Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Author : Mina Sifain Date : March 7, 2006 Title of Lesson : Normal Distribution Grade Level(s) : Grade 11 Content Area : Algebra II Unit Topic : Patterns in Variation Backward Design Overview What should students know, understand, and be able to do by the end of this lesson? The student will be able to form a conjecture based on given examples of sets of data. The student will be able to calculate the mean using a formula and approximate it from the Normal curve. The student will understand the concepts relating the Normal Distribution curve to mean and standard deviation. The student will be able to explain if a given set of data follows a Normal curve. The students will show mastery of the content through working in groups and explaining what their conjectures are from the performed experiment. What will you accept as evidence? After the students have worked individually on an experiment and they have worked on a different experiment in groups of 4, they will be able to explain in a journal that as the number of trials in an experiment increase, the data follows a Normal Distribution curve. Essential Question: Given a set of data, explain how to identify if the set of data follows a Normal curve? Standards New York State Learning Standards and Key Ideas Mathematics, Science, and Technology- Standard 3 Students will: • understand the concepts of and become proficient with the skills of mathematics; • communicate and reason mathematically; • become problem solvers by using appropriate tools and strategies; through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability. Problem Solving Strand: Students will build new mathematical knowledge through problem solving. A2.PS.3 Observe and explain patterns to formulate generalizations and conjectures Statistics and Probability Strand Students will collect, organize, display, and analyze data. A2.S.5 Know and apply the characteristics of the normal distribution Objectives Student-Centered Instructional Objectives and Bloom's Taxonomy • • • Explain the meaning of "standard deviation" and "mean" (Comprehension). Given a table of data, the student will be able to graph a histogram and a normal distribution curve (Application). Explain the characteristics of a normal distribution curve, such as symmetry and where the mean and standard deviation lie (Comprehension). Materials The materials that the teacher will need are as follow: • • 10 Overhead markers: 4 for me (Green, Black, Red, Blue), and 6 for each group (2 Green, 2 Red, Blue, Black) 24 copies of each activity sheet (1, 2, 3, and dice experiment) for each student • • • • 6 copies of a grid on overhead paper, one for each group of the students. 2 dice (a pair per group) 24 Dice experiment sheets (one per person) Homework worksheet for each student and me. Student Adaptations Adaptations One of the students in the class is blind, so one handout will be written in braille, as well as there will be a note-taker. Lesson The Anticipatory Set "Before we start with today's lesson, I'd like to ask you a quick question to think about as we discuss today's lesson. In the United States, what do you think are the average heights of a men compared to the average heights of women? (wait a moment for an answer) Well, how does that compare with the average height of just men compared to other men, or just women compared to other women? Now, let's expand a little bit. What do you think about other countries (such as Japan, France, Australia, Libya)? What's the average heights of men compared to women? (wait a moment for an answer) How does that compare with the average height of just men compared to other men, or just women compared to other women, in other countries? I just want you to think about these few questions as we go through the lesson." The Flow of Teaching and Learning Activities Sorting and Categorizing As the students enter the room, I will direct their attention to the overhead, where they will see that they have been put into 6 groups (4 students per group) that have already assigned, and ask them to get into their groups. I will hand out to each group a different activity sheet and ask them to complete it. After each group has completed the activity sheet, each student in the class will conclude/conjecture what they can see/observe from their activity sheet. Reflecting and Explaining After each student has written something, I will ask each student to discuss with his/her group their observations and to come up with one-group-conjecture. Then, I will ask for a volunteer from each group to come to the overhead to: show the group's activity sheet (the data and the histogram) to the class and to reflect on what that group has observed/conjectured on an overhead which they created when they were working with their groups. We will then have a class discussion about each activity sheet, where I will raise some questions that would lead them to further articulate their reasoning. Some questions will be: (1) How can you conclude if the graph is normally distributed? (2) If I didn't give you too much data, would you still get the same results? (3) What's better, if the data were closer to the mean or further away? (4) Explain the meaning of when your data is further away from the mean? Generalizing and Articulating With the class a whole, we will discuss that each of the data in the activity sheets fits a normal distribution curve, which will lead into the definition of the Normal Distribution Curve, its characteristics, and how it relates to the mean and standard deviation. Verifying and Refining As a class, perform one experiment, where the class takes 25 trials of the sum of two dice. I'll call up one person, who will be the dice-thrower. After the 25 trials have been performed, have each student record the data and graph the histogram on the Dice Experiment Worksheet. Then, as a class decide whether the experiment is normally distributed or not; it will be distributed, since all the data should be closer to the mean. To finish the lesson, I would make sure to tell to the whole class that any experiment could be normally distributed if there are enough trials. Conclusion On the board, I will sketch two histograms, one that is normally distributed, and one that is not, and classify them as A and B "As I draw these histograms on the board, raise your right hand if you think that histogram A is normally distributed or your left hand if you think that histogram B is normally distributed. What is the mean of the normally distributed curve?" Homework Assignment Go around in your neighborhood and take a survey of what are people's favorite beverages. Tally up your data, draw a histogram, and draw the normal distribution curve. Does that data look normally distributed? If it does, what is the mean? You must ask thirty people in your neighborhood. Name:___________________________ Algebra II Date:__________ Period:_________ Activity Sheet #1 Directions: The data below give the weights, to the nearest hundredth of a gram, of a sample of 100 new nickels. Graph the data below in a histogram and then answer the question that follows. Nickel Weight Frequency 4.87 2 4.88 1 4.89 1 4.90 2 4.91 2 4.92 4 4.93 4 4.94 4 4.95 5 4.96 5 4.97 6 4.98 6 4.99 6 5.00 7 5.01 8 5.02 7 5.03 7 5.04 6 5.05 3 5.06 4 5.07 3 5.08 2 5.09 2 5.10 1 5.11 2 What are your observations about this graph? Name:___________________________ Algebra II Date:__________ Period:_________ Activity Sheet #2 Directions: The table below gives the heights of women in a statistics class at Nazareth College. Graph the data below in a histogram and answer the question that follows. Female Heights Frequency 59 2 60 5 61 7 62 10 63 16 64 22 65 20 66 15 67 9 68 6 69 6 70 3 71 1 72 1 What are your observations about this graph? Name:___________________________ Algebra II Date:__________ Period:_________ Activity Sheet #3 Directions: For a chemistry experiment, students measured the time for a solute to dissolve. The experiment was conducted 50 times. Graph the data below in a histogram and answer the question that follows. Dissolution Time (s) Frequency 4 1 5 1 6 1 7 1 8 4 9 3 10 6 11 5 12 9 13 5 14 5 15 2 16 2 17 3 19 2 What are your observations about this graph? Name:____________________________ Algebra II Date:_________ Period:________ Dice Experiment Directions: As a whole class, conduct an experiment to check to see if the sum of rolling 2 dice 25 times is normally distributed. Record your data in the table below. Then, draw a histogram and the Normal Distribution curve, and approximate the mean using the curve. Finally, calculate the mean using the well-known formula and compare with the value received from the curve. Trial Number Sum of Dice 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 Tally Approximated Mean from Curve:___________ Calculated Mean using the formula:_____________ What’s your final conclusion from the activities seen today? Name:____________________________ Algebra II- HW Date:_________ Period:________ Directions: Go around in your neighborhood and take a survey of what are people's favorite beverages. Tally up your data, draw a histogram, and draw the normal distribution curve. Does that data look normally distributed? If it does, what is the mean? You must ask thirty people in your neighborhood. Trial Number Choice of Beverage 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 26 27 28 29 30 Tally What is the mean?