In my last blog post, I discussed the importance of using groupworthy tasks with your students. For a task to be groupworthy, it should satisfy three criteria: interdependence (the task is mathematically rich enough that students have to work together), multiple abilities (many different mathematical strengths are needed, e.g. verbal, written, spatial, visual), and multiple representations (e.g. graphical, numeric, linguistic and symbolic).
Many teachers do not have such groupworthy tasks in their curriculum, though, and do not have access to such problems. Many problems that we do have in our textbooks have potential; we just need to learn how to make them groupworthy.
When selecting a problem, it is often helpful to look at problems that involve real-world applications. Many real-world application problems, though, give step-by-step directions, which often obviates the need for a group of students to work together on the problem.
In the following example, drawn from the 6th edition of A Graphical Approach to College Algebra, students are asked to use a given equation to study the height of a ball thrown vertically on the moon with relationship to time (p. 187):
An astronaut on the moon throws a baseball upward. The astronaut is $6$ feet, $6$ inches tall and the initial velocity of the ball is 30 feet per second. The height of the ball is approximated by the function: $s(t) = -2.7t^2 + 30t + 6.5$ where t is the number of seconds after the ball was thrown.
- After how many seconds is the ball $12$ feet above the moon’s surface?
- How many seconds after it is thrown will be ball return to the surface?
- The ball will never reach a height of $100$ feet. How can this be determined analytically?
This problem uses quadratic equations, which could be mathematically rich, but due to the fact that the problem is in section 3.3: Quadratic Equations and Inequalities, the problem ends up being more of a rote exercise given that it covers only the material addressed in the section in which the problem appears.
The inclusion of a real-world context can often be a sign of a groupworthy problem, but this problem provides the equation, which removes most of the opportunity to build a model.
To make this problem groupworthy, we should start by removing the equation, the step-by-step directions, the height of the astronaut and the speed of the ball. Instead, we can let students choose their own height for the astronaut (perhaps using one of their own heights), and figure out a reasonable figure for velocity. If your students know some calculus, they could even find the formula themselves. (If you take the acceleration on the moon ($5.3$ ft/$\text{s}^2$) and integrate it twice with respect to time, you get the $2.7t^2$ from the original problem.)
It does not take calculus to understand that position has $\text{time}^2$, since velocity = acceleration times time and position equals velocity times time. Where students are likely going to have trouble without calculus is figuring out the coefficient of $t^2$ is half the value of the acceleration.
In preparing this blog post, I consulted several math teachers. The traditional way this is done is to give students the basic formulas for velocity and position of a free-falling object:
\[v = at, \qquad x = .5a t^2\]
Although I think that that formula would best be taught after students understand where the $t^2$ comes from so that it is not just a hand-waving kind of thing to students.
Another teacher suggested the following method: students could find the average velocity by taking $(v + (v-a))/2$. Since the object has no initial velocity, we can use $v = 0$. Simplifying, we get $-a / 2$ and thus we see where the $0.5 a$ comes from.
With all of these ideas in mind, here is the final version of this task that I would give to the students:
An astronaut on the moon throws a baseball upward. Choose a reasonable height for the astronaut and the velocity for the ball and find an equation to describe the position (height) of the ball at time $t$. Then demonstrate various facts of your choice about the path of the ball, such as the maximum height, and when it will reach the ground. Create a poster, using words, graphs, tables, and symbols, to explain how you found your equation and the facts about the path that you chose.
Is this task groupworthy? This task is mathematically rich in that students have to understand not only quadratic equations, but position, velocity, and acceleration. The problem encourages interdependence both by having a group product and being sufficiently challenging that a single student cannot solve it on their own. It also specifically encourages students to utilize multiple representations.
In the next blog post, I will discuss strategies for managing groupwork such as group roles, huddles, and techniques for when a group is getting stuck on a problem.