# Statistical Reasoning – S

This learning progression will be taught to a class that consists of juniors and seni ors in high school who are currently taking a college course that is taught at their high school, Math 102: Mathematical Decision Making.  The Common Core State Standards (CCSS) that will be addressed come from one domain, High School: Statistics and Probability. The CCSS clusters that will be addressed are “Make inferences and justify conclusion from sample surveys, experiments, and observational studies” and “Use probability to evaluate outcomes of decisions.”  Students will also meet Mathematical Practices 3, 5, and 7.

The central focus of this learning progression is on an introduction to statistical reasoning and the aspects of a study that may produce biased results.  The progression begins with an overview of what statistical studies are and the process that someone must go through to create one.  These foundational concepts will give the background knowledge that they will need to use throughout the remainder of the learning progression.  The students will then learn about some different types of studies and how to avoid bias while creating a study.  In the third task of this progression, the students will use all of these ideas to find their own methods of testing the validity of a study’s results.  At the end of the progression, the students will be asked to design their own study and will work on conducting this study as a project that will be added on to over time.  After completing this learning progression the students will learn how to analyze data and how to represent it graphically to be able to use the results of a study to make decisions with.  The beginning steps of this project will be the students’ assessment.

Learning Progression-Statistical Reasoning-23fwc65

# Learning Progress: Intro to Stats and Data – CCSS.Math.HS.ID

Learning Progression

HS.ID.A.1 – Represent data with plots on a real number line (dot plots, histograms, and box plots).

• Read and contextualize graphs and tables
• Analyze how data is represented (e.g. “Is this the best graph for this data?”)
• Create graphs and tables for data sets and give reasoning.

HS.ID.A.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.

• Contextualize the skew of data
• Calculate interquartile ranges, standard deviation, and distribution
• Create a graph – or data set – that represents a given description

HS.ID.A.3 – Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

• Identify outliers
• Create graphs and/or tables for two data sets and compare them

Assessments:

• Student discourse
• Mathematical reasoning
• Use of visuals
• Contextualizing to personal examples/experiences
• Hinge questions

# Creating a Pin on Your Cell Phone HSS.CP.B.9 Math Problem- How many different pins can you possibly create on your cell phone using the digits 0-9? How many different pins are possible using any given digit only once?

CCSS Math- HSS.CP.B.9 – (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

This picture will help the students relate creating a pin number for their cell phone to mathematical content. This can be used for a mathematics activity in a statistics course to influence critical thinking and understanding mathematical concepts. By having the students recognize the total number of pins possible will introduce them to more complex ideas that involve permutations and combinations. The teacher can start off the lesson by using less digits and having the students work in groups to discuss and write out all the possible pins. This will actively engage them and stimulate their ability to critically think and solve problems. For further teaching, using a graphing calculator and the alphabet to create passwords can be used also.

# High School Statistics: Summarize, Represent and Interpret data

This learning progression is designed for an 11th-grade statistics class. The CCSS-Math meet in this learning progression are

Summarize, represent, and interpret data on a single count or measurement variable.

CCSS.MATH.CONTENT.HSS.ID.A.1
Represent data with plots on the real number line (dot plots, histograms, and box plots).

CCSS.MATH.CONTENT.HSS.ID.A.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.

CCSS.MATH.CONTENT.HSS.ID.A.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 purpose of this learning segment is for students to learn to analyze, report and make inferences about data. Students will learn how to find mean, median, mode, range, and box-and-whiskers plot, as well as to write reports describing data sets. Throughout the lessons, I will give examples to help students understand the concepts. Class discussions and group work activities are designed to help students develop understanding of the concepts, create procedural fluency and elicit students reasoning.

Students will be assessed during instruction as well as formative assessment at the end of each lesson.  For the final assessment, students will show their understanding of the concepts by completing a report project.

Complete learning progression: learningprog

# What’s cool? HSF-LE.A.3

What’s cool?

For this specific lesson, the temperature prove and data collection interface and a calculator will be needed. The temperature prove is a device that will help students gather data, analyze it and graphed (in the calculator). Students will be asked to work in groups to find an exponential model for the temperature data (this will be collected by the temperature prove). By working in groups students will be able to critique and defend their own ideas and arguments, share their opinions and practice the standards. Students will develop models using the formula, Tdiff=Toe-kt. This activity is aligned with the CCSS.MATH.CONTENT.HSF-LE.A.3: Construct and compare linear, quadratic and exponential models and solve problems. Students need to record their data in the tables (see attachment) and complete several analysis questions to understand the problem correctly. # Passcode Picture Problem SP.B

You are on a mission. You’re mission, should you choose to accept it, is to unlock your best friends phone so that you can take a selfie and set it as their wallpaper. If the passcode to their phone is four digits long and you know that the first number is 7, and that 0 and 2 are never used; it is possible that numbers may be repeated. How many combinations are possible in order to unlock their phone?

CCSS.MATH.CONTENT.HSS.CP.B.9

(+) Use permutat ions and combinations to compute probabilities of compound events and solve problems.

# Lets Roll The Dice HSS.MD.B.5.A A great way to get students engaged in the mathematical concepts they are learning is to give them problems that involve real world things that they can actually relate to. Everyone at some point in time has rolled dice during some sort of game, so this picture can relate to students and get them interested in learning more about probability.

Using this picture, ask the students which numbers they think the dice will land on. This lesson can involve theoretical probability and experimental probability. They can start by calculating the theoretical probabilities of different combinations of numbers that the dice could land on. Then they can either use two actual dice, or even the dice rolling app on a graphing calculator to find the experimental probability. This activity gives students the opportunity to have a hands on experience with the concept of probability.

This problem is aligned with:

CCSS.MATH.CONTENT.HSS.MD.A.3
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

CCSS.MATH.CONTENT.HSS.MD.B.5.A
Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

# What’s the Probability HSS.MD.B.5.&7.

After introducing a new topic in mathematics, students find it difficult to take what they have learned into practice. However, keeping students interested, active, and engaged in different activities makes a significant difference in their learning experience.

Students will come up with measures of chance. One of the questions that they can ask themselves is, “how can I quantify how likely an event is?” In this case, teachers can introduce this classic activity, using a standard deck of cards. Using a deck of cards provides a concrete look at probability and chance in a hands-on math activity. A typical deck of cards has four suits of thirteen cards in each suit, twelve face cards, four aces, twenty-six red cards and twenty-six black cards. Considering this, different probability questions can be asked to practice using this concept. After introducing probability to your students, you can incorporate this activity within your lesson. Based on what they know about a standard deck of cards, students can answer questions, for instance, if you select one card randomly, what is the probability it is a heart?

CCSS.MATH.CONTENT.HSS.MD.B.5

(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.MATH.CONTENT.HSS.MD.B.7

(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Allowing students to use the deck of cards to answer probability questions will help them reinforce their understanding. They will analyze and find strategies to know the probability of the specific card. This activity allows students to interact with their peers and reinforce their mathematical thinking of finding the probability of a certain card.

# S-ID Texting Champion

A student in Ms. Smith’s class claimed that girls text faster than boys. Naturally, the boys disagreed. Therefore, Ms. Smith’s statistics class did an activity where each student calculated the average words per minute they could text. The data is recorded below.  1.    Sketch two side by side box plots to compare the average words per minute of girls and boys.   Make sure to include the minimum, maximum, first and third quartiles, and median. Additionally, calculate the mean for each data set.

2.   Why is the mean less than the median in both boys and girls?

3.      Compare and contrast the two box plots. What does this mean when we are comparing words per minute between boys and girls?

4.      If I was trying to describe the center of these distributions, would the mean or median be more appropriate?

5.    Based upon the data presented above, which gender texts faster? Support your answer with statistical data.

An assessment commentary and solution is on the attached document: IM problem

# Ice Hands: Modeling a human histogram and box & whisker plot Standard: High School: Statistics & Probability » Interpreting Categorical & Quantitative Data »                          Summarize, represent, and interpret data on a single count or measurement variable»                          Represent data with plots on the real number line (dot plots, histograms, and box plots).

In this post, teachers will learn how to incorporate the Venier Easy Temp Sensor into a statistics lesson. The lesson will be about histograms and box & whisker plots. Data for the histograms will be provided by the temperature sensor which uses Easy data on a TI 84/83 graphing calculator. The data will be collected on the calculator. The data will be shown through a projector in class, so that students are able to copy the data on to their worksheets. Teacher will set note cards across the room with different temperatures in the class. Students will form a line from coldest to warmest hand temperature. Students will then make a human histogram. Teacher and students will discuss how a histogram’s data can be manipulated. This will give students a chance to figure out how histograms and bar graphs can be distributed. This will potentially will reduce the questions that students make when making a histogram (distribution): “Can I go up by two’s? Three’s? Five’s…etc.” Teacher will then introduce the idea of a box & whisker plot. Then students with the assistance of the teacher will make a human Box & Whisker Plot and symbolizing with a rope. The Rope will be segmented according to the minimum value, lower quartile, median, upper quartile and maximum value. Teacher and students will explore the characteristics of a box whisker plot and histograms to ensure understanding of the upcoming lessons.

Teacher Notes on Worksheet

Student Worksheet

Vernier Easy Temp Sensor

Temperature Facts