Created by Evan Glazer, eglazer@coe.uga.edu

The following design document is a proposal for two Explore Math activities. The Explore Math Website (http://www.exploremath.com) currently contains over 45 interactive investigations of mathematical concepts that relate to functions, points, conic sections, and probability. Each activity has been created with Macromedia Director, and then translated into Shockwave so that it can be seen on the Web (Instructions for downloading the free Shockwave plug-in can be found at http://www.exploremath.com/about/shockerhelp.cfm). Once Shockwave is installed on a computer, users can participate in an interactive environment on the Web. In mathematics related activities, students can drag objects around the screen and observe changes in the equation or position of a graph. Hence, each activity on the Explore Math Website functions as a tool for exploring mathematics instead of as a tutor that teaches or quizzes students.

Figure 1. The Explore Math Website (http://www.exploremath.com).

Each activity on the Explore Math Website contains tools to modify the window of the graph, view coordinates on the graph, and view a table that describes values of the function on any specified interval. In addition, sliders are used to adjust variables and view adjustments in the graph as subtle changes are applied to the function. For instance, in the activity on logarithms, a student can modify the value of the base of a logarithm by dragging a slider or inputting a value and then view simultaneous changes in the graph. This feature gives students an opportunity to visualize how the base of a logarithmic function affects the steepness, asymptotes, and intercepts of the graph.

Figure 2. The logarithm activity on Explore Math without the checkbox features activated (http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=7).

Many of the Explore Math activities include show and hide features to present exploration opportunities without cluttering the screen. Checkboxes next to the graph are used to activate this feature if the user want to view its associated exponential function or the line y = x. In this case, it is useful to separate these two features so that students can hypothesize the relationship between a logarithm function and its inverse. The template uses different colors to amplify the features between the equation and graphs once the checkboxes are activated (see Figure 3).

Figure 3. The logarithm activity on Explore Math with the checkbox features activated (http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=7).

The Web-based activities at Explore Math can be used in a face to face classroom setting through a lab or presentation station, depending on the teacher's access to computers. A lab setting is ideal since it gives students a chance to explore math (excuse the pun) at their own pace, yet the design is flexible enough to be used as a presentation tool from a teacher workstation. Thus, this design can be used from two epistemological rationales to accommodate the pragmatic and philosophical perspectives of teaching and learning within the classroom. In one sense, the students construct their own representations of the mathematics they visualize by making conjectures about the patterns they see on the screen and the connections they synthesize from prior knowledge. However, the teacher can also create a guided discovery activity that leads students to follow prescribed methods and arrive at a predetermined solution. Hence, the instructional methods determined by the teacher will influence how the activities are used. The absence of a directed lesson on the site promotes flexibility among teachers to make the best use of the Explore Math activities for their own instructional situation. However, if a teacher is not sure how the activity can be used in class, the Web site provides "Explorations" that include a variety of questions related to the activity. In addition, some lesson plans have been submitted by teachers who have been using the site with their classes.

If necessary, the learning environment can be explored independently outside of a classroom setting if the student is learning at a distance, having fun exploring math, or is looking for a visual representation of an idea presented in a class. In this case, if someone is not on site to assist a student, then a teacher or another learner can be available asynchronously to answer questions, discuss ideas, or provide feedback. This possibility most closely aligns with the Institute for Distance Education's model for Independent Learning and Open Learning (see http://www.umuc.edu/ide/modldata.html for more details).

Explore Math activities are intended to be used within a mathematics classroom that has access to the Internet. They are not intended for a specific age group or math subject, but are instead to be used when teaching a particular mathematical concept (such as coordinates, functions, or rotations). Most of the content of the Explore Math Website would be most suitable for algebra, advanced algebra, precalculus, and calculus courses offered in high school. Each lesson on the Website has a set of recommended prerequisite skills to inform the teacher and learner of the activity's appropriateness. For example, a sample lesson related to logarithmic functions states that students should have the follow prerequisite knowledge:

Students are able to:

• Graph linear, quadratic, and exponential functions.
• Solve equations for a given variable.
• Evaluate exponential equations involving e.

Furthermore, each activity contains a set of objectives that the teacher can use as a tool for judging the learning outcomes. For example, in the logarithm activity, the stated objectives are:

Students will:

• Graph logarithmic equations of the form y = log_ax.
• Define logarithmic functions.
• State the restrictions of logarithmic functions.
• Explain the relationship between exponential and logarithmic functions.
• Solve application problems involving logarithms.

Explore Math takes responsibility for creating the interactive environment for learning mathematics. However, the classroom teacher is responsible for developing lessons that use the online activities. The Explore Math community encourages teachers to share their lessons with other teachers by submitting them to the Explore Math Website. Thus, the vision of the Explore Math community is to pool the talents and resources of programmers, designers, and teachers in a joint enterprise to develop a series of instructional materials.

In many classrooms, students learn about mathematical behavior and functions by examining a limited number of cases presented effortlessly by the teacher. Unfortunately, students often accept the abstract connections as fact because they do not have much opportunity to learn the material in a more meaningful and concrete fashion. However, the Explore Math activities give students the opportunity to visualize mathematical behavior by dynamically modifying characteristics of mathematical objects. Activities of this nature in a user-friendly, low memory, dynamic form are not accessible elsewhere on the Web. Their presence on the Explore Math Website is a unique way to learn mathematics that can be shared worldwide. In addition, the flexibility of the activities structured as tools allows opportunities to present mathematics in multiple ways. For instance, they can be used within a lab setting to promote student exploration or guided discovery, or as a teacher presentation tool in a one-computer setting to introduce a new concept or summarize important ideas.

The goal of this document is to create two design templates for Explore Math activities about cubic functions and a function game called "Zap It!"

Currently, the Explore Math Website contains activities for linear, quadratic, and quartic functions, but does not include anything about cubic functions. More importantly, there is a need for this type of activity because cubic functions are an integral component to algebraic and geometric problem solving. This activity will provide opportunities for students to explore and learn about mathematical concepts such as rotations, odd functions, maximizing volume, and tracing characteristics of critical points that form loci. In a mathematics lesson that uses this activity, students should be able to use the tools in the learning space to recognize patterns, formulate hypotheses, and develop arguments that incorporate mathematical reasoning in order to demonstrate comprehension of a mathematical principle.

The Explore Math Website has activities that help students visualize functions and their behaviors. However, the site does not have activities that encourage students to think about the functions before they are graphed. That is, the graph automatically appears in each of the activities and is modified when the equation changes. This activity proposes that students try to predict an equation of a graph that will contain a set of points without looking at the graph of the function. Once the equation of a linear or quadratic function is entered, the graph will appear on the screen. This activity will provide opportunities for students to engage in function modeling, evaluating functions, graphing linear and quadratic functions, and spatial reasoning in the coordinate plane. In a mathematics lesson that uses this activity, students should be able to use the tools in the learning space to predict equations, test hypotheses, modify strategies, and experiment using multiple trials.

The goal of these learning environments is to provide teachers and students with an opportunity to use a dynamic tool to learn about functions. This design is not intended to provide an instructional lesson that promotes a particular epistemological belief. Instead, it includes ideas for explorations to assist teachers in their creation of lessons that will best match their accessibility to resources and beliefs about pedagogy and learning. The rationale behind this design is to develop a product that explicitly enhances the learning environment and implicitly enhances the instructional outcomes. Even though these components are not mutually exclusive, the emphasis of this design is on the interactive activity because multiple lessons and learning outcomes can be formed from this instructional resource. Furthermore, this design encourages teachers to construct instructional materials because they are most familiar with the culture, philosophy, and conditions of their everyday learning environments. In other words, the designer's instructional strategies do not necessarily match those of every mathematics teacher. Therefore, the design of this template should be flexible enough to allow opportunities for teachers who have access to a lab and a constructivist view of learning, as well as teachers who have access to one computer and a positivist, information processing, view of learning.

The activity is intended to promote a learning event that is 30 to 90 minutes in length. After engaging with the activity in an instructional setting, the learner will be able to identify characteristics of, form conjectures about, and engage in and resolve mathematical situations related to functions. The instructional strategies employed by the teacher could include asking students to perform lower-ordered thinking, such as identifying properties, and higher-ordered thinking, such as synthesizing patterns to formulate a hypothesis about the locus of a point on the function.

A teacher interested in this learning activity should use the following criteria to determine if these Web resources are appropriate and necessary for instructional purposes:

• Cubic functions are a component of the class curriculum.
• The class has access to at least one Internet-accessible computer.
• Students are having difficulty visualizing and understanding properties of functions.
• The teacher is interested in providing a learning opportunity about cubic functions that has previously not been possible.
• The teacher is willing to create an instructional guide that will help students learn about cubic functions according to the pragmatic conditions and philosophical beliefs of the learning environment.

• Linear and quadratic functions are a component of the class curriculum.
• The class has access to at least one Internet-accessible computer.
• Students are having difficulty visualizing and understanding properties of functions.
• The teacher is interested in providing a learning opportunity about linear or quadratic functions that has previously not been possible (or not likely).
• The teacher is willing to create an instructional guide that will help students learn about linear or quadratic functions according to the pragmatic conditions and philosophical beliefs of the learning environment.
• The teacher feels comfortable using a game as a delivery method for learning mathematics.

The teacher should strongly consider using either of these design activities if all of these respective criteria apply in his or her learning environment. If one or more criteria do not apply to the learning environment and the teacher is not sure if the activity is appropriate for his or her class, then he or she should consult with another math teacher or an individual at Explore Math.

One of the activities to be developed in this document is a dynamic cubic function with equation and graph (see Figure 4). It is intended to be used within a single class lesson demonstration or exploration in an advanced algebra, precalculus, or calculus course that discusses cubic functions. The online activity contains a set of explorations for students, frequently asked questions, help, and lesson plans (optional and based on teachers' contributions to the Explore Math Website). The explorations are a series of questions about cubic functions that promote students to use the activity and develop hypotheses based on their observations. These questions will be posed in the implementation section of this design document. The frequently asked questions and help sections have already been created by Explore Math developers and are required reference links in each Explore Math activity.

The other activity to be developed in this document will be a game that asks students to apply their knowledge of graphs of linear and quadratic functions (see Figure 6). It is intended to be used within a single class lesson demonstration or exploration in an algebra or advanced algebra course that discusses linear or quadratic functions. The online activity contains a set of explorations for students, frequently asked questions, help, and lesson plans (optional and based on teachers' contributions to the Explore Math Website). The explorations are a series of questions about linear and quadratic functions that promote students to use the activity and develop hypotheses based on their observations. These questions will be posed in the implementation section of this design document. The frequently asked questions and help sections have already been created by Explore Math developers and are required reference links in each Explore Math activity.

Since these proposed activities might be used on the Explore Math Website, they will model the design style that currently exists in each of this site's activities. That is, the activities will have a graph with adjustable points or a dynamic function, a slider to modify variables (if necessary), and hide and show features (if necessary) that highlight specific components by clicking on checkboxes.

As a particular coefficient of the equation is modified in the cubic function activity, the user should see simultaneous changes in the graph. The coefficients can be modified using sliders to illustrate gradual changes in the graph, or they can be entered directly at the right of each slider to see a particular graph. The graph can be seen in different viewing windows by using the various tools in the design (see lower right corner of Figure 4).

In addition to the dynamic graph, the user will have the opportunity to explore rotational symmetry by clicking on the "show center of rotation" checkbox. If the user clicks on the second

checkbox, "show center of rotation trails," a locus of points describing the path of this point will appear on the screen as one of the coefficients is modified. For example, Figure 5 illustrates the

Figure 4. The proposed layout for a cubic function activity at Explore Math.

Figure 5. The trace of the center of rotation as the leading coefficient a varies in the cubic function activity.

locus that will be created if the leading coefficient a is varied while the other coefficients are held constant. This feature gives students an opportunity to visualize the effects of a point on a class of functions.

A student should engage in the activity for 30 to 90 minutes in one class period and walk away learning something about cubic functions and related mathematical concepts. Since one particular lesson or learning theory is not associated with this learning activity, teachers should consider the following topics as a means for exploration:

• The Fundamental Theorem of Algebra - Determine the maximum number of roots in a cubic function (algebra or advanced algebra).
• Rotational symmetry and odd functions - Determine the conditions when a cubic function is rotation symmetric about the origin (advanced algebra or precalculus).
• Modeling of realistic problems - Determine how to slice a rectangular piece of cardboard in order to produce a package (in the shape of a rectangular prism) that will hold the greatest volume of material (advanced algebra or precalculus).
• Loci of rotation - Determine the functions represented by the trail of the rotation point as each coefficient is varied (precalculus or calculus).

The user begins the game with ten points on the coordinate plane. The goal of the game is to predict the linear or quadratic equation that will pass through the most points. The points are placed on the screen at random, so there is not one particular function that will determine the best solution each time.

The user enters an equation using the sliders or inputting numbers that he or she thinks will pass through the most points on the screen. After the user presses the "Graph It!" button, he or she will see the number of points that were "zapped" in red. The user can enter either a quadratic function (see Figure 7) or a linear function (see Figure 8). In addition, the user can repeat a trial using the same points by clicking on the "Try Again" button, or view a new set of 10 points by clicking on the "Try a New Set of Points" button.

A student should engage in the activity for 30 to 90 minutes in one class period and walk away learning something about the graphs of linear or quadratic functions. Since one particular lesson or learning theory is not associated with this learning activity, teachers should consider the following topics as a means for exploration:

• Properties of functions - Determine the least number of points needed to create a linear or quadratic function (algebra or advanced algebra).
• Quadrants - Examine which functions would more likely pass through clusters of points that are primarily in two different quadrants (algebra or advanced algebra).
• Equations of lines and parabolas (systems of equations) - Find the equation of a line or parabola that passes through a set of points (algebra or advanced algebra).
• Transformations of functions - Determine the effects on the graph of a function if one of its coefficients is modified (algebra or advanced algebra).
• Modeling - Determine an equation of a line or parabola that best represents a set of data (algebra or advanced algebra).

Figure 6. The "Zap It!" game template with points scattered on the coordinate plane.

Figure 7. Testing a quadratic function in the "Zap It!" game.

Figure 8. Testing a linear function in the "Zap It!" game.

Due to different pragmatic conditions and philosophical beliefs about learning, learning event objectives will vary depending on the mathematical concepts examined and the style in which the lesson is presented. The following sets of learning objectives are examples of what a teacher might use if he or she believed that this learning experience should entail an exploration of rotational symmetry and odd functions, or an exploration about transformations of linear and quadratic functions.

During this particular learning event, the student should be able to:

• Click on the checkboxes and identify the center of rotation on the graph.
• Modify the coefficients of the equation and notice changes in behavior on the graph.
• Formulate conjectures about patterns that develop when modifying one variable of the equation.
• Modify the viewing window of the graph if the center of rotation appears outside of the standard viewing window.

 After completing this particular learning event, the student should be able to:

• State the angle of rotation about the center of rotation in any cubic function.
• Identify the position of a center of rotation on the graph of a cubic function.
• Identify the coordinates of the center of rotation on the graph (calculus only).
• State the class of cubic functions that is rotation symmetric about the origin.
• Prove that the class of cubic functions rotation symmetric about the origin are odd functions (maintains the property f(-x) = -f(x)).

In order to participate in this particular learning event, the student should be able to use their prerequisite knowledge to perform the following tasks:

• Identify the center of rotation on a graph.
• Locate a point on a graph based on its coordinate values.
• Substitute variables into an expression and simplify the result.
• Find an inflection point of a function (calculus only).

During this particular learning event, the student should be able to:

• Look at a set of points on a graph and enter the equation of a linear or quadratic function that will pass through as many possible points.
• Modify an equation so that the graph of a function passes through more points on the graph (if possible).
• Establish a set of strategies that will graph a function that passes through as many possible points on any graph.

After completing this particular learning event, the student should be able to:

• State a strategy that will find a function that passes through as many possible points on the graph.
• Explain how the graph of a function changes shape or direction when one of its equation's coefficients is modified.
• Explain why there are a minimum number of points to create a line or parabola.

In order to participate in this particular learning event, the student should be able to use their prerequisite knowledge to perform the following tasks:

• Solve a system of linear equations with two or three variables.
• Graph a line from its equation.
• Find the intercepts of a linear or quadratic function without looking at the graph.
• Distinguish the shape of a line from a parabola.

No particular instructional strategy is recommended when using these activities with an Internet-capable computer. Teachers have different pedagogical styles, and a particular instructional model may not align closely with their actual practice. For example, assume students are using the activity in a computer lab where the learning emphasis is on exploration and construction of meaning. In this scenario, Gagne's Nine Events of Instruction would not be an appropriate instructional model since the learner would not be required to process information in a particular order through a guided path. However, if the teacher is using the computer as a presentation tool, giving him or her greater control of the direction of learning, then Gagne's model might be more applicable. Thus, a basic introduction, body, and conclusion of how the activity might be used will be described along with assessment suggestions.

The teacher should emphasize the importance of functions by making connections to a real world situation and other mathematical concepts. For example, the packaging business attempts to create containers that can hold the greatest possible volume with a fixed amount of material. This example models a cubic function because solids are three-dimensional objects, and cubic functions have a degree equal to three. In addition to applications of cubic functions, the teacher should emphasize their relation to other polynomial functions, indicating that they have a separate set of properties like those of linear and quadratic functions.

The teacher should then describe the activity to the class, as well as the expected learning outcome(s) of the lesson. The teacher may need to help students recall prior knowledge or discuss particular learning strategies, depending on the nature of the learner, complexity of the concepts, and complexity of the task.

The teacher should consider creating a worksheet that provides questions, procedures, and strategies to help the students organize their thinking. The level of structure given to students in a worksheet should depend on their cognitive abilities, time available to complete the lesson, and the level of problem solving the teacher would like to use. For example, the teacher can give a set of step-by-step procedures that supports the students' thinking according to an information processing model, or the teacher can give open-ended questions that require students to establish and execute their own problem solving plan.

The teacher can ask the students to complete the task independently or in groups, depending on his or her comfort with student collaboration and views about learning. While the teacher or students are using the online activity, the students should be answering questions written on a worksheet or verbally from the teacher. These questions should guide them towards the learning objective(s). The teacher should continually assess students' understanding throughout the entire learning process. If they are in a lab, then the teacher should be rotating around the room, reading student responses, closely examining student behaviors, and probing student understanding by posing questions to individuals. Students will be able to assess each other's understanding if they are provided opportunities to work in small teams. The teacher can promote this disposition to learn by encouraging an atmosphere of collaborating, questioning, and constructing meaning throughout the learning process. If the students are in a one-computer classroom, the teacher should ask students to write down responses in order to increase participation. Similar to a lab experience, the teacher can assess understanding by rotating and questioning, as well as encourage peer assessment.

The teacher should review the learning goal established at the beginning of the class and give students an opportunity to reflect on their experience either verbally or in writing. The teacher should help the students synthesize information so that the learning outcome is evident and the learning objectives have been achieved. The teacher should also give students an indication of the next concept that will be discussed and how the learning experience in this activity will contribute and connect to the content and thinking goals or outcomes of the instructional unit.

The teacher should be continually assessing students throughout the learning process through written and verbal questioning, as well as encouraging students to assess themselves and assess their peers. Student responses to questioning indicate whether they have achieved the learning objective. A written record of these responses on a worksheet will help students keep a record of their thinking and demonstrate their understanding of the mathematical concept(s) explored. In addition to the ongoing assessment throughout the learning experience, the teacher can ask students to individually summarize the main ideas of the lesson at its conclusion. Furthermore, students can be asked to recall their learning experiences in an assessment device at a later date, such as on a project, quiz, or test.

An Internet-accessible computer with Shockwave software will run this activity. This medium has been selected so that the activity can be freely and quickly distributed around the world and run efficiently. Shockwave files typically take less time to download on the Web and have fewer bugs than other Web-based interactive environments. For example, many java applets often require the users to operate on a specific platform and a specific browser. Shockwave, on the other hand, is less restrictive and can operate on Macintosh and Windows platforms with all modern graphical Web browsers.

The proposed activity for Explore Math will require several responsibilities of the teacher prior to implementation. First, the teacher should become comfortable with the activity and become familiar with the various tools that can be used in the learning activity. If Shockwave is not installed on the computer(s), then the teacher or a technician will need to follow the procedures at http://www.exploremath.com/about/shockerhelp.cfm to obtain the software for free. Second, the teacher needs to create a lesson that utilizes students prior knowledge, integrates smoothly into the instructional unit, considers pragmatic factors in the learning environment, and accounts for philosophical views about teaching and learning. In essence, every teacher has a different instructional situation, and thus should use reflective judgment and the instructional strategies provided in this document to produce a meaningful lesson. Third, if necessary, the teacher needs to create an instructional resource, such as a worksheet, to help students record, organize, and reflect on their ideas. If a lesson and worksheet related to the activity is posted to the Explore Math Website, the teacher should consider how this resource can be altered to accommodate the learning needs of his or her students.

Fortunately, the activities will be freely available on the Web, so they can be accessed at any time and at any place with access to the Internet. However, if a teacher would like to examine other teachers' submitted lesson plans, he or she will have to register for a free membership within the community. Once a teacher is a member of the community, he or she can look at submitted lesson plans for all of the activities at the Explore Math Website.

These activities can be used in a variety of ways depending on the pragmatic conditions of the learning environment and teacher's philosophical views about teaching and learning. The situations described below relate to whether a teacher has access to a computer lab or to an individual presentation station.

If the activity is used as a tool for exploration in a lab setting, students will work alone or in pairs to address open-ended questions. For example, what class of cubic functions is rotation symmetric about the origin? Solving this problem will require students to identify as many rotation symmetric functions about the origin as possible, and then hypothesize their relationship. In the "Zap It!" game, the teacher might ask students to determine the characteristics of a function that will enable it to pass through two consecutive quadrants within a specific viewing window. Once a hypothesis is formed, students can use the online activity to check multiple cases to strengthen their confidence in their argument. At that point, students try to justify why their conjectures are true through a mathematical proof. After working in pairs, the students may check their results with those of their classmates to confirm or reject their findings. The teachers' responsibility, in this case, is to monitor and assess student understanding by rotating around the room and asking questions that raise students' problem solving awareness, such as "What are you doing?," "How will you find the solution?," and "How do you know if your solution is correct?" Once a significant majority of the students complete the activity, the teacher should assist the class in summarizing their findings, encouraging them to report multiple representations or perspectives in their solutions.

For pragmatic reasons, teachers have to balance the type of learning experiences they provide to their students with the restrictions placed on the learning environment, such as limited time and resources. Realistically, all lab activities in mathematics are likely not open-ended and based on a constructivist learning philosophy. If the teacher wants all of the students to arrive at the same conclusion more quickly, an alternate lab strategy is to provide students with a more directed discovery approach to the lesson. In this setting, students are working alone or in pairs again, but have some questions to guide their thinking. For instance, if the overall task is for students to find the class of cubic functions that is rotation symmetric about the origin, then the teacher might give the students some smaller tasks to help them process and synthesize the information. For instance, the teacher could give students a variety of carefully selected functions that incorporate counterexamples to common errors, and then ask which functions from the list are rotation symmetric. Then, the student would be asked to compare the coefficients in the cubic equation to determine which terms in the polynomial affect its rotation symmetry. The questioning would continue until the students have a proof that can be discussed with the class. If the overall task is to find characteristics of a function that that will enable it to pass through two consecutive quadrants, teachers might guide the students to check specific functions and quadrants until a pattern is established. The outcomes of these situations differ from the open-ended approach because students are directed to one particular solution without utilizing problem solving strategies or handing distracting information.

When the teacher is limited to one computer, the activities can be a helpful tool to illustrate mathematical behaviors by presenting multiple cases in a short period of time. The teacher can use a guided questioning technique to help stimulate ideas from the students. In the case of the rotation symmetric cubic function, the teacher might first modify the coefficients of the polynomial and show the graph change in response to the numerical changes. Next, the teacher might ask the students to guess an example of a function of this nature. After compiling student responses, the teacher might try a few of these functions in order to develop a discussion that will clarify some key characteristics, such as the number of odd powered terms in the expression. In the "Zap It!" game, the teacher might first graph linear functions and then quadratic functions to see if one particular function passes through more points. Then, the teacher might start changing the coefficients of the functions to illustrate different properties of a function. Hopefully, the teacher will encourage students to debate their hypotheses and then ask to have some graphs plotted on the screen for verification. A class hypothesis can be formed and students can try to generate a proof with the support of the teacher. In some instances, the teacher will guide the students through the proof, while other times the teacher will encourage the students to struggle over the proof until they have carefully reflected over the ideas generated from the class discussion. Similar to the lab situation, both types of experiences occur in mathematics classrooms depending on the time and resources available for learning this particular concept.

The teacher will need at least one Internet-accessible computer with Shockwave software to use these activities. The storyboards of the activities are illustrated in Figures 5-8 of this document. In addition, the teacher will probably need to create a worksheet that provides students with questions related to the learning objectives of the lesson. The teacher can access lessons that have been submitted by other teachers or the Explorations on the Website to help devise a resource that is most suitable for their instructional situation.

The Explorations on the Website will contain the following questions for the cubic function activity:

• What are the greatest and least number of roots in a cubic function? Set the leading coefficient a to zero so that the function is quadratic. How does your answer change? Make a conjecture about the greatest and least number of roots in any polynomial function. Explain your reasoning.
• Click on the checkbox that will show the center of rotation. Determine a general equation of a cubic function that is rotation symmetric about the origin. Are there any restrictions to this rule? Evaluate f(-x) in this case. How does the output compare to f(x)? What does this information tell you?
• Candy boxes are going to be made from a piece of rectangular cardboard with dimensions 8 inches by 10 inches. Squares of equal size are cut out of each corner and then the flaps are folded up to form a box (after the tape is applied). The manufacturer would like to create a box with largest volume so that the consumer thinks he or she is getting a lot of candy. What size squares should you cut out of the cardboard in order to make this possible? What size squares should you cut out of a piece of cardboard with dimensions x inches by y inches? Explain your reasoning.
• Click on the checkboxes that will show the center of rotation and its trail. Drag the coefficient a and observe the locus that is traced on the screen. What type of function is formed by this locus? How do you know? Click off the checkbox that shows the center of rotation trail to erase the locus, and then click it again to turn the feature back on. Drag another coefficient of the cubic function and predict the functions or relations that are modeled by the locus. How can you prove your hypotheses?

The Explorations on the Website will contain the following questions for the "Zap It!" game activity:

• Play the game several times and play against some classmates. What is the greatest number of points that a function can pass through? What is the chance that this will occur?
• What is the largest number of points that a line will always pass through, regardless of how the points are distributed on the screen? How do you know? What is the largest number of points that a parabola will always pass through, regardless of how the points are distributed on the screen? How do you know?
• If at least two-thirds of the points are distributed among two quadrants, what type of function would you graph? Explain some properties about the coefficients and constants of the function's equation, if applicable.
• If you select three points on the screen, can you find the equation of a line that passes through all three points? Can you find the equation of a parabola that passes through all three points? Explain.
• Suppose your graph of a function is slightly above a few points. If you entered a quadratic function, what part(s) of the equation would you change so that the graph would be more likely to pass through those points? What part(s) of the equation would you change if the function were linear? Explain your reasoning.

The teacher should informally assess students throughout the class period to holistically determine the extent at which the learning objectives have been attained. Based on available time and access to a lab (if necessary), the teacher needs to determine what percentage of students must perform the learning objectives by the end of the class period. Realistically, all students may not understand the content in an activity well enough to successfully achieve every learning objective. However, if many of the students are struggling with a particular learning objective by the end of the lesson, the teacher should accommodate his or her curricular schedule so that there is more time to discuss this objective more in depth in the near future. If a large majority of the class (a number determined by the teacher) has accomplished the objectives, then the teacher can move forward by summarizing the learning event and transferring the ideas and skills into new material.