Multi-Media Lesson: Area of Circles and Probability
Summary
Unit Name: Area/Probability
Lesson Name: Area of Circles and Probability
Teacher Name:Joan Carter
Grade Level:8th Grade, Pre-Algebra
Classroom Layout:Students will work individually or in pairs on wireless laptops.
Prerequisite Knowledge of Students: Students should be able to:
1)Calculate the areas of squares and circles
2)Calculate the areas of complex figures
3)Calculate the probability of a simple event.
4)Work independently on laptop with Safari (or other web-browser).
Prerequisite Knowledge of Teachers: Teachers should have:
1)Working knowledge of GeoGebra and dynamic worksheets
2)Working knowledge of wireless laptop cart
3)Introduced the topics of area of geometric and complex figures.
4)Introduced the topics of probability
Objectives of the Lesson: After completion of this lesson, students will be specifically able to:
1)Find the area of circles.
2)Find the area of complex figures.
3)Describe how the change in the radius of a circle affects it area.
4)Calculate the probability of a simple event.
5)Use technology as a tool to solve a problem.
Time Frame of the Lesson: 1 day for review of prerequisite knowledge
1 day for exploration of dynamic worksheets
List of Materials:Hardware: wireless laptop cart, teacher computer, LCD projector
Software: GeoGebra , web-browser, html worksheets
Other: calculator, paper, pencil, class set of dry erase boards and markers
Description of content: This lesson plan contains suggested uses for the use of the two html worksheets that were created for enrichment purposes. Students already have experience finding the area of circles and complex figures, as well as probability of simple events. This activity attempts to unite all three topics in an interactive activity that the students will perceive as fun.
SSS: A.1.3.1, B.1.3.1, B.1.3.3, B.3.3.1, C.3.3.1, E.3.3.2
List of Key Words:Area, square, circle, radius, complex figure, probability
Multi-Media Lesson: Area of Circles and Probability
Lesson Plan
1)Introduction to the Lesson:
The area of a polygon or circle is the measurement of the two dimensional region enclosed by the polygon or circle.
In Euclidean geometry, a circle is the set of all points in a plane that are a fixed distance from a fixed point. This fixed distance is called the radius, r; the fixed point is called the center, C. Circles are simple closed curves that split a plane into interior and exterior sections. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference. Usually, however, the circumference means the length of the circle, and the interior of the circle is called a disk.
The area of a circle is found by multiplying times the square of the radius, or A = r2 .
The properties of circles include:
1) The circle is the shape with the highest area for a given length of perimeter.
2) The circle is a highly symmetric shape, every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle groupT.
3) All circles are similar.
- A circle's circumference and radius are proportional,
- The area enclosed and the square of its radius are proportional.
4) The circle centered at the origin with radius 1 is called the unit circle.
(Excerpt From Wikipedia, the free encyclopedia), downloaded 3/13/07.
A square is a quadrilateral (4-sided polygon) that has 4 congruent sides, 4 right angles, and opposite sides parallel. All squares are similar.
The area of a square is simply, A = s2.
Probability is the chance that something will happen when these outcomes are equally likely. [1]. Probability theory is used in statistics, mathematics, and science, and philosophy to draw conclusions about the likelihood of possible events.
Probability can be calculated by forming the ratio of favorable outcomes (what you want to happen) to the total number of outcomes. Probability is always between zero and one. An impossible event has a probability of 0, and a certain event has a probability of 1. These ratios can be expressed as fractions, decimals, or percents.
Prior to teaching my students about finding the area of circles, I have them construct a large circle using a compass. They cut out the circle with scissors, and label all of the parts of the circle following my instruction, including the center, radius, diameter, and chord. Then I have them write the area and circumference formulas inside the circle. The circumference formula is written as a semi-circle around the perimeter of the circle. I tell the students, “I think of circles like their pizza pies. The circumference is like the crust on your pizza; the area is like the cheese.” I use references to pizza anytime I am talking about circles.
When teaching my students about probability, we visit the following website:
And we play the following game:
Everyone picks a card. Cards are shuffled and magically, everyone’s card is missing. Students finally realize that it’s a trick; that all the cards are subtly being changed and no one’s card remains. It’s a good lesson in the importance of reading and studying material rather than skimming and rushing to look at something. Once they realize the trick, we move on to talking about probability. For example, what’s the probability that I picked a “King of Hearts”? 1 card out of 6 total cards or 1/6.
2)Explanation of the math involved
Since this lesson is an enrichment activity, students have already learning how to calculate the area of squares and circles. The area of a circle is found by multiplying times the square of the radius, or A = r2 . The area of a square is the simply, the side squared, or A = s2.
A review of vocabulary, area formulas, and how they are used will be useful here. . A review of complex figures will also be necessary. In the first activity, students will calculate the area of complex figures (the area of the square – the area of the circle) as the radius of the circle is changing.
3) Instructional methods
I use a teacher-centered approach because I present formal notes to the class via PowerPoint presentation on my LCD projector. For this unit on area of circles, complex figures, and probability, I use the 8th Grade Glencoe Interactive Chalkboard software from Chapters 7.2, 7.3, and 8.1 respectively. This software allows for teacher led examples and student practice as well.
4)Step by step procedure:
First: I review the following via PowerPoint presentations. For a quick review of topics, I use my class set of dry-erase boards and markers. Students write their answers on the dry erase boards and I can quickly assess if the solutions are correct.
- Review of parts of circle and how to calculate area of a circle
- Review how to find the area of complex figures.
- Review probability ratios and how to find the probability of simple events.
Next: The students will work individually or in pairs on wireless laptop computers to answer the questions as set forth on the dynamic worksheets, Area of Circles and Probability and Area of Circles and Probability 2. Students should record their answers on a piece of paper or you may print out worksheets for them to write on.
The first worksheet, Area of Circles and Probability, provides students with a circle that they can manipulate using a slider. The slider changes the radius of the circle. The steps are as follows:
1)Move the slider to change the radius. How does the area change in relation to the change in radius size?
Students should study the size of the radius compared to the area. Point them to divide the area by the radius if they can’t visualize the change.
2)What is the area of the purple shaded region when the radius r = 0? when r = 1? when r = 2? when r = 3? when r = 4? when r = 5?
Students should be told that this is a complex figure. Give them time to work on this independently before telling them to subtract the area of the changing circle from the area of the square.
3)You are playing a carnival game. To win, you must land on the circle. What is the probability of winning when r = 0? r = 1? r = 2? r=3? r=4? r=5.
This question asks students to calculate the probability ratios when the radius is changed to the varying measurements. Probability is the area of the circle at the given radius divided by the area of the square (which is 100). I chose 100 to make these ratios easy to calculate.
The second worksheet, Area of Circles and Probability 2. provides the answers to question #3 above. Students can use sliders to select their pre-calculated probabilities (expressed as decimals to the hundredth place) to check their answers. If correct, they will get a “You’re right” message. If not correct, they will get a “Keep trying” message.
As students complete the worksheets using the computers, I move around the classroom addressing concerns or problems as they arise. I redirect students’ questions to the entire class.
Closure of lesson:
I review all the steps with the students and will calculate the areas and probabilities for them (even though they have checked their own answers using the second worksheet). Since these topics are so important in the Pre-Algebra curriculum, I remind students that they will see these topics again. Students will hand in their written observations and solutions from the worksheets. Since this was an enrichment activity, there will be no formal further assignments or assessments.
Assignments: None
Assessments: Informal assessment performed during activity and through written solutions from worksheets.
Feedback to students: Verbal feedback will be provided on during activity. Written feedback will be provided on the solution page that is submitted for grading.
Answer keys to assignments:
Worksheet 1: Area of Circles and Probability
1)When the radius is equal to one (r = 1), the area = 3.14 sq. units.
When the radius doubles (r = 2), the area = 12.56 sq. units, or 4X bigger.
When the radius triples (r = 3), the area = 28.26, sq. units, or 9X bigger.
When the radius is 4 X larger (r = 4), the area = 50.24 sq. units, or 16X bigger.
When the radius is 5X larger (r = 5), the area = 78.5 sq. units, or 25X bigger.
Rule: The change in the radius is 2 than the change in the area is 2 squared.
2)What is the area of the purple shaded region when the radius r = 0? 100 sq. units
when r = 1? 96.86 sq. units
when r = 2? 87.44 sq. units
when r = 3? 71.74 sq. units
when r = 4? 49.76 sq. units
when r = 5? 22.5 sq. units
3) You are playing a carnival game. To win, you must land on the circle. What is the probability of winning when r = 0? 0
r = 1? .03
r = 2? .12
r=3? .28
r=4? .5
r=5? .78
Worksheet 2: Area of Circles and Probability 2
Answers are the same as #3 above. Sliders on worksheet give students correct answers.
Supplemental activities lesson:
Be sure to pair up ESE or ESOL students with students with stronger abilities if using collaborative learning.
3) Instructional Materials
2 Interactive GeoGebra Worksheets follow.