Training Modules

 

        LTF training for Pre-AP math utilizes resources and strategies to employ the five elements of 
        math learning:·                

• Graphical: understand and interpret data, communicate data via visual representations, conceptual geometry comprehension

• Physical: utilize manipulatives and models, data collection for experiments, "real world" applications

• Verbal: interpret, explain, justify, and summarize data

• Analytical: understand and utilize functions, variables, definitions, theorems, and symbols

• Numerical: perform calculations, approximations, and data analysis

Through the modeling of lessons, teachers learn to integrate rigorous critical strands from AP Calculus and Statistics in every grade from sixth grade through high school, thus allowing students to build on concepts introduced early in middle school and reinforced every year in order to fully master concepts tested at the AP and college level. Strategies for integrating graphing calculators and manipulatives into classroom lessons represent a particularly significant component of LTF math training. Teachers also learn through demonstration the value of hands-on student involvement and ways in which it enriches existing curricula and increases levels of student engagement. LTF training in AP Calculus AB/BC and AP Statistics supports teachers in their efforts to prepare students for success on the AP exams and in their college courses. One-day seminars focusing on specific test-related topics and four-day summer institutes afford opportunities for Pre-AP and AP teachers to gain content knowledge, instructional strategies using the five elements of math learning (graphical, physical, verbal, analytical, and numerical), technology applications, and hands-on experience in planning an effective curriculum.

Pre-AP Math Training Module Descriptions:

MATH Core Training Modules—Year One

Participants will explore the AP* Calculus or AP Statistics Connection topics through manipulative-rich student lessons. Graphing calculator skills will be introduced, extended, and applied as a part of each training session. In addition, the participants will explore the LTF on-line diagnostics with both multiple choice and free response problems and will discuss scoring practices and rubrics of Pre-AP* questions related to the AP connection topic. Teachers finish every training day with lessons that are classroom-ready and with sufficient preparation to start using the lessons and assessment tools immediately.

Module One

Introduction to the LTF Guides through Analysis of Piecewise Functions

A combined group of middle school and high school Pre-AP teachers examine an overview of the entire Laying the Foundation mathematics program. One particular strand, Analysis of Functions, is used to illustrate the design of the mathematics guides. Teachers will explore how the guides can be used by vertical teams to strengthen and coordinate the middle school and high school curriculum. In the training, teachers investigate student lessons that describe and analyze piecewise functions. These lessons demonstrate how AP concepts are developed from sixth grade through pre-calculus. Participants are given password access to on-line versions of LTF guide lessons, diagnostics tests, and end-of-course examinations.

Module Two

Areas and Volumes

Middle school and high school Pre-AP teachers attend separate sessions where they explore manipulative-rich student lessons that investigate the area of two-dimensional figures, as well as surface area and volume of three-dimensional solids that result from revolving the planar figures about an axis. As the lessons progress through the vertical strand, teachers learn how students graph the original planar figure by first plotting points, then graphing equations, and finally graphing systems of inequalities. Trainers demonstrate how concepts involving area and volume are developed in Pre-AP mathematics classes from sixth grade through pre-calculus.

Module Three

Rate of Change: Average and Instantaneous

Middle school and high school Pre-AP teachers attend separate sessions where they explore student lessons connecting slope to the AP Calculus concept of rate of change. They will explore lessons that differentiate between the average and the instantaneous rate of change of a function. Middle school teachers will explore manipulative-rich lessons that introduce the concepts of constant rate of change and average rate of change. Additional lessons introduce high school teachers to the concept of a curve with a varying slope and to the calculus notation for a derivative to represent that slope. High school teachers also learn strategies for using a graphing utility in parametric mode to graph a curve that is not a function. The session further emphasizes how the concepts involving rate of change are developed in Pre-AP classes from sixth grade through pre-calculus.

Day Four

Graphical Displays; Distributions: Measures of Center, Variability, and Shape

Middle school and high school Pre-AP teachers attend separate sessions where they explore the concept of graphical displays by working student lessons that construct, compare, analyze, and interpret box-and-whisker plots, line plots (dot plots), and stem-and-leaf plots. Each lesson employs real-world data to construct, by hand and with a graphing calculator, appropriate graphical displays and to analyze the graph using measures of central tendency, variability, and shape. The training concludes with an exploration of AP Statistics free-response questions that are accessible to Pre-AP students. The session will emphasize how the concepts involving graphical displays and distributions are developed in Pre-AP mathematics classes and can be used from sixth grade through pre-calculus.

MATH Training Modules—Year Two

Participants will explore the AP* Calculus or AP Statistics Connection topics through manipulative-rich student lessons. Graphing calculator skills will be introduced, extended, and applied as a part of each training session. In addition, the participants will explore the LTF on-line diagnostics with both multiple choice and free response problems and will discuss scoring practices and rubrics of Pre-AP* questions related to the AP connection topic. Teachers finish every training day with lessons that are classroom-ready and with sufficient preparation to start using the lessons and assessment tools immediately.

Module Five

Accumulation

Middle school and high school Pre-AP teachers attend separate sessions where they explore the concept of accumulating area that leads to the concept of the definite integral in AP Calculus. Teachers will explore techniques for approximating area

of various closed regions through manipulative-rich middle school lessons. High school lessons extend these techniques to determining the area under a curve using geometric figures. In addition, non-area applications involving rates of change will be investigated. The session will emphasize how the concepts involving accumulation are developed in Pre-AP mathematics classes from sixth grade through pre-calculus.

Module Six

Probability

Middle school and high school Pre-AP teachers attend separate sessions where teachers will delve into student lessons that investigate probability. Techniques include using a sample space, conducting simulations, and collecting data. Teachers will discover and apply Pascal’s Triangle and the Binomial Theorem to probability. Additional topics include geometric probability and permutations and combinations. Trainers will emphasize how the concepts involving probability and statistics are developed in Pre-AP mathematics classes from sixth grade through pre-calculus.

Module Seven

Position/Velocity/Acceleration

Middle school and high school Pre-AP teachers attend separate sessions where teachers will explore the concepts and relationships of position, velocity, and acceleration. Participants will use physical activities and technology such as a CBR and a graphing calculator to more fully understand the concepts. Lessons include sketching a graph from a story, interpreting graphs from a verbal description, and analyzing and comparing graphs of position, velocity, and acceleration. The session will emphasize how the concepts involving position, velocity, and acceleration are developed in Pre-AP mathematics classes from sixth grade through pre-calculus.

Module Eight

Limits

Middle school and high school Pre-AP teachers attend separate sessions where teachers will explore the concept of limits from various perspectives. Student lessons use pattern recognition, perimeter and area of polygons, secant and tangent lines to circles and ellipses, and end-behavior of rational functions to lead to an informal notion of a limit. The session will emphasize how the concepts involving limits are developed in Pre-AP mathematics classes from sixth grade through pre-calculus.

MATH Training Modules—Year Three

Participants will explore the AP* Calculus or AP Statistics Connection topics through manipulative-rich student lessons. Graphing calculator skills will be introduced, extended, and applied as a part of each training session. In addition, the participants will explore the LTF on-line diagnostics with both multiple choice and free response problems and will discuss scoring practices and rubrics of Pre-AP* questions related to the AP connection topic. Teachers finish every training day with lessons that are classroom-ready and with sufficient preparation to start using the lessons and assessment tools immediately.

Module Nine

Optimization: Area and Volume Applications

Middle school and high school Pre-AP teachers attend separate sessions where they explore manipulative-rich student lessons that investigate the concept of optimization. Lessons will include maximizing and minimizing area and volume to determine an optimum solution. This session will demonstrate how concepts involving optimization are developed in Pre-AP mathematics classes from sixth grade through pre-calculus.

Module Ten

Linear and Non-linear Bivariate Data

Middle school and high school Pre-AP teachers attend separate sessions where they explore manipulative-rich student lessons that investigate the concept of linear and non-linear bivariate data. Teachers will investigate data coding, fit functions to data, use models to predict values, and determine residuals. In addition slope and intercepts will be explored in the context of the questions. The session further emphasizes how the concepts involving bivariate data are developed in Pre-AP classes from sixth grade through pre-calculus.

Module Eleven

Analysis of Functions: Transformation

Middle school and high school Pre-AP teachers attend separate sessions where teachers will delve into student lessons that investigate transformations and parent functions. They will explore translations, reflections, rotations, and dilations analytically, graphically, and numerically. This session will demonstrate how concepts involving transformations are developed in Pre-AP mathematics classes from sixth grade through pre-calculus.

Module Twelve

Rate of Change: Related Rates

Middle school and high school Pre-AP teachers attend separate sessions where they explore manipulative-rich student lessons that investigate the concept of related rates. The session will begin with a review of literal equations and how these equations can be emphasized from sixth grade through pre-calculus. Teachers will explore dynamic situations where a change in one quantity results in a change in another quantity through related rates applications involving triangles, curves, areas, and volumes.