These discussions are designed to give teachers ideas for how to introduce or explain a concept with a student or with a class. Informal and formal definitions of concepts as well as common student misconceptions are included in the dialogue. These discussions are best accessed from the lessons in which they are used, but are listed here for quick reference. Teachers who develop discussions similar to these for other topics are encouraged to submit them to the Project Interactivate. Use this form to contact the Interactivate Team for details. They are arranged according to the NCTM Principles and Standards for School Mathematics. 


Discussion:  Description: 
Integers  Introduces the concept of an Integer. 
Integer Addition and Subtraction  Introduces the Addition and Subtraction of Integers. 
Integer Multiplication  Introduces the Multiplication of Integers. 
Integer Division  Introduces the Division of Integers. 
Fractions  Discusses the introductory concept of a fraction. 
Comparing Fractions  Introduces students to the basics of reducing fractions and learning to compare fractions. 
Decimals  Deals with converting fractions into decimals. 
Multiplying Decimals and Mixed Numbers  A review of the definition of decimals and mixed numbers as well as a description of multiplying decimal numbers. 
Percents  Covers the basics of converting fractions into percents. 
Fraction Adding and Subtracting  Demonstrates how fractions are added and subtracted. 
Fraction Multiplying and Dividing  Explains multiplication and division of fractions. 
Making Estimates  Introduces students to estimation. 
Sets and Elements  Gives an introduction to sets and elements. 
Venn Diagrams  Introduces concepts needed to create Venn Diagrams. 
Algorithms  Introduces the concept of algorithms and how algorithms affect mathematics. 
What are Multiples?  Discusses integer multiples as repeated addition. 
What are Remainders?  Reviews long division of integers and modular arithmetic. 
Clocks and Modular Arithmetic  Shows how modular arithmetic can be thought of as clock arithmetic. 
Cryptography and Ciphers  Introduces the notion of using modular arithmetic to encode messages. 
Infinity and Iteration  Discusses infinity, iterations and limits by referencing fractals and sequences. 
Recursion  Discusses the idea of recursion as it pertains to fractals and sequences. 
Exponents and Logarithms  Gives an introduction to the concept of a logarithms and shows how logs can be used to calculate fractal dimension. 


Discussion:  Description: 
Quadrilaterals  Introduces students to quadrilaterals and defines the characteristics of the polygon. 
Parallelograms  Introduces students to parallelograms and rombuses and defines the characteristics necessary to determine each shape. 
Rectangles  Introduces students to rectangles and squares and defines the characteristics necessary to determine each shape. 
Squaring the Triangle  Introduces students to the Pythagorean Theorem with explanations on what it means and how it is found. 
Trapezoids  Introduces students to trapezoids and isosceles trapezoids and defines the characteristics necessary to determine each shape. 
What are Tessellations?  Examines the mathematical properties of tessellations. 
Tessellations in the World  Looks at the history of tessellations, why they are important and examines some patterns in nature and art. 
Symmetry in Tessellations  Defines symmetry and demonstrates different types of plane symmetry. 
Color in Tessellations  Explains the effect that color has on the patterns we see in tessellations. 
Optical Illusions  Looks at several optical illusions. 
Shape Explorer  Introduces students to finding areas and perimeters of irregular shapes on a grid. 
Lines, Rays, and Planes  Introduces students to lines, rays, line segments, and planes. 
Parallel Lines  Discusses Parallel Lines 
Translations, Reflections, and Rotations  Introduces students to the concepts of transformations. 
Surface Area and Volume  Introduces students to the concepts of surface area and volume. 
SelfSimilarity  Discusses how fractals are selfsimilar objects. 
Plane Figure Fractals  Compares fractals with one and two dimensional generators. 
Properties of Fractals  Reviews Mandelbrot's defining characteristics for fractal objects. 
Dimension and Scale  Discusses fractal dimension, how that dimension relates to scale, and the formula needed to calculate the fractal dimension of an object. 
Chaos  Introduces the notion of chaos as the breakdown in predictability. 
Chaos is Everywhere  Shows the wide spread use of fractals and chaos in science and nature. 
Pascal's Triangle  Introduces Pascal's Triangle in terms of probability. 
Dimension for Irregular Fractals  Discusses the problem of determining the fractal dimension of irregular fractals and how the scale is indeterminite in these fractals. 
Prisoners and Escapees  Julia Sets  Defines the notion of prisoners and escapees as they pertain to iterative functions. A prisoner ultimately changes to a constant while escapees iterate to infinity. 
The Mandelbrot Set  Shows how the set of all Julia Sets are used to create the classic Mandelbrot fractal. 


Discussion:  Description: 
Functions as Processes or Rules  Discusses the notion of functions as a "number machine" with input and output. 
More Complicated Functions  Discusses the notion of composite functions as several "number machines" with the output of one machine becoming the input of another. 
Linear Functions  Discusses functions of the form y = ___*x + ___. 
Slope and YIntercept  Discusses Slope and Yintercept and how they affect a graph. 
Introduction to the Coordinate Plane and Coordinates  Introduces coordinates through the idea of number lines. 
From Graphs to Machines and Back  Demonstrates the initial connections between functions and their graphs. 
Functions and the Vertical Line Test  Shows students why a function must pass the vertical line test to be a function. 
Gathering Information from Graphs  Interpreting graphs and their how curved lines represent velocity on a graph of distance vs. time. 
Graphing Time, Distance, Velocity and Acceleration  Analyzing graphs and creating velocity graphs from distance and acceleration from velocity. 
Impossible Graphs  Shows what makes a graph represent impossible situations and how to avoid these problems. 
Two Variable Functions  Introduces 2 variable functions as ordered pairs and how to operate perform operations on ordered pairs. 


Discussion:  Description: 
Polyhedra  Questions about dice lead to a discussion of polyhedra and geometric probability. 
Fair Choice  The proper meaning of the term "fair." 
Random Number Generators  Different methods for random fair choice between several numbers. 
Probability and Outcome  Introduction and initial discussion of the concept of probability. 
Events and Set Operations  Introduction of elementary set operations and their connections with probability. 
Tables and Combinatorics  Discussion of tables as a convenient way to store and count outcomes. 
Divisibility  The question of fairness in a game of two dice leads to the concept of divisibility. 
Trees as Data Structures  Questions about games with more than two dice lead to discussion of trees as another kind of data structure. 
StemandLeaf Plots  Introduces StemandLeaf Plots to students. 
Probability of Simultaneous Events  Computing exact probabilities for the Racing Game leads to the formula for the probability of simultaneous events. 
Expected Value  Introduction and discussion of the concept of expected value. 
Probability and Geometry  Leads the idea of probability from counting chances to measuring proportions of areas. 
Conditional Probability  Introduction of the concept of conditional probability and discussion of its application for problem solving. 
Replacement  Extends the notion of conditional probability by discussing the effects of replacement on drawing multiple objects. 
Think and Check!  Some problems are tricky; probability theory provides unique ways to check solutions. 
Internet Search and Set Operations  Introduction of elementary set operations through internet searching. 
Probability vs. Statistics  Defining, comparing and contrasting probability and statistics. 
Mean, Median and Mode  Defining and discussing the concepts. 
The Normal Distribution and the Bell Curve  An introduction to the normal distribution and the debate over the 1994 book, "The Bell Curve." 
"The Bell Curve" Revisited  Finishes up the discussion of the book as well as exploring individual differences versus group expected values. 
Standard Deviation  Introduces standard deviaton and describes how to compute it. 
Continuous Distributions  Discusses continuous versus discrete distributions. 
Class Interval: Scale and Impression  How scales help to represent or misrepresent data in histograms. 
Vertical Scale: Increase or Decrease?  How class interval size influences the look and interpretation of histograms. 
Histograms vs. Bar Graphs  Differences and similarities between the two types of graphs. 
Box Plots  How to build box plots, including the two different ways to determine interquartile range. 