Lesson 10

This warm-up prompts students to carefully analyze and compare quadratic expressions. In making comparisons, students need to look for common structure and have a reason to use language precisely (MP7, MP6). The activity also enables the teacher to hear the terminology students know and how they talk about characteristics of the different forms of expressions.

As students discuss in groups, listen for rationales that are based on the structure of the expressions. Select those students or groups to share their thinking during class discussion.

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning about why a particular item does not belong, and together find at least one reason each item doesn't belong.

Ask each group to share one reason why a particular expression doesn’t belong. 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 one does not belong, 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, factored form, linear term, or coefficient. Also, press students on unsubstantiated claims. For example, if they claim that B is the only one that cannot be written in factored form, ask them to show how they know.

Most of the factored expressions students saw were of the form \((x+m)(x+n)\) or \(x(x+m)\). In this activity, students work with expressions of the form \((px+m)(qx+n)\). They expand expressions such as \((2x+1)(x+10)\) into standard form and look for structure that would allow them to go in reverse (MP7), that is, to transform expressions of the form \(ax^2+bx+c\), where \(a\) is not 1 (such as \(2x^2+12x+10\)) into factored form.

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 \(x^2\) 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.

Remind students that they have seen quadratic expressions such as \(16t^2 + 800t + 400\) and \(5x^2 +21x -20\), where the coefficient of the squared term is not 1. Solicit some ideas from students on how to write the factored form for expressions such as these.

Arrange students in groups of 2–3 and ask them to split up the work for completing the first table, with each group member rewriting one expression into standard form.

Display the incomplete table in the first question for all to see, and then invite students to share the expanded expressions in standard form. Record the expressions in the right column, and ask students to make observations about them.

If not mentioned by students, point out that each pair of factors start with \(3x\) and \(x\), which multiply to make \(3x^2\). Each pair of factors also has constant terms that multiply to make 4. The resulting expressions in standard form are all different, however, because using different pairs of factors of 4 and arranging them in different orders produce different expanded expressions.

Ask students to keep these observations in mind as they complete the second question.

If time is limited, ask each group member to choose at least two expressions in the second table and rewrite them into factored form.

Some students may not think to check their answers to the second question and stop as soon as they think of a pair of factors that give the correct squared term and constant term. Encourage them to check their answers with a partner by giving time to do so. Consider providing a non-permanent writing surface or extra paper so students could try out their guesses and check their work without worrying about having to erase if they make a mistake the first time or two.

Display for all to see the incomplete table in the second question. Select students to complete the missing expressions in standard form and to briefly explain their strategy. To rewrite \(ax^2+bx+c\), students are likely to have tried putting different factors of \(a\) and of \(c\) in the factored expression such that when the factors are expanded, they yield a linear term with the coefficient \(b\).

Then, help students to reason about the factors more generally. Discuss questions such as:

• “To rewrite expressions such as \(x^2+bx+c\), we looked for two numbers that multiply to make \(c\) and add up to \(b\). The expressions here are of the form \(ax^2+bx+c\). Are we still looking for two numbers that multiply to make \(c\)? Why or why not?” (Yes. The constant terms in the factored expression must multiply to make \(c\).)
• “Do we need to look for factors of \(a\)? Why or why not?” (Yes. Those factors will be the coefficient of \(x\) in the factored expressions. They must multiply to make \(a\).)
• “Are we still looking for two factors of \(c\) that add up to \(b\)? Why or why not?” (No. The value of \(b\) is no longer just the sum of the two factors of \(c\) because the two factors of \(a\) are now involved.) “How does this affect the rewriting process?” (It makes it more complicated, because now there are four numbers to contend with, and there are many more possibilities to consider.)

Tell students that we’ll investigate a bit further quadratic equations in the form of \(ax^2+bx+c\) where \(a\) is not 1, and see if there are manageable ways to rewrite such equations in factored form so that they can be solved.

Earlier, students were given quadratic expressions of the form \(ax^2+bx+c\) where the \(a\) 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 \(a\) and \(c\) that would produce the value of \(b\). (Realistic quadratic functions don’t always have rational numbers for their zeros, so the quadratic expressions that define them cannot always be written in factored form. At this point, students are not yet considering rational and irrational 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.

Keep students in groups of 2. Let students become briefly frustrated by their unsuccessful attempts to find factors of the expression in standard form, but move them on to the last question after a few minutes. Provide access to devices that can run Desmos or other graphing technology. 

Ask students to share some challenges they came across when trying to rewrite the expressions in factored form. Solicit some ideas about why this equation presented those challenges. Then, discuss how they found or estimated the solution by graphing.

The approximate solution to the equation, given by the zero of the function and the \(x\)-intercept of the graph, is 2.087 seconds. Some graphing tools would give an approximation with a longer decimal expansion, giving a clue that it might be trickier to rewrite the equation in factored form. After all, when finding factors, we usually look for integers. (Some quadratic expressions containing non-integer rational numbers can still be written in factored form. For example, \(x^2 + \frac 34 + \frac18\) can be written as \((x+\frac12)(x+\frac14)\).)

Highlight that some equations are difficult or impossible to rewrite in factored form. In fact, when quadratic models appear in real life, this is usually the case. Graphing is a way to solve these equations, but there are other techniques, which students will learn over the next several lessons.

This activity is optional. It allows students to investigate another way (beside guessing and checking) for finding the factors of quadratic expressions in standard form where 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. Transitioning from, for example, \(9x^2+21x+10\) to \((3x)^2+7(3x)+10\), and then to \(N^2+7N+10\) 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 only multiply the expression by 1, but the 1 can be obtained by, say, multiplying by 5 and then by \(\frac15\).

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).

Display these expressions for all to see and ask students which expressions would be easier to write in factored form and why.

\(4x^2 - 12x + 8\)

\(t^2 - 6t + 8\)

\(d^2 + d - 20\)

\(9x^2 + 3x - 20\)

Students are likely to say that the second and the third expressions are easier because the coefficient of the squared term in each of those is 1 (or there isn’t another number that needs to be factored aside from the constant term). Tell students that they will study another strategy that can simplify the process of rewriting quadratic expressions into factored form.

Consider arranging students in groups of 2 and asking them to think quietly about at least the first couple of problems before discussing with their partner. After students have had a chance to make sense of the first worked example, pause for a class discussion. Make sure that students can follow what is happening in the shown steps before trying to apply it with new expressions.

Pause for another class discussion after students have analyzed the second worked example. Before students proceed to the last question, clarify what is happening in each step of the rewriting process when the leading coefficient is not a square number.

In the final steps of the last question, students multiply a number by a pair of factors, for example, \(\frac13 (3x+15)(3x+1)\). Some students may mistakenly apply the distributive property and multiply \(\frac13\) to both \((3x+15)\) and \((3x+1)\). Remind students that the distributive property governs multiplication over addition and subtraction, and that \((3x+15)\) and \((3x+1)\) are being multiplied together, not added or subtracted.

Invite students to share their attempts to rewrite the expressions using the method they just learned. Discuss questions such as:

• “Do you think this method is simpler or harder than guessing and checking? Is it quicker or slower?”
• “In what ways is this method simpler? In what ways is it harder?”