6fd022ee-b8a4-4379-ab5f-f782da70ff93.html

<h3>Warm Up (5 minutes)</h3>
<p>In this warm up, students consider perfect squares in standard form in which the leading coefficient is not 1. They use what they know about expanding \((x+m)^2\) to analyze the expansion of an expression of the form \((kx+m)^2\) and to identify an error. In explaining and correcting the error, students practice constructing logical arguments.</p>
<h4>Launch</h4>
<p>Display the entire task for all to see. Give students two minutes of quiet time to think individually. Select students to share their responses and how they reasoned about the error in Elena's statement.</p>
<h4>Student Activity</h4>
<p>Elena says, "\((x+3)^2\) can be expanded into \(x^2+6x+9\). Likewise, \((2x+3)^2\) can be expanded into \(4x^2+6x+9\)."</p>
<p>Find an error in Elena's statement and correct the error. Show your reasoning.</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> \((2x+3)^2\) is not equivalent to \(4x^2+6x+9\).</p>
<p>Applying the distributive property to \((2x+3)(2x+3)\) gives \(4x^2+6x+6x+9\), which is \(4x^2+12x+9\).</p>
<h4>Activity Synthesis</h4>
<p>Select one to two students to share their responses. Make sure students see that expanding the expression \((kx+m)^2\) involves the same structure as expanding \((x+m)^2\). Highlight the structure using a diagram and the distributive property.</p>
<p>When we expand \((x+3)^2\), the squared term is \(x^2\), the linear term is 2 times \((3)(x)\), which is \(6x\), and the constant term is \(3^2\). When we expand \((2x+3)^2\), the squared term is \((2x)^2\), the linear term is 2 times \((2x)(3)\), which is \(12x\), and the constant term is \(3^2\). Make sure that students rewrite \((x+3)^2\)as \((x + 3)(x + 3)\) and then perform the distributive property, rather than simply squaring each term in the binomial.</p>
<p>Students will have opportunities to transform more of such expressions later and, with practice, will better see the effects of squaring a linear expression like \(2x+3\), in which the coefficient of the variable is not 1.</p>