Day One: An Introduction to Probability
1. Clip from Two Towers is played on shooting orks.
2.Teachers asks students if Legolas hits 42 orks and there were 78 orks in the field what is the probability for that event? If Legolas takes 100 shots how many orks is he expected to hit? Which example is a theoretical probability and which example is the experimental probability and why? (assessment of prior knowledge)
3. If necessary from prior assessment: Teacher asks if students remember the difference between experimental probability and theoretical probability?
4. Teacher then discusses that experimental probability is the number of times and event occurs divided by the number of trials. (provides examples)
5.Students are divided into their groups according to social class. (assigned in history)
6. Each group is given a square sheet of paper that is divided into eight even areas.
7. Students are given eight paper clips per group.
8. Teacher asks students what is the probability of dropping a paper clip from one foot above the center of the target and it landing on the number 1, given that it hits the target?
9. Teacher asks how did you know that the answer is 1/8? (explain reasoning)
10. Teacher then directs the discussion toward the target has to be divided into sections of equal area and the theoretical probability is the area of the shaded region you are looking to hit, such as 1, over the area of the sample space.
11. Each group drops a paper clip eight times over the center of the target and only records the result if the paperclip lands on the target. (if landed on 1 three times then the number of events is 3 and number of trials is 8)
12. Students are asked to calculate their experimental probabilities for each numbers on target.
13. Teacher walks around to monitor that students calculations are correct.
14. A discussion is held of how the experimental probabilities differ from the theoretical probabilities.
15. Teacher asks theoretically how many times should the paperclip have landed on each number?
16. Teacher asks how did this differ from the results recorded from your experimental trials?
17. Teacher asks what do you think would happen if you increased the number of trials to 50? 100? 500? Explain. (45 minutes)
Day 2: Calculating Theoretical Probabilities Activity
1. Students will be placed in groups according to social classes.
2. Each group will be handed a different picture of a target with a shaded area.
3. Given that you hit the target every time, what is the probability that the student will hit the shaded area.
4. Students are expected to divide the targets into regions of equal area to the shaded area to calculate the probability (this can be done by various methods by drawing diagonals and perpendicular lines that divide shape evenly, calculating areas if area if sample space and shaded area, etc…).
5. Teacher will go over how to divide shapes into regions of equal area and provide formulas of how calculate areas of certain shapes (this has been taught previously and will only be reviewed if necessary as students are expected to apply knowledge previously learned).
6. Teacher asks students if they are calculating theoretical probability or experimental probability and why?
7. After students work on the problem together we will regroup and have each group display their target, discuss their approach to finding the solution, and explain how they calculated the theoretical probability of hitting the shaded area.
8.Teacher asks certain groups what would happen to the probability if the shaded region increased in area? What would happen if the sample space increased/decreased in area?
9. Teacher also asks students if they see a different method they could have used?
10. Teacher asks students to reflect on the question: how does geometry relate to calculating the probability of targets? (35 minutes)
1. Have students go back to their individual seats.
2. Play Lord of the Rings Two Towers clip on battle scene involving shooting a medieval crossbow.
3. Direct students to think about what factors affect calculating the probability of hitting a target when shooting a bow and arrow?
4. Also have students think about why all problems calculating theoretical probability stated: Given that you hit the target every time?
5. Tell them we will discuss these ideas tomorrow. (10 minutes)
Day 3: Exploration activity and Discussion
1. Students will meet with groups according to social class.
2. Each group will have a Velcro dartboard hanging on the wall.
3. First the group is asked to calculate the theoretical probability of hitting the bulls eye, given they hit the target every time?
4. Then each student conducts ten trials and records the experimental probability. (trial only counts if they hit the board)
5. Students regroup for a class discussion on data. (15 minutes)
1. Teacher asks what was the theoretical probability? How did you arrive at your answer? Why might the experimental probability differ from the theoretical probability?
2. Teacher asks what factors did you take into account when throwing the dart? (distractions, not aiming the same way every time, not throwing with the same force…)
3. How do these factors affect the theoretical probability?
4. When giving a problem to do in class why was the statement made: given that you hit the target every time? (aka how does this affect the sample space)
5. Why do you think the probability of landing on 1 on a target divided into 8 regions differs from hitting a target that is irregularly shaped? ( aka can you be certain to divide an irregular shape into regions of equal area)
6. Teacher re-introduces the different types of medieval weaponry (sword, war axe, and bow showing images on a PowerPoint slide) that were gone over in English.
7. Teacher asks how do these three weapons relate to one and other in terms of how they are used? (they are all used by the offense to hit a target which is the opponent)
8. Teacher introduces the idea when preparing for battle, participants in war fare would practices shooting a bow on a target(finding an experimental probability) and have an idea of the probability they would hit their opponent like the video from the day before.
9. Teacher asks when facing battle what factors are going to affect that scenario? (Think back to when you were throwing the darts)
Questions to prompt student thinking…
- What would have happened if we placed the dartboard outside?
- What features do you notice about the dartboard used and how are these features different from hitting an opponent?
- What do you notice about the placement of the dartboard and how might this placement differ from calculating the probability of hitting an opponent?
- What might have happened if the dartboard was placed further away?
- What happened in each trial that you threw the dart? Did you aim and throw the dart with the same motion every time? (aim, their opponent is an irregular shape, weather, movement, force and distance, wearing armor creates a smaller target….)
Teacher informs students that the probability distributions they were calculating the previous day are uniform distributions because the probability of hitting each section is equal.
Teacher asks students what properties did you notice a uniform probability distribution has? (P(x) is always greater or equal to zero and less than or equal to one, when you add the probabilities for all x, the sum =1)
Teacher then informs in most cases the probability distribution is not uniform.(For example the probability of hitting a target x times in n tries)
Teacher asks why is this example considered a non uniform distribution?
Teacher informs we cannot calculate the solution unless we are given the probability of hitting a target in one try.
Teacher gives formula and example.
Teacher leaves students to generalize why was it difficult to prepare for war and have a good idea of the probability they would hit their opponent? ( 15 min)