The goal of this lesson is to build the concept that probability, or chance, has to do with events that are uncertain but that have a pattern of regularity when trials are repeated many times. Students conduct experiments and record the frequency of each outcome to use in the calculation of experimental probabilities. From the experimental data they make decisions involving the predictability of the outcomes. The experiments yield a wide variation in results when conducted a small number of times. Students calculate experimental probabilities for a small number of trials as well as for a large number of trials and begin thinking about how the number of trials impacts the usefulness of the probabilities.
As students enter the classroom, hand each of them a sticky note or small slip of paper with a large dot colored on it (red, yellow, or blue), or use colored dot stickers. On the board, overhead, or chart paper display the terms experimental probability and equally likely and a spinner such as the one below.
“Think about what is most likely to happen if I spin the spinner one time. Stand up if you think your dot color is the one I am most likely to spin.” Have students who are standing hold their paper so the dot can be seen (mostly red dots should be standing).
Ask a few students to explain their thinking. Guide students to the conclusion that although there are the same number of blue and yellow spaces (one each), there are more red (two). Discuss equally likely and not equally likely outcomes.
“Another way to predict what will happen is to try spinning the spinner several times and keep track of how often each color comes up. Based on the results, we could predict what is likely to happen on future spins. This is called experimental probability since we are actually doing the activity or experimenting to find our results.”
“In today’s lesson we will be learning about experimental probability. We will look at several situations and practice calculating the experimental probabilities for the outcomes. We will also determine if a variety of actions have equally likely outcomes or not.”
Before completing the spinner activity review with students how a frequency table is used.
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Tally Marks
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Frequency
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Total
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Explain that the first column is for the “target event,” or what they are looking for, spinning, rolling, etc. The second column is where they place their tally marks and the third column is for the total number of tally marks in each row.
Spinner Activity
“I am going to spin the spinner 12 times and have you record the results. You will use a frequency table to record the results.” Hand out student copies of the Frequency Tables (M-7-1-2_Frequency Tables.doc). Instruct students to write each color in the left column, and a tally mark in the middle column for the color that comes up each time you spin. The last column is used to give the total for the number tallies.
“Before I spin, I want you to predict how many of each color you think I will spin.” Let several students share their predictions. Spin 12 times or let students spin. Use a paper clip or ruler with a hole near the end as your spinner pointer. Whichever you use, hold it in place at the center of the spinner with the point of a pencil or pen through the opening. Spin with a finger on the opposite hand. Demonstrate for students for later activities.
“Did the results turn out as you expected?” Take the time to discuss how results can be uneven especially with a small number of trials.
“I am going to spin 12 more times and have you record the results.” After students have recorded the next 12 outcomes, describe how the experimental probability is calculated as a fraction and how it is represented (number of favorable outcomes/total outcomes). After you write the following sentences on the board, say the words aloud and have students repeat them. Remind students that a fraction is really a ratio.
P(red) = ____ “The probability of spinning red is ____.”
P(blue) = ____ “The probability of spinning blue is ____.”
P(yellow) = ____ “The probability of spinning yellow is ____.”
“Since we had a total of 24 trials, the denominator for all of our probabilities is 24. The numerator will be the number of times we got the color being evaluated in our 24 spins, called the ‘favorable outcome.’ If possible, we want to simplify each ratio. Each ratio can also be represented as an equivalent percentage or decimal.” Show each of the three representations for the spins you have recorded.
“What outcomes could be expected with two spins?” Let students give suggestions. They should come up with double colors, like (blue, blue), (red, yellow), and (yellow, blue). They do not need to list every possibility at this point. Have students use one of the large 12-section frequency tables at the bottom of the sheet. Ask them to record the results with tallies as you do double spins on the spinner (or have students take turns doing the spins). Do about 25 double spins. Discuss what students can conjecture based on these 25 trials. You can do more spins if time permits. Calculate the probability of getting each of the different combinations that appeared. Ask students what they notice about the results. They may notice that the matching colors (blue, blue), (red, red), (yellow, yellow) show up more often. They may also wonder if outcomes like (red, blue) are the same as (blue, red). Ask if they think there are any combinations that did not appear during your limited number of trials (answers will vary).
Action Sorting Activity
“Who will volunteer to explain in their own words what equally likely events are?” Choose one or more volunteers. Use the Equally Likely or Not? transparency (M-7-1-2_Equally Likely Transparency Master.doc) as a discussion tool with the class.
- A baby is born…boy or girl? Equally likely.
- You shoot a basket…make it or miss it? Not equally likely unless you have an established shooting record of 50%.
- You roll a six-sided number cube…prime or composite? Not equally likely since there are three primes and only two composite numbers.
- You flip a fair coin…heads or tails? Equally likely.
- You ride the bus to school…you get a seat near the front or a seat near the back? Not equally likely; it depends on your preference and how full the bus is when you get on.
Clarify any misconceptions that come up. Encourage students to explain their thinking on the examples that appear to be the most perplexing.
Separate students into groups of three or four. Give each group a set of Action Sorting Cards and one Sorting Mat (M-7-1-2_Action Sorting Cards and KEY.doc and M-7-1-2_Sorting Mat.doc). Instruct students to discuss each action and place the card in the “Equally Likely,” “Not Equally Likely,” or “We Disagree” section of their mat. Give them about five minutes to sort all the cards. As you move about the room, observe which cards are in each group’s Disagree category. If there are several cards students in the “We Disagree” section, discuss one commonly disagreed-upon card with the entire class. Give groups a few more minutes to discuss any other cards they disagreed on. Allow them to get input from one other group if they still cannot come to a consensus.
Leave students in their small groups to complete the next activity.
Super Seconds Activity
Distribute a piece of poster board or chart paper and markers to each group. They will also need at least one copy of the Super Seconds activity sheet (M-7-1-2_Super Seconds.doc). Students must roll the number cubes 50 times and record the sum each time. Based on these results, they must calculate the experimental probability of getting each sum and answer the additional questions on the activity sheet. Explain to students that they need to prepare a poster with the results of their experiment and answers to be presented to the class. Each student will be responsible for presenting a portion of the information. A way to divide the presentation parts for groups of four would be:
- experimental data results
- how data was used to find the probabilities
- questions 1 & 2 with explanations
- question 3 and explanation
Allow approximately 25 minutes to work before presentations begin (5 minutes to roll and record sums, 5 to 10 minutes to discuss the results and answers, and 10 to 15 minutes to make a poster and determine who will present which information). Each group presentation should be about 4 to 5 minutes.
As each group presents, pose questions to lead them to correct any misconceptions they have. Encourage the group to make any necessary adjustments or corrections during their presentation or on their poster afterward. Questioning may also be used to assess the level of understanding of individual students.
After the groups finish presenting, combine the data from all of the groups’ rolls. As a class, calculate the experimental probabilities for each sum based on the class totals. Discuss how these probabilities differ from the group results, if at all, and the advantages and disadvantages of collecting a larger amount of data to calculate the probabilities (leading to a later discussion on the law of large numbers).
Approximately 5 to 10 minutes before the end of class, give each student an Exit Ticket (M-7-1-2_Lesson 2 Exit Ticket and KEY.doc). The responses provided by students can be used to assess level of understanding and for assigning additional suitable activities as suggested in the instructional strategies section.
Extension:
Use these suggestions to tailor the lesson to meet the needs of your students during the unit and throughout the year.
- Routine: Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following terms should be entered into students’ vocabulary journals: equally likely, experimental probability, frequency table, not equally likely, outcomes, and trials. Keep a supply of vocabulary journal pages on hand so students can add pages as needed. Bring up instances of chance and prediction as seen throughout the school year. Ask students to bring up examples that they see throughout the year.
- Small Group: Pyramid Sums Activity
This activity is for students who may benefit from an opportunity for additional learning and understanding of the concept or calculation of experimental probabilities. Have students roll two, four-sided number cubes 25 times and record the sums. The sums will be between 2 and 8. Using the results and the Let’s Roll Pyramids sheet (M-7-1-2_Pyramid Sums.doc), students should calculate the experimental probabilities as both fractions and percentages. Discuss the meaning of the numerator and denominator in each probability, and how using percentages is helpful. Ask students to explain how the probabilities could help to predict future rolls. Clarify any misconceptions.
- Expansion: Paint It Activity
Use this activity for students who have demonstrated solid understanding and proficiency in working with experimental probabilities. In the Paint It activity, students will be asked to spin two colored spinners. They will spin each spinner once and record the two colors they land on, along with the color of paint that would be created if their colors were mixed together. Review the color combinations (red + red = red, yellow + yellow = yellow, blue + blue = blue, red + yellow = orange, blue + yellow = green, red + blue = purple). Students will need copies of Paint It Spinners and Paint It sheet to record their data and calculate the experimental probabilities (M-7-1-2_Paint It Spinners.doc and
M-7-1-2_Paint It Record Sheet.doc).
Up to six players can play. Each student selects a color from red, yellow, blue, green, orange, and purple (or draws a slip of paper from a container with each slip listing one of these colors). Students will play the game for 10 to 15 minutes. Players take turns spinning, moving from player to player in a clockwise direction. The player who has the same paint color combination as the combination spun earns one point (i.e., if yellow and red are spun, the player with orange gets a point). If there is no player for a certain color combination, no points will be awarded for that spin, but the spin should still be recorded. Play continues until the time is up. The player with the most points wins. Have students answer the questions on the game sheet.
- Technology Connection: If computers are available, have students go to http://www.saintannsny.org/depart/math/misterg.htm. Give them time to play and experiment. The site rolls two number cubes and creates a bar graph of the results. Students will write a journal page regarding what they tried, observations from different small and large numbers of rolls, their results, and the resulting graphs. Extend the activity by asking students to physically roll two or three number cubes 20 to 30 times in the classroom and also 200 to 300 times, then make a table and graph for each situation similar to those on the Web site.