### Teaching the Common Core Statistics Standards for Grades 6

```The New Illinois Learning Standards for Grades 6 - 8
Statistics and Probability
Dana Cartier
Illinois Center for School Improvement
Julia Brenson
Lyons Township High School
Tina Dunn
Lyons Township High School
The New Illinois Learning Standards
Agenda
 Resources Available Through ISBE
 Seventh Grade – Random Sampling for
Inference and Simulation for Probability
 Eighth Grade – Bivariate Data
The New Illinois Learning Standards
ILStats
http://ilstats.weebly.com/
 All materials from this session are available at this
website.
 This website is currently under construction, but
The New Illinois Learning Standards
Standard
Focus
PBA EOY
6.SP.A.1
X
No
6.SP.A.2
X
No
6.SP.A.3
X
No
6.SP.B.4
X
Yes
6.SP.B.5
X
Yes
Calculator
Statistics Standards for



th
6.SP.A.2 Understand that a set of data collected to
answer a statistical question has a distribution which
can be described by its center, spread, and overall
shape.
6.SP.A.3 Recognize that a measure of center for a
numerical data set summarizes all of its values with a
single number, while a measure of variation describes
how its values vary with a single number.
6.SP.B.4 Display numerical data in plots on a number
line, including dot plots, histograms, and box plots.
Statistics Standards for

th
6.SP.B.5 Summarize numerical data sets in relation to their
context, such as by:
 6.SP.B.5a Reporting the number of observations.
 6.SP.B.5b Describing the nature of the attribute under
investigation, including how it was measured and its units of
measurement.
 6.SP.B.5c Giving quantitative measures of center (median
and/or mean) and variability (interquartile range and/or
mean absolute deviation), as well as describing any overall
pattern and any striking deviations from the overall pattern
with reference to the context in which the data were
gathered.
 6.SP.B.5d Relating the choice of measures of center and
variability to the shape of the data distribution and the
context in which the data were gathered
Types of Graphs
Gaps
Categories
No Gaps
BIG IDEAS:
When describing distributions, we talk about
Shape, Center, and Spread in the context of
the data.
 Try to use real life data rather than made up
data sets whenever possible.

Shape of the Distribution
Dot Plot
Shapes of Distributions
0
2
4
6
8
10 12 14 16 18
Approxim ately_Sym m etrical
20
22
Approximately Symmetrical
Dot Plot
Shapes of Distributions
Dot Plot
Shapes of Distributions
unusually large value
0
2
4
6
8
10 12 14
Skew ed_Right
16
18
20
22
Skewed
0
2
4
6
8
10 12 14
Skew ed_Left
16
18
20
22
Statistics Standards for Algebra I/Math I
Shape of the Distribution
What would the shape be for the distribution of
salaries of the 2013 Chicago Cubs?
The distribution of salaries for
the 2013 Chicago Cubs is
less than \$2 million. There are
exceptionally large salaries.
million and Edwin Jackson
Measures of Center
Mean =

Median = the center most value when
observations in the data set are ordered
BIG IDEA:
The median is a better measure of center
when the data is skewed.
Statistics Standards for Algebra I/Math I
Measures of Central Tendency
What was a typical salary for a baseball player on
the 2013 Chicago Cub Team?
Median = \$1,550,000.00
Mean = \$3,485,024.20
What is the better measure of center for this data? Why?
6th Grade & Algebra I / Math I
Measures of Center
Demonstration:
Comparing the Mean and Median
NCTM Illuminations Mean and Median Applet
http://illuminations.nctm.org/Activity.aspx?id=3576
6th Grade & Algebra I / Math I
Measures of Center
Comparing the Mean and Median
Measures of Center
The Mean as Fair Share
Dave, Sandy, Javier, and Maria have 12 cookies. How
many cookies will each student have if each student
3
3
12
3
3
Measures of Center
The Mean as Fair Share
What would each student’s fair share be if there are:
?
?
?
?
Measures of Center
From the PARCC Grade 6 EOY Evidence Table
Evidence Statement Key 6.SP.3
Rate the following statement as
True/False/Not Enough Information.
“The average height of trees in Watson Park is
65 feet. Are there any trees in Watson Park
taller than 65 feet?”
Range = maximum value – minimum value
Interquartile Range = Quartile3 – Quartile1
Interquartile Range (iqr) is the spread of the
middle 50% of the data.
Mean Absolute Deviation (MAD) = sum of the distances of
each data value from the mean divided by the total
number of observations.
Big Idea:
The mean absolute deviation (MAD) is the average
distance (deviation) of data values from the mean.
The Mean as a Balance Point (An Introduction to MAD)
From Engage NY Grade 6 Module 6 Lesson 7
Sabina wants to know how long it takes students to get to school. She
asks two students how long it takes them to get to school. It takes one
student 1 minute and the other student 11 minutes. She thinks the mean is
the balance point. What do you think?
http://www.engageny.org/sites/default/files/resource/attachments/math-g6-m6-teacher-materials.pdf
Introducing Deviations
A deviation is the distance of a piece of data from the mean.
A value that is below the mean has a negative deviation. A
value above the mean has a positive deviation.
The deviation of 1 to the
mean is 1 – 6 = - 5
The deviation of 11 to the
mean is 11 – 6 = 5
Questions:
1) What is the deviation from the mean for each of the pennies?
2) What is the sum of these two deviations?
Introducing Deviations
Sabrina wants to know what happens if there are more than two data
points. Suppose there are three students. One student lives 2 minutes from
school, and another student lives 9 minutes from school. If the mean time
for all three students is 6 minutes, she wonders how long it takes the third
student to get to school. She tapes pennies at 2 and 9.
-4
+3
+1
Questions:
1) Where should the third penny be placed to balance the ruler?
2) How can we use deviations to check this answer?
Activity: School Night Sleep
How many hours of sleep do sixth graders get on a school
night?
Let’s make some predictions:
1)
Typically, how many hours of sleep do you think a sixth
2)
How much will the number of hours of sleep vary if we
3)
What do you predict will be the fewest hours?
4)
What do you predict will be the most hours?
classmates how many hours of sleep they usually get on school
nights. He then created a dot plot of their answers.
Questions:
1) Looking at the dot plot above, typically how much sleep did the
ten sixth graders get on a school night?
2) How much did the amount of sleep vary?
3) What is the shape of this distribution?
Let’s look at another method of measuring the spread of
the data.
The mean absolute deviation (MAD) is the average distance of
the data from the mean. We find MAD by doing these steps:
1)
Calculate the mean.
2)
Find the deviation for each data value.
3)
Take the absolute value of each deviation.
4)
Find the average of these absolute deviations (distances).
Sleep Hours
Absolute
Deviation
Student on School Night
Deviation
Hours - Mean
(Hours)
|Hours - Mean|
Rachel
Gerty
Steve
Juan
Michael
Josie
Philip
Sergio
Catherine
Grace
Total
10
9
11
9
8
6.5
10
8
8
8
87.5
10 – 8.75 = 1.25
9 – 8.75 = 0.25
2.25
0.25
-0.75
-2.25
1.25
-0.75
-0.75
-0.75
0.00
1.25
0.25
2.25
0.25
0.75
2.25
1.25
0.75
0.75
0.75
10.5
ℎ
Mean =
=
87.5
10
= 8.75 hours

=
10.5
10
= 1.05 hours
mean
Mean = 8.75 hours
The number of hours of sleep on a school night
for these ten sixth graders varies1.05 hours, on
average, from the mean of 8.75 hours.
hours of sleep they typically get on a school night. Their
hours of sleep are shown on the dot plot below.
Questions:
1) What is the mean number of
hours of sleep on a school night
2) What is the median?
3) How much variability is there
4) What is the value of MAD for
this data?
Activity
Activities:

Mean, Median, Mode, and Range

Candy Bar

How Long is 30 Seconds
Statistics Education Web (STEW)
(http://www.amstat.org/education/stew/pdfs/HowLongis30Seconds.pdf)

The Mean as a Balance Point
Engage NY Grade 6 Module 6
(http://www.engageny.org/sites/default/files/resource/attachments/math-g6-m6teacher-materials.pdf)

The New Illinois Learning Standards
PBA* EOY
Standard
Focus
7.SP.A.1
Supporting
X
Yes
7.SP.A.2
Supporting
X
Yes
7.SP.B.3
X
Yes
7.SP.B.4
X
Yes
7.SP.C.5
Supporting
X
Yes
7.SP.C.6
Supporting
X
Yes
7.SP.C.7
Supporting
X
Yes
7.SP.C.8
Supporting
X
Yes
Calculator
* See Evidence
Statement 7.D.3
Micro-models from
the PARCC
Evidence Table –
Statistics Standards for


th
7.SP.A.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.
7.SP.A.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. For
example, estimate the mean word length in a book by randomly
sampling words from the book; predict the winner of a school
election based on randomly sampled survey data. Gauge how far
off the estimate or prediction might be.
Random Sampling to Draw Inferences About a Population
Part I Judgment Sample
Part II Simple Random Sample  Sampling Distribution
Random Sampling to Draw Inferences About a Population
Judgment Sample
First ask students to take a quick look at the population of
268 words and select 5 words that they think form a
representative sample of the length of words found in the
Gettysburg Address. This is a judgment sample. Students
record the five words and the number of letters in each
word in the table provided. After calculating the mean of
the sample, each student records his mean on the class dot
plot on the chalkboard.
Random Sampling to Draw Inferences About a Population
How do we ensure that we select a sample that is
representative of the population? We choose a method
that eliminates the possibility that our own preferences,
favoritism or biases impact who (or what) is selected. We
want to give all individuals an equal chance to be chosen.
We do not want the method of picking the sample to
exclude certain individuals or favors others. One method
that helps us to avoid biases is to select a simple random
sample. If we want a sample to have n individuals, we use
a method that will ensure that every possible sample from
the population of size n has an equal chance of being
selected.
Random Sampling to Draw Inferences About a Population
Which of the following would produce a simple random
sample of size 6 from the population of all students in our
classroom?
A. Select the first 6 students that enter the classroom.
B. Put every student’s name in a hat, mix and draw 6
names.
C. The classroom has 6 tables with three students per
table. Randomly select two tables. The students at
these two tables are the sample.
D. The classroom has 6 tables of students. Randomly select
one student from each table.
Random Sampling to Draw Inferences About a Population
Simple Random Sample
Use a random number generator or a random digits
table to select a simple random sample of size 5 from
the population of 268 words.
Random Sampling to Draw Inferences About a Population
Random Digits Table
Suppose, for example that we wanted a sample of size 5. There are 268
words. First select a row to use in the table. Select three digits at a time,
letting 001 represent 1, 002 represents 2, and so on. Skip 000 and
numbers that are greater than 268. Skip repeats.
Our Sample:
32, 148,
238, 128,
104
Random Sampling to Draw Inferences About a Population
Random Number Generator
Random sample of 5 numbers
representing the 5 words to be
selected.
The random number generator above is shared with permission from Beth Chance and
Allan Rossman. This applet can be found at
http://www.rossmanchance.com/applets/RandomGen/GenRandom01.htm
Random Sampling to Draw Inferences About a Population
Sampling Words – Permission to share this applet was given by Beth
Chance and Allan Rossman.
Number of Letters for all
Population
Words in the Population
Mean
Last random
sample of size
5 that was
selected.
Sample
Mean
http://www.rossmanchance.com/applets/GettysburgSampleE/GettysburgSample.html
Random Sampling to Draw Inferences About a Population
100 random samples of size 5
mean = 4.46
Random Sampling to Draw Inferences About a Population
500 random samples of size 5
Sampling Distribution
mean = 4.313
Statistics Standards for

th
7
7.SP.C.8 Find probabilities of compound events using organized
lists, tables, tree diagrams, and simulation.



7.SP.C.8a Understand that, just as with simple events, the probability
of a compound event is the fraction of outcomes in the sample space
for which the compound event occurs.
7.SP.C.8b Represent sample spaces for compound events using
methods such as organized lists, tables and tree diagrams. For an
event described in everyday language (e.g., “rolling double sixes”),
identify the outcomes in the sample space which compose the event.
7.SP.C.8c Design and use a simulation to generate frequencies for
compound events. For example, use random digits as a simulation
tool to approximate the answer to the question: If 40% of donors
have type A blood, what is the probability that it will take at least 4
donors to find one with type A blood?
Chance Processes and Probability Models
Example: Tree Diagram
Michael and Gita would like to have three children. What is
the probability that all three children will be boys?
Second Child
First Child
0.5
0.5
B
B
Third Child Possible Outcomes
0.5
0.5
0.5
0.5
G
0.5
0.5
0.5
G
0.5
B
0.5
G
0.5
0.5
0.5
B
G
B
G
B
G
B
G
BBB (0.5)(0.5)(0.5) = 0.125
BBG
BGB
BGG
GBB
GBG
GGB
GGG
Chance Processes and Probability Models
Three Children Continued…
Another way to look at this problem is to create a list of all
possible outcomes (the sample space).
(B, B, B)
(B, B, G)
(B, G, B)
(B, G, G)
(G, B, B)
(G, B, G)
(G, G, B)
(G, G, G)
This is a uniform distribution in which every outcome has an
equal chance of occurring. There are 8 outcomes and each
outcome has a 1/8 chance of occurring.
We can now answer questions like:
1)What is the probability of the couple having 3 boys?
2)What is the probability of having one boy?
Chance Processes and Probability Models
Activity: Blood Type A
If 40% of donors have type A blood, what is the
probability that it will take at least 4 donors to find one
with type A blood?
Chance Processes and Probability Models
Activity: Blood Type A





Using a random digits table, let 1, 2, 3, 4 represent
having type A blood. 0,5,6,7,8,9 represent not having
type A blood.
Select a row.
Count how many digits it takes to reach a 1,2,3, or 4.
Record this count with a tally mark in a table.
Repeat many times to determine the long-run behavior.
Chance Processes and Probability Models
Row
14
15
16
17
18
12341
58842
89104
32161
43757
56847
12111
81316
07798
26081
41089
26072
21231
30021
63824
81678
36430
30263
11234
29902
84546
46319
92049
70043
11212
35106
52699
40588
88555
29892
34561
87744
12394
24581
90515
48430
12341
89832
59894
51397
64921
11287
Continue on to
simulate the long run
behavior or combine
results with classmates.
Chance Processes and Probability Models
Blood Type A - Part II Tree Diagram
Let A = the event that a donor has blood type A
Let O = the event that a donor has some other blood
type.
OOOA
(0.6)(0.6)(0.6)(0.4) = 0.0864
Activities:




This activity is adapted from the Sampling Words activity by Beth Chance and
Allan Rossman.
Beth Chance and Allan Rossman have given permission for their Sampling
Words applet to be shared with Illinois math teachers.
(http://www.rossmanchance.com/applets/GettysburgSampleE/GettysburgSam
ple.html )
Blood Type A
The New Illinois Learning Standards
PBA* EOY
Standard
Focus
8.SP.A.1
Supporting
X
No
8.SP.A.2
Supporting
X
No
8.SP.A.3
Supporting
X
Yes
8.SP.A.4
Supporting
X
Yes
Calculator
* See Evidence Statement
8.D.2 (content from
7.SP.B) and 8.D.3
(Micro-models) from the
PARCC Evidence Table –
Statistics Standards for



th
8.SP.A.1 Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of association between
two quantities. Describe patterns such as clustering, outliers,
positive or negative association, linear association, and nonlinear
association.
8.SP.A.2 Know that straight lines are widely used to model
relationships between two quantitative variables. For scatter plots
that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the
data points to the line.
8.SP.A.3 Use the equation of a linear model to solve problems in
the context of bivariate measurement data, interpreting the slope
and intercept. For example, in a linear model for a biology
experiment, interpret a slope of 1.5 cm/hr as meaning that an
additional hour of sunlight each day is associated with an
additional 1.5 cm in mature plant height.
Investigate Patterns of Association in Bivariate Data
What is bivariate data?
bi - means two
variate – means variable
Bivariate data is data about two variables. If the two
variables are numeric, we examine the relationship
between the two variables using a scatterplot. If the two
variables are categorical, we organize the data in a twoway frequency table and look for an association.
Investigate Patterns of Association in Bivariate Data
A look at sample activities for 8.SP.1-3
 Oil Changes and Engine Repair
Adapted from an NCTM Illuminations’ activity

Bike Weights and Jump Heights
NCTM Illuminations

Animal Brains
Illustrative Mathematics

US Airports, Assessment Variation
Illustrative Mathematics

Patterns in Scatter Plots – Lesson 7 Classwork
EngageNY

Determining the Equation of a Line Fit to Data – Lesson 9 Problem Set
EngageNY
Investigate Patterns of Association in Bivariate Data
Questions
In what order would you use these activities in a unit?
What questions did you particularly like?
What questions are different than what we might
Do you have suggestions for improvement of or additions
to these activities?
Statistics Standards for

th
8
8.SP.A.4 Understand that patterns of association can
also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a
two-way table. Construct and interpret a two-way table
summarizing data on two categorical variables
collected from the same subjects. Use relative
frequencies calculated for rows or columns to describe
possible association between the two variables. For
example, collect data from students in your class on
whether or not they have a curfew on school nights and
whether or not they have assigned chores at home. Is
there evidence that those who have a curfew also tend to
have chores?
Investigate Patterns of Association in Bivariate Data
8.SP.4 Summarize categorical data in two categories
Big Ideas:
 Review the difference between numeric data and
categorical data.
 Explain that frequency refers to the count of the data.
Relative frequency is a proportion.
 Analyze relative frequencies and assess possible
associations and trends in the data.
Investigate Patterns of Association in Bivariate Data
Association of two categorical variables
 There is an association between two categorical
variables if the row (or column) conditional relative
frequencies are different from row to row (or column
to column) in the table.
 The greater the difference between the conditional
relative frequencies, the stronger the association.
Investigate Patterns of Association in Bivariate Data
Activity: Music and Sports
(http://www.illustrativemathematics.org/illustrations/1098)
Is there an association between whether a student
plays a sport and whether he or she plays a
musical instrument?
To investigate this question, each student in your
class should answer the following two questions:
1. Do you play a sport? (yes or no)
2. Do you play a musical instrument? (yes or no)
Investigate Patterns of Association in Bivariate Data
Music and Sports
Summarize the class data in a two way frequency table.
Musical Instrument
No Musical Instruments
Total
Sport
6
No Sport
7
Total
13
8
14
3
10
11
24
Questions:
1. Of those students who play a sport, what proportion play a musical
instrument?
2. Of those students who do not play a sport, what proportion play a
musical instrument?
3. Based on the class data, do you think there is an association between
playing a sport and playing an instrument?
Activities






8-SP Oil Changes and Engine Repair
Bike Weights and Jump Heights
NCTM Illuminations
8.SP Animal Brains
(http://www.illustrativemathematics.org/illustrations/1520)
8.SP US Airports, Assessment Variation
(http://www.illustrativemathematics.org/illustrations/1370)
Patterns in Scatter Plots – Lesson 7 Classwork
(www.engageny.org)
Monopoly - Determining the Equation of a Line Fit to Data – Lesson 9
Problem Set
(www.engageny.org)
More Activities

8-SP-4 Music and Sports
(http://www.illustrativemathematics.org/illustrations/1370)
Statistics Standards for Grades 6 - 8
Census at School
(http://www.amstat.org/censusatschool/)
Statistics Education Web
(http://www.amstat.org/education/stew/)
Acknowledgements and Resources
Chance, B. & Rossman, A. (Preliminary Edition). Investigating Statistical Concepts, Application
and Methods. Duxbury Press.
http://www.rossmanchance.com/.
http://chicagosports.sportsdirectinc.com/football/nflteams.aspx?page=/data/nfl/teams/rosters/roster16.html
Franklin, C., Kader, G., Mewborn, J. M., Peck, R., Perry, M. & Schaeffer, R. (2007)
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K12 Curriculum Framework. Alexandria, VA: American Statistical Association.
McCallum, B., et al. (2011, December 26). Progressions for the Common Core State
http://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_b
is.pdf.
McCallum, B., et al. (2012, April 21). Progressions for the Common Core State Standards in
Acknowledgements and Resources
Moore, D. & McCabe, P. (1989). Introduction to the Practice of Statistics. New York, NY: W.
H. Freeman.
Rossman, A. (2012). Interview With Roxy Peck. Journal of Statistics Education, 20(2). pp. 1 –
Rossman, A., Chance, B., & Von Oehsen, J. (2002). Workshop Statistics Discovery With Data
and the Graphing Calculator. New York: Key College Publishing.
Scheaffer, R., Gnanadesikan, M., Watkins, A., & Witmer, J. (1996). Activity-Based Statistics.
New York: Springer-Verlag.
Online Resources
Census at School. http://www.amstat.org/censusatschool/
http://causeweb.org/
Engage NY. http://www.engageny.org/mathematics
Illustrative Mathematics. http://www.illustrativemathematics.org/
Inside Mathematics. http://www.insidemathematics.org
Mathematics Assessment Project. http://map.mathshell.org/
Math Vision Project. http://www.mathematicsvisionproject.org/
NCSSM Statistics Institutes.
NCTM Core Math Tools – Data Sets
http://www.nctm.org/resources/content.aspx?id=32705
Online Resources
PARCC Model Content Frameworks.
http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovemb
er2012V3_FINAL.pdf
PARCC Mathematics Evidence Tables. https://www.parcconline.org/assessmentblueprints-test-specs
Smarter Balanced Assessment Consortium. http://www.smarterbalanced.org/
Statistics Education Web (STEW). http://www.amstat.org/education/STEW/
The Data and Story Library (DASL). http://lib.stat.cmu.edu/DASL/
The High School Flip Book Common Core State Standards for Mathematics.
http://www.azed.gov/azcommoncore/files/2012/11/high-school-ccss-flipbook-usd-259-2012.pdf
The New Illinois Learning Standards for Grades 6 - 8
Statistics and Probability
Thank you for joining us!
Dana Cartier
Julia Brenson
Tina Dunn
[email protected]
[email protected]
[email protected]
```