Transcript Welcome to the Seventh Grade Summer Academy!

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.
http://www.youtube.com/watch?v=gIuGqIss-N8
Use your knowledge
And the clues
To make an inference
That’s what good readers do
I know this may sound
A little crazy
Read between the lines
Don’t be lazy
Use your thinking
for comprehension
Comprehension
Comprehension
Make an inference
Use what you know
Use what you know
Use what you know
http://www.youtube.com/watch?v=GRFx5xo6MFM
Inference
From the English classroom:
Source: http://www.readingrockets.org/strategies/inference
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 2nd Amendment:
the Right to Bear Arms”? 98% of the people surveyed said “Yes”,
and 2% said “No”.
From: CCGPS Frameworks Mathematics 7th Grade Unit 4: Inferences
Task: Is It Valid?
Determine if the sample taken is representative of the population
without bias shown.
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.
From: CCGPS Frameworks Mathematics 7th 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 Plane
Bullet Holes Per Square Foot
Engine
1.11
Fuselage
1.73
Fuel system
1.55
Rest of the plane
1.8
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 we’re
interested in scores on their last math test. We know the population mean
score. But what if we didn’t know that score? How close could we get by
taking a sample from the population?
Task: How Close Can You Get?
Let’s 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):
10
1
70
11
85
23
85
2
75
12
95
24
75
3
75
13
75
25
70
7
4
99
14
75
95
6
99
82
26
5
15
16
75
27
95
6
95
17
85
28
90
7
90
18
70
29
90
8
90
19
90
30
95
3
9
99
20
77
31
80
2
10
90
21
85
32
95
22
75
9
8
5
4
1
0
70-74
75-79
80-84
85-89
90-94
95-99
Population mean: 85.03125
Task: How Close Can You Get?
Here are the scores for the class (the population):
70
11
85
23
85
2
75
12
95
24
75
3
75
13
75
25
70
4
99
14
75
26
95
15
82
27
95
16
75
17
85
28
90
18
70
29
90
19
90
30
95
20
77
31
80
21
85
32
95
22
75
1
5
6
99
95
7
90
8
90
9
99
10
90
Population mean: 85.03125
Let’s 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.
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.
10
10
8
9
6
8
4
7
2
6
0
70-74
75-79
80-84
85-89
90-94
95-99
5
4
10
3
8
2
6
1
4
0
70-74
75-79
80-84
85-89
90-94
95-99
2
0
70-74
75-79
80-84
85-89
90-94
95-99
Task: How Close Can You Get?
1
2
3
4
5
6
7
8
9
10
70
75
75
99
99
95
90
90
99
90
11
12
13
14
15
16
17
18
19
20
21
22
85
95
75
75
82
75
85
70
90
77
85
75
23
24
25
26
27
28
29
30
31
32
85
75
70
95
95
90
90
95
80
95
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.
Task: How Close Can You Get?
1
2
70
75
11
85
12
95
13
75
14
75
15
82
23
85
24
75
25
70
26
95
27
95
28
90
3
75
4
99
5
99
16
75
6
95
17
85
7
90
18
70
29
90
8
90
19
90
30
95
9
99
20
77
31
80
10
90
21
85
32
95
22
75
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, we’ve 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
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, we’ve 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.
We’ve 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 (they’ll 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 don’t 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?
What 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
What kind of distribution is this? Is it the population
distribution? Is it a sample distribution? Is it the
sampling distribution of a statistic?
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
Task: Valentine Marbles
If you had the information about the 100 samples, what
do you think the actual percentage of red marbles is?
Why?
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
Task: Valentine Marbles
The actual percentage IS 33.6%. How many samples had
more that 33.6% of red marbles? How many samples
had less?
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
Task: Valentine Marbles
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?
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
Task: Valentine Marbles
Do you think the samples were random? Why or why
not?
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
Task: Valentine Marbles
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?
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
Task: Valentine Marbles
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?
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
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/
Task: Counting Penguins
http://gpb.pbslearningmedia.org/resource/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?
I’ve 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 ¼ 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:
Color Pair
Blue
Wins
Red
Wins
Red/Blue
15
15
Blue/Green
15
Green/Red
Green
Wins
15
15
15
When you’re finished, add your totals to the table on the board.
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. 7882.
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/
Illustrative Mathematics Project
http://illustrativemathematics.org/
Mathematics Assessment Project (FALs)
Open-Ended Assessment in Math
http://map.mathshell.org
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!