Lesson 6

In this warm-up, students considers what happens if an object is launched up in the air unaffected by gravity. The work here serves two purposes. It reminds students that an object that travels at a constant speed can be described with a linear function. It also familiarizes students with a projectile context used in the next activity, in which students will investigate a quadratic function that more realistically models the movement of a projectile—with gravity in play.

Students who use a spreadsheet to complete the table practice choosing tools strategically (MP5).

Ask a student to read the opening paragraph of the activity aloud. To help students visualize the situation described, consider sketching a picture of a cannon pointing straight up, 10 feet above ground. Ask students to consider what a speed of 406 feet per second means in more concrete terms. How fast is that?

Students may be more familiar with miles per hour. Tell students that the speed of 406 feet per second is about 277 miles per hour.

Consider arranging students in groups of 2 so they can divide up the calculations needed to complete the table. Provide access to calculators, if requested.

Ask students how the values in the table are changing and what equation would describe the height of the cannonball if there were no gravity. Even without graphing, students should notice that the height of the cannonball over time is a linear function given the repeated addition of 406 feet every time \(t\) increases by 1.

Tell students that, in the next activity, they will look at some actual heights of the cannonball.

Prior to this course, students learned that an object traveling at a constant speed can be described with a linear function whose graph is a straight line. Here they see a model that accounts for the fact that an object that is launched straight up at a constant speed does not keep going at the same rate when the influence of gravity is taken into account. Adding a quadratic term to a linear function has an effect of “bending” the graph, as the output values are no longer changing at a constant rate. 

If students are unsure how to write an equation to represent the values in the table, ask them to compare how the actual heights of the cannonball at each second (when \(t\) is 1, 2, 3, etc.) differ from those in the no-gravity case (as shown in the table in the warm-up). Finding the differences between the two outputs (16, 64, 144, . . .) at the same input values should help students think of the numbers and the functions they saw in the previous lesson.

To generalize the relationship between time and distance, students reason repeatedly with numerical values and look for regularity (MP8). If students opt to use spreadsheet or graphing technology, they practice choosing appropriate tools strategically (MP5).

Arrange students in groups of 2. Give students a minute of quiet time to think about the first question, and then time to share their observations with their partner. Tell students that they will need to reference their work in the warm-up.

Some students may choose to use a spreadsheet tool to extend the pattern, and subsequently to use graphing technology to plot the data. Make these tools accessible, in case requested.

When comparing the tables, some students may make observations that lack the detail needed to write an equation for the actual height. Prompt them rewrite the outputs for the actual height in terms of the hypothetical height (\(400 = 416 - 16\), \(758 = 822 - 64\), \(1,\!084 = 1,\!228 - 144\), and so on).  Show them values of \(16t^2\) from a previous lesson to help them see and extend the pattern to write the equation.   

Invite students to share their observations about the two tables (the one from the warm-up and the one here) and how the two graphs compare. Highlight responses that suggest that the values in the second table account for the effect of gravity.

Help students see how the output for each \(t\) value varies across the two tables. When \(t\) is 1, the output in feet in the second table is 16 less than in the first table. When \(t\) is 2, there is a difference of 64 feet. When \(t\) is 3, that difference is 144 feet, and so on. The values 16, 64, 144, . . . correspond to the expression \(16t^2\) that we saw in the previous lesson (the distance fallen in feet as a function of time in seconds), so we can represent the values in the second table with the equation \(d=10+ 406t -16t^2\). Ask students:

• “What do the 10, \(406t\), and \(\text-16t^2\) mean in this situation?” (The 10 is the vertical position of the cannonball before it was launched: 10 feet above ground. In \(406t\), the 406 tells us the vertical velocity at which it was shot up. The \(\text-16t^2\) accounts for the effect of gravity on the height of the cannonball after it was shot up.)
• “Why do you think the graph that represents \(d= 10+ 406t\) changes from a straight line to a curve when \(\text-16t^2\) is added to the equation?” (Before that term was added the height increased by 406 feet every second. Adding \(\text-16t^2\) decreases how much the cannonball travels up by some amount, but that amount gets larger each successive second. Eventually the cannonball stops increasing in height and starts to fall.)

In this activity, students explore another model of a projectile motion. They graph and interpret a quadratic function in context and begin considering a reasonable domain for the function. Along the way, they practice reasoning concretely and abstractly (MP2).  By the end of the lesson, they relate the vertex of the graph to the maximum height of the cannonball and the positive zero of the function to the time when the cannonball hits the ground.

Provide access to devices that can run Desmos or other graphing technology. If needed, demonstrate how to adjust the graphing boundaries of the graphing tool.

Depending on the graphing tool available and their facility with it, students may approach the estimations in the third question in different ways (including by eyeballing). If desired, demonstrate how to use the graphing tool to trace the graph and identify the coordinates of any point on it (which may include values that are precise or values rounded to a specified decimal place). Or, first observe how students go about estimating and give additional guidance as needed.

To support students with the last question, ask students: “Is the equation a good model for predicting the height of the cannonball 10 seconds after it is fired? What about 1 minute after it is fired?”

Invite students to share their observations and interpretations of the graph. Highlight the following points:

• The graph representing a quadratic function is a parabola, which is a special kind of U shape. We will learn more about the geometry of this shape in a later course, but for now, notice that there is a point when the graph changes direction, from going up as \(x\) increases to going down (or changes from going down to going up). We call this point the vertex of the graph.
• In the previous activity, we plotted a limited set of points, so we could not tell where the vertex of the graph was. In this graph, we are able to identify the vertex. In this situation, the vertex tells us the maximum height that the cannonball reaches and the number of seconds after launch that it took before it starts to fall.
• In this graph, we can also see that the height of the cannonball is 0 when \(t\) is a little less than 20. That point, the horizontal intercept, relates to the zero of the function, or an input value that produces 0 for the output. In this situation, the zero tells us when the cannonball hits the ground.
• Even though we can continue the graph beyond \(t\) of 20, in this situation any output values beyond that point would not have any meaning. After the cannonball hits the ground, the function is no longer appropriate for modeling the movement of the cannonball. Likewise, the function is not appropriate before \(t=0\) or before the cannon is fired. In this situation, a domain between 0 and just below 20 seconds is appropriate.