a right triangle is justified by the existence f a right angle in that triangle. Fr example, explain hw the diagrams belw shw that a triangle with side lengths 6, 8, and 10, is a right triangle: First, it can easily be seen that. Then students cnsider the diagrams. Students reasn that by cnstructing a triangle in the crdinate plane s that the tw shrter legs are aligned with the axes, and their lengths being equivalent t thse frm the riginal triangle, it can be shwn that there is a right triangle that exists with thse tw lengths as legs because the axes are perpendicular and frm a right angle. Then, using rigid transfrmatins s that the third triangle is cngruent t the secnd, the lngest leg (alleged hyptenuse) is aligned with ne f the axes t verify its length. Therefre, the triangle is a right triangle. Apply the Pythagrean Therem t determine unknwn side lengths in right triangles in real-wrld and mathematical prblems in tw and three dimensins (MA10- GR.8-S.4-GLE.2-EO.b) At this stage, students are presented the Pythagrean Therem,, as a frmula representing the relatinship between the three sides f a right triangle with side lengths a, b, c where a and b represent the lengths f the legs and c represents the length f the hyptenuse. The frmula will be derived later and understanding at this pint is based in being able t slve the equatin fr any variable, after substituting in fr the ther tw. Students are given the equatin. They can slve the equatin given any tw values fr side lengths, a and b, a and c and b and c. The last tw a and c and band c generate equivalent ways f slving nly reversing the tw variables. Case 1: find the missing side length, c, the hyptenuse, when the length f bth legs are knwn: Students first apply the Pythagrean Therem and write the equatin resulting frm the frmula after substitutin as and simplify t arrive at. T slve fr c, they apply the square rt t bth sides f the equatin and determine. 8 th Grade, Mathematics 16 f 79

10 In general, students can slve the general frm fr c: Students als can adjust the frmula fr the case when right triangle is issceles s that a = b. Case 2: Find the missing side length fr ne f the legs, when the length f ther side and hyptenuse are knwn. Fr example, t find the missing side length in the fllwing triangle: Students first apply the Pythagrean Therem and write the equatin resulting frm the frmula after substitutin as and simplify t arrive at They then subtract 144 frm bth sides, giving, befre applying the square rt, which leads t an answer f a = 5 In general, students can slve the general frm fr a: A cmmn miscnceptin that arises in wrking with the Pythagrean therem is that the square rt can be applied t bth sides and reduce the frmula t a + b = c. In the triangle abve, this wuld result In a claim that, fr example,, which leads t an incrrect answer f a = 1. Students knw that if a = 1, then n triangle culd be frmed as the hyptenuse wuld equal the sum f the tw sides. Students als avid the miscnceptin that the frmula wrks n matter which side is labeled c, and s they are clear f the distinctins between the side and the hyptenuse in the frmula. 8 th Grade, Mathematics 17 f 79

11 Students recgnize the Pythagrean Therem can be used t find the missing side length f a right triangle in any cntext. Fr example, find the length f a diagnal in the given rectangle: Students recgnize that cnstructing a diagnal creates tw cngruent right triangles (and that the tw diagnals will have the same length s which ne is cnstructed is arbitrary). They therefre identify the diagnal as the hyptenuse f a right triangle whse legs have lengths equivalent t the side lengths f the rectangle and apply the Pythagrean Therem,. Or, cnsider the fllwing prblem: Daniel bught a package f chclate candies as a Mther s Day gift. The bx is shaped as shwn belw. He wants t wrap a ribbn arund the bx befre giving it t his mther and knws the length f the tw legs f the base are the same (3 inches). If he wants t leave an additinal 8 inches f ribbn fr tying a bw, hw lng f a piece f ribbn shuld Daniel cut? In rder t slve this prblem, students recgnize that the base is a right triangle and that they must determine the length f the hyptenuse f the base f the bx. They apply the Pythagrean Therem t develp the equatin and find the missing side length t be. This result can then be used t calculate the length f ribbn needed t tie arund the bx. Fr nn-perfect squares like the ne in the previus example, students understand that fr this number t be useful in measurement and real-wrld cntexts they must cnvert it t a decimal. This can easily be dne using a scientific r graphing calculatr. Students shuld recgnize that the value shwn n the calculatr is limited by the number f digits in the display and can als be runded t an apprpriate number f decimal places t give a useable estimate f the value fr a given cntext. 8 th Grade, Mathematics 18 f 79

12 Fr instance, in the previus example students use a calculatr t find that and after runding t 4.24 they determine that within human precisin fr measurement and cutting ribbn, an apprpriate estimatin wuld be 4 1 / 4 inches. Students use the Pythagrean Therem t slve prblems in three dimensins such as: A fly lands n the upper crner f an pen Kleenex bx with a base that is 4 inches by 10 inches and a height f 5 inches. The fly wants t be at the ppsite lwer crner. If it flies directly between the tw pints, what is the shrtest distance t the ppsite crner and hw much lnger wuld it be if the fly walked directly t the grund and then diagnally acrss? Students represent the prblem using the tw diagrams abve. In rder t determine the ttal distance in the right figure (walking), students first use the Pythagrean Therem t determine the length f the diagnal acrss the bttm f the bx, resulting in in. They then add 5 t get a ttal f in. Similarly, t determine the ttal distance in the left figure (flying), students use their previus result fr the length acrss the diagnal f the bttm f the bx as the length f the lnger leg in a new right triangle (shwn belw). Students us this new triangle and apply the Pythagrean Therem again t find the distance between the tw ppsite crners directly. This yields a result f in. They then subtract t find the difference in path lengths. Thrugh experience with several examples and applicatins, students determine that there are very few sets f integers that satisfy the Pythagrean Therem. They begin t recgnize and recall these sets, knwn as Pythagrean triples, and use them t easily find missing measures. Students als realize that all multiples f Pythagrean triples are als Pythagrean triples. At a minimum they shuld be familiar with the fllwing sets f integers as Pythagrean triples: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25. Students can shw hw Pythagrean triplets can be represented as lattice pints n the crdinate plane marked in units f ne. Apply the Pythagrean Therem t find the distance between tw pints in a crdinate system (MA10-GR.8-S.4-GLE.2-EO.c) Students are accustmed t decmpsing a segment riginating frm the rigin (0, 0) in the crdinate plane int its hrizntal and vertical cmpnents frm their wrk with unit ratis. This ntin is easily extended t segments that d nt cntain the rigin as an endpint using their ability t find the distance between pints in the plane that share a cmmn crdinate. (CC.8.G.8) 8 th Grade, Mathematics 19 f 79

13 Fr example, a student is asked t find the hrizntal and vertical distance between the pints (1, 2) and (25, 9): (CC.8.G.8) Students calculate the distance between (1, 2) and (25, 2) t be 24. They calculate the distance between (25, 2) and (25, 9) t be 7. Students identify the triangle frmed by an arbitrary segment between any tw pints in the crdinate plane and the segments representing the hrizntal and vertical distance between thse pints as a right triangle. They are then able t use the distances between the endpints f bth the hrizntal and vertical segments as side lengths f the legs fr the right triangle frmed t apply the Pythagrean Therem t find the length f the third side (the hyptenuse), and recgnize this length as the distance between the riginal tw pints (see Standard 8.G.7 in the Triangles and Transfrmatins LT). (CC.8.G.8) Frm the example abve, students substitute and arrive at the equatin = c 2 and slve t find the distance between the tw given pints, (1, 2) and (25, 9). (CC.8.G.8) Students recgnize that the rientatin f the right triangle frmed by cnstructing hrizntal and vertical segments des nt affect the accuracy f the Pythagrean Therem. Thus, they are able t cnstruct apprpriate hrizntal and vertical segments in multiple directins. (CC.8.G.8) Critical Language: includes the Academic and Technical vcabulary, semantics, and discurse which are particular t and necessary fr accessing a given discipline. EXAMPLE: A student in Language Arts can demnstrate the ability t apply and cmprehend critical language thrugh the fllwing statement: Mark Twain expses the hypcrisy f slavery thrugh the use f satire. A student in can demnstrate the ability t apply and cmprehend critical language thrugh the fllwing statement(s): Academic Vcabulary: Technical Vcabulary: I can use the Pythagrean Therem t determine the missing side lengths f a right triangle and calculate distances n a crdinate plane. I knw ne-seventh is a ratinal number because it is the divisin f tw integers even thugh my calculatr des nt shw hw it repeats as a decimal. Lcate, unknwn, cmpare, estimate, apply, explain, prf, prve, cnvert, distance, representatin, Pythagrean Therem, leg, hyptenuse, right angle, integer, cnverse, right triangle, indirect measurement, ratinal numbers, irratinal numbers, divisin, integers, terminate, repeat, decimals, apprximatin, number line, square rt, cube rt, perfect square, perfect cube, decimal expansin, crdinate system 8 th Grade, Mathematics 20 f 79