Lesson 14

In this warm-up, students write quadratic expressions in standard form of the type \(k^2x^2 + 2kmx + m^2\) when given values for \(k\) and \(m\). In the associated Algebra 1 lesson, students will use quadratic expressions in this form to factor it into perfect squares.

Ask students to give the general form of a quadratic expression in standard form. If students struggle to provide the general form, ask for an example of a quadratic expression in standard form. Display the general form of a quadratic expression in standard form for all to see: \(ax^2 + bx + c\).

The purpose of the discussion is for students to get familiar with using many variables and unknowns to create an expression when some of the values are known. Select students to share their solutions. Then ask students,

• “Do you notice anything special about the expression in standard form for the first question? How would you factor it?” (It is a perfect square since it can be factored as \((x+3)^2\).)
• “Do you have a good method for separating letters that represent numbers you know from letters that will remain in the final solution?” (I like to write out the general form I should work with, then immediately replace any letters with values I know. Like if I know \(k = 5\), then I would start with \(a = k^2\) and then write \(a = (5)^2\). This helps me separate which letters have known values (like \(k\)) and which do not (like \(a\)).)

In this activity, students rewrite quadratic expressions in factored form into standard form. This activity expands on previous work by requiring students to expand the expressions for which the coefficient of the quadratic term is non-zero. Students look for and make use of structure (MP7) when they use the distributive property or table to rewrite expressions.

The purpose of the discussion is for students to gain fluency in working with quadratic expressions for which the leading coefficient is non-zero. Select students to share their solutions and methods. If possible, select at least one student who used Priya’s method and one who used Tyler’s method.

In this activity, students solve for values of \(k\) and \(m\) using some given information. In the associated Algebra 1 lesson, students use similar equations to help complete the square for quadratic expressions with a non-zero leading coefficient.

The purpose of the discussion is to help students organize their thinking when combining information to solve for unknown values. Select students to share their solutions and methods. Ask students,

• “What is the order in which you used the information when solving for \(k\) and \(m\)? Could you have used the information in a different order? Explain your reasoning.” (I used the information about \(k^2\) first to determine possible values for \(k\), then used the inequality to select the exact value, then I use the information about \(2km\) to find the value for \(m\). It does not make sense to use the information about \(2km\) until I know either \(k\) or \(m\), so that will need to go last. I could’ve used the inequality first to limit the numbers I tried in the equation with \(k^2\).)
• “If the inequality was not given, how might that change your answers?” (Without the inequality, there are 2 possible values for \(k\) which leads to 2 possible values for \(m\), but \(m^2\) would be the same for these questions.)
• “If the real goal is to solve for \(m^2\) from the information given, the last equation can be transformed into \(m^2 = \frac{a^2}{4k^2}\) where \(a\) represents the constant value given in the last equation. Knowing this, the value for \(k^2\) could be substituted in and the value of \(m^2\) found without solving for \(k\) and \(m\). Explain how the equation \(2km = a\) can be transformed into \(m^2 = \frac{a^2}{4k^2}\).” (First, divide both sides of the equation by \(2k\) to rewrite the equation as \(m = \frac{a}{2k}\). Then square both sides of the equation to get \(m^2 = \frac{a^2}{4k^2}\).)