Launching the Lesson
The unit starts with a bang when Tim takes his class out to the field and launches a rocket into the air. The students all crane their necks skyward in an attempt to see the rocket travel ever higher and then to follow it as it falls gently back to earth on a parachute. As Tim repacks the rocket for another launch, he asks some questions: “How high did that rocket go? How fast did it go? Let’s launch again and see if this flight is higher or faster than the first flight.”
Another whoosh, another set of gasps, and his students are totally involved. The guesses of altitude and velocity flow freely from their mouths as they clamor to be the one who gets to chase after the rocket. Tim gathers the rocket and again asks more questions: “Did that flight go higher? Is there any way that we could figure out how high the rocket went on that flight? What about the speed? How fast did that rocket go? Oh, by the way, how does a rocket work? Let’s do one more launch and then go back into the classroom and answer some of these questions.”
The final launch, the final recovery, the final set of questions: “How many of you would like to launch rockets?” The hands of the students shoot up like rockets themselves. They are hooked. They have reached the first goal of any successful math project; they are excited to be doing it. When the students get back to the classroom, with the rocket as a prop, Tim goes over the questions again. How high did the rocket travel? Students respond with varying degrees of accuracy. The guesses for height range from 200 feet to 2000 feet; the guesses for speed range from 50 miles per hour to 1 million miles per hour. These ranges play into the lesson perfectly. The class agrees that there needs to be a more accurate way of figuring out the altitude and velocity for each rocket than mere guessing. From here, Tim turns to the whiteboard and draws a picture of a rocket launch, complete with a baseline showing where the class stood and the path of the rocket. The resulting picture looks like an L. Tim asks the class what shape they would get if he connected the top of the rocket flight (the top of the L) with the class’s location on the ground (the bottom of the “L”). Nearly everyone in the class replies that he would get a triangle.
Armed with this knowledge, Tim leads the students to the conclusion that if they knew how to find the height of the triangle, they could figure out how high the rocket traveled. The students quickly realize that it is not possible to measure the side of the triangle and that there must be another way of doing it. From previous conversations, Tim’s students understand that if they know two out of three variables in an equation, then they can find the third. Next Tim discusses what parts of the triangle we can measure and identifies the triangle in question as a right triangle. The base distance comes pretty easily Again, through guided discussion and use of a rocket-siting device (more on this later), the students find that they can measure the angle adjacent to the path the rocket took. Having established this discovery, Tim informs the students that there is a formula that they can use to find the altitude of a right triangle (see Figure 7.1). Tim points out:
It is important to note that I ask the class to agree that all of our flights will be considered right triangles, and I ask them to agree to this verbally The process of asking them to agree to something is very important as it shows the students that I value their opinions and their ideas. If a student disagrees, or cannot agree, the student feels welcome to express his/her ideas so that we can see how to address them.
After all the students have agreed, Tim introduces the formula, which they call Mr. Granger’s Formula: The height of the triangle is equal to the tangent of the angle multiplied by the base, or:
A = (tangent of the angle)(base)
Initially, Tim asks them to believe that using the formula will give them the altitude of the right triangle. (Later, he will have the students use measurement to verify the relationships.) He draws a triangle on the whiteboard and has a student measure the length of the base (14 centimeters). Next,
FIGURE 7.1 Using a Formula to Find the Altitude of a Right Triangle another student uses a protractor and measures the angle as 36 degrees. Without explanation as to its origin, Tim passes around copies of his “Great Granny’s Tangent Sheet” and directs the students to read the tangent of 36 degrees. Next, he substitutes this value into the formula, uses a calculator, and declares that the altitude is 10.2 centimeters. Expecting and noticing some skeptics in the class, he invites a student to measure the altitude with a ruler. The subsequent measurement of 10.2 centimeters convinces the class that Mr. Granger’s formula “works.”
Tim poses the question, “Do you suppose this formula works with lengths in inches as well as centimeters?” A couple of examples of formula use, followed by measurement verification using base lengths of various units, cement the ideas. Before the lesson closes, Tim asks the students when they would use this formula in daily life. Derrick offers, “To see how tall a building is.” Ashley quickly follows with, “To see if a tree is big enough to fall on my house.” Several students, recalling the opener for the unit, in chorus shout, “Rockets!” The room explodes with the same excitement as was generated by Tim’s demonstration. As a wrap-up, Tim directs the students to stand and to join his chant. Clapping together they repeat the refrain, “The altitude is equal to the tangent of the angle times the base.” The room rocks.
In order to provide the students with practice in finding the altitude of right triangles (and hence their rockets), Tim gives the students several lessons in which they actually use the formula to measure triangles. Tim takes the class out into the school’s courtyard where he has drawn large triangles on the ground with chalk. Armed with protractors, calculators, and sheets of paper listing all of the tangents for angles from 1 to 90 degrees, his students in groups of two to three figure out the height of each triangle using their formula. Then they actually measure the height using a ruler. This physical confirmation shows them that the formula they are using really does work. As the students work, Tim conducts group interviews, asking the students to explain the mathematical process used and how their answers were calculated. After they have measured triangles on the ground, he sends them to work on triangles that they cannot actually measure without the use of the formula. Tim asks them to find the height of trees that surround the school. The length of the base is given to them using a tape measure, and the angle is provided to the class using an electronic protractor mounted to a tripod. This tool will also be used as a rocket-siting device at the end of the unit. This process gives the students a great deal of practice with the formula in a way that they can relate to more than if Tim had simply given them numbers on a sheet of paper.
Tim’s next challenge is to help students discover the velocity formula (V = A/T), with V being velocity, A being altitude, and T being the time in seconds. On the playground field, he lays several courses, each with a known but different distance. The students run the courses while he times them with a stopwatch. The students then determine which student is the fastest. By having different courses, the students are made to actually find the velocity of the students instead of just saying that the fastest student was the one with the shortest time. The students record it as Trial One, Trial Two, and so on and record distance and time. They then use this information to find their velocities.
After these races are completed, the class moves inside. Working in groups of three to four, the students find the velocity of toy cars of the pullback type. The students pull them back, let them go, and time them over a known length of course. Tim’s students also time a remote-controlled car as it travels down the hallway. The use of toys keeps the interest level high while giving the students practice in finding velocity.
For further practice, the class again goes outside. With the help of another teacher standing on the school’s roof about 20 feet above the ground, objects like stones, bananas, or shoes are dropped one at a time, and the travel time of each is recorded on the students’ data sheets. The height above the pavement is measured and kept constant throughout. When the students return to the classroom, they compute the speed of each object using the distance-rate-time relationship.
Tim lets his students use calculators, and he turns on some classic rock ’n’ roll or delta blues music as background inspiration. Comments like Rocio’s, “I like dropping stuff off the roof. It’s fun to see them bounce, and we can figure out how fast they fall!” or Breland’s “The banana had a speed of 30 feet per second. Imagine!” convince Tim that his students are having fun and learning, too.
The next phase of the unit focuses on building the model rockets. Each student gets a model rocket kit to build his or her own rocket. Tim reviews the directions, highlighting any important parts. The students must have every phase of construction inspected to be certain that they have correctly completed that step (see Figure 7.2). The rockets generally take a week or so to build, paint, and dry.
Finally, it is time to launch the rockets. Tim describes the launch:
The rockets are launched on school grounds. A parent volunteer is located 150 feet from our launch site with the electronic protractor on the tripod. A second parent is located with the first as a timer and spotter. These parents also have a walkie-talkie. After a rocket is launched, the first parent sites along the top edge of the protractor (looking for the white puff of smoke that is sent out when the rocket ejects the recovery device) and then looks at the protractor. The parent reads the angle off of the protractor and then radios this information back to the launch site, where a third parent writes the angle on a large form containing the students’ names. The timer then radios back the time of flight to the launch site, which is also recorded. We generally run three different launchers in order to cut down on the time between flights.
After the launches are complete, the students come back into the classroom, and using the data collected from the field, they find the altitude and velocity of their rockets. Tim circulates around the groups as his students complete their calculations. “Did you see how fast mine went, Mr. Granger?” asks Mike. “How fast did it go, Mike?” “127.81 feet per second!” “Mr. G., mine went so high! It went 592.32 feet!” offers Jerry. The data from the class are taken and graphed using a computer spreadsheet program, and the results are compared. Next, the students write a final report telling all that they have learned about rockets, altitude, velocity, and mathematics (see Figure 7.3). Finally, the students complete a self-evaluation, describing what they did, how they did it, what they did well on, and what they would do differently the next time.
FIGURE 7.2 Rocket Quality Control Checkoff
directions: Check off when all of the following steps are finished. Once your rocket is completely finished, have a teammate check off each quality control section as well. Fix any areas that are not checked off, and resubmit for final inspection.
FIGURE 7.3 Final Report: Sample ofStudents’ Replies
FIGURE 7.3 (Continued)
The rocketry unit closes with the viewing of the movie October Sky. This movie tells the story of a group of boys who grew up in a small coal-mining town and taught themselves about rocketry Their knowledge leads them to winning a national science fair, earning college scholarships, and being able to leave their small town. Following the movie, Tim’s students write letters to the main characters telling of their own rocketry experiences.