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<h3><span>Activity (20 minutes)</span><br></h3>
<p>This activity is the first of several that draw students’ attention to the structure of linear equations in two variables, how it relates to the graphs of the equations, and what it tells us about the situations. </p>
<p>Students start by interpreting linear equations in standard form, <span class="math math-repaired">\( Ax+By=C \)</span>, and using them to answer questions and create graphs. They see that this form offers useful insights about the quantities and constraints being represented. They also notice that graphing equations in this form is fairly straightforward. We can use any two points to graph a line, but the two intercepts of the graph (where one quantity has a value of 0) can be found quickly using an equation in standard form.</p>
<p>Students then analyze the graphs to gain other insights. They determine the rate of change in each relationship and find the slope and vertical intercept of each graph. Next, they rearrange the equations to isolate <span class="math math-repaired">\( y \)</span>. They make new connections here—the rearranged equations are now in slope-intercept form, which shows the slope of the graph and its vertical intercept. These values also tell us about the rate of change and the value of one quantity when the other quantity is 0.</p>
<h3>Launch</h3>
<p>Tell students that they will now interpret an equation about games and rides. They will also use graphs to help make sense of what combinations of games and rides are possible, given certain prices and budget constraints.</p>
<p>Read the opening paragraph in the task statement and display the equation for all to see. Give students quiet time to think about what the equation means in the situation and then discuss their interpretations. Make sure students share this interpretation:</p>
<ul>
<li>Games and rides cost $1 each, and the student is spending $20 on them.</li>
</ul>
<p>Arrange students in groups of 3–4. Ask them to answer the questions for the equation.
</p>
<p>Give students quiet work time and then a few minutes to discuss their responses with their group and resolve any disagreements. Follow with a whole-class discussion.</p>
<br>
<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 2 Collect and Display: Maximize Meta-Awareness, Support Sense-Making</p>
</div>
<div class="os-raise-extrasupport-body">
<p>During the launch, listen for and collect language students use to describe the meaning of the three equations. Record a written interpretation next to each of the three equations on a visual display. Use arrows or annotations to highlight connections between specific language of the interpretations and the parts of the equations. This will provide students with a resource from which to draw language during small-group and whole-group discussions.</p>
<p class="os-raise-text-italicize">Design Principle(s): Maximize meta-awareness; Support sense-making</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</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>Demonstrate, and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to illustrate where the slope appears in each equation and corresponding graph. Continue to use colors consistently as students discuss “What do the A, B, and C represent in each equation?”</p>
<p class="os-raise-text-italicize">Supports accessibility for: Visual-spatial processing</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<img alt="Los Angeles county fair." class="img-fluid atto_image_button_text-bottom" height="366" src="https://k12.openstax.org/contents/raise/resources/e8b47724f5f8ec17be0e61287de32802b99adc9a" width="550">
<p> </p>
<p>For numbers 1–7, use the equation \(x + y = 20\) which represents the relationship between the number of games, \(x\), the number of rides, \(y\), and the dollar amount a student is spending on games and rides at an amusement park.</p>
<!--START QUESTION 1-->
<ol class="os-raise-noindent">
<li>What’s the number of rides the student could get on if they don’t play any games?</li>
</ol>
<p><strong>Answer: </strong>20.</p>
<!--END QUESTION.-->
<!--START QUESTION 2-->
<ol class="os-raise-noindent" start="2">
<li>What would the coordinate be on the graph to represent the answer to question 1?</li>
</ol>
<p><strong>Answer: </strong>\((0,20)\)</p>
<!--END QUESTION.-->
<!--START QUESTION 3-->
<ol class="os-raise-noindent" start="3">
<li>What’s the number of games the student could play if they don’t get on any rides?</li>
</ol>
<p><strong>Answer: </strong>20.</p>
<!--END QUESTION.-->
<!--START QUESTION 4-->
<ol class="os-raise-noindent" start="4">
<li>What would the coordinate be on the graph to represent the answer to question 3?</li>
</ol>
<p><strong>Answer: </strong>\((20,0)\).</p>
<!--END QUESTION.-->
<!--START QUESTION 5-->
<ol class="os-raise-noindent" start="5">
<li>On the coordinate plane, mark the points found in numbers 2 and 4. Draw a line to connect the two points you’ve drawn. Use the graphing tool or technology outside the course. Students were given access to Desmos. </li>
</ol>
<p><strong>Answer: </strong></p>
<img alt="Graph of a linear function. numbers of rides, number of games." height="194" src="https://k12.openstax.org/contents/raise/resources/584409033f473f355e3fcabc8c5bac8a2a44ad2b">
<!--END QUESTION.-->
<!--START QUESTION 6-->
<ol class="os-raise-noindent" start="6">
<li>Complete the sentence: “If the student played no games, they can get on <u>_____</u> rides.” </li>
</ol>
<p><strong>Answer: </strong>20</p>
<!--END QUESTION.-->
<!--START QUESTION 7-->
<ol class="os-raise-noindent" start="7">
<li>Complete the sentence: “For every additional game that the student plays,\(x\), the possible number of \(y\) _______________ (increases or decreases) by __________________.”</li>
</ol>
<p><strong>Answer: </strong>Decreases by 1</p>
<!--END QUESTION.-->
<!--START QUESTION 8-->
<ol class="os-raise-noindent" start="8">
<li>What is the slope of your graph? </li>
</ol>
<p><strong>Answer: </strong>-1.</p>
<!--END QUESTION.-->
<!--START QUESTION 9-->
<ol class="os-raise-noindent" start="9">
<li>At what point does the graph intersect the vertical axis?</li>
</ol>
<p><strong>Answer: </strong>\((0,20)\)</p>
<!--END QUESTION.-->
<!--START QUESTION 10-->
<ol class="os-raise-noindent" start="10">
<li>Rearrange the equation to solve for \(y\).</li>
</ol>
<p><strong>Answer: </strong>\(y = -x + 20\) or \(y = 20 -x\)</p>
<!--END QUESTION.-->
<!--START QUESTION 11-->
<ol class="os-raise-noindent" start="11">
<li>What connections, if any, do you notice between your new equation and the graph?</li>
</ol>
<p><strong>Answer: </strong>The new equation shows where the graph intersects the \(y\)-axis at 20, so if the student plays 0 games, they have the money to go on 20 rides. Since the slope of the graph is \(-1\), for every 1 fewer ride they go on, they can play 1 game.</p>
<!--END QUESTION.-->
<h3>Activity Synthesis</h3>
<p>If students continue to have trouble with the activity and need more practice, have students repeat the questions from activity 1.10.2 with one or both of the following equations:</p>
<ul>
<li>Equation 2: Games cost $2.50 each, and rides cost $1 each. The student is spending $15 on them. <br>
<p>\(2.50x + y = 15\)</p>
</li>
<li>Equation 3: Games cost $1 each, and rides cost $4 each. The student is spending $28 on them. <br>
<p>\(x + 4y = 28\)</p>
</li>
</uL>
<p>Then repeat the questioning from the activity for each equation.</p>
<br>
<h3>1.10.2: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Which of the following equations correctly solved \(2x+4y=8\) for \(y\)? </p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(y=-\frac{1}{2}x+2\)</td>
<td>That’s correct! Check yourself: To solve for \(y\), first subtract \(2x\) from both sides then divide both sides by 4. Finally, simplify any fractions.
</td>
</tr>
<tr>
<td>\( y=-2x+4 \)</td>
<td>Incorrect. Let’s try again a different way: Make sure you solve for \(y\), not \(x\). The answer is \(y=-\frac{1}{2}x+2\).</td>
</tr>
<tr>
<td>\(y=\frac{1}{2}x+2\)</td>
<td>Incorrect. Let’s try again a different way: Make sure to subtract \(2x\) from both sides of the equation when solving for \(y\). The answer is \(y=-\frac{1}{2}x+2\).</td>
</tr>
<tr>
<td>\(y=-\frac{1}{2}x+8\)</td>
<td>Incorrect. Let’s try again a different way: Make sure to divide all of the terms by 4. The answer is \(y=-\frac{1}{2}x+2\).</td>
</tr>
</tbody>
</table>
<br>
<h3>1.10.2: Additional Resources<br></h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it.</em></p>
<br>
<h4>Find the Slope of a Line</h4>
<p>When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter.</p>
<p>In mathematics, the measure of the steepness of a line is called the of <span data-schema-version="1.0" data-store="glossary-tooltip">slope</span> the line. The concept of slope has many applications in the real world. In construction the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. As you ski or jog down a hill, you definitely experience a slope. We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope is a rate of change.</p>
<img alt height="175" role="presentation" src="https://k12.openstax.org/contents/raise/resources/d487a83f81be6fa2399c5c3e7bbc5f8386db50ba" width="300">
<br><br>
<h5>Slope of a Line</h5>
<p>The slope of a line is \( m = \frac{rise}{run} \). The rise measures the vertical change and the run measures the horizontal change.
<br>To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical.
<br>To find the slope of the line, we measure the vertical distance is called the <em data-effect="italics">rise</em> and the horizontal distance is called the <em data-effect="italics">run</em>.</p>
<h5>How To Find the Slope from a Graph</h5>
<p>Find the slope of a line from its graph using \( m = \frac{rise}{run} \)</p>
<div class="os-raise-graybox">
<p><strong>Step 1 - </strong>Locate two points on the line whose coordinates are integers.<br></p>
<p><strong>Step 2 - </strong>Starting with one point, sketch a right triangle, going from the first point to the second point.<br></p>
<p><strong>Step 3 - </strong>Count the rise and the run on the legs of the triangle.<br></p>
<p><strong>Step 4 - </strong>Take the ratio of rise to run to find the slope: \( m = \frac{rise}{run} \).<br></p>
</div>
<br>
<p><strong>Example 1</strong></p>
<p>Find the slope of the line shown.</p>
<img alt class="img-fluid atto_image_button_text-bottom" height="238" role="presentation" src="https://k12.openstax.org/contents/raise/resources/0fa24bf4380c8d84b5a5b6b912c69679c5046006" width="275">
<p>Solution:</p>
<p><strong>Step 1 - </strong>Locate two points on the graph whose coordinates are integers.<br></p>
<p>\((0, 5)\) and \((3, 3)\)</p>
<p><strong>Step 2 - </strong>Starting at \((0,5)\), sketch a right triangle to \((3,3)\) as shown in this graph.<br></p>
<img height="272" src="https://k12.openstax.org/contents/raise/resources/36b71bfd1476154ebbafc505191dc879e0753fad" width="298"></p>
<p><strong>Step 3 - </strong>Count the rise(how many spaces up or down)— since it goes down, it is negative.<br></p>
<p>The rise is \(-2\).</p>
<p><strong>Step 4 - </strong>Count the run (how many spaces to the right or left).<br></p>
<p>The run is 3.</p>
<p><strong>Step 5 - </strong>Use the slope formula.<br></p>
<p>\( m = \frac{rise}{run} \)</p>
<p><strong>Step 6 - </strong>Substitute the values of the rise and run.<br></p>
<p>\( m = \frac{-2}{3} \)</p>
<p><strong>Step 7 - </strong>Simplify.<br></p>
<p>\( m = -\frac{2}{3} \)</p>
<p>So \( y \) decreases by 2 units as \( x \) increases by 3 units.</p>
<h5>Slopes of Horizontal and Vertical Lines</h5>
<p>To find the slope of the horizontal line, \( y = 4 \), we could graph the line, find two points on it, and count the rise and the run. Let’s see what happens when we do this, as shown in the graph below.<p><img alt role="presentation" src="https://k12.openstax.org/contents/raise/resources/4f4abbf823afb7298491b4688a5c84a91b9f12c2"></p>
<p><strong>Step 1 - </strong>What is the rise?<br></p>
<p>The rise is 0.</p>
<p><strong>Step 2 - </strong>What is the run?<br></p>
<p>The run is 3.</p>
<p><strong>Step 3 - </strong>What is the slope?<br></p>
<p>\( m = \frac{rise}{run} \)</p>
<p>\( m = \frac{0}{3} \)</p>
<p>\( m = 0 \)</p>
<p>So The slope of the horizontal line \( y = 4\) is 0.</p>
<p>Let's also consider a vertical line, the line \( x = 3 \), as shown in the graph below.</p>
<img alt role="presentation" src="https://k12.openstax.org/contents/raise/resources/e792c268c1a852f67cba97d3990da16d0e826003">
<p><strong>Step 1 - </strong>What is the rise?<br></p>
<p>The rise is 2.</p>
<p><strong>Step 2 - </strong>What is the run?<br></p>
<p>The run is 0.</p>
<p><strong>Step 3 - </strong>What is the slope?<br></p>
<p>\( m = \frac{rise}{run} \)</p>
<p>\( m = \frac{2}{0} \)</p>
<p>The slope is undefined since division by zero is undefined. So we say that the slope of the vertical line is undefined.</p>
<ul>
<li>All horizontal lines have slope 0. When the \(y\)–coordinates are the same, the rise is 0.</li>
<li>The slope of any vertical line is undefined. When the \(x\)–coordinates of a line are all the same, the run is 0.</li>
</ul>
<h5><strong>Quick Guide to the Slopes of Lines </strong></h5>
<img alt role="presentation" src="https://k12.openstax.org/contents/raise/resources/40350686e52f6175268fa33a48576e28e387a022">
<br>
<p>Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.<br> <br>We have seen that an ordered pair \((x,y)\) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol \((x,y)\) be used to represent two different points? Mathematicians use subscripts to distinguish the points.<br> <br>We will use (\(x\)<sub>1</sub>,\(y\)<sub>1</sub>) to identify the first point and (\(x\)<sub>2</sub>,\(y\)<sub>2</sub>) to identify the second point.<br> Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points (2,3)and (7,6), as shown in this graph.<br></p>
<img alt role="presentation" src="https://k12.openstax.org/contents/raise/resources/3c48d3120a9c7e6eb8d62ef656ca89a4b0d46a91">
<br><br>
<p><strong>Step 1 - </strong>Since we have two points, we will use subscript notation.<br></p>
<p>\( \binom {x_1, y_1} {2, 3} \)\( \binom {x_2, y_2} {7, 6} \)</p>
<p><strong>Step 2 - </strong>What is the slope?<br></p>
<p>The rise is 3. <br>
The run is 5.</p>
<p><strong>Step 3 - </strong>What is the slope?<br></p>
<p>\( m = \frac{rise}{run} \)</p>
<p>On the graph, we counted the rise of 3 and the run of 5. \( m = \frac{3}{5} \)</p>
<p>Notice that the rise of 3 can be found by subtracting the \( y \)–coordinates, 6 and 3, and the run of 5 can be found by subtracting the \( x \)–coordinates, 7 and 2.</p>
<p><strong>Step 4 - </strong>We rewrite the rise and run by putting in the coordinates.<br></p>
<p>\( m = \frac{6-3}{7-2} \)</p>
<p>But 6 is \( y \)<sub>2</sub>, the \( y \)–coordinate of the second point and 3 is \( y \)<sub>1</sub>, the \( y \)–coordinate of the first point. So we can rewrite the slope using subscript notation.<br>\( m = \frac{y_2-y_1}{7-2} \)</p>
<p>Also, 7 is \( x \)<sub>2</sub>, the \( x \)–coordinate of the second point and 2 is \( x \)<sub>1</sub>, the \( x \)–coordinate of the first point. So again, we rewrite the slope using subscript notation</p>
<p>\( m = \frac{y_2-y_1}{x_2-x_1} \)</p>
<p> We’ve shown that \( m = \frac{y_2-y_1}{x_2-x_1}\) is really another version of \( m = \frac{rise}{run} \). We can use this formula to find the slope of a line when we have two points on the line.</p>
<br>
<h5>How to Find the Slope from Two Points</h5>
<br>
<p>How do we find the slope of horizontal and vertical lines? The slope of the line between two points (\( x_1, y_1 )\) and (\(x_2, y_2\)) is: \( m = \frac{y_2-y_1}{x_2-x_1} \)</p>
<p><strong>Example 2</strong></p>
<p><strong>Step 1 - </strong>We’ll call (-2, -3) point #1 and (-7, 4) point #2.<br></p>
<p>\( \binom {x_1, y_1} {-2, -3} \) \( \binom {x_2, y_2} {-7, 4} \)</p>
<p><strong>Step 2 - </strong>Use the slope formula.<br></p>
<p>\( m = \frac{y_2-y_1}{x_2-x_1} \)</p>
<p><strong>Step 3 - </strong>Substitute the values.<br>\( y \) of the second point minus \( y \) of the first point.</p>
<p>\( x \) of the second point minus \( x \) of the first point</p>
<p>\( m = \frac{4-(-3)}{-7-(-2)} \)</p>
<p><strong>Step 4 - </strong>Simplify.<br></p>
<p>\( m = \frac{4+3}{-7+2} \)</p>
<p>\( m = \frac{7}{-5} \)<br>\( m = -\frac{7}{5} \)</p>
<p><strong>Step 5 - </strong>Let's verify this slope on the graph shown.<br></p>
<p>\( m = \frac{y_2-y_1}{x_2-x_1} \)</p>
<img alt role="presentation" src="https://k12.openstax.org/contents/raise/resources/396fc099a6ddae68d39d08ee5abb8d0a6f5d7306">
<br>
<h4>Try It: Find the Slope of a Line</h4>
<p>Find the slope of the line graph below.</p>
<img alt role="presentation" src="https://k12.openstax.org/contents/raise/resources/f14a1a85e3633cebac4f5a571179e23b88bf1e8f">
<p><strong>Answer: </strong> Your answer may vary, but here is a sample.</p>
<p><strong>Step 1 - </strong>Locate two points on the graph whose coordinates are integers.<br></p>
<p>\((0, -5) (4, 1)\)</p>
<p><strong>Step 2 - </strong>Starting at (0,-5), sketch a right triangle to (4,1) as shown in this graph.<br></p>
<img height="313" src="https://k12.openstax.org/contents/raise/resources/5c9926d108379094675a155d49d2f420062a3108" width="298">
<p><strong>Step 3 - </strong>Count the rise (how many spaces up or down)–since it goes up, it is positive.</p>
<p>The rise is -6.</p>
<p><strong>Step 4 -</strong>Count the run (how many spaces to the right or left).</p>
<p>The run is 4.</p>
<p><strong>Step 5 - </strong>Use the slope formula.<br></p>
<p>\( m = \frac{y_2-y_1}{x_2-x_1} \)</p>
<p>\( m = \frac{1-(-5)}{4-0} \)</p>
<p>\( m = \frac{1+5}{4-0} \)</p>
<p>\( m = \frac{6}{4} \)</p>
<p><strong>Step 4 - </strong>Simplify.<br></p>
<p>\( m = \frac{3}{2} \)</p>
<p>So \(y\) increases by 3 units as \(x\) increases by 2 units.</p>