This course focuses on several topics that cut across algebra and geometry. Emphasis is placed on using models to gain insight into concepts and using the graphing calculator as a problem-solving tool. Topics include the function concept, linear and quadratic functions and relations, systems of equations, Pythagorean theorem, the distance formula, and right-triangle trigonometry. Prerequisite: Math X468A (Perspectives on Algebra)

PYTHAGOREAN THEOREM AND DISTANCE FORMULA (PYTHM)
Participants carefully review the Pythagorean theorem, prove it geometrically and algebraically, and use it to solve real-world problems. Participants focus on the connection between the Pythagorean theorem and the distance formula, and use the distance formula to calculate distances between points on a square grid and in a coordinate plane.

RIGHT TRIANGLE TRIGONOMETRY (TRIG1)
Concepts of right triangle trigonometry are introduced as participants use direct and indirect measurements to examine relationships among sides and angles of right triangles. Participants then solve problems.

FUNCTIONS AND RELATIONS (FCTNS)
The lesson consists of five parts, which can be distributed over several days. The first two parts are “find a rule” activities that feature various patterns. The third part is a dice-throwing activity that introduces participants to products of sets, relations, and the vertical line property. The fourth and fifth parts are activities that illustrate the concepts of relation, graph, and function in different contexts: through a family tree and through sets in (x,y)-space.

THE TI-73 GRAPHING CALCULATOR (TI73)
Participants learn some basic functions of the TI-73 graphing calculator that will help them be successful in more advanced math topics. Participants practice using the graphing calculator by completing exercises on commonly used keystrokes, graphing functions, and plotting points. Finally, participants solve sample problems from the California Math Content Standards with paper and pencil as well as with the calculator, and reflect on the value of the exercises to them as learners and to their own students.

THE TI-84+ GRAPHING CALCULATOR (TI84+)
Participants learn some basic functions of the TI-84+ graphing calculator that will help them be successful in more advanced math topics. Participants practice using the graphing calculator by completing exercises on commonly used keystrokes, graphing functions, and plotting points. Finally, participants solve sample problems from the California Math Content Standards with paper and pencil as well as with the calculator, and reflect on the value of the exercises to them as learners and to their own students.

CIRCUMFERENCE VERSUS DIAMETER (CD)
As an introduction to linear functions, participants measure the circumference and diameter of several circular objects. Participants graph the circumference vs. the diameter of the objects on graph paper and find the line of best fit. Then, participants note how the slope of the line of best fit relates to the coefficient of the linear regression found on the graphing calculator. Finally, participants graph sets of linear functions on the calculator and examine the how the m and the b in y = mx + b affect the graph of a line.

MORE LINEAR SYSTEMS (SYS2)
Participants review techniques for solving systems of linear equations. As an introduction to applications of linear systems, participants develop linear functions to describe the cost of renting a car from three different companies. Then, participants plot the data and the find point of intersection. Finally, participants graph the three functions on their graphing calculator and find the point of intersection via ZOOM and TRACE.

FROM LINES TO PARABOLAS (QUAD1)
Participants explore the problem of finding the rectangle of a fixed perimeter with maximum area. They graph the associated quadratic area function and they interpret the meaning of the x-intercepts. Participants investigate quadratic functions that are the product of two linear functions. Participants examine how horizontal and vertical translations of the graph affect quadratic polynomials.

THE QUADRATIC FORMULA (QUAD2)
Participants apply the zero product property and the square root property to solve equations. Participants review how to model polynomial expressions and how to complete the square using math pieces. Participants then practice solving quadratic equations by completing the square. Finally, participants derive the quadratic formula by investigating a series of problems and apply it to factoring polynomials and solving equations.

PROTOPROBLEMS (PROTO)
Participants solve typical algebra textbook problems (protoproblems) using a range of problem-solving strategies. By solving these routine and nonroutine problems, participants increase their own problem-solving skills and examine alternative approaches to use with their own students. Participants present the approaches used in solving the problems and discuss implications for teaching.

FACTORING POLYNOMIALS (POLY2)
Participants examine various models for multiplication, discuss advantages and limitations of each model, and use the multiplication models to explore factoring patterns and complete the square. They make connections between the distributive property and polynomials written as sums or products.

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