Angles of Polygons Objective To develop an approach for finding the angle measurement sum for any polygon.

Transcription

1 ngles of Polygons Objective To develop an approach for finding the angle measurement sum for any polygon. epresentations etoolkit lgorithms Practice EM Facts Workshop Game Family Letters ssessment Management ommon ore State Standards urriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice ifferentiation Options Key oncepts and Skills Investigate and compare the measurement sums of interior angles of polygons. [Geometry Goal 1] Measure angles with a protractor. [Measurement and Reference Frames Goal 1] Find maimum, minimum, and median for a data set. [ata and hance Goal 2] raw conclusions based on collected data. [ata and hance Goal 2] Key ctivities Students measure to find angle sums for triangles, quadrangles, pentagons, and heagons. They use the pattern in these sums to devise a method for finding the angle sum for any polygon. Ongoing ssessment: Recognizing Student chievement Use an Eit Slip (Math Masters, page 414). [Geometry Goal 1] Materials Math Journal 1, pp Study Link 3 8 Math Masters, p. 414 transparency of Math Masters, p. 420 (optional) lass ata Pad Geometry Template (or protractor and straightedge) Practicing Epanded Notation Math Journal 1, p. 90 Student Reference ook, p. 396 Students practice place-value concepts by reading and writing large numbers and decimals in standard and epanded notation. Math oes 3 9 Math Journal 1, p. 91 Students practice and maintain skills through Math o problems. Study Link 3 9 Math Masters, p. 92 Students practice and maintain skills through Study Link activities. ENRIHMENT Tessellating Quadrangles Math Masters, p. 93 paper (8 1_ " 2 by 11") scissors tape cardstock (optional) Students investigate whether all quadrangles will tessellate. EXTR PRTIE Finding ngle Measures in Polygons Math Masters, p. 94 Students find the sums of the interior angles of polygons. ELL SUPPORT escribing Tessellations Students describe the tessellations in the Tessellation Museum. dvance Preparation This 2-day lesson begins with the Math Message on ay 1 and ividing Polygons into Triangles on ay 2. For the Math Message, draw a line plot on the board for students to record the sums of the angles they find. Label it from about 175 to 185. The Lesson 3 8 Study Link asks students to collect tessellations. These will be displayed in the Tessellation Museum. Include the class definitions for angles and triangles in this display. For the optional Enrichment activity in Part 3, do the tessellation activity yourself in advance, with conve and nonconve quadrangles, so you can help students see how the angles fit. Teacher s Reference Manual, Grades 4 6 p. 203 Lesson

2 Getting Started Mathematical Practices SMP2, SMP3, SMP5, SMP6, SMP7, SMP8 ontent Standards 5.NT.2, 5.NT.3a Mental Math and Reflees Use your slate procedures for problems such as the following: 47 10, , ,000 4, , , , , , ,000 70, ,000 79,040 Math Message Use a straightedge to draw a big triangle on a sheet of paper. Measure its angles and find the sum. Record the sum on the class line plot. Study Link 3 8 Follow-Up sk volunteers to share their tessellation eamples. Encourage them to include the names of polygons and to eplain how they identified the patterns as tessellations. 1 Teaching the Lesson On ay 1 of this lesson, students should complete the Math Message, the Study Link 3-8 Follow-Up, and eplore finding the sums of the angle measures. On ay 2 of this lesson, students begin with ividing Polygons into Triangles and eplore further how to find the sums of the angle measures in any polygon. Then have students complete Part 2 activities. Math Message Follow-Up Survey the class to complete the line plot and identify the following landmarks. What is the maimum sum of the angle measures? What is the minimum sum of the angle measures? What is the median sum of the angle measures? 175 or less or more Sample data WHOLE-LSS ISUSSION Epect a range of sums, because measurements are never eact, and because some of the sides of students drawings may not be straight or meet eactly. Eplain that if a triangle is accurately drawn, and its angles are measured with precision, the sum of the angle measures will always be 180. Eplain that students can prove this statement using their triangles as models. sk students to the three angles off their triangles as shown below. Students angle measures might seem to total slightly more or less than 180 because their original triangles might not be accurate. Net have students arrange their three angles net to each other so they line up. (See margin.) sk students what type of angle 200 Unit 3 Geometry Eplorations and the merican Tour

3 they have formed, straight angle and what the measure of a straight angle is. 180 This shows that the sum of the three angles of the triangle is 180. sk students to leave their angles arranged in a straight angle on their desks, and then check each other s triangles. When students return to their desks, ask what they observed about the angles in triangles. ll the angles will form straight angles; the sum of the angle measures in a triangle will always total 180. Record this property on the lass ata Pad. ate ngles in Quadrangles and Pentagons 1. ircle the kind of polygon your group is working on: quadrangle pentagon 2. elow, use a straightedge to carefully draw the kind of polygon your group is working on. Your polygon should look different from the ones drawn by others in your group, but it should have the same number of sides. rawings vary. NOTE Precise language would call for writing and saying: the sum of the measures of the angles instead of the sum of the angles. ut it is common in mathematics to use the shorter phrase. Finding the Sums of ngles in Polygons (Math Journal 1, pp. 85 and 86) SMLL-GROUP raw an eample of a conve polygon and a concave polygon on the board. NOTE These figures will be used later in the lesson. PROLEM SOLVING 3. Measure the angles in your polygon. Write each measure for each angle. nswers vary for individual angle measurements. 4. Find the sum of the angles in your polygon. The total sum of the angles should be close to 360 for quadrangles and 540 for pentagons. Math Journal 1, p. 85 sk students what true statements they can make about the interior angles in the two figures. Epect responses to vary, but structure your follow-up questions to guide students to recognize that the concave polygon has one angle that is a refle angle. Remind students that the names for angles refer to the angles measures, not to whether the angle is or is not an interior angle. sk students to fold a blank sheet of paper into fourths. Open it and label each bo in the top row Polygons and Not Polygons. Label the boes in the bottom row onve Polygons and oncave Polygons. ssign small groups of three to five students to work together to draw at least two eamples of each figure. s groups finish, they should eamine other students eamples. irculate and assist. Transition to the journal activity by first surveying the class for questions or observations about their drawings. Then assign groups to work on quadrangles or pentagons. sk students to circle the name of the figure they are going to work on, listed at the top of journal page 85. sk students to complete Problems 1 7 on journal pages 85 and 86. Problem 7 provides data on heagons for the net activity. s you circulate, consider asking students, who are waiting for the group to finish Problem 5, to go on to Problem 7 until the others are done. ate ngles in Quadrangles and Pentagons cont. 5. Record your group s data below. Group Member s Sketch of Sum of Name Polygon ngles 6. Find the median of the angle sums for your group. The total sum of the angles should be close to 360 for quadrangles and 540 for pentagons. 7. If you have time, draw a heagon. Measure its angles with a protractor. Find the sum. rawings vary. rawings of polygons vary. The sum of the angles varies but should be close to 360 for quadrangles and 540 for pentagons. Sum of the angles in a heagon = 720 Math Journal 1, p. 86 Lesson

4 ate ngles in Quadrangles and Pentagons cont. 8. Record the class data below. Sum of the ngles in a Quadrangle Group Group Median Sum of the ngles in a Pentagon Group Group Median Finding the Median for the Sums of ngles (Math Journal 1, p. 87) WHOLE-LSS ISUSSION The group median should be close to 360 for quadrangles and 540 for pentagons. 9. Find the class median for each polygon. For the triangle, use the median from the Math Message. Sums of Polygon ngles Polygon lass Median triangle quadrangle pentagon heagon 10. What pattern do you see in the Sums of Polygon ngles table? The group median should be close to 180 for a triangle, 360 for a quadrangle, 540 for a pentagon, and 720 for a heagon. Sample answer: s the number of sides increases by 1, the sum of the angle measures increases by 180. Math Journal 1, p. 87 ring the class together and use the board or a transparency of Math Masters, page 420 to collect the group s median angle sums; first from the quadrangle group and then from the pentagon group. sk students to record this data in the tables for Problem 8 on journal page 87. Net ask students to use the group medians to find the class median for quadrangles and pentagons. Then record this data in the table for Problem 9. For the triangle row, enter the class median from the Math Message line plot. ollect data from students who did Problem 7 on journal page 86, listing the sums on the lass ata Pad or the overhead projector. sk students to find the median of these sums and record it in the table for Problem 9. The class medians should be close to 180 for a triangle, 360 for a quadrangle, 540 for a pentagon, and 720 for a heagon. sk students to complete Problem 10 on the journal page. s they look for patterns in the Sums of Polygon ngles table, ask them to think about how the contents of each column in the table are related. sk: What are the differences between a triangle and a quadrangle? o the numbers in the class median column increase or decrease and by how much? The quadrangle has one more side than the triangle; the median sum of their angles increases by about 180. irculate and assist. ate ngles in Heptagons 1. heptagon is a polygon with 7 sides. Predict the sum of the angles in a heptagon raw a heptagon below. Measure its angles with a protractor. Write each measure in the angle. Find the sum. Sum of the angles in a heptagon 900 ividing Polygons into Triangles (Math Journal 1, pp. 87 and 88) WHOLE-LSS ISUSSION Survey the class for the patterns that students found in the Sums of Polygon ngles table. sk: Why do you think the medians for the sums of polygon angles increase by 180? Use the following points to guide the discussion. The sum of the angles of a triangle equals 180. Quadrangles divide into 2 triangles. The sum of the angles of the quadrangle equals 2 180, or a. Is your measurement close to your prediction? nswers vary. b. Why might your prediction and your measurement be different? Sample answer: ecause the angle measurement might not be eact for each angle in the heptagon Math Journal 1, p. 88 Use dotted lines to divide the polygons on the board from the earlier discussion so the two triangles can be seen. (See above.) sk: How many triangles do you think could be drawn in a pentagon? 3 What would be the sum of angles? is Unit 3 Geometry Eplorations and the merican Tour

5 In a heagon? 4; is 720. Summarize by stating that as the number of sides in a polygon increases by 1, the sum of the angle measures increases by 180. ate ngles in ny Polygon 1. raw a line segment from verte of this octagon to each of the other vertices ecept and H. H sk partners to work together to solve Problems 1 3 on journal page 88. They can use the pattern in the table or try dividing a heptagon into triangles to make their prediction. irculate and assist. ring the class together to discuss results. sk: o your predictions match your measurements? Why might they be different? The angle measurement(s) may not be eact for each angle in the heptagon. 2. How many triangles did you divide the octagon into? 3. What is the sum of the angles in this octagon? 1, Ignacio said the sum of his octagon s angles is 1,440. elow is the picture he drew to show how he found his answer. Eplain Ignacio s mistake. Ignacio should have drawn lines from one verte to each of the other vertices in his octagon, instead of drawing a line between each verte and a point in the interior of his octagon gon is a polygon with 50 sides. How could you find the sum of the angles in a 50-gon? 50-gon can be divided into 48 triangles. The sum of the angles would be Finding ngle Sums for ny Polygon (Math Journal 1, p. 89; Math Masters, p. 414) PRTNER Sum of the angles in a 50-gon = 8,640 Math Journal 1, p. 89 sk students to state the relationship between the number of sides of a polygon and the number of triangles that the polygon can be divided into. The number of triangles is 2 less than the number of sides. Eplain that some polygons are impractical to draw because they have so many sides that it s hard to draw them accurately. In these instances, the number of triangles can be determined by subtracting 2 from that polygon s number of sides. sk partners to work together to complete journal page 89. Ongoing ssessment: Recognizing Student chievement Eit Slip Use an Eit Slip (Math Masters, page 414) to assess students understanding of angle measures and relationships in polygons. Have students write a response to the following: Eplain how to find the sum of the measures of the angles in polygons without using a protractor. Students are making adequate progress if they indicate that they are able to use the sum of the measures of the angles in a triangle to calculate the angle sums for at least one other polygon. Some students may generalize finding the sum of angles for all polygons. [Geometry Goal 1] ate Practicing Epanded Notation Use the place-value chart on page 396 of the Student Reference ook to help you write the following numbers in epanded notation. 1. 6, , a. uild a 4 digit numeral. Write 3 in the hundredths place, 4 in the tens place, 6 in the ones place, and 9 in the tenths place , ,000 8, b. Write this number in epanded notation Write the following epanded notation in standard form a. uild an 8-digit number. Use these clues. The digit in the place with the greatest value is equal to 4 0. The digit in the place with the least value is equal to 3 2. The number in the hundreds place is the first counting number. The number in the tenths place multiplied by 54 is zero. The number in the tens place is the square root of 9. The number in the ones place is the square root of 4. The number in the hundredths place is the product of the number in the tens place and the number in the ones place. The number in the thousands place is equal to , b. Write this number in epanded notation. 40,000 5, Math Journal 1, p. 90 Lesson

6 ate Math oes 1. Write five names for 1,000, Use a straightedge to draw an angle Sample answers: 300, , , , ,000,000 2,000, Write or What is the measure of angle R? Q 27 R measure angle R Math Journal 1, p. 91 S that is less than 90. nswers vary. 4. I have four sides. ll opposite sides are parallel. I have no right angles. raw me in the space below. Sample answer: I am called a parallelogram. 6. Solve. 2,400,000 3, ,000 36, , ,000 72, Ongoing Learning & Practice Practicing Epanded Notation (Math Journal 1, p. 90; Student Reference ook, p. 396) INEPENENT Students practice place-value concepts by reading and writing large numbers and decimals in standard notation and in epanded notation. Students can refer to the place-value chart in the Student Reference ook, page 396. Remind students that decimals may also be written as fractions. For eample, in Problem 2, the epanded notation for may be written as _ _ 100. Math oes 3 9 (Math Journal 1, p. 91) INEPENENT Mied Practice Math oes in this lesson are paired with Math oes in Lesson 3-6. The skill in Problem 6 previews Unit 4 content. Writing/Reasoning Have students write a response to the following: John called his drawing in Problem 4 a parallelogram, and Jack called his drawing a rhombus. Who was correct? Sample answer: rhombus is a parallelogram with 4 equal sides. oth are correct. NOTE Student use of vocabulary (quadrangle, quadrilateral, parallel sides, congruent sides, and/or opposite sides) and eplanation of the concepts in their solution will provide information about how students are integrating these geometry concepts. Study Link Master Name ate Study Link 3 9 (Math Masters, p. 92) INEPENENT STUY LINK Sums of ngle Measures 1. escribe one way to find the sum of the angles in a quadrangle without using a protractor. You might want to use the quadrangle at the right to illustrate your eplanation. Sample answer: raw a line between two of the vertices to create two triangles. Since the sum of the angles in each triangle is 180, the sum of the angles in a quadrangle is The sum of the angles in a quadrangle is. 3. Follow these steps to check your answer to Problem Home onnection Students describe one way to find the sum of the angles of a quadrangle without using a protractor. They investigate finding the sum by ing off the angles and putting them together around a point. a. With a straightedge, draw a large quadrangle on a separate sheet of paper. b. raw an arc in each angle. c. ut out the quadrangle and off part of each angle. d. Tape or glue the angles onto the back of this page so that the angles touch but do not overlap. Practice 4. 3,007 1, , ,310 1,290 3, ,692 º 6 22, R3 Math Masters, p Unit 3 Geometry Eplorations and the merican Tour

7 3 ifferentiation Options ENRIHMENT INEPENENT Tessellating Quadrangles (Math Masters, p. 93) Min To apply their understanding of interior angle measures of polygons, have students investigate whether all quadrangles will tessellate. sk students whether they think the following statement is true. ecause the sum of the angles in a quadrangle is equal to the number of degrees in a circle, all quadrangles should tessellate. nswers vary. Eplain that in this activity, students will investigate whether the statement is true. sk students to tape their results onto a separate piece of paper. hallenge students to tessellate concave (nonconve) quadrangles. oncave quadrangles will tessellate, but it is more difficult to place the pieces so all four angles meet. Make sure all angles are correctly labeled to facilitate the process. Name ate Quadrangle Investigation The sum of the angles in a quadrangle is equal to 360. Since there are 360 in a circle, you might predict that every quadrangle will tessellate. Follow the procedure below to investigate this prediction. 1. Fold a piece of paper (8 1 '' by 11'') into si parts by first folding it into thirds 2 and then into halves. 2. Using a straightedge, draw a quadrangle on the top layer of the folded paper. Label each of the four vertices with a letter inside the figure for eample,,,, and. 3. ut through all si layers so that you have si identical quadrangles. Label the vertices of each quadrangle in the same manner as the quadrangle on top. 4. rrange the quadrangles so that they tessellate. 5. When you have a tessellating pattern, tape the final pattern onto a separate piece of paper. olor it if you want to. 6. Talk with other students who did this investigation. Were their quadrangles a different shape than yours? o you think that any quadrangle will tessellate? Option To make a pattern that has more than si quadrangles, draw your original quadrangle on a piece of cardstock, cut it out, and use it as a stencil. y tracing around your quadrangle, you can easily cover a half-sheet of paper with your pattern. Label the angles on your stencil so you can be sure you are placing all four angles around points in the tessellation. olor your finished pattern. Math Masters, p. 93 Teaching Master Tessellations with concave (nonconve) quadrangles EXTR PRTIE Finding ngle Measures in Polygons (Math Masters, p. 94) INEPENENT 5 15 Min Teaching Master Name ate lgebraic Thinking Students complete a table that shows the relationship between the number of the sides of a polygon and the number of interior triangles. Then they summarize the pattern by writing numerical epressions. ELL SUPPORT PRTNER escribing Tessellations 5 15 Min To provide language support for polygon angles, have students look at the Tessellations Museum and describe some of the tessellations in the museum to a partner. Encourage them to use mathematical terminology (for eample, polygons) and to describe components such as colors, shapes, and patterns. ngle Measures in Polygons The measure of the interior angles of a triangle is 180. The number of triangles within a polygon is 2 less than the number of sides of the polygon. 1. Fill in the chart below using this pattern. Number of Sides Number of Triangles Polygons Sum of ngles º º º º º 180 1, º 180 4, º 180 8, º , º , Use epressions to complete the statement. If n equals the number of sides in a polygon, of triangles within the polygon, and sum of the angles in the polygon. n 2 (n 2) 180 equals the number equals the Math Masters, p. 94 Lesson