# Grades 7–12, Geometry

Geometry is the only domain which is present in every grade, K-12, in the Common Core State Standards for Mathematics (CCSSM), but despite—perhaps because of—this, it is also under-emphasized in almost every grade. This module is designed to deepen participants’ knowledge and understanding of the CCSSM expectations for Geometry in Grades 7-12, with a brief look-back to the standards from earlier grades to emphasize the overall K-12 coherence of the geometry domain.

One important difference between the CCSSM and earlier geometry standards is an increased emphasis on transformations (translations, rotations, reflections, rotations, and dilations) and transformational arguments in the Common Core. Transformations are not necessarily studied for their own sake, but rather because they afford a quick and intuitive approach to more advanced results and applications of geometry. Nevertheless, the transformational approach is new and unfamiliar to many teachers of geometry, and this module will present teachers with a grounded introduction to transformations so that they can make an intentional decision as to the extent to which they will study transformations in their own classes.

Topics to be studied in the module include: an introduction to translations, reflections, and rotations; geometry and measurement standards in early grades (including justification of area formulas); the transformational approach to congruence and similarity; the Pythagorean theorem (proofs and applications); middle school expectations for circle measurement; and justification of volume formulas for cones and cylinders. The CCSSM Standards for Mathematical Practice, and their appearance through a geometric lens, will be addressed throughout.

## Goals

- Become familiar with the progression of standards for geometry across Grades 7-12.
- Deepen understanding of the transformational approach to geometry advocated by the CCSSM.
- Explore the nature and importance of conjecture and proof in geometry.
- Deepen understanding of the ways in which students' geometric understanding grows, and implications for classroom instruction.

# Grades 9–12, Statistics and Probability for All High School Mathematics Teachers

This module is designed to deepen participants’ understanding of the progression of statistics and probability that move students’ understanding of analyzing data as described by the standards in Grades 6 through 8 to the inferential thinking delineated in the high school standards. Statistics is an investigative problem-solving process immersed in a context and this process provides the framework for this module. This framework stresses hands-on activity learning and that statistical analysis is an investigation that turns loosely formed ideas into scientific studies by using a four-step statistical problem-solving process. The four steps are: formulating a statistical question that can be answered with data, designing a plan to collect appropriate data, analyzing the collected data by graphical and numerical methods, and interpreting the analysis with respect to the original question.

## Goals

- Understand student expectations in the Statistics and Probability Domain in Grades 6 through 8 and the High School Conceptual Category – Statistics and Probability;
- Understand the statistical problem-solving process and how this process is embedded in the Statistics and Probability Domain/Conceptual Category;
- Study statistics and probability investigations that move student understanding of analyzing data as delineated by the standards in Grades 6 through 8 to the inferential thinking delineated in the high school standards;
- Provide investigations that illustrate how classroom teachers can empower their students to become data collectors and analyzers; and
- Study the relationship between the Standards for Mathematical Practice and the Standards for Mathematical Practice Through a Statistical Lens.

* This course is a revision of the previously offered 8*–*12 Modeling with Statistics and Probability course. *

# Grades 8–12, Algebra and Functions with Modeling

A study of the CCSSM indicates that Algebra and Functions continue as pillars of the study of mathematics in high school. They are two of the six conceptual categories of the CCSSM for high school, and as such, they represent a significant focus for teachers and students in the learning of mathematics.

In addition to examining the high school standards involving Algebra and Functions, this module will examine the progression documents that outline the growth of these conceptual categories. Algebra has its roots through the earlier grade levels. The domains Operations and Algebraic Thinking are addressed in kindergarten through grade 5. The domains of Expressions and Equations are picked up in grade 6 through grade 8, and address the challenging expectations of an understanding of algebra prior to high school. High school continues the learning of algebra by expanding the role of expressions and linear equations to the study of quadratic and exponential equations, and later, to the study of logarithms, vectors, and matrices. The reasoning process continues as symbols are used in expressions, equations, and functions.

Students learn in high school that variables and operations are developed in expressions, and that expressions result in formulas and functions. This module will particularly emphasize how formulas can be derived from a situation, and how an analysis of formulas results in the understanding of equations and functions. The key components of both equations and functions remain in understanding or deriving expressions. Further, the components of understanding expressions are embedded in variables, exponents, and the operations applied to expressions.

Functions are closely related to equations, especially as high school students are learning about the composition of equations. Functions are rooted in the domains of Number and Operations of Base Ten for grades K to 5, the Number System for grades 6 to 8, Number and Operations Fractions in Grades 3–5, and Ratios and Proportional Relationships in Grades 6–7. A study of the progressions involves examining these earlier domains and how students are learning these building blocks of functions. A more formal delineation of functions is addressed in grade 8, and is continued throughout high school.

This module addresses both algebra and functions as key components of a student’s learning of mathematics in high school. These conceptual categories are connected, and build from each other as equations and functions particularly addressing new challenges and new applications for high school students. A focus of this module will be to particularly examine the algebra and function domain or clusters that are connected to the modeling conceptual category and the modeling cycle developed in the CCSSM. Modeling adds an important dimension of understanding both algebra and functions, plus adds to students’ reasoning and communication of the mathematics they learn.

The Standards for Mathematical Practice continue as key standards for high school, and are critically important in the study of algebra and functions, and their connections to modeling. Each of the investigations and problems addressed in this module will be carefully linked to the Standards for Mathematical Practice and will be equally analyzed as part of the expectations in students’ learning of mathematics.

## Goals

- Understand how the CCSSM conceptual categories of algebra and functions are connected to the conceptual category of modeling.
- Gain a deeper understanding of the CCSSM conceptual categories of Algebra and Functions.
- Investigate real-life applications of algebra and functions.
- Investigate student expectations in the area of algebra and function assessment.
- Gain insight into the CCSSM Mathematical Practices and what they look like in a high school mathematics classroom.