Activity Set 4. Trainer Guide

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1 Geometry and Measurement of Plane Figures Activity Set 4 Trainer Guide Int_PGe_04_TG

2 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 NGSSS 3.G.3.1 NGSSS 3.G.3.3 NGSSS 4.G.5.1 NGSSS 5.G.3.1 Amazing Angles In this activity, participants explore angle concepts in polygon shapes. Materials Transparency/Page: Angle Types Transparency/Page: Measuring Angles Transparency/Page: A Circle of Measure Transparency/Page: Polygon Angles Chart Transparency/Page: Polygon Angles Chart Answer Key plain 3 5 cards (4 per participant) ruler for each participant protractor for each participant scissors for each participant pens/pencils (multicolor pens) blank transparency Vocabulary degree angle vertex right angle straight angle Time: 30 minutes Int_PGe_04_TG 1

3 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 B A interior angle an angle formed by two sides of a polygon adjacent angles angles that share a common side and a common vertex between them, but that do not share any interior points exterior angle McGraw-Hill Professional Development angle types interior exterior adjacent C Transparency: Angle Types B A C an angle adjacent to, but outside of, a polygon formed by extending one side of the polygon D B GEOMETRY AND MEASUREMENT OF PLANE FIGURES/31 A C Introduce Remind participants that one aspect of geometry is the application of angles to various shapes and figures. Display Transparency: Angle Types. Go over the angle descriptions and names. Teaching Tip: It may help to clarify the definitions if you explain the meaning of adjacent having a common side or border and, in mathematics, a common endpoint. Measuring angles angle 1 angle 2 angle angle 4 angle 5 acute angle an angle less than 90º obtuse angle an angle more than 90º right angle an angle equal to 90º straight angle an angle of 180º McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF PLANE FIGURES/33 Transparency: Measuring Angles Display Transparency: Measuring Angles and have participants take out their matching pages. Take out a protractor. Demonstrate on angle 1 how to measure an angle. Align the 0 mark and line with the right-hand side of the angle, making sure that the vertex of the angle is aligned with the center mark of the 0 line. (There is usually a small hole at this location to enable you to place the vertex appropriately.) Locate the left-hand side of the angle and trace the line to the degree mark on the protractor. Have participants measure the remaining angles and write the degrees that they find in the appropriate blanks on their pages. Go over the answers with the participants and demonstrate the measurement process, if necessary, to address any questions. Review the definitions at the bottom of the page. Int_PGe_04_TG 2

4 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 Teaching Tip: If time permits, have participant volunteers come to the front to measure the angles and record the results on the transparency. Teaching Tip: If the group is advanced, have them also identify the angle type after they measure. 1 acute angle 2 acute angle and adjacent angle (adjacent to angle 3) 3 obtuse angle and adjacent angle (adjacent to angle 2) 4 straight angle (straight line) 5 right angle (formed by perpendicular lines) Ask why none of the angles are interior or exterior. (They are not part of, or adjacent to, polygons.) a circle of Measure Ask participants how many degrees are around the center of a circle. Display Transparency: A Circle of Measure. Point out to participants that the distance around the center of the circle (360 ) is the basis for all angle measure. It is a mathematics convention that the unit of angle measure (degree) is of a complete revolution around the center of a circle. McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF PLANE FIGURES/35 Transparency: A Circle of Measure Point out on the transparency that the diameter of the circle (a straight line) divides the revolution in half, creating a straight angle, or 180. Explain to participants that they will use this information to help them find the number of degrees in the interior angles of a triangle. Int_PGe_04_TG 3

5 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 Discuss and Do Distribute to each participant four 3 5 cards and a pair of scissors. Have each participant draw on one card a large triangle. Teaching Tip: Have participants use a straightedge, ruler, or card side to draw all figures. Straightedges are required to achieve accuracy for the activity. Also, no shape that they create can have overlapping edges. Have participants cut out their triangles. Have participants use a pen or pencil to color in the angles about 1 2 out from each vertex. Have them cut the triangle into 3 pieces, with each piece containing 1 angle. Tell them to lay the 3 angles together with the vertices joining and their sides touching. Point out that they now have a straight line or a straight angle, which is defined as 180. Point out that they all made different kinds of triangles. Explain that the angles of all triangles sum to 180. Have participants use their rulers to draw another triangle on one of their 3 5 cards. Have them make the triangles as large as possible for ease of measurement. Tell them to measure each angle in their triangles, using their protractors, and add the three angles together. Int_PGe_04_TG 4

6 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 Display a blank transparency. Have various volunteer participants share the angle measures within their triangles. Record on the blank transparency the angle measures as they are shared. Point out that the angles differed individually, but that the sum of the angles for any triangle was always 180. Have participants take out their third cards. Point out that the card is a rectangle. Ask them to color the 4 corners and cut the card into 4 pieces (1 corner to each piece). Ask them to arrange the corners together and tell you how many degrees there are in the angles of a rectangle. (360 ) Explain that any quadrilateral has angles that sum to 360. Draw a rectangle on a blank transparency. Draw a diagonal from one corner of the rectangle to the corner opposite. Point out that the angles of the 2 triangles thus formed also sum to 360. Have participants draw a 5-sided figure on their last cards. Have them color the corner angles, cut out the shape, and then cut it into 5 pieces 1 angle per piece. Ask participants to lay the angles together in such a way that they can tell you the total number of degrees. Suggest that, when participants complain that they cannot match all the angles, they create more than one figure. Int_PGe_04_TG 5

7 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 Ask participants how many degrees there are in the angles of a pentagon. (540 ) Have one participant come up with his or her shapes and illustrate on the overhead projector his or her solution. Draw a 5-sided polygon on a blank transparency. polygon angles chart Draw, from 1 vertex, lines to all opposing vertices for which you can make triangles. Polygon Number of Number of Sides Number of Degrees: Name Sides Minus 2 triangle quadrilateral pentagon hexagon heptagon octagon Point out that the angles of the 3 triangles thus formed also sum to 540. nonagon decagon Conclude GEOMETRY AND MEASUREMENT OF PLANE FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development TRANS_K6_PG_04 Transparency: Polygon Angles Chart Display Transparency: Polygon Angles Chart and have participants take out their matching pages. Fill in, along with the participants, the first three rows of the Polygon Angles Chart using information that they have collected during this activity. Encourage participants to create triangles of each shape to help them. Ask participants the number of degrees that they think the angles of a hexagon would total. (720 ) Complete the hexagon row on the chart. Ask participants if they recognize a pattern. (The rule is (n 2) 180.) Ask participants how this rule is derived. (n, the number of sides, less 2 is the number of non-overlapping triangles in each shape.) Int_PGe_04_TG 6

8 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon polygon angles chart Answer Key Polygon Number of Number of Sides Number of Degrees: Name Sides Minus 2 n n 2 (n 2) , , ,440 GEOMETRY AND MEASUREMENT OF PLANE FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development Transparency: Polygon Angles Chart Answer Key TRANS_K6_PG_04 Write the rule on the transparency in the fourth column heading. Go down to the last figures on the chart. Ask participants for the number of degrees at each row. Fill in the transparency at each step. Refer to Transparency: Polygon Angles Chart Answer Key, as necessary. End of Amazing Angles Int_PGe_04_TG 7

9 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 Race to Place In this activity, participants use geometric knowledge that they remember to match pictures of angles and shapes with their definitions. Materials Transparency/Page: Race to Place Directions Transparency/Page: Triangle Facts Answer Key Transparency/Page: Angle Facts Answer Key Transparency/Page: Angles in Shapes Answer Key Transparency/Page: Line Facts Answer Key Transparency/Page: Circle Facts Answer Key Race to Place Cards 5 pocket charts bell Time: 15 minutes Teaching Tip: Post the pocket charts with their definitions before the beginning of the activity. Use the Facts transparencies as a guide for the definitions that go with each title. Space the charts around the room with a lot of room between them. Int_PGe_04_TG 8

10 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 Introduce Suggest to participants that over time they have accumulated a lot of knowledge about the way lines, shapes, and angles work. Point out the five charts and their definitions. Explain to the participants that they will compete as teams to match geometric definitions with pictures that illustrate the concepts defined. Teaching Tip: If you have a large group, assign pairs instead of single people to each card. Directions Distribute your team cards evenly among the members of your team. Have team members play their cards in relay fashion. Have a player: race to place race to the chart that holds the definition of the picture on his or her card place the card next to the definition race back to the team and sit down Have the next person race to the chart and place his or her card. Have one team member race to the front and ring the bell when all the team s cards are correctly placed. GEOMETRY AND MEASUREMENT OF PLANE FIGURES AcTIvITY SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development TRANS_K6_PG_04 Transparency: Race to Place Directions DISCUSS AND DO Display Transparency: Race to Place Directions. Go over the steps of the game. Have participants move into 4 or 5 equal-sized groups. Distribute the shape cards all of one colored shape to each group, one card per person. Call, Go. Have the first group to finish send one member to the front of the room to ring the bell. Teaching Tip: If a team member cannot place his or her shape card, he or she should go to the end of the line and wait to place the card after other team members have placed their cards. Teaching Tip: If the group is inexperienced, permit them a few moments to look at the definition sheets (Answer Keys) before the game. Int_PGe_04_TG 9

11 GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 Conclude Congratulate the participants for being able to remember so many geometry concepts and definitions. Display the Answer Key transparencies in turn, quickly reviewing the definitions. Emphasize the following definitions for each key: A scalene triangle has no congruent sides and no congruent angles. An isosceles triangle has 2 congruent sides and 2 congruent angles. McGraw-Hill Professional Development triangle Facts Answer Key An equilateral triangle has 3 congruent sides and 3 congruent angles. The angles of an acute triangle are all less than 90. One angle in an obtuse triangle is greater than 90. A right triangle has one angle equal to 90. The side opposite the 90 angle is called the hypotenuse. GEOMETRY AND MEASUREMENT OF PLANE FIGURES/73 Transparencies: Triangle Facts Answer Key, Angle Facts Answer Key, Angles in Shapes Answer Key, Line Facts Answer Key, Circle Facts Answer Key Transparency: Triangle Facts Answer Key equilateral triangle right triangle (especially hypotenuse) Transparency: Angle Facts Answer Key straight angle vertical angles Transparency: Angles in Shapes Answer Key triangle equilateral triangle Transparency: Line Facts Answer Key alternate interior angles Transparency: Circle Facts Answer Key circumference End of Race to Place Int_PGe_04_TG 10

12

13 Angle Types interior exterior adjacent A C A C A C B B B D interior angle an angle formed by two sides of a polygon adjacent angles angles that share a common side and a common vertex between them, but that do not share any interior points exterior angle an angle adjacent to, but outside of, a polygon formed by extending one side of the polygon

14 Measuring Angles angle 1 angle 2 angle angle 4 angle 5 acute angle an angle less than 90º obtuse angle an angle more than 90º right angle an angle equal to 90º straight angle an angle of 180º

15 A Circle of Measure It is a mathematics convention that the unit of angle measure (degree) is of a complete revolution around the center of a circle.

16 Polygon Angles Chart Polygon Number of Number of Sides Number of Degrees: Name Sides Minus 2 triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon

17 Polygon Number of Number of Sides Number of Degrees: Name Sides Minus 2 n n 2 (n 2) 180 triangle Polygon Angles Chart Answer Key quadrilateral pentagon hexagon heptagon octagon nonagon decagon , , ,440

18 Race to Place Directions Distribute your team cards evenly among the members of your team. Have team members play their cards in relay fashion. Have a player: race to the chart that holds the definition of the picture on his or her card place the card next to the definition race back to the team and sit down Have the next person race to the chart and place his or her card. Have one team member race to the front and ring the bell when all the team s cards are correctly placed.

19 A scalene triangle has no congruent sides and no congruent angles. Triangle Facts Answer Key An isosceles triangle has 2 congruent sides and 2 congruent angles. An equilateral triangle has 3 congruent sides and 3 congruent angles. The angles of an acute triangle are all less than 90. One angle in an obtuse triangle is greater than 90. A right triangle has one angle equal to 90. The side opposite the 90 angle is called the hypotenuse.

20 Angle Facts Answer Key An acute angle is less than 90. An obtuse angle is greater than 90 and less than 180. A straight angle is equal to 180. A right angle is equal to 90. Angles that share a common side between them are adjacent. Two angles that sum to 180 are called supplementary. Nonadjacent angles formed by two intersecting lines are called vertical angles. They have the same measure.

21 Angles in Shapes Answer Key A triangle has angles that sum to 180. A rectangle has angles that sum to 360. Angles inside a shape are interior angles. Angles outside a shape are exterior angles. The base angles and opposite sides of an isosceles triangle are congruent. The sides and angles of an equilateral triangle are congruent.

22 Line Facts Answer Key A set of points, a straight path, that extends indefinitely in 2 opposite directions is a line. A line segment is 2 endpoints and the straight path between them. Perpendicular lines form right angles. If a line intersects two parallel lines, the alternate interior angles are equal. Parallel lines are equidistant from each other. 6 cm 6 cm

23 Circle Facts Answer Key A complete revolution around the center of a circle has 360º. A chord is a line segment that connects two points on the circumference of a circle. The line segment joining the center of the circle and a point on its circumference is called a radius. A diameter is a chord that passes through the center of a circle. Its length is twice that of the radius of the circle. A circle is the set of all points in a plane that are equidistant from a specified point. The distance around a circle is called its circumference.

24 Glossary Geometry and Measurement of Plane Figures angle Geometric figure made of 2 rays or 2 line segments that share the same endpoint, called a vertex. area The number of square units in a region. congruent Having the same shape, size, and/or measure. degree A unit for measuring angles. irregular polygon A polygon in which not all the sides are congruent and/or not all the angles have the same measure. line A set of points forming a straight path in 2 directions that are opposite each other. perimeter The distance around the outside of a shape or figure. plane A flat surface that extends forever in all directions. point A location in space. polygon A closed shape made up of a minimum of 3 line segments. quadrilateral A polygon with 4 sides. rectangle A quadrilateral with 4 right angles.

25 Glossary (continued) regular polygon A polygon in which all the sides are congruent and all the angles have the same measure. triangle A polygon with 3 sides.

26 Race to Place Cards (1 of 20)

27 Race to Place Cards (2 of 20)

28 6 cm 6 cm Race to Place Cards (3 of 20)

29 Race to Place Cards (4 of 20)

30 Race to Place Cards (5 of 20)

31 Race to Place Cards (6 of 20)

32 Race to Place Cards (7 of 20)

33 Race to Place Cards (8 of 20)

34 Race to Place Cards (9 of 20)

35 Race to Place Cards (10 of 20)

36 A complete revolution around the center of a circle has 360º. A chord is a line segment that connects two points on the circumference of a circle. The line segment joining the center of the circle and a point on its circumference is called a radius. Race to Place Cards (11 of 20)

37 If a line intersects two parallel lines, the alternate interior angles are equal. Parallel lines are equidistant from each other. A right angle is equal to 90. Race to Place Cards (12 of 20)

38 A set of points that extend indefinitely in 2 opposite directions is a line. A line segment has two endpoints. Perpendicular lines form right angles. Race to Place Cards (13 of 20)

39 Angles outside a shape are exterior angles. The base angles and opposite sides of an isosceles triangle are congruent. The sides and angles of an equilateral triangle are congruent. Race to Place Cards (14 of 20)

40 A triangle has angles that sum to 180. A rectangle has angles that sum to 360. Angles inside a shape are interior angles. Race to Place Cards (15 of 20)

41 Non adjacent angles formed by two intersecting lines are called vertical angles. They have the same measure. Angles that share a common side between them are adjacent. Two angles that sum to 180 are called supplementary. Race to Place Cards (16 of 20)

42 An acute angle is less than 90. An obtuse angle is greater than 90 and less than 180. A straight angle is equal to 180. Race to Place Cards (17 of 20)

43 The angles of an acute triangle are all less than 90. One angle in an obtuse triangle is greater than 90. A right triangle has one angle equal to 90. Race to Place Cards (18 of 20)

44 The diameter is a chord that passes through the center of a circle. A circle is the set of all points in a plane that are equidistant from a specified point. The distance around a circle is called its circumference. Race to Place Cards (19 of 20)

45 A scalene triangle has no congruent sides and no congruent angles. An isosceles triangle has 2 congruent sides and 2 congruent angles. An equilateral triangle has 3 congruent sides and 3 congruent angles. Race to Place Cards (20 of 20)

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