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<h4>Lesson Narrative</h4>
<p>In this lesson, students deepen their understanding of functions by comparing representations of several functions
relating the same pair of quantities. They analyze two or more graphs simultaneously, interpreting their relative
features and their average rates of change in context.</p>
<p>Students also study comparative statements in function notation, such as \(A(x)=B(x)\), \(B(10)>A(10)\), and
explain them in terms of changes in population, changes in the trends of phone ownership, and the popularity of
different television shows.</p>
<p>Students pay close attention to the intersection of two graphs in this lesson. Earlier in their study, they learned
that a solution to a system of linear equations in two variables is a point where the graphs of the equations in the
system intersect. Here, students recognize equations of the form \(f(x)=g(x)\) to mean functions \(f\) and \(g\)
having the same output value at the same input value. They see that a solution to such an equation is the
\(x\)-coordinate of a point where the graphs of \(f\) and \(g\) intersect.</p>
<p>Making comparisons involves looking beyond individual pieces of information. To accurately relate the information
from multiple representations requires careful and precise use of mathematical language and notation. Students
continue reasoning abstractly and quantitatively as they use their analyses of representations of functions to draw
conclusions about the quantities in situations.</p>
<h4>Learning Goals (Teacher Facing)</h4>
<ul>
<li>Compare key features of graphs of functions and interpret them in context.</li>
<li>Find the solutions and meanings of \(f(x)=g(x)\) in terms of a situation and a graph.</li>
<li>Interpret statements about two or more functions written in function notation.</li>
</ul>
<h4>Learning Targets (Student Facing)</h4>
<ul>
<li>Compare the features of graphs of functions and explain what they mean in the situations represented.</li>
<li>Make sense of an equation of the form \(f(x)=g(x)\) in terms of a situation and a graph, and know how to find the
solutions.</li>
<li>Explain statements about two or more functions when they are written in function notation.</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(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>
A1(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate
</td>
<td>
Math process coverage: Lesson provides content that supports this TEKS.
</td>
</tr>
<tr>
<td>
A1(F) analyze mathematical relationships to connect and communicate mathematical ideas
</td>
<td>
Math process coverage: Lesson provides content that supports this TEKS.
</td>
</tr>
<tr>
<td>
A12(B) <u>evaluate functions, expressed in function notation, given one</u> or more <u>elements</u> <u>in their domains</u>
</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>4.10.1: Analyzing Graphs and Statements in Function Notation</li>
<li>4.10.2: Interpreting Graphs and Statements in Terms of a Situation</li>
<ul>
<li>4.10.2: Self Check</li>
<li>4.10.2: Additional Resources</li>
</ul>
<li>4.10.3: Comparing Functions Represented in Separate Graphs</li>
<ul>
<li>4.10.3: Self Check</li>
<li>4.10.3: Additional Resources</li>
</ul>
<li>4.10.4: Compare Graphs and Statements that Represent Functions without a Context</li>
<li>
<ul>
<li>4.10.4: Self Check</li>
<li>4.10.4: Additional Resources</li>
</ul>
</li>
<li>4.10.5: Analyzing Functions Using Real-World Context</li>
</ul>
<p>Students will also complete a series of problems in the 4.10.6: Practice.</p>
<h4>Required Materials</h4>
<p>None</p>
<h4>Required Preparation</h4>
<p>None</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"
title="Mathematics Vocabulary Word Wall Cards">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>function</li>
<li>value</li>
<li>equation</li>
<li>proportion</li>
<li>probability</li>
</ul>
</td>
<td>
<ul>
<li>none</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/881785885/raise-unit-4-spn-vocabulary-part-1-functions-graph-characteristics-flash-cards/?i=5eauv9&x=1jq" target="_blank">Unit 4 Spanish Vocabulary, part 1</a></li>
<li><a href="https://quizlet.com/881215410/raise-unit-4-vocabulary-part-1-functions-graph-characteristics-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 4 Vocabulary, part 1</a></li>
<li><a href="https://quizlet.com/881786064/raise-unit-4-spn-vocabulary-part-2-transformations-of-graphs-sequences-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 4 Spanish Vocabulary, part 2</a></li>
<li><a href="https://quizlet.com/881222764/raise-unit-4-vocabulary-part-2-transformations-of-graphs-sequences-flash-cards/?i=5eauv9&x=1jqt" target="_blank">Unit 4 Vocabulary, part 2</a></li>
</ul>
<h4>Support for English Language Learners</h4>
<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(F) listen to and derive meaning from a variety of media such as audiotape, video, DVD, and CD-ROM to build and reinforce concept and language attainment </li>
<li>ELPS 2(I) demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs</li>
<li>ELPS 3(E) share information in cooperative learning interactions </li>
<li>ELPS 3(I) adapt spoken language appropriately for formal and informal purposes</li>
<li>ELPS 4(I) demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs</li>
</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>creativity</strong>.</p>
<p>Here are some tips to try during this lesson:</p>
<ul>
<li><a href="https://characterlab.org/tips-of-the-week/sample-then-specialize/" target="_blank"
title="Sample, Then Specialize">Sample, Then Specialize</a></li>
<li><a href="https://characterlab.org/tips-of-the-week/compound-interest/" target="_blank"
title="Compound Interest">Compound Interest</a></li>
</ul>
<p>You can find other tips located here in the <a href="https://characterlab.org/playbooks/creativity"
target="_blank">Playbook on Creativity</a> from Character Lab.</p>