Kristen Kachurek Circumference, Perimeter, and Area Grades 7-10 5 Day lesson plan Technology and Manipulatives used: TI-83 Plus calculator Area Form application (for TI-83 Plus calculator) Login application (for TI-83 Plus calculator) Learn Check application (for TI-83 Plus calculator) Learn Check Program Operating System 1.16 (for TI-83 Plus calculator) TI Navigator program 1
Objectives - Students will be able to o Determine perimeter and area of a square o Determine perimeter and area of a rectangle o Determine circumference and area of a circle o Determine area of shaded region NCTM Standards - Understand measurable attributes of objects and the units, systems, and processes of measurement - Apply appropriate techniques, tools, and formulas to determine measurements. New York State Standards Standard 3: Measurement Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by apply mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and geometry. Key Idea: Students use measurement in real-world situations. 2
Resources Calabrese-Gray, Teresa and Marrcano, Jacqueline, et al. Curriculum, Instruction, and Assesment for New York State. 18 Nov 2003. http://www.emsc.nysed.gov/ciai/home.html. Edwards, Merv. New York State Math A Semester 3. New York: Educational Design, 2002. Enterprise of Drexel University. The Math Forum at Drexel: An Online Math Education Community Center. 18 Nov 2003. http://www.mathforum.com. Ferrini-Mundy, Joan, et al. Principles and Standards for School Mathematics. Virginia: The National Council of Teachers of Mathematics, 2000. Gerver, Robert, et al. Geometry: An Integrated Approach. Cincinnati: South-Western Educational Publishing, 1998. ` Oswego City School District. Regents Exam Prep Center. 18 Nov 2003. http://www.regentsprep.org. 3
Materials and Equipment Needed Rope or string Round plastic lids (different sizes) Cardboard shapes (squares and rectangles that fit inside plastic lids) Cheerios Textbook Used for Assignments Edwards, Merv. New York State Math A Semester 3. New York: Educational Design, 2002. 4
Overview of Unit This unit will concentrate on finding perimeter and area of squares, rectangles, and circles, and finding areas indirectly such as area of a shaded region. The unit will concentrate around hands-on activities. Using the Login, Learn check applications along with TI- navigator, teachers can enter a question of their choice to be put in the student s calculator. Students then answer the question and give the calculator back to the teacher. The teacher can then download each student s response onto their personal computer. This is an effective and efficient way for the teacher to see each student s progress and comprehension of the day s lesson, and a great way to review. Lesson 1 Students will be introduced to the circle and components of the circle such as radius, diameter, and circumference with notes. Materials used will be as follows; a circular object such as a can, lid, or cup and a piece of string or rope to measure the radius and diameter of the circular object. The results of each student s measurements will be recorded on a chart which includes; the item they measured, radius, diameter, circumference and circumference/diameter. Review of an old Math A exam question threw TI-Navigator. Lesson 2 Students will be introduced to perimeter for squares, rectangles, and circles. Students will also be introduced to the area formula for squares, rectangles, and circles with notes. Materials used will be as follows; cheerios, assorted sizes of plastic lids, cardboard shapes (squares and rectangles) that fit into plastic lids. Students will use the cheerios to find the perimeter of shapes and to find the radius and diameter of the lid. Review of an old Math A exam question threw TI-Navigator. Lesson 3 Students will be introduced to shaded area with notes. Materials used will be as follows; assorted sizes of plastic lids, cardboard shapes (squares and rectangles) that fit into plastic lids. Students will place a square or rectangle of their choice that fits inside the plastic lid. They will then be asked to find the area of the region between the circle and the square or rectangle (called the shaded area). Review of an old Math A exam question threw TI-Navigator. Lesson 4 Students will review past formulas for perimeter, area, and circumference by participation. Materials used will be as follows; TI-83 plus graphing calculator, Area Form application for graphing calculator. Each student will receive a TI-83 plus calculator. Instructions for the Area form program will follow, and then students will run the program on their own. Worksheet with practice shaded area problems. Review of an old Math A exam question threw TI-Navigator. Lesson 5 Students will review past formulas for perimeter, area, and circumference by participation. Students will then be tested on perimeter and area of squares, rectangles, and circles, and shaded area. 5
Lesson Plan 1 Objectives To identify circles To identify radius, diameter, and circumference of circle Beginning to see relationship between circumference and diameter Outline of activities 1) Notes will be presented as follows: Definitions: circle set of all points in a plane equal distance from its Center. Circle Center and Radius a segment whose endpoints are the center of the circle some point on the circle. Red line is the radius Diameter a segment that contains the end points of the circle and the center of the circle. 6
Blue line is the diameter Circumference the perimeter of the circle, found by the formula C = 2p r where r is the radius. ( given p = 3.14) Area of Circle = p r ^2 Circumfere nce is in green. Area is in orange. 2) After notes, pass out materials shown below to each student (optional: students work in pairs, hence hand out one of each to the pair of students) 1 ruler 1 can, lid, or round object 1 piece of string 7
3) Before hand, make sure you label the center of each of the round objects for the students. Then, direct students to measure radius and diameter. Measure the radius by measuring the length of the string from center of object to end of the round object. You may want to provide students with tape to hold down the string while they are measuring. Measure the diameter by measuring the length of the string from one end of the circle, with the sting intersecting the center to another end. Make students use the same measure (in my example I used inches) for all measurements. Radius measurement (below) 8
Diameter measurement (below) 4) After students have found their measurements, collect their data to form a chart of all the different round objects measured. Item Radius Diameter Circumference Circumference/Diameter Lid (blue) 3.5 7 21.98 3.14 Can Lid (yellow) ETC. A sample of the other two items is shown below: Item Radius Diameter Circumference Circumference/Diameter Lid (blue) 3.5 7 21.98 3.14 Can 1.5 3 9.42 3.14 Lid (yellow) 1.25 2.5 7.85 3.14 ETC. 9
5) Discuss the relationships in the chart with the students. One of which could be who two times the radius will result in the diameter. Another of which could be how the circumference divided by the diameter will result in pi. 6) Assign homework page 295, problems #1 thru # 8 from Edwards. Pass out TI-83 calculators to students. Ask students to go under apps, then learn check, then document 1. When students press the document one key, they will be asked to enter their username. Students should enter their 3 initials as seen in the screen below. A screen will then appear with assignments. Students should select assignment 1 for lesson plan 1; assignment 2 for lesson plan 2, etc. Students should answer the question then bring up calculators to the front of the room. Download answers from all students calculators onto your personal computer. Review results. Learn check screen Document 1 screen Username screen Assignment screen Assignment 1 Assignment 1 con t 10
Assignment 1 con t Assignment 1 con t Student can use the arrow keys to scroll up and down to read the given question. The window key on the calculator which is right below tab on the screen can be used to highlight an answer. When the student highlights their answer, they then use the trace key on the calculator which is right below the CHK on the screen to check the answer they highlighted. 11
Lesson Plan 2 Objectives Find Perimeter of circles, squares, and rectangles Find Area of circles, squares, and rectangles Outline of Activities 1) Notes will be presented as follows: Area formula for circle A = 2 p r^2 (from last class) Area formula for square A = s * s (side times side) Side (s) Side (s) Area formula for rectangle A = b * h (base times height) Height (h) Base (b) Perimeter of squares and rectangles - add up all the sides Perimeter of a square P = s+s+s+s 12
Perimeter of a rectangle P = b+h+b+h Perimeter of a circle (also known as circumference) C = 2 p r 2) Pass out one plastic lid to each student. Vary the sizes of the plastic lids. I found plastic food container lids worked well. You may want to label the center of the round lid. Then pass out a cardboard shape (square and rectangle). You may also want to label the lids and shapes with numbers or colors. This way you can create an answer key for yourself. Cardboard works well, but you need to make sure that the shape s corners touch the edge of the circular lid (see picture below). Next pass out a baggie of cheerios, enough to measure around all the objects. Finally pass out a worksheet to be completed in class. 1 plastic lid 1 square 1 rectangle a baggie of cheerios 3) Have students measure the perimeter of the square, the rectangle, and the circle by placing cheerios around the outside of each object. Students should record results on given worksheet. 13
4) Next, students should measure the diameter and radius of the circular object with cheerios. Record results on worksheet. 5) After the measurements above, students should be able to figure out the area of the square, rectangle and circle. Record results on worksheet. 6) Assign homework page 219, #5 thru #7 and page 297 #21,#22. Discuss the problems on page 297. What happens if the shape we need to take the area of isn t a square or rectangle? Cut up the figure to make squares and rectangles. Then add up all the areas of all the different pieces. 7) Collect worksheets and materials. Pass out TI-83 calculators to each student. Follow steps listed in lesson plan 1 to get students to assignment 2. Students should answer the question then bring up calculators to the front of the room. Download answers from all students calculators onto your personal computer. Review results. 14
Assignment 2 Assignment 2 con t Assignment 2 con t 15
Name Date Period Perimeter, Area, Circumference, and Shaded Area Lesson 2 Perimeter, Area, Circumference Directions 1) Measure Perimeter of square, rectangle, and circle by placing cheerios all the way around the object. The amount of cheerios you have will be your measurement. Record your measurements below. 2) Measure the diameter and radius of the circle by placing a line of cheerios from one endpoint of the circle; threw the center, to another endpoint of the circle. The amount of cheerios will be your measurement for the diameter. Now, how would you find the radius using the diameter measurement? (HINT: diameter = 2radius). Record your measurements below. 3) Use your measurements to find the area of the square, rectangle, and circle. 4) After you have finished the above measurements, bring your materials up to the front of the room and pick up a TI-83 calculator. You may eat the cheerios if you wish. Measurements 1) Perimeter of your square 2) Perimeter of your rectangle 3) Perimeter (Circumference) of your circle 4) Diameter of your circle 5) Radius of your circle 16
6) Area of your square = A = s*s 7) Area of your rectangle = A = b*h 8) Area of your circle = A = 2 p r^2 Lesson 3 Shaded Area Directions 1) Place the square inside the circle. Notice the space or the area unoccupied by the square in the circle. We call this space the shaded area. Find the shaded area by filling it in with cheerios. The amount of cheerios you have will be your measurement. 2) Notice a problem? See that tiny area between the edge of the circle and the edge of the square? How can you measure that with cheerios? No, you can t cut up the cheerios, so the cheerio idea is not going to work here. You can t measure all the area with them. Well, you already know the area of the circle and the area of the square. (HINT: Use this formula: Shaded Area = - ) Once you figure out this formula, find the shaded area. Record your results below. 3) Now place the rectangle inside the circle. Use the Shaded area formula you figured out above to find the Shaded area. Record your results below. 4) After you have finished the above measurements, bring your materials up to the front of the room and pick up a TI-83 calculator. You may eat the cheerios if you wish. Measurements 1) Shaded area (with square) = 2) Shaded area (with rectangle) = 17
Lesson Plan 3 Objective - Find shaded area Outline of Activities 1) Pass out the same materials used with lesson 2 (round lid, cardboard square, cardboard rectangle, and cheerios) and worksheet from lesson 2. 2) Review past lessons 1 and 2 with students (mostly formulas). 3) Have students place one of the cardboard shapes into the circle. Tell students to measure the shaded area (that is the area unoccupied by the circle and the shape) by filling in that area with cheerios. The students will observe that the cheerios can fill in some of the shaded area, but other areas are too small for the cheerios to fill. Discuss how the shaded area can be determined? Shaded area = Area of circle Area of shape (rectangle or square) 4) Have students place the other shape inside the circle and figure out the shaded area. 5) Students should record their results on the worksheet. 6) Assign homework page 296 problems #9 thru #13. 7) Collect worksheets and materials. Pass out TI-83 calculators to each student. Follow the steps listed in lesson plan 1 to get students to assignment 3. Students should answer the question then bring up calculators to the front of the room. Download answers from all students calculators onto your personal computer. Review results. 18
Assignment 3 19
Lesson Plan 4 Objectives Practice finding area of squares, rectangles, and circles Outline of activities 1) Review of past formulas for area of square, rectangle, and triangle. 2) Pass out TI-83 calculators. Tell students to go under APPS, then under Area form. A screen like this should appear: 3) Tell students to press any key in which this screen will appear: 4) The screen below will appear with three options to choose from. 1 Definitions and formulas, 2- Area quiz, and 3- quit. Students can use option 1 if they need an extra review for formulas. If not students should select option 2 in which this screen the screen on the right will appear. After option 2 (area quiz) is selected, a screen appears as to which level to select. Students should select level 1, and can attempt level two at a later time. 20
5) Questions will then appear on the screen for the students to answer. Below is an example of the types of questions asked. Student press the window key for choice A, the zoom key for choice B, the Trace key for choice c, and the Graph key for Choice D. They keys are below the letters on the screen. Remind students that they are finding area not perimeter in this application. If students answer the question correctly, correct will appear on the screen and the next question will automatically come up. If students answer incorrectly, the incorrect answer that the student chooses will appear and then the correct answer will appear. 6) Area form does contain questions about areas of triangles and trapezoids. I had my students skip over these questions. At any time students can hit the Y= key which brings them back to the main menu where they can quit the application. 7) Since the Area form application does not contain problems on shaded area, pass out worksheet with practice problems for shaded area. 8) Assign homework practice area form application. Follow steps listed in lesson plan 1 to get students to assignment 4. Students should answer the question then bring up calculators to the front of the room. Download answers from all students calculators onto your personal computer. Review results. 21
Assignment 4 Assignment 4 con t 22
Name Date Period Shaded Area worksheet 1) Given: Rectangle ABCD with two circles removed from the rectangle. The length of the rectangle is 50 and its width is 20. The diameter of each circle is 10. A B 10 10 20 D 50 C a. What is the perimeter of the rectangle? b. What is the circumference of either circle? c. What is the area of the rectangle? d. What is the area of either circle? e. What is the area of the shaded region of the diagram? 23
2. In the accompanying figure, square ABCD is circumscribed about circle 0. The length of a side of the square and the diameter of the circle are both 12. A B O D C a. What is the circumference of the circle? b. What is the perimeter of the square? c. What is the area of the square? d. What is the area of the shaded portion of the diagram? 24
Lesson Plan 5 Objectives Test students on perimeter and area of square, rectangle, and circle. Test students on Shaded area. Outline of Activities 1) Distribute test and TI-83 calculators to students. Give students the whole class time to complete test. (Test shown below) Circumference, Perimeter, and Area Name Date Period Directions: Answer all the following questions. You may use the TI-83 calculator. 1) The pied piper and his friends are walking around a track that is shaped like a regular pentagon. Each side measures 85 feet. If they make 3 complete trips around the track, how far will they have walked? Choose: 255 ft 500 ft 1275 ft 1530 ft 25
2) Choose: Find the perimeter of this rectangle. 30 34 47 60 3). The diameter of this circular placemat is 15 inches. Find the circumference to the nearest tenth of an inch. Choose: 22.5 in. 47.1 in. 94.2 in. 176.6 in. 4) The ratio of the corresponding sides of two similar squares is 1 to 4. What is the ratio of the area of the smaller square to the area of the larger square? 1. 1:2 2. 1:4 3. 1:8 4. 1:16 5) What is the diameter of a circle whose circumference is 5? 1. A 2. B 3. C 4. D 26
6) In the figure below, ACDH and BCEF are rectangles, AH=2, GH=3, GF=4, and FE=5. What is the area of BCDG? 1. 6 2. 8 3. 10 4. 20 7) The perimeter of a rectangle is 40. One of the sides is 5. Find the lengths of the other three sides. 1. 10, 10, 10 2. 5, 10, 10 3. 5, 15, 15 4. 10, 15, 15 8) Find the area of a circle whose diameter is 10. Express answer to the nearest tenth. 1. 628.4 2. 314.2 3. 78.5 4. 39.2 9) If the lengths of the sides of a square are doubled, then the area of the sqaure would be multiplied by 1. 1 2. 2 3. 4 4. 8 27
10) The accompanied diagram shows a square with side y inside a square with side x. === y x === Which expression best represents the area of the shaded region? (1) x^2 (2) y^2 (3) y^2 x^2 (4) x^2 y^2 11) The length of the sides of two similar rectangular billboards are in the ratio 5:4. If 250 square feet is needed to cover the larger billboard, much material in square feet is needed to cover the smaller billboard? 12) An image of a building in a photograph is 6 cm wide and 11 cm tall. If the image is similar to the actual building and the actual building is 174 cm wide, how tall is the actual building? 13) In the accompanied diagram, a circle with radius 4 is inscribed in a square. 4 What is the area of the shaded region? (1) 64 16p (2) 16 16p (3) 64p - 8p (4) 16 8p 28
14) If the circumference of a circle is 10p, what is the area of the circle? (1) 10p (2) 25p (3) 50p (4) 100p 15) A target shown in the accompanied diagram consists of three circles with the same center. The radii of the circles have lengths of 3 inches, 7 inches, and 9inches. 3 7 9 What is the area of the shaded region? 29
Objectives Find perimeter of a triangle Find area of a triangle Outline of Activities Lesson Plan 5 1) Introduce students to the formula to find perimeter of a triangle and the formula to find area of a triangle. A s s Perimeter of triangle ABC = s+s+s B s C A h B s C Area of triangle = _ * h * s where h represents the height of the triangle and s represents the side to which height is drawn, also known as the base of the triangle. 23
2) Pass out cheerios, cardboard triangles, and worksheet to collect data to students. Have students measure the perimeter of the triangle by placing cheerios around the edges of the triangle. Then have students measure the area of the triangle by first placing cheerios along the base of the triangle, then placing cheerios along the height of the triangle. You may want to draw the height line in ahead of time for the students. You also may want to label the triangles using different colors or numbers in order for an easy answer key later. Perimeter of triangle Area of triangle Have students fill out worksheet with their answer for perimeter and area. 2) Pass out TI-83 calculators. Tell students to go under APPS, then AreaForm. A screen like the one below should appear. Following the prompts on the screen, students should press any key. When they do the screen below should appear. 24
Tell students to press any key. The screen below should appear. Students can use application 1, definitions and formulas for extra help and review. If students do not need the extra help and review, tell students to do under application 2, area quiz. After selecting choice 2 by pressing the number 2 key, the screen below appears. Students should first select level 1 by pressing the enter key. Inform students that they can try level 2 on their own time. After selecting level 1, the screen below should appear. Students use the WINDOW key for choice A, the ZOOM key for choice B, the Trace key for choice C, and the GRAPH key for choice D. If students select the right answer, the screen will say correct, and automatically bring up the next question. If the student is not correct, the screen will display the answer that the student chose and the correct answer, 25
then automatically bring up the next screen. Other questions such as area of squares, rectangles, and circles are in this application. Students can also attempt these problems. At the end of the quiz, a screen like the one below should appear. If students press the WINDOW key selecting problems, you will go back to the screen to select a level. 3) Assign homework from Edwards, page 244 #7 thru #15. Collect TI-83 calculators and worksheets form students. 26
Name Date Period Area and Perimeter of Triangles Your Triangle color: Perimeter of your triangle (add up all the cheerios around the sides) = Area of your triangle (multiply _ * # of cheerios on the bottom of the triangle, the base, *the # of cheerios along the height of the triangle) = Answers using the triangle in the pictures above: Perimeter = 11 + 13 + 13 = 37 Area = _* 11 * 11 = 60.5 27