MATH415
Seeing Math™: Transformations of Linear Functions (Grades 6-12) - 30hr

Learn how to help students grasp the symbolic representations of functions, while representing families of linear functions in multiple formats.

ALGEBRA

Goals and Objectives

Observe graphic and symbolic transformations of linear functions. You will be able to:

• Identify the relationships between the graphic and symbolic forms of linear functions
• Interpret changes to the graphic or symbolic forms of linear functions as transformations to the initial function

• Identify and use different symbolic forms for linear functions and equations by exploring the relevance and value of different forms in a variety of situations

Categorize, use, and represent families of linear functions in multiple formats. You will be able to:

• • Compare and contrast the symbolic forms of linear functions by identifying elements that make their graphs appear different
• • Use interactive software to link specific changes in the symbolic forms of linear functions to corresponding changes in their graphical forms, and vice versa

Interpret the concept of slope in different contexts. You will be able to:

• • Interpret slope as a rate of change, and demonstrate rate of change graphically, symbolically, and with virtual manipulatives
• • Interpret slope as a geometric ratio of rise over run

MATH420
Seeing Math™: Linear Equations (Grades 6-12) - 30hr

Develop strategies for teaching students to represent and manipulate linear equations. Examine the rationale behind symbol manipulation that maintains an equality or corresponding inequality. Use symbolic and graphic techniques to solve equations.

ALGEBRA

Goals and Objectives

Understand the rationale behind the rules of symbol manipulation that maintain an equality or corresponding inequality. The objectives for the participant are to be able to:

• Understand the relationship between variables by dynamically changing the values of x and y
• Develop solution techniques (including techniques for solving simultaneous equations) by comparing standard symbolic operations to graphic and area representations of equations

Deepen the distinction between equivalence of function and equality of value. The objectives for the participant are to be able to:

• Interpret the "=" sign in terms of equivalence, description of state, inviting a calculation, or defining a quantity

MATH425
Seeing Math™: Systems of Linear Equations (Grades 6-12) - 30hr

Learn how to help your students make the connections between the symbolic, graphic, and tabular representations of systems of linear equations. Understand each step in the solution process to help your students master the skills needed to solve these systems of equations.

ALGEBRA

Goals and Objectives

Learners will develop a deeper understanding of what it means to find the solution to a system of linear equations.

Learners will be able to:

• • Distinguish solutions to systems of linear equations from solutions to individual equations.
• • Represent the solution to a system of linear equations symbolically, graphically, and in tables.
• • Use multiple representations to interpret and understand different solution sets—one solution, no solution, or an infinite number of solutions.

Learners will recognize how valid procedures for solving systems of linear equations relate to the solution set.

Learners will be able to:

• • Predict the symbolic, graphic, and tabular results of intermediate steps in the solution process.
• • Demonstrate, in multiple representations, how valid operations may alter the solution set for individual equations in the system while preserving the solution set of the system.

Learners will recognize the steps students take when solving systems of linear equations.

Learners will be able to:

• • Interpret student thinking using multiple representations of systems of linear equations
• • Identify when students perform the procedures for solving a system of linear equations without fully understanding their meaning
• • Link their curricula and teaching practice to strategies that support student learning concepts that are fundamental to understanding systems of linear equations

MATH430
Seeing Math™: Proportional Reasoning (Grades 6-12) - 30hr

Learn effective strategies to evaluate and improve students' proportional reasoning skills. Distinguish proportional reasoning from alternative strategies, overcome stumbling blocks, connect proportional reasoning concepts.

ALGEBRA

Goals & Objectives

Explore the relationship between proportional reasoning and algebraic thinking. Learners will be able to:

• Understand why multiplicative thinking is not only basic to proportional reasoning, but essential to algebra

• Analyze the use of proportional reasoning in typical Algebra 1 problems Understand that students who appear to reason proportionally may in fact be following a procedure without understanding it. Learners will be able to:

• Identify facets of student thinking that may become stumbling blocks to proportional reasoning
• Describe student thinking in terms of four common strategies used in reasoning about proportion problems
• Learn techniques that help to identify true proportional reasoning

Understand methods for developing students' proportional reasoning. Learners will be able to:

• Identify situations that require reasoning beyond additive thinking, and create bridges to students' development of multiplicative thinking
• Identify, in curriculum materials, approaches that confirm, extend, or hinder the development of proportional reasoning skills

MATH435
Seeing Math™: Quadratic Functions (Grades 6-12) - 30hr

Acquire new ways to teach quadratic functions through modeling and problem solving. Use multiple representations — tables, graphs, and symbolic expressions — as powerful tools to model physical situations and predict patterns.

ALGEBRA

Goals & Objectives:

Search for patterns and use quadratic functions to model physical situations. Learners will be able to:

• • Engage in problem solving and understand the value of different solutions to the same problem
• • Describe a physical situation using symbols, and then interpret the symbols by mapping them back onto the physical situation
• • Distinguish between recursive and explicit patterns

Interpret the meaning and characteristics of quadratic functions as they appear in different representations. Learners will be able to:

• Interchange symbolic, graphic, numeric, and verbal representations of the same quadratic function with versatility
• • Link characteristics in one representation to corresponding characteristics in another representation
• • Use appropriate terminology to describe various characteristics

Link a personal understanding of quadratic functions to their curriculum and to students' understanding. Learners will be able to:

• • Gain skill in listening and interpreting student thinking accurately
• • Explore and generate ways to support students' understanding
• • Relate their curriculum’s treatment of quadratic equations to a thorough understanding of quadratic functions

MATH440
Seeing Math™: Transformations of Quadratic Functions (Grades 6-12) - 30hr

Learn how specific alterations to a quadratic function create an entire family of quadratic functions that is related to the original function in predictable ways. See how working with these families can help you and your students understand the traits the make a function quadratic.

ALGEBRA

Goals & Objectives

Learners predict, observe, and compare the ways that changes to graphical representations of quadratic functions mirror corresponding changes to their symbolic representations, and vice versa.

Learners will be able to:

• • Transform functions both symbolically and graphically
• • Distinguish between translations, dilations, and reflections of quadratic functions

Learners identify and characterize relationships among the three different symbolic forms for a quadratic function.

Learners will be able to:

• • Compare the three symbolic representations of quadratic functions (polynomial, vertex, and root forms) and become familiar with what information is conveyed in each
• Exploit function notation as a useful tool to visualize the relationships among related quadratic functions

Learners use transformation as a vehicle to understand the characteristics of quadratic functions in various representations, and solve related problems.

Learners will be able to:

• • Recognize the characteristics that differentiate quadratic functions from linear functions
• • Shift between the object nature and process nature of functions
• • Help students to shift between object and process viewpoints to support problem solving

MATH445
Seeing Math™: Quadratic Equations (Grades 6-12) - 30hr

Move beyond tried-and-true manipulations to examine the breadth of information available from quadratic equations. Look at the big picture: what the results reveal, how to interpret them within the context of a problem, and hot to find related information.

ALGEBRA

Goals & Objectives

Learners use quadratic equations to identify the defining characteristics of a quadratic function.

Learners will be able to:

• • Distinguish how the characteristics are represented symbolically and graphically
• Recognize patterns and relationships among certain characteristics of the function (for example, deducing the y-intercept by noticing patterns in the roots.)
• • Describe and justify the meaning of the vertex, y-intercept, and x-intercept(s) by analyzing the behavior of a quadratic function at and around those points

Learners use different symbolic forms of a quadratic function to find important information.

Learners will be able to:

• • Use the relationships between the three symbolic forms to gather information about the function at critical points
• • Solve quadratic equations by representing and analyzing the function graphically and symbolically
• • Connect traditional procedures for solving equations (for example, the quadratic formula) to symbolic and graphical representations of the function

Learners link personal understanding of quadratic equations to the curriculum and to student understanding.

Learners will be able to:

• • Gain skill in interpreting student thinking about quadratic functions
• • Interpret solutions to quadratic equations and identify information about critical points of the functions in the context of real world situations
• • Adapt curriculum materials to bring the learning from this course into your classroom

MATH455
Seeing Math™: Data Analysis (Grades 6-12) – 30 hr

What do the measures of central tendency—mean, median, and mode—tell you about the data? Teach your students the meaning of these simple measures and the ways that they characterize the data set as a whole.

ALGEBRA

Learners will understand and distinguish among tools used to analyze data.

Learners will be able to:

• • Distinguish between using an algorithm and understanding the underlying mathematical relationships among measures of center in a data set
• • Identify how different representations highlight particular characteristics of a set of data
• • Describe the relationships among the measures of center, and show how they reflect changes in individual values

Learners will guide the development of understanding from individual case to aggregate.

Learners will be able to:

• • Identify, in their work and discussion, students' developing sense of the aggregate.
• • Select graphic representations that direct students' attention to specific attributes of a data set, particularly those of the aggregate.

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