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 are given.
Common student misconceptions are pointed out as well. Teachers
might have students read them or, more commonly, use ideas
from them to guide class discussions.
They are arranged according to the
NCTM Principles and Standards for
School Mathematics.
Number and Operation Concepts: Includes
working with fractions and decimals, clock arithmetic, and finding number
patterns. (NCTM Content Standard)
Geometry and Measurement Concepts:
Includes basic notions of lines, rays and planes, working with tessellations,
fractals. (NCTM Content Standard)
Function and Algebra Concepts: Includes
an introduction to functions, special properties of linear functions, graphs
and the coordinate plane, and reading graphs.
(NCTM Content Standard)
Probability and Data Analysis Concepts:
Includes an introduction to probability, conditional probability, sampling,
expected value, statistics, histograms, boxplots, and the normal curve. Also includes statistical simulations.
(NCTM Content Standard)
Number and Operation Concepts
|
| Fractions
|
Discusses the introductory concept of a fraction.
|
| Comparing Fractions
|
Introduces students to the basics of reducing fractions and learning to identify which of two fractions is larger or smaller.
|
| 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.
|
| Sets and Elements |
Gives an introduction to sets and elements. |
| Venn Diagrams |
Introduces Venn Diagrams. |
| What are Multiples? |
Reviews multiplication of integers 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 and repeating processes infinitely many times |
| Recursion |
Discusses recursion from the point of view of fractals |
| Exponents and Logarithms |
Gives an introduction to logs, needed for calculating fractal dimension |
Geometry and Measurement Concepts
|
| 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. |
| Trapezoids |
Introduces students to trapezoids and isosceles trapezoids and
defines the characteristics necessary to determine each shape. |
| What are Tessellations? |
Examines the 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 |
Examines optical illusions. |
| Shape Explorer |
Introduces students to finding areas and perimeters of
irregular shapes. |
| Lines, Rays, and Planes
|
Introduces students to lines, rays, line segments, and planes.
|
| 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. |
| Self-Similarity |
Discusses self-similarity from the point of view of fractals |
| Plane Figure Fractals |
Compares line bending fractals to plane figure fractals |
| Properties of Fractals |
Discusses fractal properties and fractal dimension |
| Dimension and Scale |
Discusses what dimension is and how it is related to scale, giving the relationship needed to calculate fractal dimension |
| Chaos |
Introduces the notion of chaos as the breakdown in predictability |
| Chaos is Everywhere |
Mentions the wide spread use of fractals and chaos in science |
| Pascal's Triangle |
Introduces Pascal's Triangle |
| Dimension for Irregular Fractals |
Looks at random fractals and how self-similarity is generalized |
| Prisoners and Escapees -- Julia Sets |
Looks at iterating functions and starting points that stay close versus go off to infinity. |
| The Mandelbrot Set |
Shows how Julia sets for functions (a,b)^2 + (c,d) are used to make the classic Mandelbrot set. |
Function and Algebra Concepts
|
| Functions as Processes or Rules |
"Function machines" discussion |
| More Complicated Functions |
"Composite function machines" discussion |
| Linear Functions |
"Linear Function machines" discussion |
| Introduction to the Coordinate Plane and Coordinates |
First introduction to coordinates |
| From Graphs to Machines and Back |
Graphing and function rules: first connections |
| Functions and the Vertical Line Test |
The verical line test for functions |
| Gathering Information from Graphs |
From graphs to stories |
| Graphing Time, Distance, Velocity and Acceleration |
Analyzing graphs and building new graphs from old ones |
| Impossible Graphs |
What makes a graph impossible and how to avoid these problems
|
| Two-Variable Functions |
Introduces two-variable functions of the form (a,b)^2 + (c,d) |
Statistics and Probability Concepts
|
| Polyhedra |
Questions about dice lead to discussion of polyhedra |
| Fair Choice |
What does it mean exactly to play fair or to choose fair? |
| Random Number Generators |
What methods are there 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 two dice game leads to divisibility discussion |
| Trees as Data Structures |
Questions about games with more than two dice lead to discussion of trees as another kind of data structure |
| Stem-and-Leaf Plots |
Introduces Stem-and-Leaf 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 |
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 the solutions |
| Internet Search and Set Operations |
Introduction of elementary set operations through internet searching |
| Probability vs. Statistics |
Definitions of statistics and probability. Comparing and contrasting probability and statistics |
| Mean, Median, and Mode |
Mean, median, and mode |
| The Normal Distribution and the Bell Curve |
An introduction to both the normal distribution and the debate over the 1994 book |
| "The Bell Curve" revisited |
Finishes up the discussion of the book as well as exploring individual differences versus group expected values
|
| Continuous Distributions |
Discusses continuous versus discrete distributions |
| Class Interval: Scale and Impression |
How scales help to represent or mis-represent data in histograms |
| Vertical Scale: Increase or Decrease? |
Class interval size influences the look and interpretation of histograms |
| Histograms vs. Bar Graphs |
Differences and similarities
|
| Box Plots |
How to build box plots, including medians and quartiles |
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