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E-Examples from Principles and Standards for School Mathematics





Pre-Kindergarten Through Grade 2   Standards  
4.1  Creating, Describing, and Analyzing Patterns to Recognize Relationships and Make Predictions

This three-part example highlights different aspects of students' understanding and use of patterns as they analyze relationships and make predictions, as discussed in the Algebra Standard.
4.2  Investigating the Concept of Triangle and Properties of Polygons

This two-part example describes activities using interactive geoboards to help students identify simple geometric shapes, describe their properties, and develop spatial sense.
4.3  Learning Geometry and Measurement Concepts by Creating Paths and Navigating Mazes

The three-part ladybug example presents a rich computer environment in which students can use their knowledge of number, measurement, and geometry to solve interesting problems. Planning and visualizing, estimating and measuring, and testing and revising are components of the ladybug activities. These interactive figures can help students build ideas about navigation and location, as described in the Geometry Standard, and use these ideas to solve problems, as described in the Problem Solving Standard.


4.4  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 example demonstrates the potential for high-quality experiences provided by computer "shape" environments for students as they learn concepts described in the Geometry Standard. Problem-solving tasks that involve physical manipulatives as well as virtual manipulatives afford many students an entry into mathematics that they might not otherwise experience.
4.5  Learning about Number Relationships and Properties of Numbers Using Calculators and Hundred Boards

A major learning goal for students in the primary grades is to develop an understanding of properties of, and relationships among, numbers. Building on students' intuitive understandings of patterns and number relationships, teachers can further the development of number concepts and logical reasoning as described in the Number and Operations and Reasoning and Proof Standards. In this two-part example virtual 100 boards and calculators furnish a visual way of highlighting and displaying various patterns and relationships among numbers. Using calculators and hundred boards together, teachers can encourage students to communicate their thinking with others, as discussed in the Communication Standard.
Number and Operations
and Proof


4.6  Developing Estimation Strategies by Making Connections among Number, Geometry, Measurement, and Data Concepts

Estimation activities encourage students to make connections among the mathematics concepts they are learning and the skills they are developing. In this multipart video example, the decisions the teacher makes and the class discussions contribute to students' opportunities to connect their understandings of number, measurement, geometry, and data in order to make estimates. Purposeful activities together with skillful questioning by the teacher can help students understand relationships among mathematical ideas as described in the Connections Standard.
Grades 3–5   Standards  
5.1  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.

5.2  Understanding Distance, Speed, and Time Relationships Using Simulation Software

This 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. The computer simulation uses a context familiar to students, and the technology allows them to analyze the relationships more deeply because of the ease of manipulating the environment and observing the changes that occur. Activities like these can help students in the upper elementary grades understand ideas about functions and about representing change over time, as described in the Algebra Standard.
5.3  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 example, properties of rectangles and parallelograms are examined. The emphasis is on identifying what distinguishes a rectangle from a more general parallelogram. Such tasks and the software can help teachers address the Geometry Standard.
5.4  Accessing and Investigating Data Using the World Wide Web

Data sets available on the Internet are valuable resources for studying real data to address questions that interest students. Teachers and students can download data sets from the World Wide Web, collaborate in online data-collection projects, and search electronic libraries and data files. This example describes activities in which students can use census data available on the Web to examine questions about population. Working on such activities, students can also formulate their own questions and use the mathematics they are studying to address these questions. They can propose and justify conclusions that are based on data and design further studies on the basis of conclusions or predictions, as described in the Data Analysis and Probability Standard.
Data Analysis
and Probability 
5.5  Collecting, Representing, and Interpreting Data Using Spreadsheets and Graphing Software

Spreadsheets and graphing software are tools for organizing, representing, and comparing data. This activity illustrates how weather data can be collected and examined using these tools. Working on activities like these, students learn to set up a simple spreadsheet and use it in posing and solving problems, examining data, and investigating patterns, as described in the Representation Standard.
Grades 6-8   Standards  
6.1   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, students can investigate how changing the height of a rectangle with a fixed width changes its area. As discussed in the Number Standard, understanding multiplication by fractions and decimals can be challenging for middle-grades students if experiences with multiplication by whole numbers have led them to believe that "multiplication makes bigger." In these dynamic figures, the rectangle represents the familiar area model of multiplication; changing the rectangle's height can help students see the effect of multiplying a fixed positive number by numbers greater than one and less than one.

Number and Operations 
6.2  Learning about Rate of Change in Linear Functions Using Interactive Graphs

In this two-part 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). Beginning to understand the relationship between change and accumulation is a precursor to understanding calculus. This example illustrates the use of dynamic graphs to learn about change and linear relationships, as described in the Algebra Standard.
6.3  Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures

This two-part example illustrates how students can learn about the length, perimeter, area, and volume of similar objects using dynamic figures. Activities such as these can help students learn about geometric relationships among similar objects, as described in the Geometry Standard.
6.4  Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures

Rotations; translations, or slides; and reflections, or flips, are geometric transformations that change an object's position or orientation but not its shape or size. The interactive figures in this four-part example allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. Activities like these allow students to deepen their understanding of congruence, similarity, and reflection, and they also contribute to the study of transformations, as described in the Geometry Standard.

6.5  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 example presents one dynamic version of a demonstration of this relationship. Visual and dynamic demonstrations can help students analyze and explain mathematical relationships, as described in the Geometry Standard. The interactive figure in this activity can help students understand the Pythagorean relationship and gives them experience with transformations that preserve area but not shape.

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

Using interactive software, students can compare and contrast properties of measures of central tendency, specifically the influence of changes in data values on the mean and median. As 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, as discussed in the Data Analysis and Probability Standard.

Data Analysis and Probability 
Grades 9-12   Standards  
7.1  Learning about Properties of Vectors and Vector Sums Using Dynamic Software

This example illustrates how using a dynamic geometrical representation can help students develop an understanding of vectors and their properties, as described in the Number and Operations Standard. Students manipulate a velocity vector to control the movement of an object in a gamelike setting. In the first part, Components of a Vector, students will develop an understanding that vectors are composed of both magnitude and direction. In the second part, Sums of Vectors and Their Properties, students extend their knowledge of number systems to the system of vectors.
Number and Operations 
7.2  Using Graphs, Equations, and Tables to Investigate the Elimination of Medicine from the Body

This three-part 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 is adapted from High School Mathematics at Work, a publication from the National Research Council (1998, p. 80). These activities allow high school students to study modeling in greater depth, as described in the Algebra Standard. Through multiple representations of a common concept, better insight into, and a deeper understanding of, the problem situation can be achieved.

7.3  Understanding Ratios of Areas of Inscribed Figures Using Interactive Diagrams

This example illustrates how students, using dynamic and interactive geometric figures, can understand connections between algebra and geometry, as described in the Connections Standard. They can develop an understanding of how to justify geometric relationships in a technological environment, as described in the Geometry Standard.

7.4  Understanding the Least Squares Regression Line with a Visual Model

This example allows students to explore three methods for measuring how well a linear model fits a set of data points. The Data Analysis and Probability Standard calls for students to explore how residuals (the difference between a predicted and observed value) may be used to measure the "goodness of fit" of a linear model. In this example, two of the methods use residuals and the third uses the shortest distance between a data point and the line given by the model. To introduce the idea of a measure of fit, in the first tasks a line is given and you explore the effects that six data points have on three measures of error. However, it rarely happens that the model is known and the data are not. Generally, we know the data and need to find a linear model. The additional tasks provide an opportunity to suggest and evaluate a variety of linear models and methods for a particular set of data.
Data Analysis
and Probability 
7.5  Exploring Linear Functions: Representational Relationships

Technology allows the linking of multiple representations of mathematical situations and the exploration of the relationships that emerge. This example presents a series of explorations based on two linked representations of linear functions. On pages 338–40, the grades 9–12 section on the Problem Solving Standard includes an episode describing how a teacher engaged her students in problem solving and reasoning with tasks such as those presented in this example.
Problem Solving  

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