welcome to the seventh grade summer academy! (day 2) mary garner and/or sarah ledford

Welcome to the Seventh Grade Summer Academy! (Day 2) MARY GARNER AND/OR SARAH LEDFORD

Schedule Day 2 8:30 11:30 Inference 11:30 1:00 Lunch 1:00 4:00 Probability

Focus Day 2 Morning Session MCC7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. MCC7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Inference What do seventh grade students know about inference? CCSS.ELA-Literacy.RL.6.1 Cite textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.CCSS.ELA-Literacy.RL.6.1 http://www.youtube.com/watch?v=gIuGqIss-N8

Use your knowledge And the clues To make an inference Thats what good readers do I know this may sound A little crazy Read between the lines Dont be lazy Use your thinking for comprehension Comprehension Make an inference Use what you know http://www.youtube.com/watch?v=GRFx5xo6MFM

Inference From the English classroom: Source: http://www.readingrockets.org/strategies/inference

Inference Statistical inference we make use of information from a sample to draw conclusions (inferences) about the population from which the sample was taken. What do we need to make an inference? Identify the population and the parameter of interest. Figure out how to gather the sample and how large a sample we want. We need a random sample. Why? What does random mean? Examine the results and consider how sure we are of our results. Use descriptive statistics (mean absolute deviation, frequencies, means, medians) for analysis.

Inference Statistical inference we make use of information from a sample to draw conclusions (inferences) about the population from which the sample was taken. Note what statistical inference is not: We survey the students in our class about how far they travel to school and then have the students calculate the mean, median, mode, and then display how the information is distributed. (This is important, but there are no inferences being made. This is a census, not a survey see the first task in Unit 4 of the seventh grade frameworks.) Deterministic meaning we get one answer and no judgment is required. Descriptive statistics and their properties (e.g. using the median rather than the mean).

Inference Note how statistical inference is similar to inference in the English classroom: We are bombarded by statistics in the news we must be intelligent consumers of statistics and understand the power and the limitations of statistical inference. Unlike other areas of mathematics that we teach, the correct answer often cannot be known.

Task: Is It Valid? Determine if the sample taken is representative of the population without bias shown. The National Rifle Association (NRA) took a poll on their website and asked the question, Do you agree with the 2 nd Amendment: the Right to Bear Arms? 98% of the people surveyed said Yes, and 2% said No. From: CCGPS Frameworks Mathematics 7 th Grade Unit 4: Inferen ces

Task: Is It Valid? The City of Smallville wants to know how its citizens feel about a new industrial park in town. Surveyors stand in the Smallville Mall from 8 am to 11 am on a Tuesday morning and ask people their opinion. 80% of the surveyed people said they disagreed with a new Industrial park. Determine if the sample taken is representative of the population without bias shown. From: CCGPS Frameworks Mathematics 7 th Grade Unit 4: Inferences

Sampling Issues France's national railway operator placed a $20.5 billion order for 2,000 new trains, only to discover that the locomotives were too wide to fit hundreds of stations. France must now spend $68 million to narrow train platforms. Reported June 13, 2014 issue of The Week.

Sampling Issues The Statistical Research Group (SRG) was a classified group of statisticians and mathematicians assembled in Manhattan during WWII, to provide mathematical analysis of, for example, the optimal curve a fighter should trace out through the air in order to keep an enemy plane in its gun sights. The military came to SRG with data about bullet holes on aircraft that returned from engagements over Europe. They wanted to add some armor to the planes, but not so much that fuel costs would be prohibitive.

Sampling Issues Section of PlaneBullet Holes Per Square Foot Engine1.11 Fuselage1.73 Fuel system1.55 Rest of the plane1.8 Where do you think they should add the armor? From: How Not to be Wrong. The Power of Mathematical Thinking by Jordan Ellenberg

Inference From Developing Essential Understanding of Statistics Grades 6-8, published by the National Council of Teachers of Mathematics (NCTM): Four big ideas in grades 6 - 8: 1. Distributions describe variability in data. 2. Statistics can be used to compare two or more groups of data. 3. Bivariate distributions describe patterns or trends in the covariability in data on two variables. 4. Inferential statistics uses data in a sample selected from a population to describe features of the population.

Task: How Close Can You Get? One way to understanding statistical inference is to have students engage in activities that involve repeatedly taking random samples from a population, calculating a statistic, and then examining how the statistics differ across samples and how they differ from the value of the true parameter in the population. For example, suppose the population is a seventh grade class and were interested in scores on their last math test. We know the population mean score. But what if we didnt know that score? How close could we get by taking a sample from the population?

Task: How Close Can You Get? Lets say the true population mean is 85. If we took a sample, would the mean score be 85? If we took two samples, would each sample have a mean of 85? How close could we expect to be? Would we be within 2 points of the true mean? Within 4 points? How could we explore that question?

Task: How Close Can You Get? Here are the scores for the class (the whole population): Population mean: 85.03125 170 275 3 499 5 695 790 8 999 1090 1185 1295 1375 1475 1582 1675 1785 1870 1990 2077 2185 2275 2385 2475 2570 2695 2795 2890 2990 3095 3180 3295

Task: How Close Can You Get? Here are the scores for the class (the population): Population mean: 85.03125 Lets get a sample of 6 students. How? The calculator will generate random integers between 1 and 32. When I use the calculator, I get 31, 30, 5, 17, 13, 14 I take students 31, 30, 5, 17, 13, 14 and calculate the mean of 80, 95, 99, 85, 75, 75. I get 84.8 (to the nearest tenth). Do it again. I get: 26, 2, 12, 8, 28, 29 and calculate the mean of 95, 75, 95, 90, 90, 90. I get 89.2. 170 275 3 499 5 695 790 8 999 1090 1185 1295 1375 1475 1582 1675 1785 1870 1990 2077 2185 2275 2385 2475 2570 2695 2795 2890 2990 3095 3180 3295

Task: How Close Can You Get? So far, we can point to two types of distributions the population distribution and the distribution in several samples.

Task: How Close Can You Get? Please take two sets of 6 random numbers, find the associated student scores, and calculate the means. Put your mean scores on the board. Take the means and construct a distribution! Then answer the questions on the next slide. 170 275 3 499 5 695 790 8 999 1090 1185 1295 1375 1475 1582 1675 1785 1870 1990 2077 2185 2275 2385 2475 2570 2695 2795 2890 2990 3095 3180 3295

Task: How Close Can You Get? Please take two sets of 6 random numbers, find the associated student scores, and calculate the means. Put your mean scores on the board. Take the means and construct a distribution! Then answer the questions on the next slide. So, weve taken 2 samples and gotten a 90 and a 91 for an average. Here are sets of 6 random numbers (from the calculator): 1 30 4 18 28 12 16 13 8 21 20 10 32 9 4 2 24 30 26 25 15 30 20 17 1 14 10 32 27 26 11 15 26 28 9 3 20 7 32 24 10 29 23 15 16 7 4 1 8 31 30 2 4 29 21 5 22 1 20 31 23 32 21 10 3 6 9 1 30 27 24 2 32 7 13 4 8 21 8 4 24 3 20 14 29 12 4 15 32 9 16 4 13 21 12 8 18 21 30 22 26 7 24 1 13 12 8 14 18 11 7 20 29 32 6 8 26 15 11 23 4 15 10 27 32 2 23 31 24 5 13 12 19 11 8 9 29 3 7 17 11 23 31 28 170 275 3 499 5 695 790 8 999 1090 1185 1295 1375 1475 1582 1675 1785 1870 1990 2077 2185 2275 2385 2475 2570 2695 2795 2890 2990 3095 3180 3295

Task: How Close Can You Get? What do you notice about the distribution? What is its mean? How does your distribution differ from the population distribution? What percentage of means falls above the true mean? What percentage of means falls below the true mean? What percentage of the means falls within 2 points of the true mean? What percentage of the means falls within 4 points of the true mean? What if we took larger samples? How would you expect the distribution of means to change? (Note: This is tedious but may be worth doing!)

Task: How Close Can You Get? So, weve determined that if we repeatedly take random samples of 6 students, it looks like ___% will lie within 2 points of the true mean and ___% will lie within 4 points of the true mean. Weve examined three different distributions: the distribution of scores in the population, the distribution of scores in samples, and finally the distribution of the sample means. Note also that the shape of the distribution of means looks bell-shaped even though the original population is not bell-shaped. There is a theorem in statistics (theyll learn in high school) that says that such distributions of means will always be bell-shaped and the spread of the distributions is dependent on the size of the samples. Your students dont need to know this, but they do need to see that different samples produce different statistics and those statistics have a distribution (big idea #4).

Task: How Close Can You Get? What if we took samples that were not random? How would that change the distribution of means that we obtained? Sketch a possible distribution and compare it to the distribution of means that we obtained. What kind of sampling techniques would not be random?

Task: How Close Can You Get? W hat if we took samples that were not random? How would that change the distribution of means that we obtained? What kind of sampling techniques would not be random? Selecting the students as randomly as possible (what we think is random). Selecting every other student. Selecting every third student. Taking the first 5 students.

Wrap-Up: How Close Can You Get? MCC7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. MCC7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Wrap-Up: How Close Can You Get? What SMPs were addressed? What considerations might need to be made for students (scaffolding, differentiating, enhancing)? How can you make this task more relevant to your students? What changes do you need to make to the task in order to use it? 30

Task: Valentine Marbles http://www.illustrativemathematics.org/illustrations/1339 A hotel holds a Valentine's Day contest where guests are invited to estimate the percentage of red marbles in a huge clear jar containing both red marbles and white marbles. There are 11,000 total marbles in the jar. To help with the estimating, a guest is allowed to take a random sample of 16 marbles from the jar in order to come up with an estimate. (Note: when this occurs, the marbles are then returned to the jar after counting.) One of the hotel employees secretly recorded the results of the first 100 random samples. A table and dotplot of the results appears below.

Task: Valentine Marbles Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50%4 18.75%8 25.00%15 31.25%22 37.50%20 43.75%12 50.00%12 56.25%4 62.50%2 68.75%1 Total: 100 What kind of distribution is this? Is it the population distribution? Is it a sample distribution? Is it the sampling distribution of a statistic?

Task: Valentine Marbles Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50%4 18.75%8 25.00%15 31.25%22 37.50%20 43.75%12 50.00%12 56.25%4 62.50%2 68.75%1 Total: 100 If you had the information about the 100 samples, what do you think the actual percentage of red marbles is? Why?

Task: Valentine Marbles Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50%4 18.75%8 25.00%15 31.25%22 37.50%20 43.75%12 50.00%12 56.25%4 62.50%2 68.75%1 Total: 100 The actual percentage IS 33.6%. How many samples had more that 33.6% of red marbles? How many samples had less?

Task: Valentine Marbles Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50%4 18.75%8 25.00%15 31.25%22 37.50%20 43.75%12 50.00%12 56.25%4 62.50%2 68.75%1 Total: 100 What percentage of samples were within 3 points of the true percentage? What percentage of samples were within 6 points? What percentage of samples were within 9 points?

Task: Valentine Marbles Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50%4 18.75%8 25.00%15 31.25%22 37.50%20 43.75%12 50.00%12 56.25%4 62.50%2 68.75%1 Total: 100 Do you think the samples were random? Why or why not?

Task: Valentine Marbles Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50%4 18.75%8 25.00%15 31.25%22 37.50%20 43.75%12 50.00%12 56.25%4 62.50%2 68.75%1 Total: 100 Does it bother you that the true percentage is 33.6% but none of the samples had exactly 33.6% red marbles? Why or why not?

Task: Valentine Marbles Percentage of red marbles in the sample of size 16 Number of times the percentage was obtained 12.50%4 18.75%8 25.00%15 31.25%22 37.50%20 43.75%12 50.00%12 56.25%4 62.50%2 68.75%1 Total: 100 It turns out that the hotel owner wants to offer a prize to anyone who comes within nine percentage points of the true percentage will get a prize. Do you think this is a good idea? Why or why not?

Task: Counting Penguins http://gpb.pbslearningmedia.org/resourc e/mgbh.math.sp.penguincoat/estimation -from-random-sampling-worksheet/

Task: Counting Penguins http://gpb.pbslearningmedia.org/resource/mgbh.math.sp.penguincoat/estimation-from-random-sampling-worksheet/

LUNCH 11:30 1:00

Focus Day 2 Afternoon Session MCC7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Task: True or False? Ive spun an unbiased coin 3 times and got 3 heads. It is more likely to be tails than heads if I spin it again. Source: http://www.scs.sk.ca/cyber/elem/learningcommunity/sciences/science9/curr_content/science9/risks/word ocs/less1miscon.PDF

Task: True or False? I roll two dice and add the results. The probability of getting a total of 6 is 1/11 because there are 11 different possibilities and 6 is one of them. Source: http://www.scs.sk.ca/cyber/elem/learningcommunity/sciences/science9/curr_content/science9/risks/wordocs/les s1miscon.PDF

Task: True or False? A bag has 4 red marbles and 5 green marbles. The probability that I pull a red out of the bag, put it back in the bag, and then pull out a green marble from the bag is 4/9 + 5/9 = 1.

Task: True or False? Each spinner has two sections one black and one white. The probability of getting black is 50% for each spinner. Source:http://www.scs.sk.ca/cyber/elem/learningcommunity/sciences/science9/curr_content/science9/risks /wordocs/less1miscon.PDF

Task: True or False? John guesses at random on two multiple choice questions that each have 4 choices. The probability that John gets one or the other (not both) correct is because there is one right answer and 3 wrong answers.

Task: True or False? Tomorrow it will either rain or not rain. The probability that it will rain is.5. Source: http://map.mathshell.org/materials/download.php?fileid=701

Task: True or False? If you roll a six-sided number cube, and it lands on a six more than any other numbers, then the cube must be biased. Source: http://map.mathshell.org/materials/download.php?fileid=701

Task: True or False? In a true or false quiz with ten questions, you are certain to get five correct if you just guess. Source: http://map.mathshell.org/materials/download.php?fileid=701

Task: Monty Hall Problem You are a contestant in a game show in which a prize is hidden behind one of three curtains and goats are behind the other two curtains. You will win the prize if you select the correct curtain. After you have picked one curtain but before the curtain is lifted, the emcee lifts one of the other curtains, revealing a goat, and asks if you would like to switch from your current selection to the remaining curtain. Should you switch?

Task: Red, Green or Blue? This is a game for two people. You have three dice; one is red, one is green, and one is blue. These dice are different than regular six-sided dice, which show each of the numbers 1 to 6 exactly once. The red die, for example, has 3 dots on each of five sides, and 6 dots on the other. The number of dots on each side are shown in the picture below. http://www.illustrativemathematics.org/illustrations/1442

Task: Red, Green or Blue? To play the game, each person picks one of the three dice. However, they have to pick different colors. The two players both roll their dice. The highest number wins the round. The players roll their dice 30 times, keeping track of who wins each round. Whoever has won the greatest number of rounds after 30 rolls wins the game.

Task: Red, Green or Blue? What strategy do you think would win the game? Do you want to go first or second? If you went first which dice would you choose? If you had to go second, how would you chose the dice? Record your strategy.

Task: Red, Green or Blue? To play the game, each person picks one of the three dice. However, they have to pick different colors. The two players both roll their dice. The highest number wins the round. The players roll their dice 30 times, keeping track of who wins each round. Whoever has won the greatest number of rounds after 30 rolls wins the game. Please divide into groups of three or four. Assign one member of the group to be a recorder. Play three games (with each game consisting of 30 rolls), with different pairs of dice. Record your results as follows: When youre finished, add your totals to the table on the board. Color PairBlue Wins Red Wins Green Wins Red/Blue15 Blue/Green15 Green/Red15

Task: Red, Green or Blue? Analyze the results of the simulations. Who is more likely to win when a person with the red die plays against a person with the green die? What about green vs. blue? What about blue vs. red? Would you rather be the first person to pick a die or the second person? Explain.

Task: Red, Green or Blue? Find theoretical probabilities for who is more likely to win when a person with the red die plays against a person with the green die. What about green vs. blue? What about blue vs. red? How do the theoretical probabilities compare with the empirical probabilities?

Task: Red, Green or Blue? How could you alter the game to make sure each pair are equally likely to win against each other?

Task: Red, Green or Blue? What SMPs were addressed? What considerations might need to be made for students (scaffolding, differentiating, enhancing)? How can you make this task more relevant to your students? What changes do you need to make to the task in order to use it? 61

Wrap-Up Red, Green, or Blue? Other games: Hunger games: What are the Chances? By Bush and Karp in Mathematics Teaching in the Middle School Vol. 17 No. 7 March 2012 pp. 426-435 Probability Games from Diverse Cultures by McCoy, Buckner, and Munley in Mathematics Teaching in the Middle School Vol 12 No. 7 March 2007 Enriching Students Mathematical Intuitions with Probability Games and Tree Diagrams in Mathematics Teaching in the Middle School Vol 6 No 4 December 2000 pp. 214-220. Determining Probabilities by Examining Underlying Structure in Mathematics Teaching in the Middle School Vol 7 No 2 October 2001 pp. 78- 82.

Task: Waiting Time http://www.illustrativemathematics.org/illustrations/343 Suppose each box of a particular brand of cereal contains a pen as a prize. The pens come in four colors, blue, red, green and yellow. Each color of pen is equally likely to appear in any box of cereal. Design and carry out a simulation to help you answer the following questions.

Task: Waiting Time What is the probability of having to buy at least five boxes of cereal to get a blue pen? What is the mean (average) number of boxes you would have to buy to get a blue pen if you repeated the process many times?

Task: Waiting Time Complete your simulations in groups of 3 or 4. Use the materials at the front of the room. You must use a simulation technique different from all but one other group. Record you results on chart paper, showing your answer to the two questions, and a visualization of the simulation results. Be ready to present your technique and results.

Task: Waiting Time What SMPs were addressed? What considerations might need to be made for students (scaffolding, differentiating, enhancing)? How can you make this task more relevant to your students? What changes do you need to make to the task in order to use it? 66

Implementation Resources Google site (Metro RESA) https://sites.google.com/site/mathccgps/ https://sites.google.com/site/mathccgps/ Illustrative Mathematics Project http://illustrativemathematics.org/ http://illustrativemathematics.org/ Mathematics Assessment Project (FALs) http://map.mathshell.orghttp://map.mathshell.org Open-Ended Assessment in Math http://books.heinemann.com/math/ http://books.heinemann.com/math/ Developing Essential Understanding of Statistics in Grades 6-8 published in 2013 by National Council of Teachers of Mathematics

Thank You!

welcome to the seventh grade summer academy! (day 2) mary garner and/or sarah ledford

Documents

inference use

inference thats

random sample

probability slide

use of information

population generalizations

valid inferences

lazy use