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Areas of Polygons & Circumference of Circles Define: Area Area has been defined* as the following: “a two dimensional space measured by the number of non-overlapping unit squares or parts of unit squares that can fit into the space” Discuss... *State of Arizona 2008 Standards Glossary GEOBOARDS Geoboards are wonderful tools for exploring the concept of area. To change the picture all we have to do is move the geobands! If you do not have geoboards, use the grids to do the explorations. Start by making as many different sized squares as you can on your geoboard. Sketch them on the grids below. What is the area of the smallest square? What is the area of the largest square? Make as many different rectangles as you can that have an area of 4 square units. Sketch them on the grids. Find their perimeters. Are the perimeters all the same? GEOBOARDS Make as many different rectangles as you can that have an area of 4 square units. Sketch them on the grids. Find their perimeters. Are the perimeters all the same? Finding Triangles: Make as many different triangles as you can that have an area of 2 square units. Sketch them on the grids. Explain, in your own words, how your formed them. Area Find the area of each polygon by counting unit squares. Areas of Irregular Figures Find the area of each of the figures. Make sure to keep track of your work and/or the process you took as you found the area. Areas on a Geoboard Addition method divide an area into smaller pieces and then add the areas. What is the area of this figure? 19 square units A 36 complete squares 36 complete squares 2 from 4 halves 36 complete squares 2 from 4 halves 1½ from 1 x 3 triangle 36 complete squares 2 from 4 halves 1½ from 1 x 3 triangle 3½ from 1 x 7 triangle 36 + 2 + 1.5 + 3.5 = 43 B Triangle Area = ½ bh Base = 5 Height = 10 ½ (5)(10) = 25 C. 9 x 8 = 72 TAKE OUT (4x3)/2 = 6 5x3 = 15 1x4 = 4 (1x1)/2 = 0.5 (3x1)/2 = 1.5 72 – 6-12.5-8-15-4-0.5-1.5 = 24.5 D 18 complete squares 18 complete squares 1 from 1 x 2 triangle 18 complete squares 1 from 1 x 2 triangle 7 from 7 x 2 triangle 18 + 1 + 7 = 26 Rectangle method – construct a rectangle encompassing the entire figure and then subtract the areas of the unshaded regions. E Area = 16 – (3 + 1 + 1 + 1 + 1) = 9 F Find the area of the figure. F The area of the hexagon equals the area of the surrounding rectangle minus the sum of the areas of figures a, b, c, d, e, f, and g. Area and Perimeter Connections Consider a rectangle that has length and width measurements that are whole numbers. Given the below conditions determine the length and width measurements for two examples. If it is not possible to create such a rectangle, explain why. 1. The area is 30 square units. 2. The perimeter is 30 units. 3. The area is 25 square units. 4. The perimeter is 25 units. 5. The area is an even whole number 6. The perimeter is an even whole number. 7. The area is an odd whole number. 8. The perimeter is an odd whole number. 9. The area is a prime number. 10. The perimeter is a prime number. 11. What generalizations can be made regarding area and perimeter of rectangles? Consider only whole numbers in your generalizations. 12. What generalizations can be made about the relationship between area and perimeter of a rectangle? Consider only whole numbers in your generalizations. Finding Area by Dissection 1. How do you compute the area of a rectangle? Area = length x width = lw 2. Illustrate a concrete method of finding the area of a rectangle. 3 6 6 x 3 = 18 Use the rectangle you created to find the formula for the area of a triangle. If the formula for the area of the triangle is half of the rectangle, why is the formula ½ bh rather than ½ lw? Figure One a. Can you make a non-rectangular parallelogram with these two pieces? b. Describe the process from part a. A right triangle was cut from one end of the rectangle and slid to the other side to create a nonrectangular parallelogram. c. Based on your observation, write a sentence describing the area of a parallelogram. The area of the rectangle is equal to the area of the parallelogram. The width of the rectangle is equal to the height of the parallelogram and the length is equal to the base. d. Write a formula for the area of a parallelogram. Area = base x height Area = bh Figure Two Cut out both figure two shapes from your material sheet. a. What are the two shapes? Trapezoids What word describes the relationship between the Congruent two shapes? b. Put the two shapes together to form a parallelogram. c. Describe the process from part b. Two congruent trapezoids were put together by rotating one of them 180o to form a parallelogram Figure Two continued d. Based on your observations, write a sentence describing the area of one of these shapes. Top Bottom Bottom Top The area of the trapezoid is half the area of the parallelogram (½bh). The base of the parallelogram is equal to the top + bottom of the trapezoid. Area = ½ (top + bottom) height Area = ½ (a + b)h = ½ (b1 + b2)h Problem Solving Application. You have an unusually shaped pool and you need to buy a pool cover. The pool cover cost $4.25 per square foot. How much will it cost to cover your pool? Triangle: ½ bh = 3 ft 8 ft 8 ft 12 ft 10 ft 4 ft ½ (3 x 8) = 12 ft2 Rectangle: lw = 10 x 8 = 80 ft2 Parallelogram: bh = 12 x 4 = 48 ft2 Pool Area = 12 + 80 + 48 = 140ft2 Cost = 140 x $4.25 = $595 Find the areas: 20 cm2 56 cm2 Area of Polygons Triangle Rectangle Square Parallelogram Trapezoid Odd Shape Polygon Area of Polygons Triangle A 1 2 bh Rectangle A = lw Square A = s2 Parallelogram A = bh Trapezoid A = ½ (b1 + b2)h Odd Shape Polygon Break into known polygons How can you use these shapes to come up with the formula for the area of a: rectangle, parallelogram, triangle, and parallelogram? What is a circle? • Share definitions. A collection of points equidistant from a given point What does a compass do? • Draw a circle with your compass. Circumference What is a circumference? The total distance around a circle. • What is a diameter? Diameter explorations Take one of the circular objects and piece of string. Mark the length of the diameter on your piece of string How many of those diameters fit around the circumference of your circular object? Understanding Circles Locate at least 4 round objects and measure the diameter and measure the circumference of each. Record your results in the table below. Be sure to include the units you used in the measuring process. Object measured Circumference Diameter Circumference ÷ Diameter 3.14 = pi ( ) Circumference Pi = Circumference ÷ diameter Circumference = d Circumference = 2 r Circumference of a Circle Circle – the set of all points in a plane that are the same distance from a given point, the center. Circumference – the perimeter of a circle. Pi – the ratio between the circumference of a circle and the length of its diameter. Find each of the following: a. The circumference of a circle with radius 2 m. Leave answer in terms of pi 4π m b.The radius of a circle with circumference 15π m 7.5 m Which is the shorter route? Discovering and Relating Area Formulas Using Dot Paper Area is a spatial concept – a covering of two-dimensional space. Complete the tables What do you need to watch for with your students when doing the triangle, parallelogram, and trapezoid? Generalize patterns in the table Write the generalized formula for each Alpha Shapes Sort the alpha shapes into two different categories Capture the Quadrilaterals You will need the quadrilaterals from the alphashapes and a partner! Cut on the dotted lines. Make 2 piles of cards: (1) one pile contains all attributes referring to angles and (2) the other pile contains all attributes referring to sides. Piles should be placed upside down. Upon your turn, take one card from each pile. You then look for all the quadrilateral alphashapes that match both the attributes on the side and angle cards. Those quadrilaterals that match both become “captured” by you. It is than your opponents turn. He/she will follow the same procedures. Once a quadrilateral is “captured” it may not be taken unless the “WILD CARD” is used. The player(s) with the most “captured” quadrilaterals at the end is the winner! What is My Shape? Use one set of Shapes – spread them out in the middle of the table. Group members take turns being the chooser. The chooser chooses one of the Shapes while group members need to find the shape that matches the shape the chooser choose. Group members are only allowed to ask the chooser “yes” or “no” questions to help narrow down the possibilities. Group members are not allowed to point to a shape and ask, “Is this the one?” Also, group members are not allowed to ask questions about the letter on the shape. Rather, they must continue to ask questions that reduce the choices to one shape by using different attributes of the Shapes. Once a group has an idea what shape was chosen, they ask the chooser if they are correct. Alpha Shapes and Area Using the centimeter grid paper, find the area of shape W, you may need to estimate. Comparing Shapes – Two V’s What is the area of V? How can you use V to help find the area of C? Comparing Shapes 1) Compare the area of each piece to the areas of W and Q. Sort the shapes into five piles, as shown: a) Area less than W b) Area same as W c) Area greater than W but less than Q d) Area same as Q e) Area greater than Q 2) Write about your results and explain how you compared areas. Class Discussion “How did you decide what strategy to use to find the area of each shape?” “Which area measurements are rough estimates? Which ones are exact? Why?” “Did you find shapes that are not congruent that have the same area?” Good Questions Each group will work through 3 - 5 of the problems to come up with an answer key for the entire set of problems *Questions from Good Questions for Math Teaching Grades 5 – 8 and Good Questions for Math Teaching Grades K – 6. Who Put the Tan in Tangram? Find the area of each of the shapes from a set of tangrams Continue working through the packet from Georgia Department of Education.