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Geometry and Measurement of Plane Figures Activity Set 6 Trainer Guide geometry and measurement of Plane figures—Activity Set 6 Mid_PGe_06_TG Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF Plane FIGURES Activity Set 6 NGSSS 6.G.4.2 NGSSS 7.A.5.2 GeoJam In this activity, participants will apply computation skills and their understanding of plane geometry to various problem-solving situations. Materials • Transparency/Page: In the Swim • Transparency/Page: In the Swim Answer Key • Transparency/Page: Absolutely Floored • Transparency/Page: Absolutely Floored Answer Key A • Transparency/Page: Absolutely Floored Answer Key B • Transparency/Page: Angle Puzzle • Transparency/Page: Angle Puzzle Answer Key • calculator (1 per pair) • blank transparency (1 per group) Vocabulary • area • perimeter • angle Time: 30 minutes TEACHING TIP: This activity can be modified to address specific content skills. For example, to focus on angles, have all participants work on the angle puzzle. geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_TG 1 GEOMETRY AND MEASUREMENT OF Plane FIGURES Activity Set 6 Introduce •Tell participants that they are going to apply some of their knowledge about area and perimeter to real-life problem-solving situations. in the swim The community recreation center is building a new swimming pool and deck area. The deck will measure 150 feet wide and 250 feet long, including a 3 feet wide walkway that will wrap around the pool. The pool, which is 82 feet wide and 164 feet long, will be centered in the deck area. 1. How much area will the deck cover? (including walkway) 2. How much area will the walkway cover? geometry and measurement of Plane figures—activity set 8 TRANS_MS_PG_08 Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development Transparency: In the Swim •Display Transparency: In the Swim and have participants take out their matching pages. •Tell participants that their local recreation center is building a new pool and deck area. Participants are on the committee to help choose the material that will be used for the deck. Participants will need to know how much area the deck will cover before the material to be used can be determined. •Tell participants that the length of the total area will be 250 feet and the width 150 feet. In this area, a pool, measuring 164 feet long and 82 feet wide, will be centered. •Ask a participant volunteer to come up and label the transparency with the facts that are known about the pool and deck areas. TEACHING TIP: Emphasize that the deck includes the walkway and that the calculations concerning the deck should reflect this fact. •Have a volunteer participant suggest the first step in the solution process for question 1. (Find the area of the total deck, including the area that the pool will cover.) •Ask participants what the length of the area is. (250 feet) •Ask participants what the width of the area is. (150 feet) geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_TG 2 GEOMETRY AND MEASUREMENT OF Plane FIGURES Activity Set 6 •Ask participants how they would find the area of the deck. (Multiply the length times the width of the total area and subtract the area of the pool.) Note: This is one approach. There may be others. •Write the problem on the transparency. (250 • 150 = __) •Direct participants to find the total area. (250 • 150 = 37,500 square feet) •Fill in the correct answer to the problem on the transparency. (37,500 square feet) •Ask a volunteer participant to suggest the next step in the problem. (Find the area of the pool.) •Ask participants what the length of the pool is. (164 feet) •Ask participants what the width of the pool is. (82 feet) •Ask participants how they would find the area of the pool. (Multiply the length times the width.) •Write the problem on the transparency. (164 • 82 = __) •Direct participants to find the area of the pool. (164 • 82 = 13,448 square feet) •Fill in the correct answer to the problem on the transparency. (13,448 square feet) •Ask participants to suggest the next step in solving the problem. (Subtract the area of the pool from the area of the total deck.) •Ask participants what the total area is. (37,500 square feet) •Ask participants what the area of the pool is. (13,448 square feet) •Write the problem on the transparency. (37,500 – 13,448 = __) geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_TG 3 GEOMETRY AND MEASUREMENT OF Plane FIGURES Activity Set 6 •Direct participants to solve the problem. (total area – pool area = deck area, 24,052 square feet) •Ask participants how they could determine the total area of the walkway and the pool. •Explain, if they do not mention it, that the dimensions of the area of the walkway and the pool together are 6 feet wider and 6 feet longer than the dimensions of the pool alone. •Ask participants to determine the total area of the walkway and the pool. (88 • 170 = 14,960 square feet) absolutely Floored The Andagans are buying new carpet for their living room, family room, and dining room. The carpet that they have selected costs $36.00 per square yard. 12' Use the picture to find how much they will have to pay for the carpet. (Round to the nearest dollar.) 12' 20' 8' Living Room 10' 16' Dining Room Family Room •Ask participants how to use the information they now have to find the area of the walkway alone. •Explain, if they do not mention it, that they can subtract the area of the pool from the area of the pool and walkway combined. 12' •Fill in the corresponding numbers and solve the problem. (14,960 – 13,448 = 1,512 square feet of walkway) geometry and measurement of Plane figures—activity set 8 TRANS_MS_PG_08 Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development Transparency: Absolutely Floored angle Puzzle Use the figure below to find the measure of each shaded angle. (Matching hash marks indicate congruent lines.) angle 1 angle 4 angle 2 angle 5 angle 3 3° 20° �2 Discuss and Do •Tell participants that they will now do some problem solving of their own. •Have participants work in pairs or groups of three within grade-level groups (as appropriate). �5 •Give each pair or group a blank transparency. 125° �3 125° �4 �1 •Display Transparency: Absolutely Floored and Transparency: Angle Puzzle and have participants take out their matching pages. geometry and measurement of Plane figures—activity set 8 TRANS_MS_PG_08 Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development Transparency: Angle Puzzle geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_TG 4 GEOMETRY AND MEASUREMENT OF Plane FIGURES Activity Set 6 •Assign each group one of the following pages: Absolutely Floored absolutely Floored ◆ Answer Key A The Andagans are buying new carpet for their living room, family room, and dining room. The carpet that they have selected costs $36.00 per square yard. 12' Use the picture to find how much they will have to pay for the carpet. (Round to the nearest dollar.) 12' 20' 8' Living Room 10' Dining Room •Ask participants to use the blank transparencies to show their step-by-step solutions to the problems. •Give participants 5–7 minutes to work and record their solutions. Family Room 16' Angle Puzzle ◆ 12' area of living room = 12 • 20 = 240 sq ft area of dining room = 12 • 8 = 96 sq ft area of family room = 16 (10 + 12) = (16)(22) = 352 sq ft total area = 240 + 96 + 352 = 688 sq ft total area in square yards = 688 ÷ 9 = 76.44 (1 sq yd = 9 sq ft) cost = total square yards • $36.00 cost = 76.44 • $36.00 = $2,752.00 Conclude (rounded to the nearest dollar) geometry and measurement of Plane figures—activity set 6 TRANS_MS_PG_06 Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development Transparency: Absolutely Floored Answer Key A absolutely Floored Answer Key B The Andagans are buying new carpet for their living room, family room, and dining room. The carpet that they have selected costs $36.00 per square yard. 12' Use the picture to find how much they will have to pay for the carpet. (Round to the nearest dollar.) 12' 16' •Ask for a volunteer from a Absolutely Floored group to share its step-by-step solution with the whole group. •Ask if any other Absolutely Floored group had a different solution or a different approach for solving the problem. 20' 8' 10' •Call the groups together. Living Room Dining Room Family Room 12' area of family and dining rooms (including empty space) = 24 • 22 = 528 sq ft area of empty space = 10 • 8 = 80 sq ft area of family and dining rooms (not including empty space) = 528 – 80 = 448 sq ft area of living room = 12 • 20 = 240 sq ft total area = 448 + 240 = 688 sq ft •Have any group that had a different solution or approach to the problem share its solution. •Display, if participants do not suggest both solutions, Transparencies: Absolutely Floored Answer Keys A and B to show two approaches for solving this problem. total area in sq yd = 688 ÷ 9 = 76.44 (1 sq yd = 9 sq ft) cost = total sq yd • $36.00 cost = 76.44 • $36.00 = $2,752.00 (rounded to the nearest dollar) geometry and measurement of Plane figures—activity set 8 TRANS_MS_PG_08 Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development Transparency: Absolutely Floored Answer Key B •Ask participants who solved Angle Puzzle how their problem differed from the flooring problem. (It involved angles instead of area and perimeter.) •Ask participants what additional knowledge they needed to solve this problem. Some possible answers may include: The sum of all the angles in a triangle is 180˚. ◆ A straight angle is equal to 180˚. ◆ A right angle is equal to 90º. ◆ The sides and angles of an equilateral triangle are congruent. ◆ geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_TG 5 GEOMETRY AND MEASUREMENT OF Plane FIGURES Activity Set 6 •Ask for a volunteer from an Angle Puzzle group to share a step-by-step solution with the whole group. •Refer to Transparency: Angle Puzzle Answer Key to resolve any questions. •Display a blank transparency. •Ask all participants to name some skills that students need, in addition to knowledge about area, perimeter, and angles, in order to solve real-life problems similar to these. •Record participant responses on the blank transparency. Some possible answers include the ability to read for understanding ◆ solve multistep problems ◆ sequence information ◆ choose the correct operation ◆ use addition ◆ use subtraction ◆ use multiplication ◆ use division ◆ use rounding ◆ compute the number of square feet in a square yard ◆ End of GeoJam geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_TG 6 In the Swim The community recreation center is building a new swimming pool and deck area. The deck will measure 150 feet wide and 250 feet long, including a 3 feet wide walkway that will wrap around the pool. The pool, which is 82 feet wide and 164 feet long, will be centered in the deck area. 1. How much area will the deck cover? (including walkway) 2. How much area will the walkway cover? geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_PM In the Swim Answer Key The community recreation center is building a new swimming pool and deck area. The deck will measure 150 feet wide and 250 feet long, including a 3 feet wide walkway that will wrap around the pool. The pool, which is 82 feet wide and 164 feet long, will be centered in the deck area. 1. How much area will the deck cover? (including walkway) deck = 150 • 250 = 37,500 square feet pool = 82 • 164 = 13,448 square feet deck – pool = 24,052 square feet of decking 2. How much area will the walkway cover? walkway = 88 • 170 = 14,960 square feet walkway – pool = 1,512 square feet of walkway geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_PM Absolutely Floored The Andagans are buying new carpet for their living room, family room, and dining room. The carpet that they have selected costs $36.00 per square yard. 12' Use the picture to find how much they will have to pay for the carpet. (Round to the nearest dollar.) 12' 20' 8' 10' 16' Living Room Dining Room Family Room 12' geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_PM Absolutely Floored Answer Key A The Andagans are buying new carpet for their living room, family room, and dining room. The carpet that they have selected costs $36.00 per square yard. 12' Use the picture to find how much they will have to pay for the carpet. (Round to the nearest dollar.) 12' 20' 8' 10' 16' Living Room Dining Room Family Room 12' area of living room = 12 • 20 = 240 sq ft area of dining room = 12 • 8 = 96 sq ft area of family room = 16 (10 + 12) = (16)(22) = 352 sq ft total area = 240 + 96 + 352 = 688 sq ft total area in square yards = 688 ÷ 9 = 76.44 (1 sq yd = 9 sq ft) cost = total square yards • $36.00 cost = 76.44 • $36.00 = $2,752.00 (rounded to the nearest dollar) geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_PM Absolutely Floored Answer Key B The Andagans are buying new carpet for their living room, family room, and dining room. The carpet that they have selected costs $36.00 per square yard. 12' Use the picture to find how much they will have to pay for the carpet. (Round to the nearest dollar.) 12' 20' 8' 10' 16' Living Room Dining Room Family Room 12' area of family and dining rooms (including empty space) = 24 • 22 = 528 sq ft area of empty space = 10 • 8 = 80 sq ft rea of family and dining rooms a (not including empty space) = 528 – 80 = 448 sq ft area of living room = 12 • 20 = 240 sq ft total area = 448 + 240 = 688 sq ft total area in sq yd = 688 ÷ 9 = 76.44 (1 sq yd = 9 sq ft) cost = total sq yd • $36.00 cost = 76.44 • $36.00 = $2,752.00 (rounded to the nearest dollar) geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_PM Angle Puzzle Use the figure below to find the measure of each shaded angle. (Matching hash marks indicate congruent lines.) angle 1 angle 4 angle 2 angle 5 angle 3 3° 20° �2 125° �3 �5 125° �4 �1 geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_PM Angle Puzzle Answer Key Step 6 Step 4 Angle 1 and angle 2 are formed by a line that crosses parallel lines. Therefore they are equal to their alternate interior angles: 20º and 35º. This means that angle 3 is 180º – 20º – 35º = 125º. Subtract angle 4 and 125º from 180º to find the remaining angle (5) of this triangle. 180º – 125º – 33º = 22º 20° �2 3° �5 Step 2 Angle b is equal to 180º – 90º – 3º. (A triangle has 180º, so subtract the known angles to find the unknown.) �3 125° 125° �4 b a c �1 Step 1 Angle a is part of an equilateral triangle—all angles are equal. So, angle a equals 60º. Step 5 180º – 125º – 20º = 35º (The angles of a triangle sum to 180º.) Step 3 Angle a plus angle b plus angle 4 = 180º—a straight line. 180º – a – b = angle 4 180º – 60º – 87º = 33º geometry and measurement of Plane figures—Activity Set 6 Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development Mid_PGe_06_PM Glossary Geometry and Measurement of Plane Figures acute angle An angle with a measure less than 90degrees (°). angle A geometric figure composed of two rays or line segments that share the same endpoint, called a vertex. area The number of square units in a region. circle The set of all points in a plane that are the same distance from a fixed point (the center of the circle). circumference The perimeter of (distance around) a circle. The circumference can be found using the formula C = 2πr, where C is the circumference of the circle and r is the radius of the circle. congruent figures Two figures that have identical size and shape so that when one is placed over the other, they coincide exactly. coordinate pair An ordered pair of numbers that indicates the position of a point on a plane. The first number of a coordinate pair gives the point’s location in relation to the x-axis. The second number in a coordinate pair gives the point’s location in relation to the y-axis. coordinate plane A plane containing an x-axis and a y-axis. Every point on the plane can be described using a coordinate pair. degree (°) A unit of measure for angles. 1° is around a point. 1 360 of a complete revolution equilateral The property of having equal, or congruent, sides. equilateral triangle A three-sided polygon with all sides and with all angles congruent. hexagon A six-sided polygon. irregular polygon A polygon in which not all the sides are congruent and not all the angles have the same measure. geometry and and measurement measurement of of Plane Plane figures—activity figures—Activityset Set16 BLM_MS_PG_01 Mid_PGe_06_PM Copyright© thebyMcGraw-Hill Companies—McGraw-Hill Professional Development Copyright© by 2002 the McGraw-Hill Companies—McGraw-Hill Professional Development Glossary (continued) isosceles triangle A triangle that has two congruent sides and two congruent angles. line The set of all contiguous (touching) points that form a straight path extending indefinitely in two directions opposite each other. line segment A part of a straight line that has two end points and a fixed length; a straight line segment marks the shortest distance between two points. linear unit A unit of measure for elements of a single dimension—length. obtuse angle An angle with a measure greater than 90° and less than 180°. parallel lines Lines that do not intersect and that are everywhere equidistant from each other. parallelogram A quadrilateral in which both pairs of opposite sides are parallel. pentagon A five-sided polygon. perimeter The distance around the outside of a plane shape or figure. perpendicular At right angles to. Two lines are perpendicular if their intersection creates right angles. pi (π) The ratio of the circumference of any circle to its diameter (3.141592653. . .). Pi is usually represented by the Greek letter, π. plane A flat surface that extends forever in all directions. plane figure A figure that lies entirely in one plane. point A location in space. polygon A simple, closed plane shape composed of a minimum of three straight-line segments. geometry and and measurement measurement of of Plane Plane figures—activity figures—Activityset Set16 BLM_MS_PG_01 Mid_PGe_06_PM Copyright© thebyMcGraw-Hill Companies—McGraw-Hill Professional Development Copyright© by 2002 the McGraw-Hill Companies—McGraw-Hill Professional Development Glossary (continued) quadrilateral A four-sided polygon. radius A segment connecting the center of a circle to any point on the circle; the length of the radius. ray A subset of a line that includes one endpoint and that extends infinitely from that endpoint in one direction. rectangle A quadrilateral that includes four interior right angles. regular polygon A polygon in which all the sides are congruent and all the angles have the same measure. rhombus A parallelogram in which all sides are congruent. right angle An angle with a measure of 90°. right triangle A triangle with one right angle. scalene triangle A triangle in which no sides are congruent and no angles have the same measure. similar figures Figures that have congruent corresponding angles and in which corresponding sides are proportional. square A quadrilateral in which all sides and all angles are congruent. square unit A unit of measure used to describe the surface (area) of figures of two dimensions—length and width. straight angle An angle with a measure of 180°. trapezoid A quadrilateral in which only one pair of sides is parallel. triangle A three-sided polygon. vertex (pl. vertices) The intersection point shared by two sides of a polygon or the two sides (rays) of an angle. Also the intersection point shared by three or more edges of a polyhedron. geometry and and measurement measurement of of Plane Plane figures—activity figures—Activityset Set16 BLM_MS_PG_01 Mid_PGe_06_PM Copyright© thebyMcGraw-Hill Companies—McGraw-Hill Professional Development Copyright© by 2002 the McGraw-Hill Companies—McGraw-Hill Professional Development