1. Numbers and Operations
   1. 6.1 Learning about Multiplication Using Dynamic Sketches of an Area Model
      1. 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.
2. Algebra
   1. 6.2 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.
3. Geometry
   1. 6.3 Learning about Length, Perimeter, Area, and Volume of Similar Objects Using Interactive Figures
      1. 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.
   2. 6.4 Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures
      1. 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
   3. 6.5 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
4. Data Analysis and Probability
   1. 6.6 Comparing Properties of the Mean and the Median through the use of Technology
      1. 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.