Lesson 17

Previously, students have seen that some quadratic functions have no zeros and that some quadratic equations have no solutions. In this warm-up, they recall that when a solution is a square root of a negative number, the equation has no solutions. Note that students don’t yet know about any numbers that aren’t real at this point, so it is unnecessary to specify “no real solutions.” (To students, the word “real” would seem like an extra word added for no reason.)

Arrange students in groups of 2. Give students quiet think time, and then time to share their thinking with a partner.

Select students to share their responses and reasoning.

If not mentioned in students’ explanations, point out that we can look at and reason about the structure of these equations to tell which ones have no solutions, without taking any steps to solve. For example, the equation \((x+3)^2 + 9=0\) means “a square plus 9 equals 0.” Here are two ways to reason about its structure:

• For two numbers to add up to 0, they have to be opposites. Since the number 9 is positive, the other number must be negative, which a square cannot be.
• A square is always positive, so a square plus a positive number must also be positive and can never equal zero.

These lines of reasoning also allow us to see that \(49+x^2=0\) has no solutions.

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 writing equations of the form \(\text{expression in factored form}=0\) and applying the zero product property.

Students have also used graphs to estimate input values that yield non-zero 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.

Remind students that in earlier lessons they encountered two functions that modeled the height of a launched object as a function of time. They solved problems about the functions by graphing. Tell students that they now have additional strategies at their disposal and ask them to solve these problems without graphing. Provide access to calculators. Tell students to use them only for numerical computations. If time is limited, consider asking students to answer only the first set of questions.

Watch for students who incorrectly substitute 40 for \(t\) and evaluate \(h(40)\), or who write \(g(t)=8\) instead of evaluating \(g(8)\). 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 \(h(t)\) represents the output for input \(t\).

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 non-zero 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 is 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 understandings around writing and solving quadratic equations. They are now ready to formulate the problem effectively and solve it methodically.

Students engage in aspects of mathematical modeling (MP4) as they identify the constraints in a situation, formulate a problem, construct a model, and interpret their solutions in context.

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.

Ask students to recall the framing problem from the beginning of the unit, how they tried to solve it, and what challenges they encountered. Consider preparing a copy of the picture and framing material from that lesson to serve as a visual aid. Tell students that they now have enough knowledge and skills to solve the problem more effectively and no longer have to rely on guessing and checking.

Arrange students in groups of 2. Give students a moment of quiet think time to interpret the equation \((7 + 2x)(4 + 2x) = 38\) in the first question and then discuss it with a partner. Display some sentence stems to help students articulate their interpretation. For example:

• The two factors in the equation represent . . .
• The number 38 represents . . .
• The variable \(x\) represents . . .
• Solving the equation means finding . . . . Pause afterward for a brief discussion. Make sure students understand the meaning of the equation before they proceed to solve it.

This activity was designed to be completed without graphing, so ask students to put away any graphing devices. Provide continued access to calculators for numerical computations.

If time is limited, consider asking students to complete only the first question.

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.)