This progression is designed for a high school statistics and probability course. The progression meets standards S-ID.1 “Represent data with plots on the real number line (dot plots, histograms, and box plots)”, 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.” and 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).” The lesson plan focuses on interpreting data from a parallel box plot and histogram to find the median/mode and interquartile range/standard deviation.
Learning Progression
Lesson Plan

This learning progression is a focused on statistics and probability in a 7th grade middle school mathematics classroom. There are many ways probabilities can be taught, this progression will use models and data collection. There are three levels of functions that are taught at the high school level: Compare probabilities from a model to observed frequencies, Develop a uniform probability model by assigned equal probability to all outcomes and use the model to determine probability events, and Develop a probability model by observing frequencies in data generated from a chance process. Probabilities describe the extent to which something is probable; the likelihood of something happening or being the case. In the progression of Interpreting Statistics and Probability students learn how to understand and develop probability events, how to observe and  interpret collected data, and formulating questions, designing studies, and collecting data about a population through random sampling allow us to make inferences and compare data.

 

Learning Progression-Probability Modeling Statistics and Probability Lesson Plan

Grade Level: 12^th

Textbook: Discovering Statistics

Cluster: Interpret Linear Models

• CCSS.Math.Content.HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
• CCSS.Math.Content.HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
• CCSS.Math.Content.HSS-ID.C.9 Distinguish between correlation and causation.

 

In this learning progression students will learn about a line of regression for a set of data. They will learn about the different components of the equation, how to find the line of regression for a set of data both by and by using a graphing calculator, and the correlation coefficient.  Finally the students will apply what they have learned to a given set of data and explain the difference between correlation and causation in terms of a real life situation.

 

To learn the concept outlined in this standard the students will be presented with simple linear and non-linear functions that have real worlds application to practice identifying the meaning behind the different significant features of the graph and its equation. Then the transition will be made from simple linear equations to equations that are the linear regression of a set of data.	CCSS.Math.Content.HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.   Example problem:
To learn this procedural fluency students must learn the individual processes involved with the end product. I will first introduce students to the concept of a linear regression. Then I will teach the students how to calculate this by hands. Once they have practice with this process for small sets of data I will teach them how to use their calculators to find this line for larger sets of data.	CCSS.Math.Content.HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.     Example problem: Find the linear regression and correlation coefficient for the following data set. Weight (kilograms) Age (Years) Blood fat content  1  1  84  46  354  2  1  73  20  190  3  1  65  52  405  4  1  70  30  263  5  1  76  57  451  6  1  69  25  302  7  1  63  28  288  8  1  72  36  385  9  1  79  57  402 10  1  75  44  365 11  1  27  24  209 12  1  89  31  290 13  1  65  52  346 14  1  57  23  254 15  1  59  60  395 16  1  69  48  434 17  1  60  34  220 18  1  79  51  374 19  1  75  50  308 20  1  82  34  220 21  1  59  46  311 22  1  67  23  181 23  1  85  37  274 24  1  55  40  303 25  1  63  30  244   Data From https://orion.math.iastate.edu /burkardt/data/regression/x09.txt
To meet this standard the students will work with examples of sets of data where there is a large correlation between two factors but neither is caused by the other. Some of these examples the variables will be related and some will be random. The students must experience this relational phenomenon for themselves to solidify this difference in their minds.	CCSS.Math.Content.HSS-ID.C.9 Distinguish between correlation and causation.   Example problem: The day of the year with the most reports of domestic violence is Super bowl Sunday. This means that there is a large correlation between these two factors. Does this mean that the Super Bowl causes domestic violence or vice a versa? Why or why not

 

 

 

Lesson Title: Interpreting the Data

Unit Title: Linear Regression

Teacher Candidate: Albany Thompson

Subject, Grade Level, and Date: Statistics, 12^th Grade,  2/27/14

 

This lesson will be the final lesson in this sequence. The student will only be able to perform these calculations and interpret the given data after completing the previous lesson in this section.

Central Focus: Finding the linear regression and correlation coefficient for a set of data.

Essential Question: Does a strong correlation imply causation? Does a weak correlation imply no causation?

• CCSS.Math.Content.HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
• CCSS.Math.Content.HSS-ID.C.9 Distinguish between correlation and causation.

Learning Outcomes	Assessment
By the end of this lesson students should be able to use their calculators to find a line of regression for a large set of data. They should also be able to find and interpret the correlation coefficient in terms of the original question.	I will assess the students’ mastery of the material by speaking with the students during the group work time and observing their process. I will also assess them informally during the end discussion when the students present their findings. The students will also be assessed based on the rubric found below.

 

Learning Targets	Student Voice
I will be able to find a line of regression. I will be able to calculate the correlation coefficient for a set of data. I will be able to explain the difference between correlation and causation.	The students will have the opportunity to explain their understanding to both the other students and myself during their group work time and the class discussion at the end of the lesson.

 

The students will be comfortable with the concept of linear regression and correlation from the previous lesson. I will assess their understanding of these topics through their answers to homework and quiz questions.

Academic Language Demands
Vocabulary & Symbols	Language Functions	Precision, Syntax & Discourse
linear regression correlation coefficient correlation causation	The language in this lesson functions to give the students the vocabulary to talk about data in a quantitative sense in order to make comparisons.	Mathematical Precision: The students will be able to discuss their findings with precision as a result of their calculations for comparison.   Syntax:   Discourse: Students will be able to talk about data in terms of its correlation to a straight line.

 

Language Target	Language Support	Assessment of Language Target
My goal for the students in this lesson is that they will be able to discuss their finding from their calculations with the precisions required for statistical analysis.	I will support the students’ use of appropriate language in a mathematical sense by conversing with them throughout the lesson and by targeting words that students have found confusing based on their prior exit slips.	I will assess the students’ mastery of the language target informally through their responses to their exit slips about the vocabulary that was taught in the lesson.

 

This lesson will be the culminating project for this section. It will incorporate all of the components that they have been learning about in this unit and give them an opportunity to apply their knowledge to model real life data and present their discoveries.

I will group students according to ability level to give al students the opportunity to be challenged in one way or another. I will also provide extra support to any students that have missed instruction or are just struggling in this section by assisting them with their calculations when need be.

• computer lab
• graphing calculators
• projector
• whiteboard
• whiteboard markers

Teaching & Instructional Activities
Time	Teacher Activity	Student Activity	Purpose
5 minutes	Group students and assign the lesson’s project Instructions below	Get in groupings and listen to instruction	Prepare students for their group work time.
35 minutes	Be available to answer student questions	Create model and prepare presentation	Group work time
10 minutes	Listen to presentations to assess student understanding	Listen to and give presentations of findings	Present findings, assess through products
5 minutes	Hand out and then collect exit slips	Fill out exit slips	Formatively assess student learning

 

 

Data taken from http://the-numbers.com/movie/budgets/

 

Instructions:

Choose one of the data charts from the website above to model with a linear regression. Then calculate the coefficient of correlation for this model. If necessary remove outliers from your data set to improve your model. Be prepared to give a short presentation about your findings and what they mean in the real world by the end of the period.

 

Modeling Activity Rubric

	0	1	2	3
Work time was used productively
The correct linear regression and coefficient were found
Explanation was given for the meaning of the findings any data points removed
The distinction between correlation and causation was made

 

Exit slip:

List 3 concepts that we covered today and indicate whether you would like to cover any of them again.

 

 

 

 

 

 

List two new terms from the lesson today and if possible define them

 

 

 

 

 

List one thing you found interesting from the lesson today.

Learning Progression 2

This learning progression could be used for multiple classes; however, I am using it for a Trigonometry Honors class.

The following segment of the learning progression for high school statistics and probability utilizes Precalculus Ninth Edition (Michael Sullivan, Prentice Hall, Boston, MA 2012) as the textbook for instruction.  For this class of 16 junior and senior high school students, the lessons and homework problems herein provide a tool for the teacher to help their students’ reach the four Common Core State Standards contained in this math cluster.

Diego Mendoza Learning Progression

Diego Mendoza LearningProgression LessonPlan

LEARNING PROGRESSION
The class that I am using for this learning progression would be the alternative high school program that is through Ellensburg High School. This alternative program is for students who do not learn well in a traditional classroom. The students in the class are seniors and are trying to finish up their collective of evidence and take the SAT/ACT to graduate. When these students work hard and take the time to learn the concepts we teach, we reward them with teaching them a math that they want to learn like statistics.
This Learning Progression is about the basics of probability; sets, subsets, union, intersections, complements, independences, and conditional probability. The Common Core Standard for this Learning Progression are HSS-CP.A.1, HSS-CP.A.2, and HSS-CP.A3. Each standard will be covered in a day or two but after each standard there will be a benchmark assessment to check for student understanding and to see if the standard was reached.
At the beginning of each class, students will try to define a term that they will learn that day. The teacher will start the class by asking students for their definitions to be written on the board, and then the teacher will go through each of the definitions with the class to identify the correct one. At the end of each class, the students will take a formative exam (exit slip) to check for student understanding and to determine if the teacher needs to re-teach the lesson the next day. The first activity (HSS-CP.A.1), will be a concrete example that the students will use to learn about sets and subsets. The students will receive skittles and M&Ms that will be defined as the set. The students will have to separate the candy into subset. The last activity the students will receive a mathematical model and will have to create subsets from that set. The next activity the students will be doing is an activity on union, intersections, and complements. This activity will be using the previous mathematical model from the previous activity. The students will need to be able to identify unions of the subsets they created, intersections and complements. There will be a benchmark assessment after this activity for HSS-CP.A.1. The third activity (HSS-CP.A.2 and HSS-CP.A.3) that the students will be doing will be working with Conditional probability and how to use conditional probability to prove sets that are independent of each other. A worksheet will follow this activity because the students will need to practice and be able to connect the two concepts together. There will be a bench mark assessment for HSS-CP-A.2 and HSS-CP.A.3. The final activity will be a summative assessment based on the two benchmark assessment.

High School: Statistics and Probability
Cluster: S-CP: Understand Independent and Conditional Probability
CCSS.Math.Content.HSS-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
CCSS.Math.Content.HSS-CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
CCSS.Math.Content.HSS-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

The terms entrance tasks will have the students work together to figure out how to define a word. Each group will get two of the words to define and then the teacher will assist the students to find the correct definition.

The exit exam or slip will give the students information about what they covered in the lesson. Some of these exit slips will benchmark assessments to see if the students are reaching the standards and to see if the students are aligned to the standards.
• Example problems for an exit slip would be:
o the set of odd integers has a subset of the multiples of 3 in it. True or False?
o given the set U={1,2,3,4,5,6,7,8,9} and the subsets of A={1,2,3,5,6,8}, B={2,5,6,7,8}, and C={1,4,5,6,7,9}. Find the union of A and B, the intersection of B and C, and the complement of C.
o Using the same sets as above, determine if A is independent of B.
o Write the formula for conditional probability. Find the probability of C given that B is true.

The M&M and Skittles activity could have subsets like colors. This would allow the students to create unions, intersections, and complements.

A mathematical model that could be used is the set of integers from 1 to 10. The subsets could be even, odd, prime, composite.

LESSON PLAN FOR LEARNING PROGRESSION:
Lesson Title: The Sets of Candy
Unit Title: Probability
Teacher Candidate: Megan Turner
Subject, Grade Level, and Date: High School, Statistics, 2/6/14

Placement of Lesson in Sequence
This lesson is the introductory lesson to sets and subsets. It is the concrete example for students to define a set and subsets. This lesson will be referred to during the rest of the unit as a concrete example.
Central Focus and Essential Questions
The central focus of this lesson is defining a set and being able to identify a subset within the set. The students will learn how to identify a subset given the universal set. The students will also be able to explain what a subset is and how to find it in the set. The students will also create a subset from a given set and then defend it with their knowledge and reasoning skills.
Essential questions: What is a set? What is a subset? How can I find a subset within a set?
Content Standards
CCSS.Math.Content.HSS-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Learning Outcomes Assessment
The Learning Outcomes for the students will be
• Defining a set
• Creating subsets within the set • A worksheet that will be graded from the Skittles and M&Ms activity
• Exit slip with a true or false question

Learning Targets Student Voice
I will be able to
• Correctly identify what a set is
• Find a subset within the set given The students will work with pairs for this activity and will partake in a class discussion to find the correct definition of the terms set and subset.

Prior Content Knowledge and Pre-Assessment
The students may have prior knowledge for this lesson, but it is not required since this is the beginning of a unit.
The pre-assessment will be the entry task, where the students will have to work together be come up with a definition of set and subset.
Academic Language Demands
Vocabulary & Symbols Language Functions Precision, Syntax & Discourse
I will be able to define the terms:
• Set
• Subset

• Define and be able to provide examples of a set and subset.
Mathematical Precision:
The students will practice with Skittles and M&Ms to form subsets
Syntax:
The students will have to be able to explain their answer using correct vocabulary.
Discourse:
Students will have to work with each other to find all of the subsets.

Language Target Language Support Assessment of Language Target
I will be able to define and explain:
• A set
• A subset within a given set The students will be working pairs on this activity and will have to use the correct vocabulary and be able to explain and defend their choice. The students will have write down their choice of a subset and explain why that works.

Lesson Rationale (Connection to previous instruction and Objective Standards)
This lesson covers the first part of the Standard. The second part of the standard with the students using union, intersection, and complements will be the next lesson taught. This is a new unit so there is no connection to previous instruction. The exit slip will be graded and returned to the students before the next lesson. The exit slips will help the teacher with the data to determine if the lesson needs to be re-taught or the misconception was caught by all of the students.
Differentiation, Cultural Responsiveness and/or Accommodation for Individual Differences
The accommodation that is being made is that the students get to work with a partners during the activity.
Materials – Instructional and Technological Needs (attach worksheets used)
Skittles
M&Ms
Worksheet
White board
Markers
Exit Slip

Entry Task: Work with a partner and come up with the definition of set and subset.

Worksheet:
Name:

1. Sort the candy into colors and then write down then take a picture on your phone.
2. Then sort the candy into relative size and take a picture on your phone.
3. Create your own subset. And take a picture of it.

Given the set of even integers, find three subsets.

Exit Slip:
Name:
Given the set of odd integers. True or false? The set of all the multiples of three are in the set of odd integers.

Why?

Teaching & Instructional Activities
Time Teacher Activity Student Activity Purpose
10 minutes Entry Task on the whiteboard Entry Task To define the term set and subset.
30-40 minutes Skittle and M&M activity; assisting students Skittles and M&Ms activity To practice finding the subsets and getting a concrete example of a set and subset.
10 minutes Exit Slip and final questions Exit Slip To determine whether re-teaching is needed and to see if misconceptions are found.