This lesson refreshes high school students on geometry of triangles, squares, and rectangles. The students will be reminded of the angle total in these figures. They will also be refreshed on perimeter and area. Mary Koerber Geometry in two dimensional and three dimensional objects Grade 9 5 day lesson plan Day 1 Day 2 Day 3 Day 4 Day 5 This lesson This lesson This lesson follows the follows the introduces lesson on the areas of lesson on proving that students to explore triangles, parallelograms relationships squares, and exist and among classes rectangles. calculating the of two and This lesson area of a three introduces the rhombus and a dimensional students to the trapezoid. In geometric proof that the last lesson objects, make parallelograms we saw that the and test exist in area of a conjectures geometry. We rhombus is onehalf about them and also proof that the product solve problems rhombus have of the involving them. an area that we diagonals, and can calculate. the area of a From the trapezoid is opening activity one-half the proof we can height times the see that a sum of the rhombus is a bases. parallelogram and we can find its area. This lesson will allow students to become familiar with nets and have students come up with Euler s Formula. Students should experiment with different nets and various sided dice.
Overall Objectives: Objectives Day 1: To refresh students on angle measurement and use of protractor. To show the students the different types of triangles formed with different angles, and how they add up to 180 degrees. (Square 360 degrees.) To review the calculations of perimeter and area in a triangle, square, and rectangle. Objectives Day 2: To show students an introduction to proof. To show the students the proof that a rhombus exists and to calculate its area. To show students the formula to calculate the area of a trapezoid. Objectives Day 3: To show students how to use different shapes to find the area of polygons. To show the students what the total sum of the angles are in certain polygons. Objectives Day 4: To have students learn how to calculate surface area. To have students understand how surface area is used in real life situations. Objectives Day 5: Students will understand what nets are and how to create them. Students will be able to create three dimensional models using nets. Koeber Page 2
Day 1 Name: Mary Koerber Mathematical Concepts Addressed: The perimeter and area of triangles, squares, and rectangles. Grade Level: Grade 9 Textbook: Geometry Tools for Changing a World (Prentice Hall) This lesson refreshes high school students on geometry of triangles, squares, and rectangles. The students will be reminded of the angle total in these figures. They will also be refreshed on perimeter and area. Objectives: To refresh students on angle measurement and use of protractor. To show the students the different types of triangles formed with different angles, and how they add up to 180 degrees. (Square 360 degrees.) To review the calculations of perimeter and area in a triangle, square, and rectangle. Prerequisites: Students should have a firm grasp on algebra and angle manipulation. Students should know how to use a protractor. Students should know what triangles and squares are. Materials and Facilities: A classroom with desks, overhead projector, and Geometers Sketchpad. Students will be using protractors for a handout. Students will be given a sheet of paper to fold for the closing activity. Performance Standards: The NYS Learning Standards for mathematics that are addressed in this topic is Number & Numeration. This is used because the students must measure the angles of triangles and squares and then come up with the idea that the total interior angle measurement of a triangle is 180 degrees and the total angle measurement for a square is 360 degrees. Koeber Page 3
The NCTM Principles and Standards for School Mathematics states that students should use trigonometric relationships to determine lengths and angle measures. This correlates to this lesson because the students must determine the angles of triangles and squares by measuring them. Opening Activity: For the opening activity the students will receive a bunch of different shapes that include triangles, squares, and rectangles. The students will also get protractors so they can measure the angles of the figures. They will split up into groups of four to work together. The students will then measure the interior angles of the triangles using the protractor and they will write down the addition of the angles. They should discover from their calculations that when they add all of the interior angles together they get 180 degrees. The students will then move on to measuring the interior angles of the squares and rectangles. They will do the same thing as the triangles by adding these measurements together. By adding these together they should discover that the total angle measurement is equal to 360 degrees and that all of the angles are right angles. We will also show students that if we have two angles in the triangle and three angles in the square and rectangles we can find the third and fourth sides respectively. We can also find the exterior angles outside of the triangles, squares, and rectangles. Developmental Activity: In the developmental activity we will be focusing on measurement of the sides of the figures that we are focusing on. With these measurements we can calculate perimeter and area of those figures. Finding the midpoint of a side is also important, because we use it in the closing activity. We will start by using the same shapes that we had in the opening activity. The students will first take the square that is the largest and measure two of the sides; they should be equal. Now ask the students what the length is around the square. They should come up with a perimeter that is four times one side. Tell them that they just calculated the perimeter and have them do the same with one of the isosceles triangles. After they have done this have them go back to the square. Ask them how they would find the area of the square. Show the students on the blackboard how to calculate the area of a square and have them calculate the areas of the squares and rectangles in front of them. When they are done ask them what they think the area of the triangle will be; they should say that it is one-half of the area of the square or rectangle depending on what triangle they use. After they figure this out write both formulas on the board and tell them to find the perimeters and areas of the rest of their shapes. Use Geometers Sketchpad to check areas and perimeters, Closing Activity: For the closing activity the students will be given a large scalene triangle. The teacher will read off a set of instructions on how to fold our triangle into a rectangle. The students will then be asked a series of questions relating the folds of the triangle that occur to the rectangle. In the questions the students will be asked what type of quadrilateral is formed, how the first fold compares to the base of the triangle, and the segment that joins the midpoints of the sides. Koeber Page 4
Developmental Activity Ditto handout 1) 8cm 12cm Perimeter: Area: 2) BC = 3.32 cm C CA = 6.85 cm CD = 2.40 cm D DB = 2.29 cm B AB = 4.13 cm A Perimeter: Area: Koeber Page 5
3) Perimeter: A Area: m AB = 14.97 cm m AE = 6.18 cm m CA = 8.86 cm B C D 4) Square: m BC = 7.27 cm m DC = 6.46 cm 4cm Area: Perimeter: Koeber Page 6
Review on Geometers Sketchpad. i = 7 ft j = 25 ft h = 24 ft P = 56 ft A = 84 ft2 1.) s = 15 cm r = 8 cm t = 17 cm q = 20 cm v = 32 cm P = A= 2.) a = 5 m c = 13 m f = 24 m e = c P = A = Koeber Page 7
3.) i = 8 m j = 17 m h = 15 m P = A= Find the area and perimeter of the rectangle. 1. 6 ft 7 ft P = ft A = square ft 2. 6 km 9 km P = km A = square km 3.) 11 mm 6 mm P = mm A = square mm 4.) 18.4 mi 14.8 mi P = mi A = square mi Koeber Page 8
Opening Activity For the opening activity all triangles add up to 180 all squares add up to 360 Developmental Activity 1) 8cm 12cm Perimeter: 8+8+12+12=40cm Area: 12*8=96 2) Perimeter: 3.32+6.85+4.13=14.3 Area: =4.956 3) Perimeter: 7.27+8.86+14.97= 31.1cm Area: =22.4643 4) Square: 4cm Area: 4+4+4+4=16cm Perimeter:4*4=16 Closing Activity A photocopied handout is given to accompany the closing activity to show how to fold the triangle. Homework Assignment given P1 P = 56 ft A = 84 ft 2 1.) P = 69 cm A = 300 cm? 2 2.) P = 50 m A = 120 m 2 3.) P = 40 m A = 120 m 2 1.) P = 26 ft A = 42 ft 2 2.) P = 30 km A = 54 km 2 3.) P = 34 mm A = 66 mm 2 4.) P = 66.4 mi A = 33.2 mi 2 Koeber Page 9
Day 2 Name : Mary Koerber Mathematical Concepts Addressed : Proof that parallelograms exist and the area of a rhombus Grade Level : Grade 9 Textbook : Geometry Tools for Changing a World This lesson follows the lesson on the areas of triangles, squares, and rectangles. This lesson introduces the students to the proof that parallelograms exist in geometry. We also proof that rhombus have an area that we can calculate. From the opening activity proof we can see that a rhombus is a parallelogram and we can find its area. Objectives: To show students an introduction to proof. To show the students the proof that a rhombus exists and to calculate its area. To show students the formula to calculate the area of a trapezoid. To see the angular measurements of a rhombus and how we use them to obtain the area equation. Prerequisites: Students should know logical steps of proof. Students should know what a parallelogram is. Students should have a firm grasp on algebra and angles. Materials and Facilities: A classroom with desks, overhead projector, and chalkboard are needed. Students will be receiving a handout for developmental and closing activities. Koeber Page 10
Performance Standards: The NYS Learning Standards for mathematics that are addressed in this topic is Mathematical Reasoning. This is used because we prove that a parallelogram exists and then we use this conjecture to show the equation of a rhombus. The NCTM Principles and Standards for School Mathematics states that students must establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others. In this lesson we prove the geometric figure of a parallelogram and make conjectures to make the equation of a rhombus. Opening Activity: In the opening activity the students will be guided through the proof that a parallelogram exists with the teacher. We are given that segment AC and segment BD bisect each other a E (view teacher s notes). Then there are several other conjectures that we can say, such as angle AEB is congruent to angle CED, because vertical angles are congruent. By the same reasoning, we can say that angle BEC is congruent to angle DEA. We can also see that segment AE is congruent to segment CE, because of the definition of a segment bisector. From here we can say that triangle AEB is congruent to triangle CED and triangle BEC is congruent to triangle DEA; both by side- angle- side. By corresponding parts of congruent triangles are congruent, we have that angle BAE is congruent to angle DCE and angle ECB is congruent to angle EAD. So if we have congruent alterior angles then we have that segment AB is parallel to segment CD and segment BC is parallel to segment AD. After all of these steps we can finally say that ABCD is a parallelogram. Developmental Activity: In the beginning of the developmental activity we ask students several questions about a certain rhombus. These questions expand on what the students know about proof techniques. The students will be required to know how to answer questions about side angle side, corresponding parts of congruent triangles are congruent, and angle measurement. In the second part of the developmental activity we use the proof that segment AC is perpendicular to segment BD to show that the area of a rhombus is equal to one-half the product of the diagonals. We are first given that ABCD is a rhombus, and by definition of a rhombus segment AB is congruent to segment AD. Since segment AC is a diagonal of a rhombus it bisects angle BAD, and therefore angle BAE and DAE are congruent. By the Reflexsive Property of Congruence we have that segment AE is congruent to segment AE; and by side angle - side we see that triangle ABE is congruent to triangle ADE. By corresponding parts of congruent triangles are congruent we get that angle AEB is congruent to angle AED; and because these angles are both congruent and supplementary, they are right angles. We can now finish that by the definition of perpendicular we have that segment AC is perpendicular to segment BD. Once we have proven this we can now develop another proof that allows us to use a formula to calculate the area. The proof will be given as a homework assignment, so that the students can see where the formula comes from. It is essential that the teacher goes over the homework with the students so that anyone who gets the proof wrong knows where they went wrong. Closing Activity: In the closing activity we tell the students the formula to the area of a rhombus; and we state that a square is also a rhombus. Later in the homework the students will see why the area is one-half the product of the diagonals. The teacher will also show the students the formula to find the area of a trapezoid. The students will learn that in a trapezoid the parallel sides are the bases, and the non-parallel sides are the legs. Also, the height is the perpendicular distance between the two parallel bases. Koeber Page 11
Opening Activity (done on overhead) B C Given: AC and BD bisect each other at E. E Prove: ABCD is a parallelogram. A D AC and BD bisect each other at E Given <AEB@<CED AE@CE and BE@DE <BEC@<DEA vertical angles are @ Def. Of seg. Bisector vertical angles are @ DAEB@DCED SAS DBEC@DDEA SAS <BAE@<DCE CPCTC <ECB@<EAD CPCTC AB CD BC AD If @ alt. Int. < s, then lines are. If @ alt. Int. < s, then lines are. ABCD is a parallelogram Def. Of parallelogram. Koeber Page 12
Developmental Activity Day 2 A) ABCD is a rhombus. Why is D ABC@ DADC? SSS or SAS What is the relationship between <1 and <2? Between <3 and <4? <1@<2;<3@<4; Corresponding parts of congruent triangles are congruent try this: if m<b =120, find the measures of the numbered angles. 30;30;30;30 Given: ABCD is a rhombus. Prove: AC ^ BD. Paragraph Proof: By the definition of rhombus, AB@AD. Because AC is a diagonal of a rhombus, it bisects <BAD. Therefore <1 @ <2. AE @ AE by the Reflexive Property of Congruence. By the SAS Postulate, DABE @DADE. By CPCTC, <AEB @<AED. Because <AEB and <AED are both congruent and supplementary, they are right angles. By the definition of perpendicular, AC ^ BD. Area of a rhombus: d1* d2 2 Homework: from textbook Koeber Page 13
Day 3 Name : Mary Koerber Mathematical Concepts Addressed : Finding the area of polygons by using the areas of previously known areas and measurement of the interior angles in polygons. Grade Level : Grade 9 Textbook : Geometry Tools for Changing a World This lesson follows the lesson on proving that parallelograms exist and calculating the area of a rhombus and a trapezoid. In the last lesson we saw that the area of a rhombus is one-half the product of the diagonals, and the area of a trapezoid is one-half the height times the sum of the bases. Objectives: To show students how to use different shapes to find the area of polygons. To show the students what the total sum of the angles are in certain polygons. Prerequisites: Students should know what polygons are. Students should know how to calculate area in polygons from previous lesson plans. Students should have a firm grasp on algebra and angles. Materials and Facilities: A classroom with desks, overhead projector, and chalkboard are needed. Students will be receiving a handout for developmental and closing activities. Students need a ruler and will receive triangles, squares, rectangles, pentagons, hexagons, and octagons from the teacher. Performance Standards: The NYS Learning Standards for mathematics that are addressed in this topic is Modeling/ Multiple Representations to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships. This standard is used because we try to fit the areas of figures that we already know into figures that we don t know to find their areas. The NCTM Principles and Standards for School Mathematics states that students must explore relationships ( including congruence and similarity) amount classes of two- and three- dimensional geometric objects, make and test conjectures about them, and solve Koeber Page 14
problems using them. As stated before we correlate this to the lesson by take the areas of figures that we already know to find the areas of figures that we don t know. Opening Activity : For the opening activity the students will break up into groups of four and receive different sized triangles, squares, rectangles, pentagons, hexagons, and octagons. Each group will discuss how they can fit their shapes into the polygon that is provided. By using this technique the students can find the area of the unknown polygon from the areas of the known polygons. Developmental Activity: The developmental activity builds upon the opening activity. Again we use this same concept of using the areas of previously known shapes to find the area of shapes that we do not have a formula for. In this activity we first start by breaking a hexagon down into three pieces. We have two of the same triangles and one rectangle. The base of the hexagon is 15 cm and the height is 25 cm. With a little measurement we find that the height of the triangles is 5 cm. We now have enough information to find the area of the two triangles and the square. The area of one triangle is 62.5 cm, so the total area of both triangles 125 cm. The area of the rectangle is 375 cm, so the area of the hexagon is 125 cm plus 375 cm, which is 500 cm 2. We then move on to finding the area of an octagon. This is about the same procedure, except we will have two trapezoids and one rectangle. First have the students cut up the octagon into the shapes stated above. When we break up the octagon we will get the dimensions of 5 cm for the width of the rectangle and 10 cm for the height. When we calculate the area we get 50 cm 2. For the trapezoids we get that the bases are 3 cm and 10 cm and the height is 2 cm. When we calculate the area of one trapezoid we get 13 cm 2, and for two we get 26 cm 2. So the total area of the octagon is 76 cm 2 when we add the areas together. When we are done with this activity have the students work one other shapes and see if they can find the areas. Closing Activity: In the closing activity the teacher will have the students draw polygons that have 4,5,6,7,and 8 sides on their papers. The teacher will then instruct the students to use a ruler to make triangles within each polygon by drawing all the diagonals from one vertex to the middle of the polygon. This is an easy way to find the total interior angle measurement within multi-sided polygons. Have the students count up the number of triangles within the polygons and multiply that number by 180 degrees to get the total interior angle measurement. By following a pattern the students should see that this works for any type of polygon. Koeber Page 15
Opening Activity (day 3) Go over finding the area of a trapezoid. Example: 24.3 cm 8.5cm Area: 144.5 cm 2 9.7cm Hand out different sized triangles, squares, rectangles, pentagons, hexagons, and octagons to each group. Have each group discuss how they can find the area of these shapes without knowing a formula for these specific shapes. Students should come up with that the area can be found by adding up the area of already known shapes that fit inside the new shape. Koeber Page 16
Developmental Activity ( teachers notes) Find the area of the polygon: 5cm 25cm 15cm 5cm First find the area of the triangles: (25 cm * 5 cm) / 2 = 62.5cm 2 There is two of them so 2*62.5 cm =125cm 2 Now find the area of the rectangle: 25 cm *15 cm = 375cm 2 The total sum of the area of the hexagon is; 375 cm 2 +125 cm 2 = 500cm 2 Have students in groups draw polygons with 4,5,6,7,and 8 sides. Divide each polygon into triangles by drawing all the diagonals from one vertex Multiply the number of triangles by 180 to find the sum of the measures of the interior angles of each polygon. Have students try to come up with the polygon interior angle-sum Theorem by inductive reasoning. Hint: tell them to look for a hint in the pattern in their table. Koeber Page 17
This will be part of homework. Opening Activity In the opening activity we must recall area of a triangle = (1/2)b?h area of a rectangle = b?h area of a rhombus = (length of diagonal 1? length of diagonal 2/2) area of a trapezoid = ((length of base 1 + length of base 2)?h/2) We can use these areas to find the areas of teh shapes we hand out to the students (pentagon, hexagon, octagon). Developmental Activity Area of the triangles = (25 cm?5 cm /2) = 62.5 cm(^2) = 62.5 cm(^2)? 2 = 125 cm(^2) Area of rectangle = 25 cm? 15 cm = 375 cm(^2) Total sum of areas = 375 cm(^2) + 125 cm(^2) = 500 cm(^2) Closing Activity 3 -sided total interior angle measurement - 180deg 4 - sided - 360deg 5 - sided - 540deg 6 - sided - 720deg 7 - sided - 900deg 8 - sided - 1080deg The students should come up with the Polygon Interior Angle - Sum Theorem from doing this The Polygon Interior Angle - Sum Theorem states that the sum of the measures of the interior angles of an n-gon is (n - 2)? 180deg Homework Assignment -pg 80 Problems 10,11,12,14,15,16,19,20,21,23,24,25 10.) 108;72 11.) 140;40 12.) 160;20 14.) 5 15.) 10 16.) 20 19.) 102 20.) 145 21.) y = 103 ; z = 70 23.) x = 69 ; w = 111 Koeber Page 18
Day 4 Name : Mary Koerber Mathematical Concepts Addressed : Surface area in 3-D objects. Grade Level : Grade 9 Textbook : Geometry Tools for Changing a World Objectives: Have students learn how to calculate Surface area. Have students understand how to use surface area in real life situations. Equipment: This lesson should be taught in a classroom equipped with a writing board or overhead. Materials needed are scissors, pipe cleaners, and rulers. Performance Standards: Analyze properties and determine attributes of two and three dimensional objects. Explore relationships among classes of two and three dimensional geometric objects, make and test conjectures about them, and solve problems involving them. NY State: Modeling / Representation and Measurement. Opening Activity: Show students a three dimensional figure such as a pyramid and ask how they think they can find the area of the three dimensional shape. Have discussion until idea is understood. Developmental Activity: Have students working in groups creating their own pyramid out of pipe cleaners. Have students use the ruler to make the measurements and have them calculate the surface area of their polygon. Closing Activity: Review with a real world situation. Koeber Page 19
Opening Activity Introduce a pyramid to students. Ask students how they think you can find the surface areas of this pyramid. Expected response: find the areas of the different parts and add them up. Discuss the difference between bases that are of a regular shape and bases that are not. Ask students which kind of shapes would be quicker to find out the surface area and ask them why. Expected response: A regular shape with a regular base. This way the area of one lateral side is needed since they will all be the same. Introduce how to calculate surface area and go into the developmental activity. Developmental Activity (Surface Areas) Pyramids Have students make their own regular pyramids out of pipe cleaners. Have students measure the slant height and the length of an edge of a base. The surface area of a regular pyramid is the sum of the lateral area and the area of the base. s = side. l = slanted s - *l height The area of each lateral face is 2. Multiply this area by the number of sides on your base to get the lateral area. Find the area of the base and add it to the lateral area to get the surface area. After students have completed this activity, ask a few volunteers to discuss their pyramid they made with the lengths and explain how they got the surface area. Koeber Page 20
Developmental Activity (Surface Areas) Pyramids Have students make their own regular pyramids out of pipe cleaners. Have students measure the slant height and the length of an edge of a base. The surface area of a regular pyramid is the sum of the lateral area and the area of the base. s = side. l = slanted s - *l height The area of each lateral face is 2. Multiply this area by the number of sides on your base to get the lateral area. Find the area of the base and add it to the lateral area to get the surface area. After students have completed this activity, ask a few volunteers to discuss their pyramid they made with the lengths and explain how they got the surface area. Closing Activity Relating to the real world The Great Pyramid at Giza, Egypt, was built about 2580 B.C. as a final resting place for Pharoe Khufu. At the time it was built, its height was about 481 ft. Each edge of the square base was about 756 ft long. What was the lateral area of the pyrami?? Koeber Page 21
A B C. The legs of DABC are the height of the pyramid and the apothem 756 of the base. The height of the pyramid is 481 ft. The apothem of the base is or 378 ft. You can use Pythagorean Theorem to find the slant height. l 2 = 481 2 + 378 2 l = (481 2 + 378 2 ) l = 611.75567 Now use the formula for the lateral area of a pyramid. The perimeter of the base is approximately 4 * 756, or 3024 ft. L.A.= 3024*611.75567 2 =924974.57. The L.A was about 925,000 ft 2. 2 Koeber Page 22
Homework Show all WORK! The Transamerica building in San Fransisco is a pyramid. The length of each edge of the square base is 149 ft and the slant height of the pyramid is 800 ft. What is the Lateral Area of the pyramid??? 2) Make up your own shape with measurements and find the surface area of your shape: Koeber Page 23
Assignment Teacher Notes The Transamerica building in San Fransisco is a pyramid. The length of each edge of the square base is 149 ft and the slant height of the pyramid is 800 ft. What is the Lateral Area of the pyramid??? 800*149 2 * 4 = 238, 400 ft 2 2) Make up your own shape with measurements and find the surface area of your shape: Koeber Page 24
Day 5 Lesson Plan Objectives: Students will understand what nets are and how to create them. Students will be able to create three dimensional models using nets. Equipment: Different sided dice, overhead, scissors, tape, and dittos. Performance Standards: Analyze characteristics and properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships. Use visualization, spatial reasoning, and geometric modeling to solve problems. NY State: Modeling/Representation Prerequisites: Students should know how to find perimeter, area, and know basic attributes about shapes. Opening Activity: Have Students work in groups of four. Pass out opening activity ditto to students. Have students cut the nets out of the ditto to make three dimensional models. Developmental Activity: Using a cube, a pyramid, and the other three models the students made in the opening activity, we are going to work together to make a table containing the Polyhedron, Number of faces (F), Number of vertices (V), and Number of edges (E). Polyhedron Cube Pyramid Figure 1 Figure 2 Figure 3 Number of faces (F) Number of vertices (V) Number of edges (E) Koeber Page 25
Closing Activity: Have students try do draw their own net that has to fold to form a three dimensional figure. Have students use the Euler s Formula they came up with to make sure it works with their figure. Students should have various figures and notice that Euler s formula works for all of them. Opening Activity Euler s Formula Space figures and Nets: Review: Most buildings are polyhedrons. A polyhedron is a three dimensional figure whose surfaces are polygons. The Polygons are the faces of the polyhedron. An edge is a segment that is the intersection of two faces. A vertex is a point where edges intersect. A net is a two dimensional pattern that you can fold to form a three dimensional figure. Packagers use nets to design boxes. Have Students work in groups of four. Pass out opening activity ditto to students. Have students cut the nets out of the ditto to make three dimensional models. Ask students if the nets can look different to make the three dimensional shape. Koeber Page 26
Developmental Activity Using a cube, a pyramid, and the other three models the students made in the opening activity, we are going to work together to make a table containing the Polyhedron, Number of faces (F), Number of vertices (V), and Number of edges (E). Polyhedron Cube Pyramid Figure 1 Figure 2 Figure 3 Number of faces (F) Number of vertices (V) Number of edges (E) Look for a pattern in the pattern. Write a formula E in terms of F and V. Students should be able to come up with this formula. Discuss how this relationship is true for any polyhedron. This formula is known as Euler s Formula. Closing Activity Hand out different sided dice to check Euler s Formula. Have students try do draw their own net that has to fold to form a three dimensional figure. Have students use the Euler s Formula they came up with to make sure it works with their figure. Students should have various figures and notice that Euler s formula works for all of them. Homework: from textbook. Koeber Page 27
Opening Activity handout Koeber Page 28
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