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# Copy of Revised Version of CCGPS Advanced Algebra

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## jennifer greer

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#### Transcript of Copy of Revised Version of CCGPS Advanced Algebra

2 days...
12 hours...
Better Understanding

Unit 1 Statistics
Our Goals:
3) Build a Collaborative Network
2) Share Ideas for Resources
Professional Learning Objectives: Participants in this 12 hour professional learning session will investigate each of the six units in the third high school CCGPS math course, Advanced Algebra. During this course each unit will be summarized with key concepts being addressed through the use of video resources, GADOE resources, quality external resources, and strategically chosen learning tasks. Teachers will also examine clear “take aways” from the unit exploration culminating with participant ideas for instruction and allowing a venue for clearing up any remaining questions about the course or unit of focus.
Resources to consider: Georgia Virtual School Shared Resources are open access resources for teachers, students, and parents. These resources do not require joining or a log in and may be used in your classroom http://www.gavirtuallearning.org/Resources/CCGPSAdvancedAlgebra.aspx
Sample work and group discussion
of the Learning Task "Why Randomize"
Table Talk:
1. What are your key "take aways" from our discussion of Unit 1? (chart paper 1)

3. Do you know where YOUR state level WIKI is?
http://www.livebinders.com/play/play/643707
Unit 4 Exponentials and Logs
Sample work and group discussion
of the Learning Task will happen next
Table Talk:
1. What are your key "take aways" from our discussion of Unit 4? (chart paper 1)

3. Do you know where YOUR state level WIKI is?
In unit 4 students will:
• Review exponential functions and their graphs
• Explore exponential growth
• Develop the concept of a logarithm as an exponent along with the inverse relationship with exponents
• Define logarithms and natural logarithms
• Develop the change of base formula
• Develop the concept of logarithmic function
• Solving problems relating to exponential functions and logarithms
http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/log_exp/logandexponentialindex.html
Handout Dr. Evil Activity
Handout Investigating Exp Activity
http://learnzillion.com/lessons/3286-use-an-exponential-formula-to-solve-a-population-problem
Hand out "What is a Logarithm?"
http://www.livebinders.com/play/play/643763

• Construct appropriate graphical displays (dot plots, histogram, and box plot) to represent sets of data values.

• Describe a distribution using shape, center and spread and use the correct measure appropriate to the distribution

• Compare two or more different data sets using center and spread

• Recognize data that is described well by a normal distribution

• Estimate probabilities for normal distributions using area under the normal curve using

• Design a method to select a sample that represents a variable of interest from a population

• Design simulations of random sampling and explain the outcomes in context of population and know proportions or means

• Use sample means and proportions to estimate population values and calculate margins of error

• Read and explain in context data from real-world reports
By the conclusion of Unit 1, students should be able to demonstrate the following competencies:
Great Video and Unit Resource
http://www.learner.org/courses/againstallodds/index.html
Unit 6 Mathematical Modeling
In Unit 6 Modeling students will:
• Synthesize and generalize what they have learned about a variety of function families

• Explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying functions

• Identify appropriate types of functions to model a situation, adjust parameters to improve the model, compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit

• Determine whether it is best to model with multiple functions creating a piecewise function.
http://mrmeyer.com/threeacts/hotcoffee/
Three Acts Math
http://blog.mrmeyer.com/?p=10285
http://www.livebinders.com/play/play?present=true&id=643771
Bonus Rounds
Formative Item Bank
in OAS

AKA TRL

Mastery Connect
Livebinders
Engage New York
High School Flipbook
http://www.engageny.org/mathematics
http://katm.org/wp/common-core/
https://www.masteryconnect.com/
http://www.livebinders.com/shelf/search?search=&terms=Math&type=3
Math Assessment Project (MARS)
http://map.mathshell.org/materials/index.php
Professional Learning Objectives
Participants in this 12 hour profession learning session will:
1. Examine the Key Concepts for each unit in CCGPS Advanced Algebra

2. Address Challenging Standards through the use of video resources, GADOE resources, quality external resources, and strategically chosen learning tasks.

3. Discuss clear “take aways” from the unit exploration culminating with participant ideas for instruction.

4. Investigate a venue for clearing up any remaining questions about the course, a particular unit, or a specific standard.

Learnzillion video on this standard
http://learnzillion.com/lessonsets/458-use-mean-and-standard-deviation-to-fit-a-data-set-to-a-normal-distribution-when-appropriate-estimate-population-percentages-using-tools

Our sample mean is ________.

Our sample standard deviation is __________.

Our best prediction for the population mean is _________.

But, the standard error of our mean is:
(our standard deviation/square root of n)___________

How confident do you want to be???

Implications of the Central Limit Theorem

What do you know?

What do you need to know?

How confident are you of this answer?

What is the average number of years that a random Georgia Teacher has been teaching?

Although there is no one right choice of the classes in a histogram, too few classes will give a “skyscraper” graph, with all values in a few classes with tall bars and too many will produce a “pancake” graph, with most classes having one or no observations.

Neither choice will give a good picture of the shape of the distribution.

Five classes is a good minimum.

Be sure classes all have the same width. Then area is determined by height and all classes are fairly represented.

If you use a computer or graphing calculator, beware of letting the device choose the classes.

Histogram Tips

In other words, if the number is important – the mean and median have a meaning, then you need a histogram.
If the number merely denotes a category, you need a bar graph.

Histograms

Is the first result a histogram?
How do you know?
Is the second result a histogram?
How do you know?

Histograms

(Sports) Average height of a high school basketball player
(Food) Number of chips in a chocolate chip cookie
(Cars) Price for a gallon of gas
(Social) Average number of Facebook hits a day

What might our students like to know?

Multiply our z-score (which gives us the desired interval) by the standard error of our mean (which gives us how far off we might be) to get our margin of error.

Add and subtract that from our mean.

How high? How low?

Is the number of years of teaching experience of the participants in this room normal?

Why or why not?

Is the number of years of teaching experience of math teachers in Georgia normal?

Why or why not?

What type of sample is this?

If we took another such sample, would we get similar results?
Why or why not?

Is this Normal?

You are getting 2 sticky notes.
On one of them, write the unit with which you are most uncomfortable. On the other, write the number of years you have been teaching.

Who are you?

What percent of the scores will be below our upper bound?

What is the z-score associated with that percentage (percentile rank)?

This number is our z-score.

What is the average number of years that a random Georgia Teacher has been teaching?

Place your sticky note on the
appropriate location on the
chart paper
MCC9-12.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. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.★

Understand and evaluate random processes underlying statistical experiments

MCC9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.★

MCC9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

MCC9-12.S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.★

MCC9-12.S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.★

MCC9-12.S.IC.6 Evaluate reports based on data.★
Snapshot of the Progression Document on Probability and Statistics Standards for Common Core
Snap shot of Annenberg Unit 15
Free Annenberg Interactive Tools
Key Standards Addressed in "Who are You?"
Is the data quantitative or categorical?
The Progression Documents offer
insight on the learning path of the
statistics standards
http://www.illustrativemathematics.org/illustrations/191
Be prepared to report your group results
Unit 6
Mathematical Modeling
: In this unit students synthesize and generalize what they have learned about a variety of function families.
Unit 1
Inferences and Conclusions from Data
: Students apply methods from probability and statistics to draw inferences and conclusions from data.
Unit 2
Polynomial Functions
: This unit develops the structural similarities between the system of polynomials and the system of integers. Students will multiply, divide, identify zeros, make connections and culminate with the Fundamental Theorem of Algebra
Unit 5
Trigonometric Functions
: Building on their previous work with functions, and on their work with trigonometric ratios and circles in Analytic Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena.
Unit 3
: Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Similarly, radical expressions follow the rules governed by irrational numbers
Unit 4
Exponential and Logarithmic
Functions: Students extend their work with exponential functions to include solving exponential equations with logarithms. They analyze the relationship between these two functions
Locate the S-IC Why Randomize Task
Work in small groups to
*Be prepared to report
Locate the Sample Assessment from GAVS
What kind of Mindset do you have?
Can you relate?
How do we categorize our students?
Math Boot camp http://www.mathbootcamps.com/interpreting-confidence-intervals/
http://www.prenhall.com/esm/app/calc_v2/calculator/medialib/Technology/Documents/TI-83/desc_pages/confidence_interval.html
Calculator help https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=93641

http://www.occc.edu/math/statistics/StatPDF/invnmdist.pdf
TKES connections to SMP
complete document

Video from Unit 15 http://www.learner.org/courses/againstallodds/unitpages/unit15.html
Unit 5 Webinar - August 29, 2013

Link to Unit 5 Webinar recording:
https://sas.elluminate.com/mr.jnlp?suid=M.EAF68C60E4BAF92ED3C37FC62B6CF0&sid=2012003

Unit 5 Webinar PowerPoint and Links:

Aug29_800amAGBAAUnit5.pptxAug29_800amAGBAAUnit5.pptx

Inside Mathematics- http://www.insidemathematics.org/
Annenberg Learner - http://www.learner.org/index.html
Edutopia – http://www.edutopia.org
Teaching Channel - http://www.teachingchannel.org

http://bit.ly/17QDmw9
http://www.schooltube.com/video/81f35b2779ef8d4727fd/

Robert Kaplinsky
http://robertkaplinsky.com/
Unit 1 Big Ideas:
Summarize, represent, and interpret data
on a single count or measurement variable

Understand and evaluate random processes underlying statistical experiments

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Unit 5 AAG is Unit 1 AA
From www.georgiastandards.org
unit by unit webinars
6th Grade Statistics and Probability (unit 6)
• Develop understanding of statistical variability.
• Summarize and describe distributions

MCC6.SP.1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

MCC6.SP.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.

MCC7.SP.4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
MCC7.SP.3. Informally assess the degree of visual overlap of two numerical data distributions with similar variability, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
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. 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.
• Use random sampling to draw inferences about a population.
• Draw informal comparative inferences about two populations.
• Investigate chance processes and develop, use, and evaluate probability models.

Apply and extend previous understandings of measurement and interpreting data.
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.

MCC8.SP.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.
MCC8.SP.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.
MCC8.SP.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.
• Investigate patterns of association in bivariate data.

MCC8.SP.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.

Coordinate Algebra
Interpreting Categorical and Quantitative Data S.ID
Summarize, represent, and interpret data on a single count or measurement variable
MCC9-12.S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
• Choose appropriate graphs to be consistent with numerical data: dot plots, histograms, and boxplots.
MCC9-12.S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range) of two or more different data sets.
• Include review of Mean Absolute Deviation as a measure of variation.
MCC9-12.S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
• Students will examine graphical representations to determine if data are symmetric, skewed left, or skewed right and how the shape of the data affects descriptive statistics.
MCC9-12.S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data
MCC9-12.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

MCC9-12.S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

MCC9-12.S.ID.6b Informally assess the fit of a function by plotting and analyzing residuals.

MCC9-12.S.ID.6c Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models
MCC9-12.S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

MCC9-12.S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

MCC9-12.S.ID.9 Distinguish between correlation and causation.
CCGPS Analytic Geometry Statistics Standards

Students will
represent bivariate data on a scatter plot;
describe relationships for bivariate data;
understand conditional probability, including independence;
use the rules of probability to compute probabilities of compound events.

Interpreting Categorical and Quantitative Data S.ID
Summarize, represent, and interpret data on two categorical and quantitative variables

MCC9-12.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

MCC9-12.S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize quadratic models.
MCC9-12.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. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Summarize, represent, and interpret data on a single count or measurement variable

MCC9-12.S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
Understand and evaluate random processes underlying statistical experiments
MCC9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
MCC9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
MCC9-12.S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

MCC9-12.S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

MCC9-12.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

MCC9-12.S.IC.6 Evaluate reports based on data.
What is the progression of statistics standards
Let's look back.......
Teams of 3-4 will trace the path of our Adv Alg Stats Standards beginning with 6th grade
Standards for Mathematical Practice Look Fors
Student Behaviors
1. Make sense of problems and persevere in solving them.
Students are:
• Working and reading rich problems carefully (TKES 3.7)
• Drawing pictures, diagrams, tables, or using objects to make sense of the problem (TKES 3.7)
• Discussing the meaning of the problem with classmates (TKES 4.3)
• Making choices about which solution path to take (TKES 5.2)
• Trying out potential solution paths and making changes as needed (TKES 8.2)
• Checking answers and making sure solutions are reasonable and make sense (TKES 6.7)
• Exploring other ways to solve the problem (TKES 8.7)
• Persisting in efforts to solve challenging problems, even after reaching a point of frustration. (TKES 8.5 & 8.6)

Standards for Mathematical Practice Teacher Behaviors
1. Make sense of problems and persevere in solving them.
Teachers are:
• Providing rich problems aligned to the standards (TKES 1.2)
• Providing appropriate time for students to engage in the productive struggle of problem solving (TKES 8.6)
• What information do you have? What do you need to find out? What do you think the answer might be?
• Can you draw a picture? How could you make this problem easier to solve?
• How is ___’s way of solving the problem like/different from yours? Does your plan make sense? Why or why not?
• What tools/manipulatives might help you? What are you having trouble with? How can you check this?

What's in the Comprehensive Teaching Guide?
Overview of Domains
Overview of Standards
Unit Overviews
Curriculum Map
www.georgiastandards.org
Student
Editions
MCC6.SP.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.

MCC6.SP.4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
MCC6.SP.5. Summarize numerical data sets in relation to their context, such as by:
MCC6.SP.5.a. Reporting the number of observations.
MCC6.SP. 5.b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement
MCC6.SP.5.c. 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.
MCC6.SP.5.d. 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.
Webinars: https://www.georgiastandards.org/Common-Core/Pages/Math-PL-Sessions.aspx

Village
https://www.georgiastandards.org/resources/Pages/Tools/LearningVillage.aspx
For AG Scaffolded Item
How many of you have
used one of the tasks we've
done so far in your classroom?
Unit 2 Polynomial Functions
This unit develops the structural similarities between the system of polynomials and the system of integers.
Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property.
Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers.
Students identify zeros of polynomials and make connections between zeros of polynomials and solutions of polynomial equations.
The unit culminates with the fundamental theorem of algebra.
Unit 3 Rational and Radical Relationships OVERVIEW
In this unit students will:
• Explore Rational and Radical Functions
• Determine rational numbers extend the arithmetic of integers by allowing division by all numbers except zero. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial
• Notice the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers
• Investigate the properties of simple rational and radical functions and then expand their knowledge of the graphical behavior and characteristics of more complex rational functions
• Recall and make use of their knowledge of polynomial functions as well as compositions of functions to investigate the characteristics of these more complex rational functions

• Apply these rational and radical functions with an emphasis on interpretation of real world phenomena as it relates to certain characteristics of the rational expressions
• Understand that not all solutions generated algebraically are actually solutions to the equations and extraneous solutions will be explored
• Solve equations and inequalities involving rational and radical functions
continued....
Unit 3 Continued
Extension of Jelly Blubbers
confidence Intervals

1. How much money does Fry have in his bank account right now?
2. Write down a guess.
3. Write down an answer you know is too high. Too low.
Sequel Ideas.....

4. It took Fry 1,000 years to get that much money. How many more years will it take him to double it?

5. How long will it take him to get a trillion dollars?
from Robert Kaplinsky....
Let’s investigate g(x) = x2 + 3x – 10. What facts can you write about g(x)?

What is the Domain?
How do you determine the Domain?
What is the Range?
How do you determine the Range?
Where are the Roots or Zeros found? What are some different ways you know to find them?
What is the End Behavior?
How do you know?
Let’s investigate f(x) = x + 1.
What facts can you write about f(x)?
What is the Domain?
What is the Range?
What are the Roots or Zeros?
What is the End Behavior? How do you know?
And a "touch of
Units 2 and 3...
Now let’s consider the case of the rational function
r(x) = f(x) / g(x) where f and g are the polynomial functions above. Write the expression for the function r(x).
What is the domain of r(x)? Which function, f or g, affects the domain the most? Why?
What do you think the range of r(x) will be? Why is this so difficult to determine?
What are the roots or zeros of r(x)? Which function helps you find them?
What do you think the end behavior will be? Why?
Where will r(x) intersect the y-axis? How do you know?
Now let’s look at the graph of r(x) using your calculator.
In Unit 5 Trigonometric Functions students will:
• Expand their understanding of angle with the concept of a rotation angle
• Explore the definition of radian
• Define angles in standard position and consider them in relationship to the unit circle
• Make connections to see how a real number is connected to the radian measure of an angle in standard position which is connected to an intercepted arc on the unit circle which is connected to a terminal point of this arc whose coordinates are connected to the sine and cosine functions

• Gain a better understanding of the unit circle and its connection to trigonometric functions. Develop an understanding of the graphs of the sine and cosine functions and learn to recognize the basic characteristics of their graphs
• Realize transformations of y = sin (x) and y = cos (x) behave just as transformations of other parent functions
• Learn that the concepts of amplitude, midline, frequency, and period are related to the transformations of trigonometric functions
• Learn how to look at a graph of a transformed sine or cosine function and to write a function to represent that graph explore several real-world settings and represent the situation with a trigonometric function that can be used to answer questions about the situation.
• Develop and use the Pythagorean identity for sine (x) and cosine (x)
And a Touch of
Unit 5 Trig
Full transcript