People are always looking for good math activities. Mathematics is applied in science classes on a regular basis, so often in fact that mathematics is deemed an essential tool for teaching science. Students use the skills that they learn in math classes to solve equations in chemistry and physics and to analyze data in science experiments. Often science teachers find themselves re-teaching the math skills that students need to do the science. Mathematics is an important tool in science classes, but science is not generally considered a tool for teaching mathematics; it should be. With science-related experiments, I use math labs in my Advanced Algebra classes to effectively cement mathematics concepts and skills while illustrating real-world applications. Just as laboratory work is central to teaching a sound science course, math labs enhance any sound math course.

Students in my Advanced Algebra class begin every unit of study with a math lab in which they perform an experiment, gather data, graph the data, and determine an equation that models the data. They make poster-size graphs that remain on display in the classroom throughout the unit. I refer to the graphs on a daily basis as new concepts are introduced. In addition to motivating the students and giving them real-world applications of math concepts, math labs provide the "hook" that will help the students retain the concepts.

While the students are working in their small groups on a lab, I walk around the room so that I can watch students work, evaluate their individual learning needs, ask them questions about the lab, and answer any questions they may have. Math labs require all students to be actively engaged in learning as they make predictions, take measurements, analyze their data, and make decisions about presenting their work. What continues to amaze me about math labs is the high level of excitement and involvement of all the students! The lab challenges students to display their individual talents and math abilities in real-world problem solving situations.

What follows are four math labs that demonstrate the use of science experiments to introduce math concepts. The first activity is explained in great detail to demonstrate how students move through the scientific method as they perform the math lab, how they analyze their data, and how they evaluate their completed lab. Special attention is given to the multifaceted representation of data using tables, graphs, and equations. Students see how the different modes of data representation are related and attain a concrete real-world understanding of abstract concepts such as domain and range, independent and dependent variables, and discrete and continuous functions. Each lab reinforces these crucial concepts.

I. Multiplying Money with Mirrors (Hyperbolas)

Figure 1: Arrangement of mirrors and penny for a 120° angle.

Money may not grow on trees, but in this math lab, mirrors multiply money! Two small mirrors are taped together along one edge so that they can stand upright and the angle between them can be measured. A penny is placed between the two mirrors. First the mirrors are set so that the angle between the front surfaces of the mirrors is 180º. I tell the students to place the penny about an inch from the mirrors. Then I ask the students how many pennies they see. Someone volunteers that he sees one penny. Another student agrees that she sees one penny, too. After a few more identical responses, one student bravely counters that she sees two pennies, the reflection in the mirror and the actual penny. I tell the students that they will be changing the angle between the mirrors and counting the number of pennies that they see, reflections plus the actual penny. Figure 1 shows the arrangement of the mirrors and the penny for an angle of 120º; you can see three pennies, the actual penny and two reflections.

After explaining the experimental procedure, I ask the students what the independent variable is (angle between the mirrors) and what the dependent variable is (number of pennies). Then they determine that the angle will be graphed horizontally and the number of pennies will be graphed on the vertical axis. I have the students make a sketch of what they think the graph will look like. We then discuss several of their predictions as a class. I make sure that different predictions are presented so that students have the opportunity to see and hear a variety of perspectives on this problem. After discussing the predictions the students work in pairs to collect their data.

The students plot the points on a graph with the angle between the mirrors on the horizontal axis and the number of pennies on the vertical axis. Then the students decide which class of relations, inverse or direct, is modeled by their data. They conclude that this is an inverse relation because as the angle between the mirrors decreases, the number of pennies that they see increases. Invariably at this point in the discussion, some students have started to wonder if the graph should be continuous or discrete. I ask the students what the domain is, that is, what angles they can have between the two mirrors.

They respond that the angle can be any real number of degrees between 0 and 360, and that the graph is therefore continuous. Their curiosity gets the better of most students at this point and they try to place the mirrors at angles other than those that they originally had-that gave an integer number of pennies. The students are fascinated to see that the number of pennies can be a fraction. Some even realize that I didn't have them use other angles because the fractions of pennies are difficult to measure. A student graph produced using a computer program is shown in Figure 3.

Figure 3: Graph of number of pennies vs. angle between the mirrors.

After students have graphed their data, they determine an equation that models the data. In this activity I challenge the students to determine the equation by looking at their table of values to see if they can see patterns between the two columns of numbers. Working with their partners, the students realize that the product of the angle measure and the number of pennies is 360. The equation that models the data is therefore A*P = 360, where A is the angle between the mirrors and P is the number of pennies. I ask the students to solve this equation for the dependent variable and to then write the equation in the form of a function, P(A) = 360/A, a hyperbola. Since the domain is restricted to positive angles between 0 and 360, the hyperbola is in the first quadrant only. Next the students use the equation to predict how many pennies they would see at a particular angle, say 50º, and then place the mirrors at this angle and count the number of pennies, a non-integer number in this case. They also determine the angle at which they will see seven pennies and verify their calculation by placing the mirrors at that angle and counting the number of pennies.

As groups finish the activity they use a rubric to evaluate their work. Then they ask someone from another group to check their work. After the peer evaluation is complete, the students make any changes/corrections before asking me to evaluate it. I evaluate the work in front of the students so that I am able to give them verbal feedback and ask them questions about their work, checking their understanding and whether the instructional goals for this activity have been achieved. This activity gives the students an excellent opportunity to study the reflection of light as well as the properties of hyperbolas. The review of graphing techniques and discussions of domain and range, independent variables and dependent variables, continuous and discrete functions, inverse and direct relations tie in extremely well to this activity.

Figure 4: A student steadies the cup before measuring the distance.

II. Hooke's Law with a Suspended Slinky® (Linear Equations) In this math lab a Slinky (actually, one half of aSlinky, since I cut them in half to get more lab stations) is tucked under the ceiling tile supports. Two paper clips are unfolded and attached to the opposite sides of a small paper cup. Then the paper clips are taped to the bottom ring of the Slinky. Students measure the distance from the bottom of the cup to the tabletop with the empty cup. I have the students use Texas Instruments Calculator-Based Rangers® (CBRs) to measure the distance, but they could just as easily use meter sticks or yardsticks. The CBRs attach directly to the students' graphing calculators and give a distance vs. time table of values as well as a graph. Since the cup at the end of the Slinky is stationary, the distance vs. time graph is a horizontal line. Students add pennies to the cup, five at a time, and measure the distance of the cup from the table. In Figure 4 a student has added pennies to the cup and is stopping the cup from bouncing before measuring the distance.

After plotting their data (shown in Figure 5), students draw a best line (by eye) through their data and then find the slope of the line using two points that are on the line (not necessarily data points). Then they find the intercept directly on the graph where the best line crosses the vertical axis. Next they write an equation for the line in slope-intercept form using appropriate variables for distance (the dependent variable) and number of pennies in the cup (independent variable). The students discover that the slope of the line represents the amount that the spring will stretch if you add one penny at a time. This lab is an example of a linear/inverse relationship since as the number of pennies in the cup increases, the distance from the table decreases.

Figure 5: Graph of distance vs. number of pennies in the cup for the Slinky experiment.

III. Measuring the Density of Liquids (Linear Equations) In most physical science classes, students areintroduced to the concept of density as the mass divided by the volume. A typical experiment involves the students measuring the mass of a specified volume of a substance in a graduated cylinder. Then they subtract the mass of the empty cylinder, and divide the resulting mass of the substance by the volume of the substance. The units of density will be grams per milliliter or grams per cubic centimeter. Usually students measure the density of water first and then the density of some other liquid that is more or less dense than water. In this lab, however, students change the volume of the liquid in a graduated cylinder and measure the mass of the cylinder and the liquid together, as shown in Figure 6. Then they graph the mass vs. the volume, draw a best line (by eye) through the data points and determine the slope of the line using two points that are on the line (not necessarily data points). They find the intercept directly from the graph by determining where the line crosses the vertical axis. Finally, they use the slope and intercept to write an equation for the line. Students quickly discover that the slope of the line, which has units of gram per milliliter, represents the density of the liquid and the intercept, which has units of grams, represents the mass of the empty cylinder. Many of them are amazed to realize that they do not have to subtract the mass of the empty cylinder to get the density of the liquid.

Figure 6: Students measure the mass of 40 ml of water in a plastic graduated cylinder.

Figure 7: Graph of mass vs. volume for water (blue) and saturated salt water (red).

The students repeat the lab using saturated salt water, which has a density of about 1.2 grams per milliliter. When they plot the data, they discover that the slope is greater than it was for water (since the density of salt water is greater than water), but the intercept is still the same (since the empty cylinder has the same mass). Students use the information from this lab to predict what the graph would look like if they were to use cooking oil, which has a density less than water. A typical table of values, graph and equations for this math lab are shown in Figure 7.

IV. Bouncing Ball (Geometric Series and Exponential Functions) In this lab students study how the height of a golf ball bouncing from the floor varies with the number of bounces. Students can use meter sticks to measure the height of the golf ball from the floor. They drop the ball from a predetermined height and then catch the ball after its first bounce, freeze their hand at the position of the catch, and then measure the height of the ball from the floor. They measure the height in this way three times and then take the average of the three trials to get the height of the ball after one bounce. Then they drop the ball from the original predetermined height, but this time they let it bounce twice, and then measure the distance from the floor. They measure the height of the second bounce three times and then take the average. The process is repeated for three bounces, four bounces, and so on, until it becomes difficult to get reproducible results because the ball is too close to the floor.

Figure 8: Students use a CBR attached to a calculator t measure the distance of a bouncing golf ball from the floor.

Another method of measuring the height of the golf ball from the floor is to use a CBR linked to a graphing calculator, as illustrated in Figure 8. The CBR has a program called "Ball Bounce" that shows the distance from the floor vs. time graph when the motion of a bouncing ball is measured. Students can then use the graph and the TRACE feature on their calculator to determine the height of each bounce. When using the CBR, repeated trials are not necessary; the results from just one trial are very accurate.

Once students have a table of values of bounce number and height of the ball from the floor, they can determine recursive and explicit formulas for the geometric sequence that models their data. They determine the constant ratio between successive bounces by dividing the height of the second bounce by the height of the first, the third bounce by the second, and so on. Then they determine the average ratio and use it in the explicit and recursive formulas. The recursive formula for the data in Figure 9 is: g1= 1.42 gn = (0.815)gn-1 for integers n >2 The explicit formula for the data is: gn = 1.42(0.815)n-1 for integers n >1 Note that the term number in the equations and the bounce number in the table are not the same.

n (bounce number) h (meters) 0 1.42 1 1.16 2 0.95 3 0.78 4 0.63 5 0.51 Figure 9: Data from bouncing golf ball.

I have the students repeat the experiment using a super ball, so that they can compare and contrast the recursive and explicit formulas. Typically they find that the growth factor for the super ball is larger than the growth factor for the golf ball, indicating that the super ball bounces higher on each successive bounce than the golf ball.

Several weeks after studying the bouncing golf ball in a unit on geometric sequences, students use the same data to explore exponential functions. They graph the data from the bouncing ball experiment, with the bounce number on the horizontal axis and the bounce height on the vertical axis, as shown in Figure 10. They draw a "best curve" by eye using a dotted line to show that the data are discrete. Then they use the intercept of the graph to determine the value of "a" in the general equation for an exponential function, h = a*bⁿ. In this equation h is the height of the ball from the floor and n is the bounce number. Students find a point on the curve and substitute its coordinates into the equation along with the intercept value they have measured, and then solve the equation for the growth factor, b. After finding the equation that models the golf ball data, students use the same method to find a similar equation for the super ball. Finally the students compare the exponential equations to the recursive and explicit formulas they derived in the unit on geometric sequences; they are intrigued to discover that the growth factors are the same, even though they used very different methods to get them.

Conclusion

After observing the active participation of students in math labs and listening to the mathematics discussions that take place in lab groups, I am convinced that they are an effective way to teach mathematics. I even use the math lab format with activities that do not involve science concepts, but still utilize the scientific method, requiring the students to make predictions, gather data, graph the data, and determine the equation that models that data. Prediction, measuring, and analyzing skills used in the math labs, science-based or not, will aid students in endeavors outside the mathematics classroom. It is clearly evident that math labs help students understand mathematics concepts. The math labs foster understanding of abstract concepts, such as domain and range, independent and dependent variables, and continuous and discrete functions. After completing all of the math labs in my Advanced Algebra class, students have no doubt in their minds that science is an essential tool for learning math.