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<div class="activities">
<div class="hero">
<h2><span>Warm Up (10 minutes)</span></h2>
</div>
<div class="body">
<p>This Math Talk encourages students to look for connections between the features of graphs and of linear equations that each represent a system. Given two graphs on an unlabeled coordinate plane, students must rely on what they know about horizontal and vertical lines, intercepts, and slope to determine if the graphs could represent each pair of equations. The equations presented and the reasoning elicited here will be helpful later in the lesson, when students solve systems of equations by substitution.</p>
<p>All four systems include an equation for either a horizontal or a vertical line. Some students may remember that the equation for such lines can be written as <span class="math math-repaired">\( x = a \)</span> or <span class="math math-repaired">\( y=b \)</span>, where <span class="math math-repaired">\( a \)</span> and <span class="math math-repaired">\( b \)</span> are constants. (In each of the first three systems, one equation is already in this form. In the last system, a simple rearrangement to one equation would put it in this form.) Activating this knowledge would enable students to quickly tell whether a system matches the given graphs.</p>
<p>Those who don’t recall it can still reason about the system structurally. For instance, given a system with <span class="math math-repaired">\( x=-5 \)</span> as one of the equations, they may reason that any point that has a negative <span class="math math-repaired">\( x \)</span>-value will be to the left of the vertical axis. The solution (if there is one) to this system would have to have \(-5\) for the <span class="math math-repaired">\( x \)</span>-value. The intersection of the given graphs is a point to the right of the vertical axis (and therefore has a positive <span class="math math-repaired">\( x \)</span>-value), so the graphs cannot represent that system.</p>
<p>To match graphs and equations, students need to look for and make use of structure in both representations. In explaining their strategies, students need to be precise in their word choice and use of language. </p>
<p>Because the warm up is intended to promote reasoning, discourage the use of graphing technology to graph the systems.</p>
</div>
<h3>Launch</h3>
<div class="body">
<p>Display one system at a time. Give students quiet time to think individually for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.</p>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR 8 Discussion Supports: Speaking</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.</p>
<p class="os-raise-text-italicize">Design Principle(s): Optimize output (for explanation)</p>
</div>
</div>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Representation: Internalize Comprehension</p>
</div>
<div class="os-raise-extrasupport-body">
<p>To support working memory, provide students with sticky notes or mini whiteboards.</p>
<p class="os-raise-text-italicize">Supports accessibility for: Memory; Organization</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Use the following information to answer questions 1–8. Here are graphs of two equations in a system. </p>
<blockquote>
<div> <img alt="Graph of 2 intersecting lines." height="96" src="https://k12.openstax.org/contents/raise/resources/f623edcf858d0cd5915a0df984d3fc5aafdd9b66" width="122"></div>
<div></div>
<p>Determine if each of these systems could be represented by the graphs. Be prepared to explain how you know.</p>
</blockquote>
<ol class="os-raise-noindent">
<li>\( \begin{cases} x + 2y = 8 \\x = -5 \end{cases} \) </li>
</ol>
<p><Strong>Answer: </Strong>No</p>
<!--Q#-->
<ol class="os-raise-noindent" start="2">
<li>How do you know?</li>
</ol>
<p><Strong>Answer: </Strong>No</p>
<p> Your answer may vary, but here is a sample. No, it cannot be represented by the graph. The graph for \(x=−5\) would be on the left side of the vertical axis, which is not what is shown.</p>
<ol class="os-raise-noindent" start="3">
<li>\( \begin{cases} y = -7x + 13 \\y = -1 \end{cases} \) </li>
</ol>
<p><Strong>Answer: </Strong>No</p>
<ol class="os-raise-noindent" start="4">
<li>How do you know?</li>
</ol>
<p><Strong>Answer: </Strong></p>
<p> Your answer may vary, but here is a sample. No, it cannot be represented by the graph. The graph for \(y=−1\) would be a horizontal line, but no horizontal line is shown.</p>
<ol class="os-raise-noindent" start="5">
<li>\( \begin{cases} 3x = 8\\3x + y = 15 \end{cases} \) </li>
</ol>
<ol class="os-raise-noindent" start="7">
<li>How do you know?</li>
</ol>
<p><Strong>Answer: </Strong>No</p>
<p> Your answer may vary, but here is a sample.
Yes. \(3x = 8\) can be rearranged into \(x=\frac83\). The graph for \(x=\frac83\) is a vertical line to the right of the vertical axis. \(3x + y = 15\) can be rewritten as \(y = -3x + 15\). Its graph has a negative slope and a positive \(y\)-value for the vertical intercept.</p>
<ol class="os-raise-noindent" start="7">
<li>\( \begin{cases} y = 2x - 7\\4 + y = 12 \end{cases} \)</li>
</ol>
<ol class="os-raise-noindent" start="8">
<li>How do you know?</li>
</ol>
<p><Strong>Answer: </Strong>No</p>
<p> Your answer may vary, but here is a sample.
No. The graph for \(y = 2x - 7\) would have a positive slope and a negative \(y\)-value for the vertical intercept, neither of which matches the given graphs.</p>
<!--Text Entry Interaction Start -->
<h3>Activity Synthesis</h3>
<div class="body">
<p>Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:</p>
<ul>
<li>“Who can restate <span class="math math-repaired">\( \underline{\hspace{.5in}} \)’</span>s reasoning in a different way?”</li>
<li>“Did anyone have the same strategy but would explain it differently?”</li>
<li>“Did anyone solve the problem in a different way?”</li>
<li>“Does anyone want to add on to <span class="math math-repaired">\( \underline{\hspace{.5in}} \)’</span>s strategy?”</li>
<li>“Do you agree or disagree? Why?”</li>
</ul>
<p>If no students mentioned solving the systems and then checking to see if the solution could match the graphs, ask if anyone approached it that way. For instance, ask: “How could we find the solution to the second system without graphing?” Give students a moment to discuss their ideas with a partner and then proceed to the next activity.</p>
</div>
</div>