ALGEBRA 2
USING DATA TO MAKE INFERENCES
AND JUSTIFY CONCLUSIONS
Summer 2015
College and Career-Readiness Conference
Introductions


Mike Parker – Algebra & Stats Teacher at Patterson
Mill High School (Harford County)
Brett Parker – Algebra & Geometry Teacher at C.
Milton Wright High School (Harford County)
TODAY’S OUTCOMES
Participants will:
1. Briefly review the instructional shift, COHERENCE,
as related to making inferences and justifying
conclusions.
2. Look at the PARCC Model Content Framework and
Evidence Statements for the high school statistics
and probability standards.
3. Take an in-depth look at the S-ID.4 standard and the
S-IC standards taught in Algebra 2.
4. Share best practices and identify muddy points.
OUTCOME 1
Participants will:
1. Review the instructional shift of
COHERENCE.
A purposeful placement of standards to create
logical sequences of content topics that bridge
across the grades and courses, as well as across
standards within each grade/course.
In what grade/subject do students:
Use relative frequencies calculated for rows or
columns to describe possible association between
the two variables.
Relate the choice of measures of center and
variability to the shape of the data distribution
Use measures of center and measures of
variability for numerical data from random
samples to draw informal comparative inferences
about two populations.
In what grade/subject do students:
Use relative frequencies calculated for rows or
columns to describe possible association between the
two variables. – Grade 8/Algebra 1
Relate the choice of measures of center and
variability to the shape of the data distribution.
– Grade 6
Use measures of center and measures of variability
for numerical data from random samples to draw
informal comparative inferences about two
populations. – Grade 7
OUTCOME 2
Participants will:
2.
Look at the PARCC Model Content
Framework and Evidence
Statements for the high school
statistics and probability
standards.
PARCC Model Content Framework
Algebra 2
PARCC Evidence Statements
Algebra 2
 Refer to the Algebra 2 PARCC Evidence Statements.
 Find the following Evidence Statement Keys and
read at the “Clarifications”:
 S-ID.4
 S-IC.2
 S-IC.3-1

What do notice about the “Clarifications” provided
by the PARRC Evidence Statement?
OUTCOME 3
Participants will:
3.
Take an in-depth look at the S-ID.4:
Use the mean and standard deviation
of a data set to fit it to a normal
distribution and to estimate
population percentages.
Essential Skills and Knowledge



Ability to construct, interpret and use normal curves,
based on standard deviation
Ability to identify data sets as approximately
normal or not.
Ability to estimate and interpret area under curves
using the Empirical Rule.
Density Curves and
The Normal Distribution
Examples:
Heights of People
Blood Pressure
Test Scores
IQ Scores
*originlab.com
Density Curves and
The Normal Distribution
Always remains on or above the horizontal axis
 The total area under the curve is 1.
 An area under a density curve gives the
proportion of observations that fall in a range
of values.
 It is a description of the overall pattern of the
distribution.

Standard Deviation



The mean is the balance point of the curve. (As if it
were a solid object.)
The median divides the curve into two equal areas.
It is difficult to locate the standard deviation by
eye.
http://condor.depaul.edu
Normal Distributions



Normal curves are bell-shaped symmetric density
curves.
Has all the properties of density curves.
There are 3 rules for working with Normal
distributions:
1.
2.
3.
Make a picture,
Make a picture,
Make a picture!
68-95-99.7 Rule (Empirical Rule)



68% of the observations fall within 1 standard
deviation of the mean (mean ± 1s)
95% of the observations fall within 2 standard
deviations of the mean. (mean ± 2s)
99.7% of the observations fall within 3 standard
deviations of the mean. (mean ± 3s)
Breaking Down the Empirical Rule
Guiding Questions:
 Where is the mean located?
 Where would you locate the standard deviations from
the mean?
 What area of the curve represents 68% of the data?
 What happens if we only want +1 standard deviation
from the mean?
 How can we determine the area between +1 and +2
standard deviations?
 Is there data more than three standard deviations from
the mean?
Empirical Rule: Breaking it down
Label the area/percentages (68-95-99.7) between 1, 2, and 3
standard deviations on either side of the mean on the normal curve at
the top of your paper using what you have learned from the Empirical
Rule.
Solution
http://sites.stat.psu.edu/~ajw13/stat500_su_res/notes/lesson02/lesson02_03.html
Solution
Released PARCC PBA - #17

The heights of the male students at a college are
approximately normally distributed. Within this
curve, 95% of the heights, centered about the
mean, are between 62 and 78 inches. The
standard deviation is 4 inches. Use this information
to estimate the mean height of the males.
 Approximate
the probability that a male student is
taller than 74 inches. Explain how you determined your
answers.
PARCC – EOY #16 Part A
PARCC – EOY #16 Part B
S.IC - Making Inferences & Justifying
Conclusions in Algebra 2






IC.A.1- Making inferences about population parameters
based on random sample.
IC.A.2- Decide if model is consistent with results of
data-generating process: using simulation.
IC.B.3- Purposes of and differences among sample
surveys, experiments, and observational studies.
IC.B.4- Use data from a survey to estimate population
mean or proportion.
IC.B.5- Use data from randomized experiment to
compare two treatments; use simulations to decide if
differences between parameters are significant.
IC.B.6- Evaluate reports based on data.
OUTCOME 3
Participants will:
3. Take an in-depth look at the SIC.A.2: Decide if a specified model
is consistent with results from a
given data-generating process,
e.g., using simulation.
Essential Skills and Knowledge



Ability to calculate and analyze theoretical and
experimental probabilities accurately.
Ability to design, conduct, and interpret the results
of simulations.
Ability to explain and use the Law of Large
Numbers. (The average of a large number of trials
should approach the expected value.)
PARCC Evidence Statements
Algebra 2 - EOY
S-IC.2
Decide if a specified model is consistent with results
from a given data-generating process, e.g., using
simulation. For example, a model says a spinning coin
falls heads up with probability 0.5. Would a result of
5 tails in a row cause you to question the model?
MP.2, MP.4 i.) Tasks might ask the students to look at the results
of a simulation and decide how plausible the
observed value is with respect to the simulation. For
an example, see question 7 on the calculator section
of the online practice test
(http://practice.parcc.testnav.com/#).
Simulation
A simulation imitates a real situation.
 Is supposed to give similar results.
 Acts as a predictor of what should actually
happen.
 It is a model in which repeated experiments
are carried out for the purpose of estimating in
real life.

Monty Hall

http://stayorswitch.com/
Monte Hall Problem

http://www.shodor.org/interactivate/activities/Sim
pleMontyHall/
HSA


3.1.3: The student will calculate theoretical probability or use
simulations or statistical inferences from data to estimate the
probability of an event.
Sample problem:
In a simulation designed to represent families with two children, two coins
are tossed to model the gender of each child. The results of 50 trials are
shown in the table below.
Based on the results in the table, what is the
probability that a family with two children have
at least one boy?
A. 0.30
B. 0.44
C. 0.58
D. 0.74
PARCC- EOY
PARCC- EOY
OUTCOME 3
Participants will:
4. Take an in-depth look at the SIC.B.3: Recognize the purposes of
and differences among sample
surveys, experiments, and
observational studies; explain how
randomization relates to each.
Essential Skills and Knowledge



Ability to construct sample surveys, experiments,
and observational studies.
Understanding of the limitations of observational
studies.
Ability to recognize and avoid bias
PARCC Evidence Statements
Algebra 2 - EOY
S-IC.3-1
Recognize the purposes of and differences among
sample surveys, experiments, and observational studies.
MP.4
i.) The "explain" part of standard S-IC.3 is not assessed here;
ii.) Purposes and distinctions are as follows:
a. Survey: To estimate or make a decision about a
characteristic of a population based on random sample.
b. Experiment: To estimate or compare the effects of different
treatments based on randomized assignment of treatments to
units for the purpose of establishing a cause and effect
relationship.
c. .Observational study: To suggest patterns and/or
associations among variables where treatments or conditions
are inherent and not assigned to units.
Surveys, Observational Studies, &
Experiments
Activity

Sort the Research Questions into three categories of
which would be the best method of data collection:
 Sample
Survey
 Experiment
 Observational Study
Sample
Survey
Experiment
Observational
Study
Solutions
Sample Survey
Experiment
Observational Study
Who is going to win next
election?
Do students learn better in
online courses?
Is there an association
between eating processed
foods and life expectancy?
Is Justin Bieber more or less Do running shoes versus
popular than 2 years ago? other tennis shoes really
help performance?
Do a majority of
Marylanders support gay
marriage?
Are cruises more fun than
hotel vacations?
Is smoking related to heart
disease?
Does eating chocolate help Is binge drinking
you do better on a test?
associated with
depression?
PARCC – EOY
(#33 Paper, #25 Computer)
PARCC - EOY
Best Practices
What have you done that works?
Additional Resources



Illustrative Mathematics
PARCC Practice Test
American Statistical Association
What are the muddiest points?
Record any question
you still have after
today’s presentation
on your post-it note.
Please provide your
name and email
address.
Stick your post-it on the door as you leave
today, and we will respond. Thank you!