College and Career-Readiness Conference Summer 2015 ALGEBRA 2 USING DATA TO MAKE INFERENCES AND JUSTIFY CONCLUSIONS

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 *originlab.com Examples: Heights of People Blood Pressure Test Scores IQ Scores

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.

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. Make a picture, 2. Make a picture, 3. Make a picture!

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 ( ) 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

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 S- IC.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 (

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 

Monte Hall Problem  pleMontyHall/ 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

OUTCOME 3 Participants will: 4.Take an in-depth look at the S- IC.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 Experiment Observational Study Sample Survey

Solutions Sample SurveyExperimentObservational 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 popular than 2 years ago? Do running shoes versus other tennis shoes really help performance? Is smoking related to heart disease? Do a majority of Marylanders support gay marriage? Does eating chocolate help you do better on a test? Is binge drinking associated with depression? Are cruises more fun than hotel vacations?

PARCC – EOY (#33 Paper, #25 Computer)

PARCC - EOY

What have you done that works? Best Practices

Additional Resources  Illustrative Mathematics Illustrative Mathematics  PARCC Practice Test PARCC Practice Test  American Statistical Association 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 address. Stick your post-it on the door as you leave today, and we will respond. Thank you!