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html/000396d2-561e-40f3-aecf-86d8f79bb28f.html

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<div class="os-raise-extrasupport">
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<div class="os-raise-extrasupport-header">
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<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
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<p class="os-raise-extrasupport-name">MLR 5 Co-Craft Questions: Speaking, Reading</p>
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</div>
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<div class="os-raise-extrasupport-body">
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<p>Use this routine to help students compare and interpret two functions. Display the student task statement and
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intervals, and the intersection point of the two graphs. The process of creating mathematical questions, without
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the pressure of producing answers or solutions, prompts students to make sense of the given information and to
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activate the language of mathematical questions. This work helps prepare students to process the actual questions.
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<p class="os-raise-text-italicize">Design Principle(s): Maximize meta-awareness; Support sense-making</p>
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</p>
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</div>
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</div>
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<br>
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<div class="os-raise-extrasupport">
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<div class="os-raise-extrasupport-header">
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<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
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<p class="os-raise-extrasupport-name">Engagement: Develop Effort and Persistence</p>
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</div>
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<div class="os-raise-extrasupport-body">
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<p>Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students
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to share their work with a partner. Display sentence frames to support student conversation, such as: &ldquo;One
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thing that is the same is . . .,&rdquo; &ldquo;One thing that is different is . . .,&rdquo; and &ldquo;_____
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represents _____.&rdquo;</p>
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<p class="os-raise-text-italicize">Supports accessibility for: Language; Social-emotional skills</p>
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</div>
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</div>
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<br>
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with the self check. Students will not automatically have access to this content, so you may wish to share it
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with those who could benefit from it.</em></p>
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<h4>Comparing Populations</h4>
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<p>Graphs are very useful for comparing two or more functions. Here are graphs of functions \(C\) and \(T\), which
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give the populations (in millions) of California and Texas in year \(x\).</p>
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<p><img alt="Graph of the populations of two states over time on a coordinate plane."
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src="https://k12.openstax.org/contents/raise/resources/127fe8074872a6c9489127ecd8679d6c4f86ec31"></p>
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<p>Examine the graph given above that displays the populations of California and Texas over time.</p>
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<table class="os-raise-wideequaltable">
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<thead>
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<tr>
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</tbody>
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</table>
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<br>
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<h4>Try It: Comparing Populations</h4>
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<br>
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<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
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<br>
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<p>Which state had a greater population in 1920?</p>
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<p>Use the graphing tool here or technology outside the course. Then select the <strong>solution</strong> button to
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compare your work.</p>
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<h5>Solution</h5>
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<p>Here is how to read the graph to find the larger population:</p>
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<p>On the horizontal axis, at 1920, the graph for Texas touches at about 5 million, while the graph for California
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touches at about 4 million.</p>
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<p>In 1920, Texas had a larger population than California.</p>
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<h4>Comparing Populations</h4>
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<p>Graphs are very useful for comparing two or more functions. Here are graphs of functions \(C\) and \(T\), which give the populations (in millions) of California and Texas in year \(x\).</p>
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<p><img alt="Graph of the populations of two states over time on a coordinate plane."
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src="https://k12.openstax.org/contents/raise/resources/127fe8074872a6c9489127ecd8679d6c4f86ec31"></p>
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<h4>Try It: Comparing Populations</h4>
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<br>
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<p>Examine the graph below that displays the population of California and Texas over time. Here the graphs of functions C and T give the population in millions of California and Texas (respectively) in year x.</p>
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<p>Examine the graph given above that displays the populations of California and Texas over time.</p>
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<ol class="os-raise-noindent" start="1">
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<li>Which state had a greater population in 1920?</li>
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<p><strong>Answer:</strong> Texas</p>
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<p><strong>Solution:</strong> Texas</p>
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<p>Here is how to read the graph to find the larger population:
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On the horizontal axis, at 1920, the graph for Texas touches at about 5 million, while the graph for California touches at about 4 million. In 1920, Texas had a larger population than California.
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</p>
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<li>Write a true mathematical statement using function notation to describe which state has the greater population in 1920.</li>
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<p><strong>Answer:</strong>C(1920) &lt; T(1920)</p>
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<p><strong>Solution: </strong>\(C(1920)\;\lt\;T(1920)\)</p>
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</ol>
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html/0151560d-a4ab-4859-9b31-ef8d060fe9b2.html

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<h3>Activity (15 minutes)</h3>
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<p>The goal of this activity is to introduce students to the formula to determine the nth of a geometric sequence. It also sets the stage for the upcoming unit on exponential functions.</p>
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<p>As students work to determine when the population exceeds 500 members, circulate around the room. This is an opportunity for students to choose which strategy or tool they prefer to use in order to make this prediction. Identify and select students who have used different representations (table, graph, or equation) to arrive at the answer and have them present their findings to the class. </p>
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<p>
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Also, use this time to monitor student use of vocabulary terms. Students should be comfortable using terms such as initial value, first term, starting value, common difference, common ratio, etc. New terms they may be reluctant to use include recursive formula and explicit formula. Encourage the use of these terms as students interact with others.
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Launch
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<p>Also, use this time to monitor student use of vocabulary terms. Students should be comfortable using terms such as initial value, first term, starting value, common difference, common ratio, etc. New terms they may be reluctant to use include recursive formula and explicit formula. Encourage the use of these terms as students interact with others.
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</p>
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<h4>Launch</h4>
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<p>Arrange students in groups of 2. Ask students if they have ever seen a wildcat in the wild or in a zoo. In some regions of the world, wildcats are endangered and their populations are closely monitored. Ask students why it might be difficult to get accurate measurements of the population of wildcats. </p>
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<p>As they begin the activity with a partner, provide them with paper or graph paper on which to complete their tables and graphs. Alert students that they may be chosen to present their method for determining when the wildcat population exceeds 500. Encourage students to add notes or details to their displays to help communicate their reasoning.</p>
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html/026a024d-dd99-485c-b670-446a3b8156a3.html

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</tr>
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</table>
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<br>
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<ol class="os-raise-noindent" start="6">
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<li>Use the graphing tool or technology outside the course. Graph the data from your table of values that represents this scenario using the Desmos tool below.</li>
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</ol>
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<br>
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<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
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<p>(Students were provided access to Desmos.)</p>
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<p>When you have finished graphing, select the <strong>solution</strong> button to compare your work.</p>
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<p class="os-raise-text-bold">Answer:</p>
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<p>Compare your answer:</p>
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<img src="https://k12.openstax.org/contents/raise/resources/3178249882a1aef70117a4dac6649419754ec4e6"/>
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<br><br>
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<ol class="os-raise-noindent" start="7">
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<li> Now, in the Desmos window graph \(y=x\). </li>
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</ol>
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<p>When you have finished graphing, select the <strong>solution </strong>button to compare your work.</p>
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<p class="os-raise-text-bold">Answer:</p>
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<p>Compare your answer:</p>
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<img src="https://k12.openstax.org/contents/raise/resources/5a0c1abe93bea10c7497c81c538f68938dbd02c7"/>
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<br><br>
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<ol class="os-raise-noindent" start="8">
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<li>How does the line need to move to capture all of the points?</li>
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</ol>

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</table>
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<br>
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<h4>Additional Resources</h4>
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<h3>4.5.2: Additional Resources</h3>
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<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on
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their experience with the self check. Students will not automatically have access to this content, so you may wish
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to share it with those who could benefit from it. </em></p>

html/07c539b4-2f56-4bfc-8ca4-fad2f7ee620f.html

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<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/6b6a6694b54997af858d05e4d65298543c67fe0e" srclang="en_us">https://k12.openstax.org/contents/raise/resources/befe818ae5a7e5b370b57698151b35c39b80849f
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</video></div>
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</div>
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<br>
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<h3>4.15.3: Self Check</h3>
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<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>
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<p class="os-raise-text-bold">QUESTION:</p>

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sort the collected language into three groups, one for language used to describe arithmetic sequences, a second for
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geometric sequences, and the third for neither.</p>
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<p class="os-raise-text-italicize">Design Principle(s): Support sense-making; Maximize meta-awareness</p>
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<p class="os-raise-extrasupport-title">Provide support for students</p>
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<p>
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<a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
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to the students to assist them with participating in this routine.
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</p>
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</div>
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</div>
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<br>

html/12702f20-bb71-4741-9a07-3175a8c2e3d5.html

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<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/92159e35ce10b76ecda963022a98325d7722afac " srclang="en_us">https://k12.openstax.org/contents/raise/resources/ce9b7554a5f6c2231ff7bae0a6c1e09f8ad55e39
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</video></div>
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</div>
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<br>
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<h4>Activity Synthesis</h4>
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<p>The goal of the discussion is to make sure students understand that a solution to an equation in two variables
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is any pair of values that, when substituted into the equation and evaluated, make the equation true. Discuss

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<h4>Launch</h4>
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<p>Arrange students in groups of 2&ndash;4. Display the graphs for all to see. Give students 1 minute of quiet time to think individually and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn&rsquo;t belong.</p>
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<h4>Student Activity</h4>
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<h4>Warm Up Activity</h4>
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<p>Examine the following 4 graphs and think about why 1 of the graphs does not belong in the same group as the other 3. <br></p>
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<div class="os-raise-d-flex os-raise-align-items-center">

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<div class="os-raise-extrasupport-body">
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<p>Before the whole-class discussion, give students time to discuss their reasoning for determining which phone falls in value more quickly. Listen for and collect vocabulary and phrases students use to describe depreciation. Write the students&rsquo; words on a visual display and update it throughout the remainder of the lesson. Be sure to emphasize the term &ldquo;growth factor&rdquo; along with associated words used to calculate it, such as &ldquo;consecutive values&rdquo; and &ldquo;dividing.&rdquo; Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.</p>
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<p class="os-raise-text-italicize">Design Principle(s): Support sense-making</p>
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<p class="os-raise-extrasupport-title">Provide support for students</p>
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<p>
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<a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
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to the students to assist them with participating in this routine.
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</p>
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</div>
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</div>
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<br>

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<div class="os-raise-extrasupport">
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<div class="os-raise-extrasupport-header">
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<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
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<p class="os-raise-extrasupport-name">MLR 2: Speaking, Representing, Collect, and Display </p>
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<p class="os-raise-extrasupport-name">MLR 2 Collect and Display: Speaking, Representing</p>
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</div>
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<div class="os-raise-extrasupport-body">
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<p>Use this routine to support whole-class discussion. As students share their responses, amplify and record any words and phrases that describe the meaning of each equation or inequality on a display. Call students&rsquo; attention to the different ways the constraints are represented by language in context and equations or inequalities of different forms.</p>

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<div class="os-raise-extrasupport-header">
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<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
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<p class="os-raise-extrasupport-name">Representation: Internalize Comprehension </p>
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<div class="os-raise-extrasupport-body">
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<p>Demonstrate and encourage students to use color coding and annotation to highlight connections between representations in a problem, for example, highlighting \(f(4)\) as written in the question and then highlighting the corresponding output value on the graph. Some students may benefit from reviewing that the \(x\)-coordinate is 4 and \(f(4)\) is the \(y\)-coordinate.</p>
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<p class="os-raise-text-italicize">Supports accessibility for: Visual-spatial processing</p>
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</div>
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</div>
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</div>
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<div class="os-raise-extrasupport-body">
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<p>Demonstrate and encourage students to use color coding and annotation to highlight connections between representations in a problem, for example, highlighting \(f(4)\) as written in the question and then highlighting the corresponding output value on the graph. Some students may benefit from reviewing that the \(x\)-coordinate is 4 and \(f(4)\) is the \(y\)-coordinate.</p>
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<p class="os-raise-text-italicize">Supports accessibility for: Visual-spatial processing</p>
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</div>
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</div>
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<br>
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<h4>Student Activity</h4>
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<p>The cost, in dollars, to produce 1 watt of solar energy is a function of the number of years since 1977, \(t\).</p>

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<h3>Cool Down (5 minutes)</h3>
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<h4>Student Activity</h4>
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<p>Using the equation \(m=\frac{x-4}{6}\), complete the table of values.</p>
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<table class="os-raise-horizontaltable">
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<thead> </thead>

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