0 0 Locker LSSON. Subdividing a Segment in a Given Ratio ommon ore Math Standards The student is expected to: G-G.6 ind the point on a directed line segment between two given points that partitions the segment in a given ratio. lso G-O. Mathematical ractices M. Using Tools Language Objective Work in groups to find ratios of subdivided segments. NGG ssential Question: How do you find the point on a directed line segment that partitions the given segment in a given ratio? If the segment lies on a number line, subtract the coordinates to find the distance between the endpoints. Then multiply the length by the ratio to find the coordinate of the point that divides the segment in that ratio. If there are no coordinates, use a compass and straightedge to divide the given segment into equal parts, and identify the point that divides the segment in the given ratio. RVIW: LSSON RORMN TSK View the ngage section online. iscuss the photograph and point out that the Golden Ratio is found throughout nature even, some say, in the ratio of the length of a person s forearm to the length of the person s hand. Then preview the Lesson erformance Task. Houghton Mifflin Harcourt ublishing ompany Image redits: ondor 6/Shutterstock Name lass ate. Subdividing a Segment in a Given Ratio ssential Question: How do you find the point on a directed line segment that partitions the given segment in a given ratio? xplore 0 artitioning a Segment in a One-imensional oordinate System It takes just one number to specify an exact location on a number line. or this reason, a number line is sometimes called a one-dimensional coordinate system. The mile markers on a straight stretch of a highway turn that part of the highway into a one-dimensional coordinate system. On a straight highway, the exit for rthur venue is at mile marker. The exit for ollingwood Road is at mile marker. The state highway administration plans to put an exit for riar Street at a point that is _ of the distance from rthur venue to ollingwood Road. ollow these steps to determine where the new exit should be placed. Mark rthur venue (point ) and ollingwood Road (point ) on the number line. 0 0 0 What is the distance from rthur venue to ollingwood Road? xplain. 0 miles; find the absolute value of the difference of the coordinates: - = 0. How far will _ the riar Street exit be from rthur venue? xplain. 0 miles; 0 miles = 0 miles What is the mile marker number for the riar Street exit? Why? mile marker ; + 0 = lot and label the riar Street exit (point ) on the number line. heck students number lines. 0 Resource Locker Module 6 Lesson Name lass ate. Subdividing a Segment in a Given Ratio ssential Question: How do you find the point on a directed line segment that partitions the given segment in a given ratio? G-G.6 ind the point on a directed line segment between two given points that partitions the segment in a given ratio. lso G-O. Houghton Mifflin Harcourt ublishing ompany Image redits: ondor 6/Shutterstock xplore artitioning a Segment in a One-imensional oordinate System It takes just one number to specify an exact location on a number line. or this reason, a number line is sometimes called a one-dimensional coordinate system. The mile markers on a straight stretch of a highway turn that part of the highway into a one-dimensional coordinate system. On a straight highway, the exit for rthur venue is at mile marker. The exit for ollingwood Road is at mile marker. The state highway administration plans to put an exit for riar Street at a point that is _ of the distance from rthur venue to ollingwood Road. ollow these steps to determine where the new exit should be placed. Mark rthur venue (point ) and ollingwood Road (point ) on the number line. 0 0 0 What is the distance from rthur venue to ollingwood Road? xplain. How far will the riar Street exit be from rthur venue? xplain. What is the mile marker number for the riar Street exit? Why? lot and label the riar Street exit (point ) on the number line. Resource 0 miles; find the absolute value of the difference of the coordinates: - = 0. _ 0 miles; 0 miles = 0 miles mile marker ; + 0 = heck students number lines. Module 6 Lesson HROVR Turn to Lesson. in the hardcover edition. 6 Lesson.
The highway administration also plans to put an exit for akota Lane at a point that divides the highway from rthur venue to ollingwood Road in a ratio of to. What is the mile marker number for akota Lane? Why? (Hint: Let the distance from rthur venue to akota Lane be x and let the distance from akota Lane to ollingwood Road be x.) mile marker 6; since the distance from rthur venue to ollingwood Road is 0 miles, x + x = 0, x = 0, and x = 6. Therefore, the distance from rthur venue to akota Lane is x = miles. lot and label the akota Lane exit (point ) on the number line. heck students number lines. Reflect. How can you tell that the location at which you plotted point is reasonable? oint appears to be about _ of the way (or about 67% of the way) from point to point.. Would your answer in Step be different if the exit for akota Lane divided the highway from rthur venue to ollingwood Road in a ratio of to? xplain. Yes; in this case, the exit would be closer to ollingwood venue. It would be at mile marker. XLOR artitioning a Segment in a One- imensional oordinate System INTGRT THNOLOGY Show how to use geometry software to partition a segment. QUSTIONING STRTGIS What does it mean to say that point _ divides in the ratio to? oint divides the segment into two segments, and the length of one is three times the length of the other. xplain artitioning a Segment in a Two-imensional oordinate System directed line segment is a segment between two points and with a specified direction, from to or from to. To partition a directed line segment is to divide it into two segments with a given ratio. xample ind the coordinates of the point that divides the directed line segment from to in the given ratio. (-8, -7), (8, ) ; to Step Write a ratio that expresses the distance of point along the segment from to. oint is _ + = _ of the distance from to. Step ind the run and the rise of the directed y 8 line segment. (8, ) run = 8 - (-8) = 6 rise = - (-7) = -8-0 - Rise (-8, -7) -8 Run (8, -7) Module 6 Lesson x Houghton Mifflin Harcourt ublishing ompany XLIN artitioning a Segment in a Two- imensional oordinate System INTGRT MTHMTIL RTIS ocus on atterns M.8 Have students draw a segment and find the midpoint to divide it into two segments with a ratio of to. Then have them find the midpoint of one of those segments to divide the segment with a ratio of to. sk students what other ratios they could divide the segments into by using midpoints. ROSSIONL VLOMNT Learning rogressions Students have previously studied slope and learned to find the slope of a line. They also used slope to write the equation of a line, either in slope-intercept form (y = mx + b) or in point-slope form (y y = m (x x )). Here students use the definition of slope (the ratio of the rise to the run) to help them find a point that divides a given line segment in a given ratio. Subdividing a Segment in a Given Ratio 6
QUSTIONING STRTGIS How can you predict whether the point to divide a line in a given ratio will be closer to one end or the other? If the ratio is less than onehalf, the point will be closer to the first endpoint, and if it is greater than one-half, the point will be closer to the second endpoint. an a proportion compare measurements that have different units? xplain. Yes; if the numerators are both in one unit and the denominators in another, or if one fraction is in one unit and the other fraction is in another Step oint is _ of the distance from point to point, so find _ of both the rise and the run. (6) = _ of rise = _ () = 9 Step To find the coordinates of point, add the values from Step to the coordinates of point. x-coordinate of point = -8 + = y-coordinate of point = -7 + 9 = The coordinates of point are (, ). (-, ), (, ) ; to -8-0 - - 8 y O (-8, -7) -8 Run (8, -7) Step Write a ratio that expresses the distance of point along the segment from to. oint is = _ of the distance from to. y + Step Graph the directed line segment. ind the rise and the run of the directed line segment. run = - (-) = 6-0 - (8, ) Rise x x rise = - = - - Step oint is _ of the distance from point to point. (6) = ( ) = - - Houghton Mifflin Harcourt ublishing ompany Reflect Step To find the coordinates of point, add the values from Step to the coordinates of point. x-coordinate of point = - + = - y-coordinate of point = + - = The coordinates of point are ( -, ). lot point on the above graph.. In art, show how you can use the istance ormula to check that point partitions the directed line segment in the correct ratio. = ; = (8 - ) + ( - ) = =. = ( - (-8)) + ( - (-7)) = The ratio of to is to or to, which is the correct ratio.. iscussion What can you conclude about a point that partitions a segment in the ratio to? How can you find the coordinates of such a point? The point is the midpoint of the segment. Use the Midpoint ormula. Module 6 Lesson OLLORTIV LRNING eer-to-eer ctivity Have students work in pairs to draw and label a line, a line segment, and a directed line segment. Have them discuss their similarities and differences. 6 Lesson.
Your Turn ind the coordinates of the point that divides the directed line segment from to in the given ratio.. (-6, ), (, -); to 6. (, ), (-6, -); to oint is _ + = of the distance oint is _ 8 + = of the distance from to. from to. run = - (-6) = 8; rise = - - = -8 run = -6 - = 0; rise = - = 8 ; of rise = - 8-6; of rise = -9 x-coordinate of point = -6 + = ; x-coordinate of point = - 6 = ; xplain _ y-coordinate of point = + (-) = 0 The coordinates of point are (, 0). _ y-coordinate of point = + (-9) = -7 The coordinates of point are (, -7). onstructing a artition of a Segment XLIN onstructing a artition of a Segment QUSTIONING STRTGIS an you partition a line (as opposed to a line segment)? xplain. No; because a line continues infinitely in both directions, it cannot be divided into segments with a definite ratio. xample Given the directed line segment from to, construct the point that divides the segment in the given ratio from to. to Step Use a straightedge to draw. The exact measure of the angle is not important, but the construction is easiest for angles from about 0 to 60. Step lace the compass point on and draw an arc through. Label the intersection. Using the same compass setting, draw an arc centered on and label the intersection. Using the same compass setting, draw an arc centered on and label the intersection. Step Use the straightedge to connect points and. onstruct an angle congruent to with as its vertex. onstruct an angle congruent to with as its vertex. Step The construction partitions _ into equal parts. Label point at the point that divides the segment in the ratio to from to. Houghton Mifflin Harcourt ublishing ompany Module 6 Lesson IRNTIT INSTRUTION Multiple Representations iscuss whether it matters if the segment with endpoints and is named as or as _. oth are correct. The order does not matter as long as and are endpoints. If the segment is a directed line segment, the order does matter. Subdividing a Segment in a Given Ratio 6
VOI OMMON RRORS Remind students to pay attention to the order of the letters in the problem in order to ensure they find the correct point for the given direction. to G Step Use a straightedge to draw _. Step lace the compass point on and draw an arc through _. Label the intersection. Using the same compass setting, draw an arc centered on and label the intersection. Using the same compass setting, draw an arc centered on and label the intersection. Using the same compass setting, draw an arc centered on and label the intersection G. Step Use the straightedge to connect points and G. onstruct angles congruent to G with,, and as the vertices. Step The construction partitions _ into equal parts. Label point at the point that divides the segment in the ratio to from to. Reflect 7. In art, why is _ is parallel to _? was constructed to be congruent to. These are congruent corresponding angles, so _ _. Houghton Mifflin Harcourt ublishing ompany 8. How can you use the Triangle roportionality Theorem to explain why this construction method works? The construction ensures that the segments of the ray you constructed are congruent. lso, the segments that are drawn in Step are all parallel. y the Triangle roportionality Theorem, you can conclude that the segments along _ are congruent to each other. Your Turn Given the directed line segment from to, construct the point that divides the segment in the given ratio from to. 9. to 0. to G H Module 6 Lesson LNGUG SUORT onnect Vocabulary Help students develop an understanding of the meaning of dimension by considering one-dimensional figures, such as a line, whose only dimension is length, and two-dimensional figures, such as a square. The two dimensions of a square are length and width. 6 Lesson.
laborate. How is a one-dimensional coordinate system similar to a two-dimensional coordinate system? How is it different? In both types of coordinate systems, numbers are used to specify the locations of points. In a one-dimensional coordinate system, a single number is used to specify the location of a point on a line. In a two-dimensional coordinate system, two numbers are used to specify the location of a point on a plane.. Is finding a point that is _ of the distance from point to point the same as finding a point that divides _ in the ratio to? xplain. No; the point that is _ of the distance from point to point is 80% of the distance from point to point ; the point that divides _ in the ratio to is _ or approximately % of 9 the distance from point to point.. ssential Question heck-in What are some different ways to divide a segment in the ratio to? If the segment lies on a number line, subtract the coordinates to find the distance between the endpoints. Then find the point that is _ of the distance from one endpoint to the other. If the segment is on a coordinate plane, use the run and the rise to find the point that is _ of the distance from one endpoint to the other. If there are no coordinates, use a compass and straightedge to divide the given segment into equal parts. Then identify the point that divides the segment in the ratio to. valuate: Homework and ractice choreographer uses a number line to position dancers for a ballet. ancers and have coordinates and, respectively. In xercises, find the coordinate for each of the following dancers based on the given locations. Online Homework Hints and Help xtra ractice. ancer stands at a point that is _ of the. ancer stands at a point that is of the 6 distance from ancer to ancer. distance from ancer to ancer. = - = 8 = - = 8 _ 8 = 6 8 = 6 ancer is units from ancer, so the ancer is 6 units from ancer, so the coordinate for ancer is + = 0. coordinate for ancer is + 6 =. Module 66 Lesson Houghton Mifflin Harcourt ublishing ompany Image redits: mcpix/istockhoto.com LORT QUSTIONING STRTGIS an a proportion compare measurements that have different units? xplain. Yes, but each ratio must compare measurements with the same unit; for example, you can write a proportion with a ratio of feet to feet equal to a ratio of meters to meters. SUMMRIZ TH LSSON How can you divide a directed line segment in a given ratio? irected line segments are always read in the order in which the points are given. segment partition divides the segment from a given point to another point in a ratio of a : b. In a coordinate plane, the rise and run can be used to find the point dividing the segment in the ratio a : b. Segment partitions can also be constructed using a compass and straightedge. Subdividing a Segment in a Given Ratio 66
VLUT. ancer stands at a point that divides the line segment from ancer to ancer in a ratio of to. = - = 8 Let the distance from ancer to ancer be x and let the distance from ancer to ancer be x. Then x + x = 8, x = 8, and x = 6. So the distance from ancer to ancer is x = (6) =. The coordinate for ancer is + = 7. SSIGNMNT GUI oncepts and Skills xplore artitioning a Segment in a One- imensional oordinate System xample artitioning a Segment in a Two- imensional oordinate System xample onstructing a artition of a Segment ractice xercises xercises 8 xercises 9. ancer stands at a point that divides the line segment from ancer to ancer in a ratio of to. = - = 8 Let the distance from ancer to ancer be x and let the distance from ancer to ancer be x. Then x + x = 8, 6x = 8, and x =. So the distance from ancer to ancer is x =. The coordinate for ancer is + = 8. ind the coordinates of the point that divides the directed line segment from to in the given ratio.. (, ), (, ) ; to 6. (, ), (7, ) ; 7 to is + = _ of the distance from to. is 7 7 + = 7_ of the distance from to. 8 run = - ( ) = ; rise = - ( ) = run = 7 - ( ) = 8; rise = - = -8 () = 9; _ of rise = _ () = 7_ 7_ (8) = 7; 7_ of rise = 7_ (-8) = -7 8 8 8 8 x-coordinate of point = + 9 = 6; x-coordinate of point = + 7 = 6; y-coordinate of point = + = y-coordinate of point = + (-7) = The coordinates of point are (6, ). The coordinates of point are (6, ). INTGRT MTHMTIL RTIS ocus on Modeling M. iscuss the importance of careful constructions and labels in the portioning of segments. Houghton Mifflin Harcourt ublishing ompany 7. (, ), (-9, 0) ; to 8. (7, ), (-7, ) ; to is + = of the distance from to. is + = _ of the distance from to. 7 run = -9 - ( ) = -8; rise = 0 - = - run = -7-7 = ; rise = - ( ) = 7 (-8) = -; of rise = (-) = - (-) = -6; _ of rise = _ 7 7 7 7 (7) = x-coordinate of point = + ( ) = ; y-coordinate of point = + ( ) = The coordinates of point are (, ). Given the directed line segment from to, construct the point that divides the segment in the given ratio from to. 9. to 0. to G x-coordinate of point = 7 + (-6) = ; y-coordinate of point = + = 0 The coordinates of point are (, 0). G H Module 67 Lesson xercise epth of Knowledge (.O.K.) Mathematical ractices Skills/oncepts M. Modeling 8 Skills/oncepts M. Reasoning 9 Skills/oncepts M. Using Tools 6 Skills/oncepts M. Modeling 7 0 Skills/oncepts M. Modeling Skills/oncepts M. Reasoning Strategic Thinking M. Logic 67 Lesson. Strategic Thinking M. Logic
Given the directed line segment from to, construct the point that divides the segment in the given ratio from to.. to. to G H G VOI OMMON RRORS Some students may confuse various ratios. Remind them that the similarity ratio refers only to the ratio of the lengths of the sides. It is equal to the perimeter ratio and the square root of the area ratio. H ind the coordinate of the point that divides each directed line segment in the given ratio. J K L M N -0-0 0 0 0. from J to M; to 9. from K to L; to JM = 0 - (-) = KL = - (-6) = Let J = x and let M = 9x. Let K = x and let L = x. Then x + 9x =, 0x =, and x =.. Then x + x =, x =, and x =.. The coordinate of point is +. =. The coordinate of point is -6 +. = 0.. from N to K; to 6. from K to J; 7 to NK = -6-8 = Let N = x and let K = x. Then x + x =, 8x =, and x =. So N = () = 9. The coordinate of point is 8-9 = 9. KJ = - - (-6) = 9 Let K = 7x and let J = x. Then 7x + x = 9, 8x = 9, x = 0.. So K = 7 (0.) =.. The coordinate of point is -6 -. = -9.. 7. ommunicate Mathematical Ideas Leon constructed a point that divides the directed segment from to in the ratio to. helsea constructed a point Q that divides the directed segment from to in the ratio to. How are points and Q related? xplain. oints and Q are the same point. Sample explanation: oint is _ of the distance from to. oint Q is of the distance from to. This means the points lie at the same location along the line segment. Houghton Mifflin Harcourt ublishing ompany Module 68 Lesson xercise epth of Knowledge (.O.K.) Mathematical ractices Strategic Thinking M. Reasoning Strategic Thinking M. Reasoning Subdividing a Segment in a Given Ratio 68
8. ity planners use a number line to place landmarks along a new street. ach unit of the number line represents 00 feet. fountain is located at coordinate and a plaza is located at coordinate. The city planners place two benches along the street at points that divide the segment from to in the ratios to and to. What is the distance between the benches? = - (-) = ; Then x + x =, x =, and x = 8. Let the distance from to the first bench be x and let the distance from the first bench to be x. The coordinate for the first bench is + 8 =. Let the distance from to the second bench be x and let the distance from the second bench to be x. Then x + x =, x =, x = 6. So the distance from to the second bench is x = (6) = 8 units. The coordinate for the second bench is + 8 =. The distance between the benches is - = 0 units or 000 feet. 9. The course for a marathon includes a straight segment from city hall to the main library. The planning committee wants to put water stations along this part of the course so that the stations divide the segment into three equal parts. ind the coordinates of the points at the which the water stations should be placed. Houghton Mifflin Harcourt ublishing ompany Image redits: sportgraphic/otolia - - 0 - ity Hall - y Main Library M rom to M, run = - ( ) = 6; rise = - ( ) = Let the water stations be at points and Q. oint is of the distance from to M. (6) = ; of rise = () = _ x-coordinate of = + = ; y-coordinate of = - + _ = - The coordinates of point are (-, - ). oint Q is _ of the distance from to M. (6) = ; _ of rise = _ () = x-coordinate of Q = + = ; y-coordinate of Q = - + = The coordinates of point Q are (, ). The water stations should be placed at (-, - x ) and (, ). Module 69 Lesson 69 Lesson.
0. Multi-Step arlos is driving on a straight section of highway from shford to Lincoln. shford is at mile marker and Lincoln is at mile marker. rest stop is located along the highway _ of the distance from shford to Lincoln. ssuming arlos drives at a constant rate of 60 miles per hour, how long will it take him to drive from shford to the rest stop? The distance from shford to Lincoln is - = 0 miles. _ 0 = 80, so arlos must drive 80 miles from shford to the rest stop. d = rt, where d is the distance, r is the rate, and t is the time, so 80 = 60t, and t =. So it will take hours ( hour and 0 minutes) to drive from shford to the rest stop.. The directed segment from J to K is shown in the figure. y oints divide the segment from J to K in the each of the following J ratios. Which points have integer coordinates? Select all that apply. to. to. to. to. to - - 0 - K x VOI OMMON RRORS Students may forget to use the correct direction to partition line segments. Remind students to doublecheck direction in order to choose the correct partition point. rom J to K, run = 0 - ( ) = ; rise = - = -6. The point is + = of the distance from J to K. () = ; of rise = (-6) = -. x-coordinate of the point is - + = - ; y-coordinate of the point is + ( ) = 0. The point does not have integer coordinates.. The point is + = _ of the distance from J to K. () = ; _ (-6) = -. x-coordinate of the point is - + = -; y-coordinate of the of rise = _ point is + (-) =. The point has integer coordinates. + = _. The point is of rise = _ (-6) = - _. x-coordinate of the point is - + = - _ ; y-coordinate of the point is + _ (- ) = _. The point does not have integer coordinates.. The point is + = of the distance from J to K. () = _ ; of rise = (-6) = -. x-coordinate of the point is - + _ = - ; y-coordinate of the point is + (- ) =. The point does not have integer coordinates.. The point is + = of the distance from J to K. () = ; (-6) = -. x-coordinate of the point is - + = -; y-coordinate of the of rise = of the distance from J to K. () = ; _ point is + ( ) =. The point has integer coordinates. Houghton Mifflin Harcourt ublishing ompany Module 60 Lesson Subdividing a Segment in a Given Ratio 60
JOURNL Have students summarize the process for finding the point on a directed line segment that partitions the segment in a given ratio. H.O.T. ocus on Higher Order Thinking. ritique Reasoning Jeffrey was given a directed line segment and was asked to use a compass and straightedge to construct the point that divides the segment in the ratio to. He said he would have to draw a ray and then construct 6 congruent segments along the ray. Tamara said it is not necessary to construct 6 congruent segments along the ray. o you agree? If so, explain Tamara s shortcut. If not, explain why not. Yes; the ratio to is equivalent to the ratio to. To construct a point that divides a segment in the ratio to, it is only necessary to construct congruent segments along the ray. Houghton Mifflin Harcourt ublishing ompany. xplain the rror oint has coordinate -9 and point has coordinate 9. student was asked to find the coordinate of the point that is _ of the distance from to. The student said the coordinate of point is. a. Without doing any calculations, how can you tell that the student made an error? oint must be closer to point than to point, so the coordinate of point should be positive. b. What error do you think the student made? Sample answer: The student found the coordinate of the point that is _ of the distance from to.. nalyze Relationships oint divides the directed segment from to in the ratio to. The coordinates of point are (-, -) and the coordinates of point are (, ). ind the coordinates of point. + = _ oint is of the distance from to. The run from to is - (-) = 6. Let the run from to be x. Then 6 = _ x and x = 0. The rise from to is - (-) =. Let the rise from to be y. Then = _ y and y =. x-coordinate of point = - + 0 = 6; y-coordinate of point = - + = The coordinates of point are (6, ) _. ritical Thinking RS passes through R (, ) and S (, ). ind a point on RS _ such that the ratio of R to S is to. Is there more than one possibility? xplain. If point is on _ RS, then point is + = _ of the distance from R to S. 9 run = - (-) = 7; rise = - = 9 9 (7) = 8_ 9 ; _ of rise = _ 9 9 () = 9 x-coordinate of point = - + 8_ 9 = 8_ In this case, the coordinates of point are ( 8_ 9, 9 ; y-coordinate of point = + 9 = 9 There is also a point, not on RS, that lies beyond point S. Let have coordinates (x, y). Then rise of R rise of S = _ so x - (-) = _, (x + ) = (x - ), x + = x - 0, = x - 0, x - and x =. lso, rise of R rise of S = _, so y - y - = _, (y - ) = (y - ), y - = y -, - = y -, and y =. 9). In this case, the coordinates of point are (, ). Module 6 Lesson 6 Lesson.
Lesson erformance Task In this lesson you will subdivide line segments in given ratios. The diagram shows a line segment divided into two parts in such a way that the longer part divided by the shorter part equals the entire length divided by the longer part: a_ b = _ a + a b ach of these ratios is called the Golden Ratio. To find the point on a line segment that divides the segment this way, study this figure: a a + b b L M S R Q In the figure, LMQS is a square. LN equals the Golden Ratio (the entire segment length divided LM by the longer part).. escribe how, starting with line segment _ LM, you can find the location of point N.. Letting LM equal, find LN LM = LN = LN, the Golden Ratio. escribe your method.. Using _ LM as one side, construct square LMQS. onstruct R, the midpoint of _ SQ. lace the compass point on R and use _ RM as a radius to construct an arc intersecting line SQ. Mark point where _ RM intersects SQ. onstruct a perpendicular through intersecting line RM. Mark point N where the perpendicular intersects LM.. RM equals the length of the hypotenuse of a right triangle with sides and 0.. y the ythagorean Theorem, RM.8. So, R.8. SR = 0., so S.8 + 0..68. LN = S, so LN, the Golden Ratio, equals approximately.68. N Houghton Mifflin Harcourt ublishing ompany INTGRT MTHMTIL RTIS ocus on Modeling M. The opening of the Lesson erformance Task states that for the divided line segment a + b, the longer part divided by the shorter part equals the entire length divided by the longer part. So, a b = _ a + b a. Use that same relationship to complete this equation relating to line segment LN _ LM =?. LN MN LM INTGRT MTHMTIL RTIS ocus on Math onnections M. The Lesson erformance Task shows a line segment divided into two parts a and b such that a b = _ a + b a. If b =, then a = _ a + a. Solve the equation for a. ompare your results with your results in the Lesson erformance Task and make a conjecture about the value of a. + a =.68; a appears to equal the Golden Ratio. Module 6 Lesson XTNSION TIVITY The first two numbers in the ibonacci sequence of natural numbers are and. To find each new term in the sequence, add the two previous terms. So, the first few terms are,,,,, and 8. Have students use calculators to write the ratios of each of the first ten numbers in the sequence to the previous number, rounding to the nearest ten-thousandth. Here are the first four ratios: _ = ; = ; =.;.. fter students complete their calculations, have them compare their results with their results in the Lesson erformance Task, and then make a conjecture about the ratios they calculated. The ratios get closer and closer to the Golden Ratio the farther the sequence of ratios continues. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Subdividing a Segment in a Given Ratio 6