Probability - the likelihood that an event will occur

Area - the entire surface of an object/figure

Radius - the segment of a circle that connects the center to a point on the circle

Diameter - the segment in a circle that passes through the center and connects two points on the circle

 Concentric Circles - two circles that have different length radii but share a center

 Sector of a Circle - a region of a circle that includes created by two radii 

Theoretical data - data that is created mathematically (i.e. a formula)

Experimental Data - acutal results from completing trials

Students will explore and analyze the properties of geometric areas and the qualities of probability that are controlled by the two dimensional objects.  Students will demonstrate these concepts using an interactive spinner and dartboard.  Students will be able to:

• find the area of circle to the nearest tenth;
• find the area of a sector of a circle to the nearest tenth;
• find the area of non overlapping parts of concentric circles to the nearest tenth;
• find the probability using the given formula for each of these areas creating the theoretical data rounded to the nearest whole percent;
• find the probability using the given formula for each of these areas by creating experimental data rounded to the nearest whole percent;
• compare and contrast factors that contribute to the theoretical data as opposed to experimental data.

80 to 120 minutes

PowerPoint Presentations:  Reviewing the terms and vocabulary students will need in order to complete the lesson plan on geometric probability. 

Record Worksheet : The worksheet students will use to formulate their theoretical data, complete and record the trial results, formulate the experimental data statistics, and then compare the results between theoretical and experimental data.

http://illuminations.nctm.org/ActivityDetail.aspx?id=79 : An online interactive spinner that allows the educator or student to manipulate the size of the sectors, provides the theoretical data, and also allows educator or student to gather the experimental data by interactively spinning as many trials as desired.  Additionally, the spinner indicates the comparison of the results of the theoretical and experimental data.

https://www.mathopenref.com/concentric.html: An online interactive tool to look at concentric circles.  Students can practice finding the area of the non overlapping parts of concentric circles (identifying areas and then calculating the difference between the circles when the radius of each circle is given).

Dart board measurements worksheet:  A worksheet used to give the measurements of a standard dartboard in order to guide students in finding the areas of the parts of the dart board. 

http://www.greatdaygames.com/games/darts.aspx :An online interactive dart board that students can use to conduct the experimental data gathering.  If using the website as opposed to an actual dartboard, students will need the supplemental dart board measurement worksheet. 

Other Materials:  Dependent on availability of technology - interactive whiteboards, computers, spinners (could be hand made, from board games, or from the internet source), dartboards (real or from the internet source), pencils, rulers, protractors, provided worksheets with dimensions (optional), notes on probability (PowerPoint).

W: In this lesson goals will be that students are able to use area of figures to find probability.  This lesson is framed to reinforce finding areas of figures (primarily circles and parts of circles) and finding probability.  The main objective of this lesson is to combine the two concepts in a real world application using geometric probability.  Using real world objects and activities (i.e. spinner and dartboard) helps motivate students with this concept.

H:  The students will be actively participating in the lesson because of the level of engagement provided.  The hook is that the lesson does apply real world applications in a way that is fun for students as they will be spinning the spinners and throwing the darts.  They need to complete the theoretical data in order to compare it the experimental data (the more motivating part of the lesson for students).  They can also determine and discuss the validity of the mathematical formulas they are using to come up with the theoretical data.   

E:  Students will be introduced to online tools (interactive spinner and interactive dartboard) and mathematically finding the theoretical probability (landing in a region with either spinner or dart), but then also being able to participate in the experimental trials which will lead to actual experimental probability.  They can then compare and contrast the differences between the two sets of data, coming up with conclusions as to why there were similarities and differences in the data sets.

R:  Students will have to reflect on how each of their trials went, thus effecting the outcomes of their data.  After they gather the experimental trials and complete the experimental probability, they will have to revisit their theoretical probability to compare and contrast the two sets of data.  Students will have to rethink how the conclusions they reached may differ from other students' sets of data.  They will have to rethink the factors that resulted in the differences not only between their own theoretical and experimental data sets but also between other students' data sets

E:  Students will express their understandings of this lesson with in-class discussion of results and a paragraph that is required to be written on the worksheet on which they are recording the data. 

T: Differentiated possibilities for this lesson may include:

O:  The lesson will be split into two days of activities.  The first day will deal with the spinner.  The second day will deal with the dartboard.  The breakdown of the two days is as follows:

Requisite Prior Knowledge - Previous knowledge is needed of how to find and solve for area of a circle and area of a sector of a circle (reviewed in power point presentation).

Day 1:

1. Students will be shown the Thinkfinity online spinner tool and asked "Is there any way to tell where the spinner will land?" <Student Response> Then the teacher will explain that this concept is probability and that we will need certain vocabulary to discuss probability.  Students will be introduced to probability concepts with vocabulary and definitions that shall be entered into student notebooks. (PowerPoint Presentations)
2. Students will practice vocabulary while participating in guided instruction as they learn how to use the areas of a circle and sectors of a circle to construct the theoretical probability of spinners landing on each of the sectors. Website: http://illuminations.nctm.org/ActivityDetail.aspx?id=79 (Thinkfinity Resource)
3. Students will perform 10 trial spins (either on the website or on own given spinners) and record where the spinner landed for each trial.  Students will then translate the data into resulting experimental  probability data.  Using the handout (Record Worksheets that is a chart) they will record the differences in the theoretical data and experimental data. 
4. Students will write a paragraph in their notebooks (worksheets) on the factors that contribute to the difference  between the two forms of data
5. Students worksheets will be used as formative assessment.

Day 2:

1. Students will review the activities from day one
2. Students will be introduced to strategy of finding area of the non overlapping area of concentric circles using the internet source: (or provided worksheet given available technology).  The strategy is taking the area of the bigger circle and subtracting the area of the smaller circle. Website: https://www.mathopenref.com/concentric.html  (Thinkfinity resource)
3. Students will be given a worksheet on parts of a dartboard and its measurements in order to solve for the areas of each of the concentric circles.  (Dart board measurements worksheet)
4. Students will calculate and record the theoretical data onto the worksheet provided using the dimensions of a dartboard.  Record Worksheets
5. Students will be performing 10 trials of throwing a dart (either an actual dartboard or using the provided internet source dart board) and record the results as experimental probability data. Website:  http://www.greatdaygames.com/games/darts.aspx
6. Students will record the experimental data into the chart in order to compare and contrast the theoretical and experimental data.
7. Students will write a paragraph in their notebooks on the worksheets regarding the factors that contribute to the difference between the two forms of data.
8. Students worksheets will be used as formative assessment.

Formative Assessments will be determined by evaluating worksheet used for each day for recording relevant information (with any additional work attached to the worksheet) and the summative paragraph completed on the worksheet.  The work includes:

1.  The correct use of the formulas for area of a sector of a circle and concentric circles. 

2.  The correct use of the probability formulas to find the theoretical data and the experimental data

3.  A 4-6 sentence paragraph that summarizes the results, compares and contrasts the theoretical data to the experimental data, and the possible contributing factors that could vary these results. 

Formative assessment may also include student participation in discussion of results recorded by them as well as the discussion regarding factors in the similarities and differences in students' data sets. 

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=8269 :  A Similar Thinkfinity lesson on geometric probability using dartboards with worksheets provided to compare and contrast theoretical data to experimental data. (Thinkfinity Resource)