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i-Math Investigations are ready-to-use, online, interactive, multimedia math investigations. Complete i-Maths include student investigations, teacher notes, answers, and related professional development activities. (Not every i-Math is currently complete, but they are all ready to be used. To get an idea of what a complete i-Math looks like, see Shedding Light on the Subject: Function Models of Light Decay.)

Student i-Maths: For each i-Math Investigation, there is a specially-formatted student version that is classroom-ready. Student versions contain no irrelevant links and all teacher notes, answers, and professional development activities have been removed. All student i-Maths are listed on the Student i-Math page. This is the only page that needs to be bookmarked for classroom use.

e-Math Investigations are selected e-examples from the electronic version of the NCTM Principles and Standards for School Mathematics. These investigations can be identified by the icon to the right. Given their interactive nature and focused discussion tied to the Principles and Standards document, the e-examples are natural companions to the i-Math Investigations. Some of the e-examples will be expanded to complete i-Math Investigations. We'd love to hear which ones you'd like to see done first.


Developing Geometry Concepts Using Computer Programming Environments

Computer programming environments can be used to help children understand geometric concepts. The interactive tool in this i-Math investigation illustrates how LOGO can be used to foster creative problem solving and encourage young students to estimate length and angle measures.

Understanding a Child's Development of Number Sense

This i-Math Investigation is coming soon. You can see the related Reflection on Teaching activity now.

Creating, Describing, and Analyzing Patterns to Recognize Relationships and Make Predictions

This three-part e-example highlights different aspects of students' understanding and use of patterns as they analyze relationships and make predictions. The example includes an interactive figure for creating, comparing, and viewing multiple repetitions of patterns.

Investigating the Concept of Triangle and Properties of Polygons

This two-part e-example describes activities using interactive geoboards to help students identify simple geometric shapes, describe their properties, and develop spatial sense. A virtual geoboard is available for constructing triangles and polygons.

Developing Geometry Understandings and Spatial Skills through Puzzlelike Problems with Tangrams

Describing figures and visualizing what they look like when they are transformed through rotations or flips or are put together or taken apart in different ways are important aspects of geometry in the lower grades. This two-part tangram e-example demonstrates the potential for high-quality experiences provided by computer "shape" environments for students as they learn concepts.

Learning about Number Relationships and Properties of Numbers Using Calculators and Hundred Boards

Building on students' intuitive understandings of patterns and number relationships, teachers can further the development of number concepts and logical reasoning. In this two-part e-example virtual 100 boards and calculators furnish a visual way of highlighting and displaying various patterns and relationships among numbers.


Exploring Geometric Solids and Their Properties

Investigating and then reasoning about the relationships within and between three-dimensional shapes is important for students in grades 3-5 as they continue to develop understanding about geometry and spatial sense. The interactive tools in this i-Math investigation are designed to allow students to explore geometric solids and their properties.

Gathering Evidence About Students' Understanding of Volume

This i-Math Investigation is coming soon. You can see the related Reflection on Teaching activity now.

Teaching, Learning, and Communicating About Fractions

This i-Math Investigation is coming soon. You can see the related Reflection on Teaching activity now.

Communicating about Mathematics Using Games

Mathematical games can foster mathematical communication as students explain and justify their moves to one another. In addition, games can motivate students and engage them in thinking about and applying concepts and skills. This e-example contains an interactive version of a game that can be used in the grades 3-5 classroom to support students' learning about fractions.

Understanding Distance, Speed, and Time Relationships Using Simulation Software

This e-example includes a software simulation of two runners along a track. Students can control the speeds and starting points of the runners, watch the race, and examine a graph of the time-versus-distance relationship. This kind of experience helps students understand ideas about functions and about representing change over time.

Exploring Properties of Rectangles and Parallelograms Using Dynamic Software

Dynamic geometry software provides an environment in which students can explore geometric relationships and make and test conjectures. In this e-example, properties of rectangles and parallelograms are examined. The emphasis is on identifying what distinguishes a rectangle from a more general parallelogram.

Collecting, Representing, and Interpreting Data Using Spreadsheets and Graphing Software

Spreadsheets and graphing software are tools for organizing, representing, and comparing data. This e-example illustrates how weather data can be collected and examined using these tools.


Analyzing Numeric and Geometric Patterns of Paper Pool

The interactive paper pool game in this i-Math investigation provides an opportunity for students to further develop their understanding of ratio, proportion and least common multiple. Students develop rules for predicting the corner (pocket), number of bounces and the length of the path for a cue ball on rectangular pool tables of different sizes.

Simulating Probability Situations Using Box Models

Investigations that allow students to explore the relationship between theoretical and experimental probabilities are important in the study of probability and statistics. A "box model" can help students in such investigations. This i-Math investigation contains an interactive tool that simulates a statistical "box model" as well as ideas for activities.

Exploring Histograms

A histogram is a standard way of representing a collection of data. The interactive tool contained in this i-Math investigation allows students to create their own sets of data and examine how various statistical functions such as mean, median, and standard deviation depend on the choice of data.

Gathering Evidence About Students' Understanding of Volume

This i-Math Investigation is coming soon. You can see the related Reflection on Teaching activity now.

Learning about Multiplication Using Dynamic Sketches of an Area Model

Students can learn to visualize the effects of multiplying a fixed positive number by positive numbers greater than 1 and less than 1 with this tool. Using interactive figures within this e-example, students can investigate how changing the height of a rectangle with a fixed width changes its area.

Learning about Rate of Change in Linear Functions Using Interactive Graphs

In this two-part e-example, users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). The investigation explores the case when the cost per minute for phone use remains constant over time and the case when the cost per minute for phone use changes after the first sixty minutes of calls.

Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures

This two-part e-example illustrates how students can learn about the length, perimeter, area, and volume of similar objects using dynamic figures.

Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures

The interactive figures in this four-part e-example allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations.

Understanding the Pythagorean Relationship Using Interactive Figures

The Pythagorean relationship, a2 + b2 = c2 (where a and b are the lengths of the legs of a right triangle and c is the hypotenuse), can be demonstrated in many ways, including with visual "proofs" that require little or no symbolism or explanation. The activity in this e-example presents one dynamic version of a demonstration of this relationship.

Comparing Properties of the Mean and the Median through the use of Technology

Using interactive software in this e-example, students can compare and contrast properties of measures of center, specifically these tasks illustrate how changes in data values influence the mean and median. When students change the data values, the interactive figure immediately displays the mean and median of the new data set. Experimenting with this software helps students compare the utility of the mean and the median as measures of center for different data sets.


Put the Heart into Mathematics: Cardiac Output, Rates of Change and Accumulation

This activity explores the measurement of the amount of blood being pumped by a heart. The goal of this i-Math investigation is a rich exploration of concentrations (mixture problems), rates of change and accumulation in context. A multitude of resources are available to help understand the science and context of this investigation.

Investigating Linear Relationships: The Regression Line and Correlation

Interactive computer-based tools allow students the opportunity to manipulate numbers easily and investigate the relationships between data points and sets. This i-Math Investigation uses an interactive linear regression tool to help students develop a conceptual understanding of the least-squares regression line and correlation.

Using Algebra and Discrete Mathematics to Investigate Population Changes in a Trout Pond

This investigation illustrates the use of iteration, recursion, and algebra to model and analyze a changing fish population. Graphs, equations, tables, and technological tools are used to investigate the effect of varying parameters on the long-term population.

Shedding Light on the Subject: Function Models of Light Decay

In this investigation students develop and analyze exponential models for the behavior of light passing through water. Accompanying video clips show the i-Math in action in real classrooms.

Whelk-Come to Mathematics: Using Rational Functions to Investigate the Behavior of Northwestern Crows

In this investigation students use rational functions to explore the possible reasons behind the observation that northwestern crows consistently drop a type of mollusk called a whelk from a height of 5 meters.

Learning about Properties of Vectors and Vector Sums Using Dynamic Software

This e-example illustrates how using a dynamic geometrical representation can help students develop an understanding of vectors and their properties. Students manipulate a velocity vector to control the movement of an object in a gamelike setting to develop an understanding that vectors are composed of both magnitude and direction and to explore the sums of two vectors.

Using Graphs, Equations, and Tables to Investigate the Elimination of Medicine from the Body

This three-part e-example illustrates the use of iteration, recursion, and algebra to model and analyze the changing amount of medicine in an athlete's body. This example includes: (1) an interactive environment to become familiar with the parameters involved and the range of results that can be obtained, (2) an interactive environment is used to investigate how changing parameter values affects the stabilization level of medicine in the body, and (3) an interactive graphical analysis provides a visual interpretation of the results.

Understanding Ratios of Areas of Inscribed Figures Using Interactive Diagrams

This e-example illustrates how students, using dynamic and interactive geometric figures, can understand connections between algebra and geometry. They can develop an understanding of how to justify geometric relationships in a technological environment.

Understanding the Least Squares Regression Line with a Visual Model

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Exploring Linear Functions: Representational Relationships

Technology allows the linking of multiple representations of mathematical situations and the exploration of the relationships that emerge. This e-example presents a series of explorations based on two linked representations of linear functions using an interactive figure.


Exploring Connections Between Mathematics and Art
- Under Construction

In this i-Math investigation you will be able to take a virtual tour of the San Francisco Museum of Modern Art and explore connections between art and mathematics. You can take the tour and do a few activities now. More coming soon!


© 2000 - National Council of Teachers of Mathematics

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Last updated: February 3, 2001