6 Chapters

## Chapter 6 Statistics and Probability

Edward C. Nolan Solution Tree Press ePub

How do you define statistics and probability? Statistics is the collection, analysis, and interpretation of data, often in numeric form. Probability involves examining the fraction of successful outcomes out of the total number of possible outcomes and is an important factor in understanding random processes. This chapter explores how to build conceptual understanding of statistics and probability in the middle grades. You will reason with problem-solving situations that involve statistical investigations, data analysis, and probability.

One important aspect of statistical reasoning in the middle grades is comparing data sets. What are the different ways that you can compare data sets? How do you know which way to use for particular data sets? Think about your answers to these questions as you complete the task shown in figure 6.1.

Mr. Richard and Ms. Chutto decide to compare the grades in their two science classes on the last quiz.

The grades in Mr. Richard’s class on the twenty-point quiz were:

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## Chapter 6: Statistics and Probability

Nolan, Edward C.; Dixon, Juli K.; Safi, Farshid; Haciomeroglu, Erhan Selcuk Solution Tree Press ePub

CHAPTER 6

Statistics and Probability

This chapter connects statistics and probability concepts to the reasoning and sense making you have explored from algebra and geometry. Some tasks will connect with the reasoning and justifying aspects of the previous chapters, such as reasoning how to make predictions using information from a real-world situation, justifying decisions made from the analysis of a given data set, or examining how sample surveys, experiments, and observational studies can be used as mathematical models. Other tasks will look at how the analysis of a data set connects to the shape of the data display and measures how well a function models the pattern of a data set.

By definition, statistics is the collection, analysis, and interpretation of data. Probability is the ratio of successful outcomes to the total number of outcomes. These two topics, along with the use of randomness, are important concepts in mathematical modeling and simulating real-world events.

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## Chapter 1: Equations and Functions

Nolan, Edward C.; Dixon, Juli K.; Safi, Farshid; Haciomeroglu, Erhan Selcuk Solution Tree Press ePub

CHAPTER 1

Equations and Functions

What is the purpose of teaching functions in secondary mathematics? The concept of functions is one of the most important concepts in mathematics; it enables students to make connections within and across mathematical topics. Students need to understand patterns and the link to functional relationships, learning that mathematics is more than just manipulating algebraic expressions. As students use functions to represent relationships in multiple ways, they begin to recognize mathematical ideas in real-world situations and form notions of mathematical modeling (see chapters 5 and 6 for discussions on modeling). In order to emphasize the importance of exploring and connecting different types of functional relationships, different function types, such as linear and quadratic functions, and systems of linear equations are explored through real-world situations in this chapter and in chapter 2. Connections are made among verbal, numerical, graphical, and algebraic representations of mathematical concepts to illustrate the role and significance of multiple representations in understanding functions.

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## Chapter 4: Types of Functions

Nolan, Edward C.; Dixon, Juli K.; Safi, Farshid; Haciomeroglu, Erhan Selcuk Solution Tree Press ePub

CHAPTER 4

Types of Functions

How does the understanding of functions evolve as students experience different aspects of high school mathematics? As students transition from early coursework in high school mathematics to more advanced courses, they use their understanding of linear and quadratic functions (see chapters 1 and 2) to make sense of exponential, logarithmic, and rational functions, among others. As with earlier work with functions, these more complex functions are explored contextually, numerically, graphically, and algebraically. This chapter will focus on rate of change and other characteristics of various function types. Furthermore, the role of parameters in function transformations and the connections between functions and their inverses are examined, while highlighting the role of symmetry and geometric relationships. In chapter 5, tasks that use these mathematical concepts will be explored with a focus on modeling.

The Challenge

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## CHAPTER ONE: The psychic birth of emotional experience: the mental equipment for making contact and understanding the psychic reality

Pistiner de Cortinas, Lia Karnac Books ePub

“We are such stuff as dreams are made on
and our little life is rounded with a sleep”

(Shakespeare, The Tempest)

In this chapter, I am approaching the problems with which we are confronted in psychoanalytical treatments of patients who have serious difficulties in the psychic transformation of their emotional experiences*. Most of these problems are related to failures in the process of symbolic transformation. I am interested in the specific symbolic failure which is related to the obstruction of the development of phantasies, dreams, dream thoughts, etc. I want to focus attention on a frozen area of the mind, which, through immobility and emotional isolation, avoids falling into states of traumatic helplessness. I also want to differentiate bombardment of stimuli, or chaotic states of excitement, from psychically elaborated emotional experiences.

In my clinical experience with severely disturbed patients, I found myself differentiating a pathology that I frame in what Bion (1967, p. 43) calls hypertrophy of the apparatus for projectiveidentification from another in which a detention of projective identifications and an emotional isolation prevail.

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