Wednesday’s Lesson
The Starburst Activity

Mathematics is a tool that models the relationships we see in everyday life, and many of these relationships can be described by functions. For example, a situation as simple as an after school job that pays $7 per hour can be described symbolically by the linear function y = 7x.

There is power in looking at functions in different representations – words, numeric (or tabular), symbolic, and graphic – and understanding that a change in one is mirrored by a change in the others. Once students practice comparing representations and observing the connections between them, they are less likely to rely on point-by-point graphing every time they encounter a new function. And they are more likely to ultimately understand how functions can model real-world problems.

Figure 1.Typical Textbook Transformation.

Comparing Representations

The Starburst activity below focuses on comparing changes in the graphs of linear functions with corresponding changes in the symbols used to describe them. Students can gain insights into the connections between symbolic and graphic representations of linear functions by transforming them.

Most algebra textbooks approach transformations geometrically. A typical problem might challenge students to start with a set of coordinate points that form a figure, such as a square, and “transform” the figure by moving it, one point at a time, to a new position (Figure 1). This transformation develops skill in interpreting Cartesian coordinates. It leads students to focus on changing a set of points, rather than on moving the object itself.

Transformations of Linear Functions

To transform a linear function, students must shift their perception. They must view the graph of the function (a line) as an object.

Often when they are asked to graph two different linear functions, such as y = 2x and y = 2x + 3, students start by identifying a few values of the function (values of y) for a few input values (values of x), then list them in a table and graph each function. When they do this, students see the functions as unrelated to each other. However, the functions y = 2x and y = 2x + 3 are related: the line that represents the graph of the function y = 2x + 3 is simply the graph of y = 2x moved up 3 units.

Figure 2.RTT Linear Transformer.

The Starburst Activity

How does changing the slope of a line change the symbolic expression of the function? How does changing the symbolic expression (or equation) “move” the graph of the function on the coordinate plane?

The Concord Consortium’s interactive tool called the Ready to Teach (RTT) Linear Transformer (Figure 2) gives students the power to change different representations of a function and then instantly observe what happens. In the Starburst activity, students can use the Linear Transformer to observe the relationships between different representations of a function by manipulating the functions that make up a “starburst” pattern.

Getting to Know the RTT Linear Transformer

Give students a chance to familiarize themselves with the software before doing the activity.

To access the Linear Transformer, go to: http: //rtt.pbs.org/rtt/interactives.cfm

Note: You need Java 1.3.1 or higher to run the interactives. Check your Java by using the RTT wizard (http: //rtt.concord.org/Wizard), or install or update Java Software for the Desktop (http://java.com).

Open the Linear Transformer and explore the features.

1. Make a line.
2. Grab the line, move it, and watch how the equation that describes the line changes.
3. Make another line. Notice the color coding.
4. Change values in the symbolic expression of the line and observe how the graph of the line changes.
5. Explore the Reflect feature.
6. Explore the Pushpin feature.
7. Experiment with other features. When you are satisfied that you understand how the Linear Transformer works, try the challenges below.

Challenge 1: The Starburst

Figure 3.The Starburst.

Using the Linear Transformer, make the starburst pattern (Figure 3). All lines intersect at the origin (0,0).

Hint: The angles do not need to be equal.

• Write the equations of the lines that make up the starburst pattern.
• Which elements do the equations have in common? Which elements are different?

Challenge 2: Lift the Starburst

Figure 4.Lift the Starburst.

Make a similar starburst pattern in which all the lines intersect at the point (0, 4) (Figure 4).

• Write the equations of the lines that make up this starburst pattern.
• Compare the equations of the lines in this starburst to the original starburst. Which elements of the equations are similar? Which are different?

Challenge 3: Shift the Starburst

Figure 5.Shift the Starburst.

Make a similar starburst pattern in which all the lines intersect at the point (4, 2) (Figure 5).

• Write the equations of the lines that make up this starburst pattern.
• Compare these equations to the original starburst. Which elements are similar? Which are different?
• Which form of the equations did you find most helpful in doing these challenges (slope-intercept or point-slope form)? Explain.
• While working through these activities, when did you choose to manipulate the symbolic or graphical representations?

Challenge 4: Make Your Own Starburst

Figure 6.Starburst Activity Extension.

Describe how you would make a starburst pattern anywhere on the coordinate plane.

Extensions to the Starburst Activity

Giving students new challenges can help you gauge your students’ observations and understanding. Here are a few examples:

1. Write the equations that make up a starburst pattern that passes through (-2, 0). Try it first without graphing. Then test your prediction by making the pattern on the Linear Transformer.
2. Write the equations that make up a starburst pattern that lies in the third quadrant.
3. Make the diamond pattern (Figure 6).

In the Classroom

Though students were not asked explicitly to discuss or define variables or unknowns in this activity, they explored different ways to express functions symbolically and graphically. This activity, especially through the use of the Linear Transformer, highlights two very common conceptual hurdles:

• Distinguishing between a variable (a quantity that changes) and an unknown (a fixed quantity)
• Knowing when to use different symbolic forms to represent a linear function: slope-intercept (y = mx + b) vs. point-slope (y – y1) = m(x – x1) 

Have students use the Linear Transformer to explore different forms of a linear equation. Encourage them to observe the relationships between the symbolic and graphic representations. Once they have had many experiences comparing different representations of functions, students will be better equipped to generalize about the characteristics of linear functions.