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<h3>Lesson Narrative</h3>
<p>In a previous lesson, students solved systems of linear equations by graphing. Here, they transition to solving systems algebraically–by substitution–and to reasoning about systems without context.</p>
<p>The lesson activates and builds on what students have learned in grade 8 about solving by substitution. Students see that a system can be solved by replacing a variable with a number or with an expression and that various substitutions can be done to solve the same system. They also begin to build an awareness of the kinds of systems that are conducive to being solved by substitution.<p>
<p>Students practice looking for and making use of structure as they identify the variables or expressions to substitute and ways to perform substitutions efficiently.</p>
<h4>Learning Goals (Teacher Facing)</h4>
<ul>
<li>Recognize that a system can be efficiently solved by substitution if one variable is already isolated or can easily be isolated.</li>
<li>Recognize that there are multiple ways to perform substitution to solve a system of equations.</li>
<li>Solve systems of linear equations by substituting a variable with a number or an expression and check solutions by substituting them back into the equations.</li>
</ul>
<h4>Learning Targets (Student Facing)</h4>
<ul>
<li>Solve systems of equations by substituting a variable or an expression.</li>
<li>Identify more than one way to perform substitution and decide how to substitute based on how the given equations are written.</li>
</ul>
<table class="os-raise-textheavytable">
<caption> Texas Essential Knowledge and Skills (TEKS)</caption>
<thead>
<tr>
<th scope="col">TEKS</th>
<th scope="col">Explanation of Coverage</th>
</tr>
</thead>
<tbody>
<tr>
<td>A1(A) apply mathematics to problems arising in everyday life, society, and the workplace</td>
<td>Math process coverage: Lesson provides content that supports this TEKS. </td>
</tr>
<tr>
<td>A1(B) use a problem-solving model that incorporates analyzing given information, formulating a plan or
strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the
reasonableness of the solution
</td>
<td>Math process coverage: Lesson provides content that supports this TEKS.
</td>
</tr>
<tr>
<td>A1(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems
</td>
<td>Math process coverage: Lesson provides content that supports this TEKS.
</td>
</tr>
<tr>
<td>A2(I) <u>write systems of two linear equations given</u> a table of values, <u>a graph</u>, and a verbal
description
</td>
<td>Partial coverage: Lesson provides content that covers part of this TEKS. The parts that are covered have been
underlined.
</td>
</tr>
</tbody>
</table>
<br>
<h4>Lesson Activities</h4>
<p>Here are the instructional activities for the lesson:</p>
<ul>
<li>2.3.1: Finding Connections between Graphs and Equations</li>
<li>2.3.2: Checking Solutions in Systems
<ul>
<li>2.3.2: Self Check</li>
</ul>
<ul>
<li>2.3.2: Additional Resources</li>
</ul>
</li>
<li>2.3.3: Solving Equations Using Substitution
<ul>
<li>2.3.3: Self Check</li>
</ul>
<ul>
<li>2.3.3: Additional Resources</li>
</ul>
</li>
<li>2.3.4: Solving More Equations</li>
</ul>
<p>Students will also complete a series of problems in the 2.3.5: Practice.</p>
<h4>Required Materials</h4>
<ul>
<li>Graphing technology</li>
</ul>
<h4>Required Preparation</h4>
<p>Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their
own device.</p>
<h4>Lesson Vocabulary</h4>
<p>During this lesson, it is important to:</p>
<ul>
<li>Familiarize students with the vocabulary words they will see throughout the lesson. Encourage students to look for
these words and notice their use and meanings.</li>
<li>Encourage students to use key vocabulary words in “math talk” and their written and oral explanations.
</li>
<li>Utilize a word wall. Sample cards are located here: <a
href="https://k12.openstax.org/contents/raise/resources/8ed3aa68763498713b44d3c2e537ffe61b4755f0"
target="_blank">Mathematics Vocabulary Word Wall Cards</a></li>
</ul>
<p>Vocabulary words that are emphasized in this lesson include:</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col"> Previous Vocabulary </th>
<th scope="col"> New Vocabulary </th>
</tr>
</thead>
<tbody>
<tr>
<td>
<ul>
<li>linear equation</li>
<li>constraint</li>
</ul>
</td>
<td>
<ul>
<li>solutions to a system of equations</li>
<li>system of equations</li>
<li>system of linear equations</li>
</ul>
</td>
</tr>
</tbody>
</table>
<br>
<p>To support newcomers or students identified at the beginning level of language proficiency, share the following Quizlet links to help students gain an understanding of the academic vocabulary. Use the Spanish versions to anchor student understanding before bridging to the English versions. </p>
<ul>
<li><a href="https://quizlet.com/881194743/raise-unit-2-spn-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 2 Spanish Vocabulary</a></li>
<li><a href="https://quizlet.com/881145916/raise-unit-2-vocabulary-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 2 Vocabulary</a></li>
</ul>
<h5>Support for English Language Learners</h5>
<p>Throughout this lesson, activities are incorporated that align to the following ELPS. The suggested activities are
only a sampling of the types of support and scaffolding that can extend the learning for English language learners.
Continue to find additional opportunities as you build your own set of ELL learning routines.
</p>
<ul>
<li>ELPS 2(A) distinguish sounds and intonation patterns of English with increasing ease </li>
<li>ELPS 2(B) recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters</li>
<li>ELPS 3(A) practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible</li>
<li>ELPS 4(A) learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words </li>
<li>ELPS 4(B) recognize directionality of English reading such as left to right and top to bottom </li>
<li>ELPS 4(H) read silently with increasing ease and comprehension for longer periods </li>
<li>ELPS 4(J) demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs</li>
<li>ELPS 5(C) spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired </li>
<li>ELPS 5(D) edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired </li>
<li>ELPS 5(E) employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as</li>
<ul>
<li>ELPS 5(Ei) using correct verbs, tenses, and pronouns/antecedents</li>
</ul>
</ul>
<br>
<h4>Support for Building Character</h4>
<p>Throughout this unit, find ways to encourage and support students to work on cultivating their <strong>growth
mindse</strong><strong>t</strong>.</p>
<p>Here are some tips to try during this lesson:</p>
<ul>
<li><a href="https://characterlab.org/tips-of-the-week/step-back/" target="_blank">Step Back</a></li>
<li><a href="https://characterlab.org/tips-of-the-week/myth-of-the-lazy-genius/" target="_blank">Myth of the Lazy Genius</a></li>
</ul>
<p>You can find other tips located here in the <a href="https://characterlab.org/playbooks/growth-mindset/"
target="_blank">Playbook on Growth Mindset</a> from Character Lab.</p>