By now, students are pretty familiar with quadratic functions that model the height of objects as a function of time after being launched up or being dropped. They have found the zeros of such functions graphically, by identifying horizontal intercepts, as well as algebraically, by applying the zero product property.

Students have also used graphs to estimate input values that yield nonzero output values. Prior to learning about completing the square or the quadratic formula, however, they did not have the tools to solve for such inputs algebraically. In this activity, students apply the knowledge and skills they recently developed to solve contextual problems that they couldn’t previously solve without graphing.

Because algebraic reasoning is the aim of this activity, graphing technology should not be used to solve the equations. It could be used to verify solutions during the Activity Synthesis, however.

If time is limited, assign the first question to half of the class and the second question to the other half. Alternatively, ask students to choose one question to answer.

Watch for students who incorrectly substitute 40 for and evaluate , or who write instead of evaluating . These students may be having trouble distinguishing between the input and output because the meaning of words such as “the height is a function of time” is still unclear to them. Remind these students that “height is a function of time” means that the variable in the formula given represents time, and that the formula can be used to find the height. Point out that represents the output for input .

Make sure students understand what equations to write and what it means to solve each equation in the given contexts. Then, focus the discussion on how the solutions can be found, interpreted, and verified. Ask questions such as:

• “Once you have the equations, how did you find the solutions when the equation has a 0 on one side?” (by using the quadratic formula or by completing the square)
• “What about when the equation has a nonzero number on one side?” (Rearrange it so that it is in standard form and has 0 on one side, and then use one of the solving methods.)
• “How many solutions did you get?” (two for each equation) “Do both solutions make sense in the given context?” (No. Only positive solutions are meaningful because they represent time after the objects were launched.)
• “How can we check if our solutions are accurate?” (Solve with a different method, substitute them back into the equation, and see if the equation is true, or check by graphing.)

Consider displaying the graphs of the two functions to verify students’ solutions.

In this activity, students return to the framing problem they encountered when starting the unit. In that first lesson, they were challenged to use an entire sheet of paper to frame a picture and ensure that the thickness was uniform all around. At that time, students did not have adequate knowledge to solve the problem methodically, so they relied mainly on guessing and checking. Since then, students have developed their understanding about writing and solving quadratic equations. They are now ready to formulate the problem effectively and solve it methodically.

Students must reason abstractly and quantitatively to interpret the equations in the situation (MP2).

Because algebraic reasoning is the aim of this activity, graphing technology should not be used to solve the equations. It could be used to verify solutions during the Activity Synthesis, however.

Invite students to share their solution strategies. If not mentioned in students’ explanations, make sure to discuss:

• Why negative solutions aren’t meaningful in this context.
• Which strategy they found effective and efficient for solving the equations. (The quadratic formula might not be the speediest way to solve the last equation, for example, as it can be solved by rewriting the equation in factored form and by completing the square.)

If time permits, consider asking students to verify their solution to the first question using the picture and framing materials from the blackline master from the first lesson of the unit. Ask students to cut the paper into strips that are as thick as the solution they calculated and arrange the strips around the picture. (If their solution is correct, there should be no leftover framing material and the frame should be uniform all around the picture.)