Probability and Statistics Activity: TEKS: Mystery Bags (3.13) Probability and statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data. The student is expected to: (C) use data to describe events as more likely than, less likely than, or equally likely as. (3.14) Underlying processes and mathematical tools. The student applies Grade 3 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to: (A) identify the mathematics in everyday situations; and (C) select or develop an appropriate problem-solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and (3.15) Underlying processes and mathematical tools. The student communicates about Grade 3 mathematics using informal language. The student is expected to: (A) explain and record observations using objects, words, pictures, numbers, and technology; and Note: Portions of this lesson address TEKS at other grade levels as well; however, the intent of the lesson fits most appropriately at the grade level indicated. Overview: Materials: Students will be given four Mystery Bags, each containing 10 tiles. Each bag will have a certain number of blue tiles and a certain number of red tiles. Students will draw twenty-five times (with replacement) from each bag and record their data. Then, students will share their data with the other students in their group. Next, each group will create a likelihood line and write down where each bag belongs. They will also explain why they placed the bags where they did. Groups will then discuss their results with the class. Color tiles (red and blue) Paper bags (4 per group See Procedure 1 for details on preparing the bags.) Markers Chart paper What s in the Bag? Handout/Transparency 1a Mystery Bags Page 1
What s in the Bag? (Sample Data) Handout/Transparency 1b Mystery Bags Group Recording Sheet Handout/Transparency 2 I m Seeing Red! Handout/Transparency 3 I m So Blue! Handout/Transparency 4 Our Predictions! Handout/Transparency 5 Our Predictions, Revisited! Handout/Transparency 6 It s Your Turn to Spin Handout/Transparency 7 Grouping: Time: 4 students per group 1 to 2 hours Lesson: Procedures 1. Prior to the lesson, create a set of mystery bags for each group of students. Notes Each set should be prepared as follows: Bag 1: Place 10 blue tiles Bag 2: Place 8 blue tiles and 2 red tiles Bag 3: Place 5 blue and 5 red tiles Bag 4: Place 2 blue and 8 red tiles 2. Tell the students: I have assembled a set of Mystery Bags for each of the groups. The four bags contain 10 tiles each, in combinations of red and blue. The actual distribution of red tiles and blue tiles in each bag is a secret, though. Your job is to collect some data to try to predict how many blue tiles and how many red tiles are in each bag. You are NOT allowed to look in the bags! Ask for a volunteer to help you model the activity (drawing with replacement) for the class. Give the volunteer one of the Mystery Bags (1, 2, 3, or 4). The student should draw one tile from the bag, and then record the outcome in the appropriate Mystery Bag s column on Handout/Transparency 1a. Then, the student should place the tile back into the bag and shake the bag to randomly distribute the tiles. Go through two or three draws to make sure that each student in the class Mystery Bags Page 2
Procedures understands that the tiles are to be replaced after each draw and that the outcome from each draw should be recorded on the recording sheet. Notes 3. Have students get into groups of 4. Distribute a copy of What s in the Bag? (see Handout/Transparency 1a) to each student. Distribute a set of four Mystery Bags to each group. See Procedure 1 for notes on assembly of the bags. Each student in the group should get a Mystery Bag and draw (with replacement) 25 times from the bag, recording the outcome each time on the recording sheet. Then, each student in the group should get a different bag and repeat until he or she has completed 25 draws from each bag in the set. A sample of student data is provided on Handout/Transparency 1b. 4. After each student has recorded the results from all four bags, distribute one copy of Mystery Bags Group Recording Sheet Handout/Transparency 2 to each group of students. Have all the students within each small group record their data on this group recording sheet. 5. Discuss with students how to describe the results of the experiment using the following labels. Impossible Less likely than Equally likely as More likely than Certain The probability of an event is a number between 0 and 1. The number represents the chance that an event will happen. If an event is certain to occur, its probability is 1. If an event is impossible, then the probability is 0. Terms applied to the comparison of the probabilities of two events: Mystery Bags Page 3
Procedures Notes An event A is more likely to occur than another event, B, if the probability of the first event, P(A), is greater than the probability of the second, P(B). An event A is less likely to occur than another event, B, if the probability of the first event, P(A), is less than the probability of the second, P(B). An event A is equally likely to occur as another event, B, if the probability of the first event, P(A), is equal to the probability of the second, P(B). Example: A bag contains 12 balls (3 red, 1 blue, 2 green, 4 yellow, and 2 orange). If you reach into the bag without looking: (1) it is more likely that you will pick a yellow ball than a blue ball since P(yellow) > P(blue) 4 1 or > ; 12 12 (2) it is less likely that you will pick a green ball than a red ball since P(green) < P(red) or 2 3 < ; 12 12 (3) it is equally likely that you will pick a green or orange ball since P(green) = P(orange) or 2 2 =. 12 12 6. Distribute Handouts/Transparencies 3 and 4 (I m Seeing Red! and I m So Blue!). For each of the four Mystery Bags, have each group write the number of the bag Mystery Bags Page 4
Procedures above each appropriate label on the likelihood lines on Handouts/Transparencies 3 and 4. Then, have each group complete the sentences to justify their classifications. Notes Finally, have each group of students predict how many tiles of each color are in the bags. Groups should record their predictions on Handout/Transparency 5. Remind groups that they are NOT allowed to look into the bags yet! 7. Have each group present their likelihood lines on Handouts/Transparencies 3 and 4 to the class. 8. After each group has presented, ask the entire class the following questions: 1) How are the results for the two likelihood lines the same? Different? 2) Were the individual results the same as the group results? Why or why not? 3) Did all the groups place the bags under the same labels? Why or why not? 4) What might cause the results to be different? Same? 5) Can you predict exactly how many tiles of each color are in the bags based on the class data? 9. Ask: Now that you have heard each group present its results to the class, do you wish to revise your predictions for the number of red tiles and the number of blue tiles in each bag? If so, record your new predictions on Handout/Transparency 6. 10. Finally, have the students look in the bags to check their predictions. How close was the entire class to the exact answer? Discuss the results. 11. As students reflect on the lesson, have them write a sentence or two explaining each of Mystery Bags Page 5
Procedures the following descriptions: Notes impossible less likely than equally likely as more likely than certain Homework: Assessment: Extensions: Resources: Have students complete It s Your Turn to Spin (Handout/Transparency 7). Students will color in each spinner on the homework sheet and write a sentence explaining what description each spinner might represent (impossible, less likely than, equally likely as, more likely than, or certain) using the colors they have selected. Have the students give real-life examples of events that might be described in each of the following ways: impossible less likely than equally likely as more likely than certain Have students get into groups and create activities to demonstrate each of the following descriptions: impossible less likely than equally likely as more likely than certain North Carolina Department of Public Instruction, I. (2006, August1). Mathematics: Instructional Resources. Retrieved September 21, 2006, from http://community.learnnc.org/dpi/math/archives/instructional_resources Mystery Bags Page 6
What s In the Bag? Select a bag. Draw one tile from the bag, record its color on the chart below, and then replace the tile. Repeat the process until you have drawn 25 times from that bag. Then, select another bag and repeat the process. Continue until you have drawn 25 times from each of the mystery bags in your set. Draw # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Totals 1 1 2 2 3 Handout/Transparency 1a 3 4 4 Mystery Bags Page 7
What s In the Bag? (Sample Data) Select a bag. Draw one tile from the bag, record its color on the chart below, and then replace the tile. Repeat the process until you have drawn 25 times from that bag. Then, select another bag and repeat the process. Continue until you have drawn 25 times from each of the mystery bags in your set. Draw # 1 1 2 2 3 3 4 4 1 X X X X 2 X X X X 3 X X X X 4 X X X X 5 X X X X 6 X X X X 7 X X X X 8 X X X X 9 X X X X 10 X X X X 11 X X X X 12 X X X X 13 X X X X 14 X X X X 15 X X X X 16 X X X X 17 X X X X 18 X X X X 19 X X X X 20 X X X X 21 X X X X 22 X X X X 23 X X X X 24 X X X X 25 X X X X Totals 25 0 20 5 11 14 4 21 Handout/Transparency 1b Mystery Bags Page 8
Mystery Bags Group Recording Sheet Record the number of blue tiles and the number of red tiles in each bag for each person in your group. Bag 1 Bag 2 Bag 3 Bag 4 STUDENT NAME TOTAL Handout/Transparency 2 Mystery Bags Page 9
I m Seeing Red! After recording your group s data, identify which label on the likelihood line best describes the probably of drawing a red tile compared to the probability of drawing a blue tile for each of the four Mystery Bags. You may have some labels that are not used at all. In addition, any single label might describe more than one bag. Red is Impossible Red is Less Likely than Blue Red is Equally Likely as Blue Red is More Likely than Blue Red is Certain Justifications for our Answers: We placed 1 where we did because. We placed 2 where we did because. We placed 3 where we did because. We placed 4 where we did because. Handout/Transparency 3 Mystery Bags Page 10
I m So Blue! After recording your group s data, identify which label on the likelihood line best describes the probably of drawing a blue tile compared to the probability of drawing a red tile for each of the four Mystery Bags. You may have some labels that are not used at all. In addition, any single label might describe more than one bag. Blue is Impossible Blue is Less Likely than Red Blue is Equally Likely as Red Blue is More Likely than Red Blue is Certain Justifications for our Answers: We placed 1 where we did because. We placed 2 where we did because. We placed 3 where we did because. We placed 4 where we did because. Handout/Transparency 4 Mystery Bags Page 11
Our Predictions Based on your group s data, predict how many blue tiles and red tiles are contained in each of the four Mystery Bags. Predicted number of blue tiles Bag 1 Bag 2 Bag 3 Bag 4 Predicted number of red tiles Total number of tiles 10 10 10 10 Handout/Transparency 5 Mystery Bags Page 12
Our Predictions, Revisited After hearing each group in your class present its results, do you wish to revise your group s prediction about the number of blue tiles and the number of red tiles in each bag? If so, record your new predictions below. Predicted number of blue tiles Bag 1 Bag 2 Bag 3 Bag 4 Predicted number of red tiles Total number of tiles 10 10 10 10 Handout/Transparency 6 Mystery Bags Page 13
NAME: It s Your Turn to Spin For each spinner, color it with your choice of colors and then write a sentence explaining what description each spinner might represent (impossible, less likely than, equally likely as, more likely than, or certain) using the colors you selected. Handout/Transparency 7 Mystery Bags Page 14