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. The first part, Making Patterns, includes an interactive figure for creating, comparing, and viewing multiple repetitions of patterns. The interactive figure illustrates how students can create pattern units of squares then predict how patterns with different numbers of squares, will appear when repeated in a grid, and check their predictions. In the second part, Describing Patterns, examples of various ways students might interpret the same sequence of cubes are given. This illustrates the importance of discussing and analyzing patterns in the classroom. The third part, Extending Pattern Understandings, demonstrates ways in which students begin to create a "unit of units," or a grouping that can be repeated, and begin to relate two patterns in a functional relationship.
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. The first part, Making Triangles, focuses on the concept of triangle, helping students understand the mathematical meaning of a triangle and the idea of congruence, or sameness, in geometry. In the second part, Creating Polygons, students make and compare a various polygons, describing the salient properties of the shapes they create. Both parts help students reach the goals described in the Geometry Standard.
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. In the first part, Hiding Ladybug, students create a path that enables the ladybug to hide under a leaf. In the second part, Making Rectangles, students plan the steps necessary for the ladybug to draw rectangles of different sizes. In the last part, Ladybug Mazes, students plan a series of moves that will take the ladybug through a maze.
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. In Tangram Puzzles, students can choose a picture and use all seven pieces to fill in the outline. In the second part, Tangram Challenges, students can use tangram pieces to form given polygons.
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. In the first part, Displaying Number Patterns, the same patterns are displayed on a calculator and on a hundred board simultaneously. In the second part, Patterns to 100 and Beyond, students examine number patterns, using a calculator to move beyond 100.
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. In the first part, Estimating Scoops, the teacher presents an estimation task (estimate the number of scoops of cranberries in a jar) to the second-grade students and talks about the teaching decisions she is making. In the second part, Discussing Strategies, the students work in groups to share their ideas and reach a reasoned consensus about their estimates. In the third part, Estimating Cranberries, the students estimate the number of cranberries rather than the number of scoops.
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. The first part of this example, Playing Fraction Tracks, contains an interactive version of a game (based on the work of Akers, Tierney, Evans, and Murray 1998) that can be used in the grades 3–5 classroom to support students' learning about fractions. By working on this activity, students have opportunities to think about how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions, as discussed in the Number and Operations Standard. In the second part, The Role of the Teacher, two video clips illustrate communication about mathematics among a teacher and her students. The third part, Communication among Students, shows how activities like this allow students to use communication as a tool to deepen their understanding of mathematics, as described in the Communication Standard. In the fourth part, Reflecting on Practice, the teacher reflects on her own mathematical learning that occurs as a result of using activities like this game with her 5th-grade students.
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.
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. In the first part, Collecting and Examining Weather Data, students organize and then examine data that has been collected over a period of time in a spreadsheet. In the second part, Representing and Interpreting Data, students use the graphing functions of a spreadsheet to help them interpret data. 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.
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 1 and less than 1.
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). In the first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes 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. In the first part, Side Length and Area of Similar Figures, the user can manipulate the side lengths of one of two similar rectangles and the scale factor to learn about how the side lengths, perimeters, and areas of the two rectangles are related. In the second part, Side Length, Volume, and Surface Area of Similar Solids, the user can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships among edge lengths, surface areas, and volumes. 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. In the first part, Visualizing Transformations, one can choose a transformation and apply it to a shape to observe the resulting image. In the next part, Identifying Unknown Transformations, the user is challenged to identify the transformation that has been used. In Composing Reflections, users can examine the result of reflecting a shape successively through two different lines. And in the fourth part, Composing Transformations, the users are challenged to compose equivalent transformations in two different ways. 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 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, as discussed in the Data Analysis and Probability Standard.
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.
7.2 Using Graphs, Equations, and Tables to Investigate the Elimination of Medicine from the Body
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. In the
first part, Modeling the Situation, an interactive environment is
used to become familiar with the parameters involved and the range
of results that can be obtained. In the second part, Long-Term Effect,
the interactive environment is used to investigate how changing
parameter values affects the stabilization level of medicine in
the body. In the third part, Graphing the Situation, an interactive
graphical analysis provides a visual interpretation of the results.
Through multiple representations of a common concept, better insight
into, and a deeper understanding of, the problem situation can be
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
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.
7.5 Exploring Linear Functions: Representational Relationships
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 33840, the grades 912
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.