In this interactive, students must determine how many bubbles are left under Okta's shell after some have been added or taken away.

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. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The e-examples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math investigations.

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 from Illuminations, virtual hundred 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 this 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. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The e-examples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math investigations.

In this two-part example from Illuminations, 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 this 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. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The e-examples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math investigations.

In this task, students are given an opportunity to practice composing and decomposing shapes by creating their own puzzle. Students are given a square colored sheet of paper and an envelope. Students cut the square into four pieces and place the pieces in an envelope. Next, students trade puzzles and try to solve each other’s puzzles by reassembling the shapes into a square.

This task encourages students to brainstorm and share the different strategies they can use to solve the problem. The purpose of this task is to help enhance students' ability to fluently add by assisting them in finding the most effective strategies to use when adding.

Students to enter and compare numeric or algebraic expressions in this interactive pan balance, from Illuminations, . They can "weigh" the expressions they want to compare by entering them on either side of the balance, allowing them to practice arithmetic and algebraic skills, as well as to investigate the concept of equivalence.

The interactive pan balance uses multiple shapes with different values, allowing students to investigate what happens as different shapes are placed on the balance. This student interactive, from Illuminations, provides an interesting environment in which students can consider the concept of equivalence.

Students carry out an interactive, geometric "proof without words" for the algebraic technique of completing the square in this interactive. The page also includes directions and a link to the final solution.

Students explore the relationship between theoretical and experimental probabilitiesin this student interactive, from Illuminations. Students use this "box model" as a statistical device to simulate standard probability experiments such as flipping a coin or rolling a die.

Students investigate how changes in the scale factor influence the ratio of perimeters and the ratio of areas between two figures.

Composing Transformations--This is part four of a four-part e-example from Illuminations that features interactive figures that allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. In this part, Composing Transformations, the users are challenged to compose equivalent transformations in two different ways. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards for School Mathematics (PSSM). Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math Investigations.