The Mysterious Polygonians

Transcription

1 The Mysterious Polygonians MA.C ,.2, and.3 LESSON FOCUS Creating and describing two-dimensional figures. COMPANION ANCHORS LESSONS Lines, Line Segments, Rays, and Angles; Recognizing Polygons; Visualizing Polygons MATERIALS Excursions student pages or 5 drinking straws for each student Nickel-size lump of clay for each student Masking tape Twist ties (optional) LANGUAGE DEVELOPMENT angle attribute congruent line segments endpoint heptagon hexagon intersecting lines intersection line line segment model octagon parallel lines LESSON OVERVIEW parallelogram pentagon perpendicular lines point polygon quadrilateral ray rectangle rhombus right angle square trapezoid triangle vertex Students learn about geometric concepts and twodimensional figures by modeling them, drawing them, and describing their attributes. Then students decipher messages written with an alphabet based on the same geometric concepts and figures. Finally, students use the same alphabet to create their own riddles. SETTING THE STAGE Learn an ancient language. Recount the following situation (or a similar one) to the class: While excavating the site of the ancient Polygonian kingdom in the vast Heptagonia Desert, archaeologists have discovered several stones with carved inscriptions. The archaeologists have managed to interpret some of the inscriptions, which are based on geometric concepts and figures. They now call on you, a crack team of third-grade mathematicians, to help them translate the rest of the inscriptions and thus answer important questions about the Polygonian culture. First, you will need to learn the Polygonian alphabet. BUILDING CONCEPTUAL KNOWLEDGE Model geometric figures. Distribute the straws, a small piece of clay, and a length of masking tape to each student. Explain to students that they will be creating models of geometric concepts and figures, and then drawing their models and describing them. Tell students to tear the masking tape into approximate 1-inch lengths, which they should be able to reuse several times. Model how to measure and cut two straws exactly in half, how to measure and cut one straw into 3 equal pieces, and how to measure and cut one straw into 4 equal pieces. Have students open their books to The Polygonian Alphabet, pages Display a small ball of clay on the overhead and explain that such a dot can represent a point. Have the class make similar balls of clay. Point out that in the Polygonian alphabet, a point is the same as the A of the English alphabet. Talk about the attributes of points a point is a location but it has no size. Make sure students understand that they must give a point some size to draw it. Continue the same procedure for the remaining letters of the Polygonian alphabet. Use straws, clay, and/or tape to model the geometric figure; have students model the figure and discuss the figure s attributes; and make sure students draw the model correctly in the column labeled Polygonian. 67

2 Working together, the class will complete the table of the Polygonian alphabet (so that everyone can translate the mysterious inscriptions on the stones). Below are some suggestions for ways to model the figures and guidelines for discussing the figures attributes. Depending on your class s abilities and comprehension, you may wish to model the figures differently or present the attributes in a different manner. line Explain that a line goes on forever. Start taping straws together. After connecting several straws, tell the class that this is going to take too long, so instead, you are just going to draw an arrow at each end of the line. Place a section of a straw on the overhead and draw arrows at each end. Students can draw arrows on a section of straw or cut the ends to make points. line segment Tell the class that a line segment is part of a line. Erase the arrows of the line on the overhead. Wonder aloud whether people will understand that the line segment stops at those points. Draw a dot at each end the endpoints of the line segment. Point out that line segments have definite, measurable lengths. Have students model line segments by attaching small balls of clay to a section of straw. ray Explain that a ray is one side of a line. Erase one of the endpoints of your line segment and draw an arrow. angle Model an angle by making two rays sections of straw with a ball of clay at one end and an arrow or point at the other end and then joining them at their endpoints. Model different angles. Show how to draw an angle on the overhead. right angle Have students use an existing right angle such as the corner of an Excursions book to make a right angle. Encourage students to draw the symbol for a right angle ( ). intersecting lines Use straws to model two lines on the overhead and point out that they can cross each other or not. Explain that lines that cross or meet are called intersecting lines. parallel lines Show how, if your two lines are going in exactly the same direction, they will never meet. You may want to demonstrate this with lines The Mysterious Polygonians THE POLYGONIAN ALPHABET Follow your teacher s instructions. Use straws, tape, and clay to make the polygonian letter. Then draw it and complete the list of its attributes. Drawings of Polygonian letters will vary. English Polygonian Geometrical Attributes English Polygonian Geometrical Attributes G right angle A right angle forms a square corner. You can use the symbol to show a right angle. A B C D E point line line segment ray angle A point is a location. A point has no size. A line is an endless straight set of points. A line extends in two directions. A line segment is part of a line. A line segment has two endpoints. A ray is part of a line. A ray has one endpoint. A ray extends forever in one direction. An angle is a figure formed by two rays with the same endpoint. The endpoint is called the angle s vertex. H I K L M intersecting lines parallel lines perpendicular lines congruent line segments triangle Intersecting lines are lines that cross. or meet Parallel lines are lines that never intersect. or cross, or meet Perpendicular lines are lines that intersect and form four right angles. Congruent line segments are the same length. A triangle is a polygon with three sides and three Page 91 Page 92 68

3 made of several straws each (or yardsticks), so you can show how even a small difference in the lines directions will make them intersect eventually. perpendicular lines Have students use an existing right angle such as the corner of an Excursions book to model the intersection of two perpendicular lines. Make sure everyone sees that perpendicular lines must form four right angles; therefore, students only need to draw one rightangle symbol ( ). congruent line segments Have students model congruent line segments with two sections of straws of equal length. Most students will find it easier to draw congruent line segments that are parallel, but make sure they understand that congruent line segments don t have to be parallel. You may wish to show students how to draw a mark across each line segment to demonstrate congruence. You may also wish to have them tie twists to their line segments to demonstrate congruence. BUILDING SKILLS AND STRATEGIES Model polygons. triangle Display three line segments (sections of straws with clay endpoints) on the overhead. Show how you can make a figure by connecting all the line segments endpoint to endpoint. Make sure everyone understands that each endpoint must be connected to another endpoint. quadrilateral Display four line segments on the overhead (use unequal lengths of straws to help students understand that a quadrilateral isn t necessarily a parallelogram). Show how you can make a figure by connecting all the line segments endpoint to endpoint. Make sure everyone understands that each endpoint must be connected to another endpoint. Tell the class that this quadrilateral and the triangle they just made are examples of polygons. Explain that a polygon is a figure made with line segments, that the line segments connect only at English Polygonian Geometrical Attributes English Polygonian Geometrical Attributes N quadrilateral A quadrilateral is a polygon with four sides and four T square A square is a rhombus. A square has four right angles. So a square is also a rectangle. O P R S trapezoid parallelogram rectangle rhombus A trapezoid is a quadrilateral with one, and only one, pair of parallel sides. A parallelogram is a quadrilateral with two pairs of parallel sides. The parallel sides are also congruent. A rectangle is a parallelogram. A rectangle has four right angles. A rhombus is a parallelogram. A rhombus has four congruent sides. U V W Y pentagon hexagon heptagon octagon A pentagon has five sides and five A hexagon has six sides and six A heptagon has seven sides and seven An octagon has eight sides and eight Page 93 Page 94 69

4 their endpoints, and that each endpoint is connected to another endpoint. Model some examples of polygons and figures that aren t polygons. trapezoid Model a trapezoid on the overhead. Ask volunteers to describe the line segments. Someone should point out that one pair of sides is parallel. You may wish to show students how to mark the parallel sides of a polygon (or any parallel line segments or lines). You may also wish to have students tie twists to the parallel sides of their trapezoids. Polygons Not polygons Explain to the class that polygons are named by the number of sides they have. Make sure students understand that the number of sides of a triangle (or of a quadrilateral or of any other polygon) is the same as the number of parallelogram Use two pairs of equal lengths of straws to model a parallelogram. Make sure everyone understands that a parallelogram has two pairs of sides that are congruent and parallel. rectangle Change your parallelogram into a rectangle by making the angles right angles (you can form the angles on an existing right angle). Make sure everyone understands that their rectangles are still parallelograms. MYSTERIOUS POLYGONIAN MESSAGES In the vast Heptagonia Desert, archaeolgists have discovered several stones with messages written in the Polygonian alphabet. The archaeologists have already translated the messages of one stone, a series of questions. Now they have found the stone with the answers! 5. What did the queen say when her parrot flew away? polygon 6. What part of a line can t you trust? the lying segment Use your knowledge of the Polygonian alphabet to translate the answers to these ancient questions. 1. What do you call it when two bad rays get together? a wrong angle 2. What do acorns say when they grow up? geometry 3. When lines want to meet, where do they go? the intersection 4. What became of the the square that the king sat upon? a rhombus 7. What is the king s name? Ray 8. What is the name of the king s brother? Parallel Ray Invent your own Polygonian riddle. Write the question with English letters. Write the answer with Polygonian letters. 9. Question: Answer: 10. Question: Answer: Responses will vary Page 95 Page 96 70

5 rhombus Use four equal lengths of straws to model a rhombus. square Change your rhombus into a square by making the angles right angles. Make sure everyone understands that their squares are still rhombuses. You may wish to ask students whether their squares are parallelograms. (yes) pentagon, hexagon, heptagon, and octagon Model these polygons in the same manner as the previous ones. You may wish to model regular versions of these polygons. If so, point out that the sides are congruent, and make sure students can identify the parallel sides. Also make sure students understand that these polygons don t have to be regular. HOME CONNECTION Draw or describe polygons at home. Have students take home Polygons at Home, pages 97 and 98. Tell students that if they can t find examples of all the figures at home, they can look for them elsewhere (for example, at a store, in a playground, or at the library). Encourage students to bring their completed worksheets to class. Have volunteers share some of their geometric examples by drawing them on the board or describing them to the class. regular hexagon irregular hexagon PUTTING IT INTO ACTION Decipher Polygonian messages. Have students open their books to Mysterious Polygonian Messages, pages 95 and 96. Tell them that now they can use their completed Polygonian alphabet to translate the messages into English. Problems 9 and 10 If students have difficulty inventing riddles, encourage them simply to write questions about geometry. Point out that the Polygonian alphabet doesn t include all of the English letters. You may wish to have students invent Polygonian letters for the missing English letters. 71

6 The Mysterious Polygonians THE POLYGONIAN ALPHABET Follow your teacher s instructions. Use straws, tape, and clay to make the polygonian letter. Then draw it and complete the list of its attributes. Drawings of Polygonian letters will vary. English Polygonian Geometrical Attributes A B C D E point line line segment ray angle A point is a location. A point has no. A line is an endless two line one points straight set of. A line extends in directions. A line segment is part of a. A line segment has endpoints. line A ray is part of a. A ray has A ray extends forever in one direction. An angle is a figure formed by two rays with the same. The endpoint is called the angle s. endpoint. 91

7 English Polygonian Geometrical Attributes G right angle A right angle forms a corner. You can use the symbol to show a right angle. H intersecting lines Intersecting lines are lines cross that. or meet I parallel lines Parallel lines are lines that never intersect. or cross, or meet K perpendicular lines Perpendicular lines are lines that intersect and form four right angles. L congruent line segments Congruent line segments are the same. M triangle A triangle is a polygon with three sides and 92

8 English Polygonian Geometrical Attributes A quadrilateral is a polygon N quadrilateral with sides and O P R S trapezoid parallelogram rectangle rhombus A trapezoid is a quadrilateral with one one, and only, pair of parallel sides. A parallelogram is a quadrilateral with pairs of parallel sides. A rectangle is a parallelogram. A rectangle has right angles. A rhombus is a parallelogram. A rhombus has congruent sides. two The parallel sides are also congruent. 93

9 English Polygonian Geometrical Attributes T square A square is a rhombus. A square has right angles. four So a square is also a. U pentagon A pentagon has sides and five five V hexagon A hexagon has sides and six six W heptagon A heptagon has sides and seven seven Y octagon An octagon has sides and 94

10 MYSTERIOUS POLYGONIAN MESSAGES In the vast Heptagonia Desert, archaeolgists have discovered several stones with messages written in the Polygonian alphabet. The archaeologists have already translated the messages of one stone, a series of questions. Now they have found the stone with the answers! Use your knowledge of the Polygonian alphabet to translate the answers to these ancient questions. 1. What do you call it when two bad rays get together? a wrong angle 2. What do acorns say when they grow up? geometry 3. When lines want to meet, where do they go? the intersection 4. What became of the the square that the king sat upon? a rhombus 95

11 5. What did the queen say when her parrot flew away? polygon 6. What part of a line can t you trust? the lying segment 7. What is the king s name? Ray 8. What is the name of the king s brother? Parallel Ray Invent your own Polygonian riddle. Write the question with English letters. Write the answer with Polygonian letters. 9. Question: Responses will vary. Answer: 10. Question: Answer: 96

12 HOME CONNECTION: POLYGONS AT HOME Dear Parent or Guardian: Your child has been learning about geometric figures (such as lines, rays, and angles) and polygons (such as triangles, rectangles, and pentagons). In this assignment, your child will find examples of these figures and polygons in your home. For example, the corner of a door could represent an angle, while two adjoining sides of a tile could represent perpendicular lines. Draw or describe an example of the figure or polygon. 1. point 2. line 3. line segment 4. angle 5. right angle 6. intersecting lines 7. parallel lines 8. perpendicular lines 97