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Download Grade 7 Mathematics Module 6, Topic B, Lesson 15
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Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’6 Lesson 15: Using Unique Triangles to Solve Real-World and Mathematical Problems Student Outcomes ο§ Students use conditions that determine a unique triangle to construct viable arguments that angle measures and lengths are equal between triangles. Lesson Notes In Lesson 15, students continue to apply their understanding of the conditions that determine a unique triangle. In Lesson 14, they were introduced to diagrams of triangles that had pre-existing relationships, as opposed to the diagrams in Lesson 13 that showed distinct triangles with three matching, marked parts. This added a new challenge to the task of determining whether triangles were identical because some information had to be assessed from the diagram in order to establish a condition that would determine triangles as identical. In Lesson 15, students are exposed to yet another challenge where they are asked to determine whether triangles are identical and to show how this information can lead to further conclusions about the diagram (i.e., showing why a given point must be the midpoint of a segment). Problems in this lesson are both real-world and mathematical. All problems require an explanation that logically links given knowledge, a correspondence, and a condition that determines triangles to be identical; some problems require these links to yield one more conclusion. This lesson is an opportunity to highlight Mathematical Practice 1, giving students an opportunity to build perseverance in solving problems. Classwork Example 1 (5 minutes) Example 1 MP.1 A triangular fence with two equal angles, β πΊ = β π», is used to enclose some sheep. A fence is constructed inside the triangle that exactly cuts the other angle into two equal angles: β πΊπΉπΎ = β π»πΉπΎ. Show that the gates, represented by πΊπΎ and πΎπ», are the same width. There is a correspondence β³ πΊπΉπΎ ββ³ π»πΉπΎ that matches two pairs of angles of equal measurement, β πΊ = β π» and β πΊπΉπΎ = β π»πΉπΎ, and one pair of sides of equal lengths shared side πΉπΎ. The triangles satisfy the two angles and side opposite a given angle condition. From the correspondence, we can conclude that πΊπΎ = πΎπ», or that the gates are of equal width. Lesson 15: Date: © 2014 Common Core, Inc. Some rights reserved. commoncore.org Using Unique Triangles to Solve Real-World and Mathematical Problems 5/8/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 145 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’6 Example 2 (5 minutes) Scaffolding: As students work through Example 2, remind them that this question is addressed in an easier format in Grade 4, when students folded the triangle so that π΄πΆ folded onto π΅πΆ. Example 2 In β³ π¨π©πͺ, π¨πͺ = π©πͺ. John says that the triangle correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ matches two sides and the included angle and shows that β π¨ = β π©. Is John correct? C C A B A ο§ If needed, provide a hint: Draw the segments π΄πΆ and π΅πΆ. ο§ Also, provide students with compasses so that they can mimic the construction marks in the diagram. B L esson Four t een, U sing U nique Tr iangles t o Solve R eal-W or ld and M at hem at ical We are told that π¨πͺ = π©πͺ. The correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ tells us that π©πͺ β π¨πͺ, πͺπ¨ β πͺπ©, and β πͺ β β πͺ, which Pr oblems means β³ π¨π©πͺ is identical to β³ π©π¨πͺ by the two sides and included angle condition. From the correspondence, we can St udent Out comes: St udent s use condit ions t hat det ermine a unique t riangle t o const ruct viable conclude that β π¨ = β π©; therefore, John is correct. argument s t hat angle measures of lengt hs equal. heyC,crit ique t\ he of otInhers. t riangle correspondence4.between (Using aa ttriangle riangle and correspondence itare self.) In 4 T between AB \ aA t=riangle B .reasoning and Jill it self.) 4 AB C, \ A = \ B . Jill t he t riangle correspondencesays 4 AB t hat C$ t he4t riangle B AC mat correspondence ches two angles 4 AB andCt he $ included 4 B AC mat ches two angles and t he included Exercises 1β4 (20 minutes) January plain why AC = B C. Is Jill side right t o ?explain why AC = B C. Is Jill right ? 1. Mary put s t he cent er of her compass at t he vert ex O of t he angle and locat es point s A and B onExercises t he sides 1β4 of t he angle. She t hen cent ers her compass at each of A and B t o locat e point #» C. She t hen const ruct softher hecompass ray OC. why \ B = points \ AOC. : sides Draw t he line 1. Mary puts the center at the Explain vertex πΆ of the angle andOC locates π¨ and π©Hint on the of the ββββββ . segment sangle. AC Next, andshe BC centers her compass at each of π¨ and π© to locate point C C πͺ. Finally, she constructs the ray πΆπͺ Explain why β π©πΆπͺ = β π¨πΆπͺ. B B O A A B C O B A O A C B A Since Mary uses one compass adjustment to determine points π¨ and π©, πΆπ¨ = πΆπ©. Mary also uses the same compass adjustment from π© and π¨ to find point πͺ; this means π©πͺ = π¨πͺ. Side πΆπͺ is common to both the triangles β³ πΆπ©πͺ and β³ πΆπ¨πͺ. Therefore, there is a correspondence β³ πΆπ©πͺ ββ³ πΆπ¨πͺ that matches three pairs of equal sides, A t riangular wit two equal angles, \ S = \ From T , is t o enclose some that sheep. and thefence triangles arehidentical by the three sides condition. theused correspondence, we conclude β π©πΆπͺA= fence β π¨πΆπͺ. is a model 5. of Quadrilat a kit e. T eral he diagonals ACB D is AB a model and CD of a represent kit e. T he t he diagonals st icks AB and CD represent is const ruct ed inside t he t riangle t hat exact ly cut s t he ot her angle int o two equal angles; i.e., 2. eral ACB D keep t he kit e\ rigid. hat .help keep rigid. SRW = \ TtRW Show t hatt he Wkit iset he midpoint of t he part of t he fence from S t o T . t he st icks says t hat \ ACD = \ B CD(a) . Can Johnyou says uset hat ident \ ical ACDt riangles =R \ B CD t o .show Can tyou hat use Johnident is ical t riangles t o show t hat John is ect ? correct ? ays t hat t he two st icks are perpendicular (b) Jill says t hat t o each t he two ot her. st icks Useare t heperpendicular fact t hat \ ACD t o each = ot her. Use t he fact t hat \ ACD = CD and ident ical t riangles t o show \ B \CD AEand C =ident 90β¦ .ical t riangles t o show \ AE C = 90β¦ . says t hat Jillβs t riangle correspondence (c) John sayst hat t hatshows Jillβs t riangle he st icks correspondence are perpendicular t hat t shows o t he st icks are perpendicular t o ot her also shows t hat t he t he st each icksotcross her also at t he shows midpoint t hat t he of tthe he horizont st icks cross al stat ick.t he midpoint of t he horizont al st ick. t do you t hink of Johnβs st at ement What ? do you t hink of Johnβs st at ement ? Lesson 15: Using Unique Triangles to Solve Real-World and Mathematical 146 Problems Date: © 2014 Common Core, Inc. Some rights reserved. commoncore.org 5/8/17 S T W This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 2. 7β’6 Quadrilateral π¨πͺπ©π« is a model of a kite. The diagonals π¨π© and πͺπ« represent the sticks that help keep the kite rigid. a. John says that β π¨πͺπ« = β π©πͺπ«. Can you use identical triangles to show that John is correct? From the diagram, we see that π¨πͺ = π©πͺ, and π¨π« = π©π«. πͺπ« is a common side to both triangles β³ π¨πͺπ« and β³ π©πͺπ«. There is a correspondence β³ π¨πͺπ« ββ³ π©πͺπ« that matches three pairs of equal sides; the two triangles are identical by the three sides condition. From the correspondence, we conclude that β π¨πͺπ« = β π©πͺπ«. John is correct. b. Jill says that the two sticks are perpendicular to each other. Use the fact that β π¨πͺπ« = β π©πͺπ« and what you know about identical triangles to show β π¨π¬πͺ = ππ°. Since we know that π¨πͺ = π©πͺ and β π¨πͺπ« = β π©πͺπ«, and that triangles β³ π¨πͺπ¬ and β³ π©πͺπ¬ share a common side, πͺπ¬, we can find a correspondence that matches two pairs of equal sides and a pair of equal, included angles. The triangles are identical by the two sides and included angle condition. We can then conclude that β π¨π¬πͺ = β π©π¬πͺ. Since both angles are adjacent to each other on a straight line, we also know they must sum to πππ°. We can then conclude that each angle measures ππ°. c. John says that Jillβs triangle correspondence that shows the sticks are perpendicular to each other also shows that the sticks cross at the midpoint of the horizontal stick. Is John correct? Explain. Since we have established that β³ π¨πͺπ¬ and β³ π©πͺπ¬ are adjacent to each other, we know that π¨π¬ = π©π¬. This means that π¬ is the midpoint of π¨π©, by definition. 3. In β³ π¨π©πͺ, β π¨ = β π©. Jill says that the triangle correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ matches two sides and the included angle and shows that π¨πͺ = π©πͺ. Is Jill correct? We are told that β π¨ = β π©. The correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ tells us that β π¨ = β π©, β π© = β π¨, and π¨π© = π©π¨, which means β³ π¨π©πͺ is identical to β³ π©π¨πͺ by the two angles and included side condition. From the correspondence, we can conclude that π¨πͺ = π©πͺ; therefore, Jill is correct. Lesson 15: Date: © 2014 Common Core, Inc. Some rights reserved. commoncore.org Using Unique Triangles to Solve Real-World and Mathematical Problems 5/8/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 147 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 4. 7β’6 Right triangular corner flags are used to mark a soccer field. The vinyl flags have a base of ππ cm and a height of ππ cm. a. Mary says that the two flags can be obtained by cutting a rectangle that is ππ cm × ππ cm on the diagonal. Will that create two identical flags? Explain. If the flag is to be cut from a rectangle, both triangles will have a side of length ππ cm, a length of ππ cm, and a right angle. There is a correspondence that matches two pairs of equal sides and an included pair of equal angles to the corner flag; the two triangles are identical to the corner flag as well as to each other. b. Will measures the two non-right angles on a flag and adds the measurements together. Can you explain, without measuring the angles, why his answer is ππ°? The two non-right angles of the flags are adjacent angles that together form one angle of the four angles of the rectangle. We know that a rectangle has four right angles, so it must be that the two non-right angles of the flag together sum to ππ°. Discussion (8 minutes) Consider a gallery walk as a means of reviewing responses to each exercise. ο§ Hold students accountable for providing evidence in their responses that logically progresses to a conclusion. ο§ Offer opportunities for students to share and compare their solution methods. Closing (2 minutes) ο§ In deciding whether two triangles are identical, examine the structure of the diagram of the two triangles to look for a relationship that might reveal information about corresponding parts of the triangles. This information may determine whether the parts of the triangle satisfy a particular condition, which might determine whether the triangles are identical. ο§ Be sure to identify and label all known measurements and then determine if any other measurements can be established based on knowledge of geometric relationships. Exit Ticket (5 minutes) Lesson 15: Date: © 2014 Common Core, Inc. Some rights reserved. commoncore.org Using Unique Triangles to Solve Real-World and Mathematical Problems 5/8/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 148 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM Name 7β’6 Date Lesson 15: Using Unique Triangles to Solve Real-World and Mathematical Problems Exit Ticket Alice is cutting wrapping paper to size to fit a package. How should she cut the rectangular paper into two triangles to ensure that each piece of wrapping paper is the same? Use your knowledge of conditions that determine unique triangles to justify that the resulting pieces from the cut are the same. A B D C Lesson 15: Date: © 2014 Common Core, Inc. Some rights reserved. commoncore.org A D B D B C Using Unique Triangles to Solve Real-World and Mathematical Problems 5/8/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 149 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’6 Exit Ticket Sample Solutions Alice is cutting wrapping paper to size to fit a package. How should she cut the rectangular paper into two triangles to ensure that each piece of wrapping paper is the same? Use your knowledge of conditions that determine unique triangles to prove that the resulting pieces from the cut are the same. A B D C A D B D B C Alice should cut along the diagonal of rectangle π¨π©πͺπ«. Since π¨π©πͺπ« is a rectangle, the opposite sides will be equal in length, or, π¨π© = π«πͺ and π¨π« = π©πͺ. A rectangle also has four right angles, which means a cut along the diagonal will result in each triangle with one ππ° angle. The correspondence β³ π¨π©π« ββ³ πͺπ«π© matches two equal pairs of sides and an equal, included pair of angles; the triangles are identical by the two sides and included angle condition. Problem Set Sample Solutions 1. Jack is asked to cut a cake into π equal pieces. He first cuts it into equal fourths in the shape of rectangles, and then he cuts each rectangle along a diagonal. Did he cut the cake into π equal pieces? Explain. Yes, Jack cut the cake into π equal pieces. Since the first series of cuts divided the cake into equal fourths in the shape of rectangles, we know that the opposite sides of the rectangles are equal in length; that means all π triangles have two sides that are equal in length to each other. Each of the triangular pieces also has one right angle because we know that rectangles have four right angles. Therefore, there is a correspondence between all π triangles that matches two pairs of equal sides and an equal, ππ° non-included angle, determining π identical pieces of cake. Lesson 15: Date: © 2014 Common Core, Inc. Some rights reserved. commoncore.org Using Unique Triangles to Solve Real-World and Mathematical Problems 5/8/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 150 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 2. 7β’6 The bridge below, which crosses a river, is built out of two triangular supports. The point π΄ lies on segment π©πͺ. The beams represented by π¨π΄ and π«π΄ are equal in length, and the beams represented by π¨π© and π«πͺ are equal in length. If the supports were constructed so that β π¨ and β π« are equal in measurement, is point π΄ the midpoint of π©πͺ? Explain. B M C A D Yes, π΄ is the midpoint of π©πͺ. The triangles are identical by the two sides and included angle condition. The correspondence β³ π¨π©π΄ ββ³ π«πͺπ΄ matches two pairs of equal sides and one pair of included, equal angles. Since the triangles are identical, we can use the correspondence to conclude that π©π΄ = πͺπ΄, which makes π΄ the midpoint, by definition. 3. In β³ π¨π©πͺ, π¨πͺ = π©πͺ. Bill says that triangle correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ matches three equal sides and shows that β π¨ = β π©. Is Bill correct? C A B We are told that π¨πͺ = π©πͺ. The correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ matches three equal sides: π¨π© = π©π¨, π©πͺ = π¨πͺ, and π¨πͺ = π©πͺ; this means β³ π¨π©πͺ is identical to β³ π©π¨πͺ by the three sides condition. From the correspondence, we can conclude that β π¨ = β π©; therefore, Bill is correct. 4. In the previous problem, Jill says that triangle correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ matches two equal sides and the included angle. This also shows that β π¨ = β π©. Is Jill correct? We are told that π¨πͺ = π©πͺ. The correspondence β³ π¨π©πͺ ββ³ π©π¨πͺ matches two equal sides and the included angle: π©πͺ = π¨πͺ, π¨πͺ = π©πͺ, and β πͺ = β πͺ; this means β³ π¨π©πͺ is identical to β³ π©π¨πͺ by the two sides and included angle condition. From the correspondence, we can conclude that β π¨ = β π©; therefore, Jill is correct. Lesson 15: Date: © 2014 Common Core, Inc. Some rights reserved. commoncore.org Using Unique Triangles to Solve Real-World and Mathematical Problems 5/8/17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 151
