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Slide 1
Welcome to the Seventh Grade Summer Academy! (Day 2) MARY
GARNER AND/OR SARAH LEDFORD
Slide 2
Schedule Day 2 8:30 11:30 Inference 11:30 1:00 Lunch 1:00 4:00
Probability
Slide 3
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.
Slide 4
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
Slide 5
Slide 6
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
Slide 7
Inference From the English classroom: Source:
http://www.readingrockets.org/strategies/inference
Slide 8
Inference From the English classroom: Source:
http://www.readingrockets.org/strategies/inference
Slide 9
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.
Slide 10
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).
Slide 11
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.
Slide 12
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
Slide 13
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
Slide 14
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.
Slide 15
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.
Slide 16
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
Slide 17
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.
Slide 18
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?
Slide 19
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?
Slide 20
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
Slide 21
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
Slide 22
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.
Slide 23
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
Slide 24
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
Slide 25
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!)
Slide 26
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).
Slide 27
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?
Slide 28
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.
Slide 29
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.
Slide 30
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
Slide 31
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.
Slide 32
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?
Slide 33
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?
Slide 34
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?
Slide 35
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?
Slide 36
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?
Slide 37
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?
Slide 38
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?
Focus Day 2 Afternoon Session MCC7.SP.8 Find probabilities of
compound events using organized lists, tables, tree diagrams, and
simulation.
Slide 45
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
Slide 46
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
Slide 47
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.
Slide 48
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
Slide 49
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.
Slide 50
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
Slide 51
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
Slide 52
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
Slide 53
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?
Slide 54
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
Slide 55
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.
Slide 56
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.
Slide 57
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
Slide 58
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.
Slide 59
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?
Slide 60
Task: Red, Green or Blue? How could you alter the game to make
sure each pair are equally likely to win against each other?
Slide 61
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
Slide 62
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.
Slide 63
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.
Slide 64
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?
Slide 65
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.
Slide 66
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
Slide 67
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