LESSON PLAN #1. Name: Kevin Laley Date: March 1, NYS Mathematics, Science, and Technology Learning Standards Addressed


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1 LESSON PLAN #1 Name: Kevin Laley Date: March 1, 2005 Today s Lesson: Circumference Exploration Unit Topic: Applying Fractions Course: Math 7 NYS Mathematics, Science, and Technology Learning Standards Addressed Objectives: Standard 1: Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions. Standard 3: Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in realworld settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry. Standard 6: Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning. Standard 7: Students will apply the knowledge and thinking skills of mathematics, science, and technology to address reallife problems and make informed decisions. The student will measure the diameters and circumferences of various circular objects and use their data to discover a relationship between the diameter and the circumference of a given circle. (Synthesis) The student will be able to use a calculator with the formula C = π d to find the circumference of a circle given the circle s diameter or radius. (Knowledge) Materials: 8 Circumference Kits, each containing 5 different circular objects (each kit should contain the same 5 objects), about ten pieces of string, a metric ruler, and a pair of scissors 2 overheads of the tasksheets and one for each student (21) 1 Group Report Sheet for each group (7) Calculators (at least 8, but preferably one for each student, if available) One homework sheet for each student (21 total) Overhead ruler Overhead Pens and Marker Board Markers
2 Anticipatory Set: I will begin class by drawing the following rectangle on the front markerboard: 5 ft 3 ft Ask Can anybody tell me how we can calculate the perimeter of this rectangle? (ans: = 16 feet) Next, I would draw the following figure on the board: 2 in. 5 in. 1 in 3 in. 5 in. 1 in 3 in. Again, ask Can someone explain to me how I can calculate the perimeter of this object? (ans: add = 20 inches). Lastly, I would draw the following figure: 4 cm 10 cm 10 cm 4 cm Ask What makes this figure different from the other two figures I ve just drawn? (I d expect answers such as, it has rounded edges, we do not know how to calculate the perimeter of the rounded edges, etc..) Explain to the class that we re going to learn a couple techniques for
3 solving a problem such as this in our exploration today. (Anticipatory set should take about 5 minutes.) Lesson Body: SetUp Before dividing the class into groups, explain to the students what will be expected of them in the class period. First, I will give one task sheet ( Circumference Exploration ) to each student. I will then tell them that they will be working in groups of three, which have been preassigned ahead of class, to measure the diameter and circumference of the five objects in their Circumference Kits. (These are the bags containing five circular objects, a metric ruler, string, and a pair of scissors that have been prepared before class.) Next, I will take one of these objects from that bag and model how the students are to measure the diameter and circumference. (Using a piece of string and an object from the bag, hold one end of the string on one edge of the circle, and pull the string tight across the object so that it passes through the center of the circle and touches the opposite edge of the circle. Holding both ends of the string where they meet the edge of the circle, place the string up to the metric overhead ruler and show how measuring the length of the string between these two points gives us a measurement for the diameter of the object. Similarly, pull the string tight so that it goes around the circumference, or edge, of the circle, holding the string at its starting and end point. Then show that measuring the distance between these two points of the string gives us a measurement for the circumference of the circle. This should take about 5 minutes. Experimenting Break the class into groups of three people. (For our class of 21, there will thus be 7 groups). Make sure each team pulls their desks together so that they are facing one another. Ask one member of each team to come to the front of the room and get a Circumference Kit and group report sheet for the team and another person from each group to get enough calculators for each individual in their team. Once this is done, instruct each team to designate one person to measure the diameter of each object, another person to measure the circumference of each object, and a third person to record all the values. (This third person will also copy their data on the group report sheet.) If one group finishes before the others, have them switch roles so that they check each others measurements. This should take approximately 10 minutes and then another 2 minutes to discuss what students found in the exploration. (Ask if anyone experienced any difficulties, how they overcame these difficulties, or if they had a chance to check each other s measurements. Some students may have trouble eyeballing the center of the circle for the diameters or getting the string to stretch around the circumference.) Reflecting and Explaining With the teams, now assembled as a whole class group, put the task sheet on the overhead and ask one group what they got for the circumference of object A, the diameter of object A, and the ratio of the circumference to the diameter of object A. Then, ask another group to volunteer their
4 data for Object B and so on until one set of measurements for each of the objects is recorded. (If time is beginning to run short, you may choose to only collect data from the first few objects.) At this time, collect the group report sheets from each group. The students do not need to copy down the data from the overhead, because I will type up a sheet with each group s data along with the class averages for the measurements. This summary sheet will be handed out at the beginning of the following class and will serve as a quick review of the material covered today and will allow each student to have a complete report of our findings. This class reflection should take no longer than 10 minutes. Hypothesizing and Articulating Ask, (1) What do you notice about the quotients you calculated? (Hopefully, students will notice that all of them are relatively close to 3 or 3.14.) If for some reason the data is off, explain to the class that the values are supposed to all be equal to Next, ask students (2) Why are the ratios we calculated not exactly equal to 3.14? or equivalently, What are some possible sources of error in the exploration. (Error in estimating the center of the circle, not correctly measuring the length of the string with the ruler, using stretchy string etc ). This should take about 5 minutes. For homework, I will ask the students to write a short journal entry detailing their experience with today s exploration and what they gained from it. Verifying and Refining Now that the class has been told that the circumference of any circle divided by its diameter always equals approximately 3.14, tell them that mathematicians have given a special name to this number, Pi (π ). Stress that 3.14 is merely an approximation for this number and that it actually goes on forever, never repeating or terminating. (This is called an irrational number.) (3) Ask the students Can you write an equation for the circumference of a circle given its diameter? (Give them one or two minutes for contemplation before asking if any of them have a suggestion.) The correct response should look something like the following: circumference diamter This should take about 57 minutes. = π or circumference= π d or C = π d.
5 Closure Say, We already said that the ratio of the circumference to the diameter of any given circle is approximately equal to Ask, (4) What relationship between the circumference and diameter does this tell us? (The circumference is approximately 3.14 times as large as the diameter.) Return to the previously unsolvable problem from the beginning of class, and say we now have enough information to calculate the perimeter of this object, which you ll be asked to do on your homework. Also, let the students know that we will spend the first part of class tomorrow sharing their thoughts and/or solutions to this problem. Accommodations for IEP For the student with ADD, I have included very specific instructions throughout the exploration to ensure that each member of the group has a defined task to complete. Since no team can function without every person s contribution, the other members of the team will presumably make sure that the student will stay on task. I will also keep a close watch on this student as I roam about the groups during the exploration. If he seems to be losing focus, I will attempt to ask questions that will motivate him to stay on task. Homework/Assessment Hand out the worksheet labeled Circumference Homework. Since all tasks on this sheet have been referred to throughout the lesson, this should require no additional explanation. Remind students that they are each expected to attempt the unsolvable problem on the bottom of the sheet, but urge them not to become frustrated if they can not answer it. Let them know that we will discuss their findings on this problem as a class tomorrow. Extensions If one group finishes the exploration early or if time permits at the end of class, give them a bag of quarters to complete the Extra Fun! exploration on the task sheet. Have them measure the distance around one quarter with the string, and then stretch the string on a piece of paper, marking the length of the circumference of the quarter. Then ask them how many quarters they can lay sidebyside along the length of this string. (They should be able to fit three quarters, with a little bit of the string left over. This illustrates that the circumference of the quarter is equal to approximately 3.14 times its diameter, because the three quarters sidebyside represent three diameters.)
6 Name Partners Circumference Exploration Date Directions: Using the string and metric ruler provided, measure the circumferences and diameters of each of the five circular objects given to you and record these measurements in the table below. When all measurements have been taken, find the ratio of the circumference to the diameter of each object by dividing its diameter by its circumference. What do you notice about the ratios you calculated? Object Circumference (cm) Diameter (cm) Ratio (C D) Extra Fun! Measure the circumference of a quarter. Draw a line equal to the length of the circumference below: Now, how many quarters can you fit across this line, if the line must go through the center of each quarter? Can you say anything about the relationship between the circumference of one quarter and the diameter of a quarter?
7 Group Members Date: Circumference Exploration Group Report Sheet Directions: Have the data recorder from your group copy the measurements you obtained from your exploration in the table below. This sheet will be handed it at the end of the class. Object Circumference (cm) Diameter (cm) Ratio (C D)
8 Name: Date: Circumference Homework What number is symbol π approximately equal to? Explain what the significance of this number is in regards to the circumference and the diameter of a circle. Based on our findings from class, write a formula for finding the circumference of a circle: Calculate the circumference of the following circles: r = 15 r = 1 Try to find the perimeter of this object: 4 cm 10 cm 10 cm 4 cm
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