This Warm-up prompts students to compare four quadratic expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as “standard form,” “factors,” and “coefficient.” Also, press students on unsubstantiated claims.

Most of the factored expressions students saw were of the form or . In this activity, students work with expressions of the form . They expand expressions such as into standard form and look for structure that would allow them to go in reverse—that is, to transform expressions of the form , where is not 1 (such as ) into factored form (MP7).

Going from factored form to standard form is fairly straightforward given students’ experience with the distributive property. Going in reverse, however, is a bit more challenging when the coefficient of is not 1. With some guessing and checking, students should be able to find the factored form of the expressions in the second question, but they should also notice that this process is not straightforward.

Earlier, students were given quadratic expressions of the form , where is not 1. They found that the rewriting process was a bit more involved but was not impossible, at least for the problems at hand.

In this activity, they encounter a real-world example in which they struggle to find a combination of rational factors of and that would produce the value of .

Realistic quadratic functions don’t always have rational numbers for their zeros, so the quadratic expressions that define them cannot always be easily written in factored form without finding the zeros in another way. At this point, students are not yet considering noninteger solutions and are not expected to know why some expressions cannot be easily written in factored form.

By graphing the function, students discover that the horizontal intercepts are decimals (rounded to different decimal places, depending on the graphing technology used). The graph allows them to estimate the solution, but that is as far as they could go. The challenges in this activity set the stage for introducing a more productive technique for solving quadratic equations.

This activity is optional. It allows students to investigate another way (besides guessing and checking) to find the factors of a quadratic expression given in standard form in which the leading coefficient is not 1. The method involves temporarily substituting the squared term with another variable so that the leading coefficient is 1 (which makes it easier to spot the factors of the expression), and then substituting the original variable back once the expression is in factored form.

Students may find the reasoning and substitution processes challenging. For example, transitioning from to and then to requires abstract reasoning. When the leading coefficient is not a square number (as in the second half of the activity), an additional step of multiplying is needed to make the leading coefficient a square number. To keep the value of an expression unchanged, students need to remember that they can multiply the expression only by 1, but the 1 can be obtained by, say, multiplying by 5 and then by .

Before offering support, allow students to make sense of these processes and to discuss with their partner. The work encourages students to persevere in sense making and problem solving (MP1) and to make use of structure (MP7). It also prompts students to attend to precision (MP6). As they make symbolic substitutions, students need to be clear about what the variables stand for and how the substitutions transform the expressions.

Later in the unit, students will encounter the use of a placeholder, such as done here, as a way to derive the quadratic formula. If desired, this activity can be done at that point to warm students up to the idea of using a placeholder.

If time is limited and if desired, the activity could be split into two halves and done separately (over two class periods).