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<h4>Activity (10 minutes)</h4>
<p>In earlier activities, students wrote expressions to describe visual patterns by first creating tables and reasoning
repeatedly about how the pattern changes at incrementally greater step numbers. Here they write an equation to
describe a pattern without creating a table of values. They look more directly at the structure of the visual pattern
and its connections to the parts of the equations, and then they reason in the other direction: given an equation,
they generate a visual pattern. Students also begin to frame the relationship between the quantities as a function
with inputs and outputs leading to a definition of quadratic function.</p>
<p>Some students are likely to view the nth step of the given pattern as an \(n\)-by-\(n\) square with four small
squares added at the corners, leading to the expression \(n^2+4\) for the total number of small squares. Others may
view this as an \(n+2\) by \(n+2\) square with 4 rectangles removed, each consisting of \(n\) squares. The latter
leads to the (equivalent) expression \((n+2)^2-4n\). In the next activity, students will look closely at how
equivalent expressions arise when analyzing geometric patterns, but if equivalent expressions come up in this
activity, consider inviting students to share them.</p>
<p>This prompt gives students opportunities to see and make use of structure when writing an equation from a visual
pattern. The specific structure they might notice is an inner square with a side length equal to the step number along
with a small square on each corner. Later, generating a pattern given an equation requires them to reason abstractly
and concretely.</p>
<h4>Launch</h4>
<p>Give students a moment to observe the pattern from the activity and ask them what they notice and what they wonder.
Then, ask students to sketch the next step in the pattern and share their sketch with a partner.</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 2 Collect and Display: Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p> Prepare students for the discussion by inviting them to first
explain to a partner how each part of their equation relates to the pattern. Record the words, phrases, and diagrams
students use to explain the equations they wrote to represent the pattern. Display the collected language for all to
see, and invite students to borrow from, or add more language to the display throughout the remainder of the lesson.
This will help students read and use mathematical language during partner and whole-class discussions.</p>
<p class="os-raise-text-italicize"> Design
Principle(s): Optimize output (for explanation); Maximize meta-awareness</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>
<h4>Student Activity </h4>
<p><img alt="Three steps of a growing pattern." height="168"
src="https://k12.openstax.org/contents/raise/resources/cdcca2de2d8650c3f470bc20faffa7b1228fea96" width="422"></p>
<ol class="os-raise-noindent">
<li> If the pattern continues, what will we see in Step 5 and Step 18? </li>
<ol class="os-raise-noindent" type="a">
<li> Sketch or describe the figure in each of these steps. </li>
</ol>
</ol>
<p>Write down your answer, then select the <strong>solution </strong>button to compare your work.</p>
<p><strong>Answer:</strong> Step 5 is a collection of 25 small squares, in a 5-by-5 arrangement, surrounded by 4 other
small squares at the corners. Step 18 is a collection of \(18^2\) small squares, in an 18 by 18 arrangement,
surrounded by 4 other small squares at the corners.</p>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> How many small squares are in each of these steps? Explain how you know. </li>
</ol>
</ol>
<p>Write down your answer, then select the <strong>solution </strong>button to compare your work.</p>
<p><strong>Answer:</strong> Step 5: 29, because it is \(5^2+4\) or \((5 \cdot 5)+4\). Step 18: 328, because it is
\(18^2+4\) or \((18 \cdot 18)+4\).</p>
<ol class="os-raise-noindent" start="2">
<li> Write an equation to represent the relationship between the step number \(n\) and the number of squares \(y\). Be
prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.)
</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong>\(y=n^2+4\) or equivalent. The \(n^2\) term represents the number of small squares in
the central \(n\)-by-\(n\) large square. The 4 represents the 4 small squares that are added to the large square at
its 4 corners.</p>
<ol class="os-raise-noindent" start="3">
<li> Sketch the first three steps of a pattern that can be represented by the equation \(y=n^2-1\). </li>
</ol>
<p>Write down your answer, then select the<strong> solution </strong>button to compare your work.</p>
<p><strong>Answer: </strong>An \(n\)-by-\(n\) array of squares with one square missing at the top right. The first step
(for \(n=1\)) will have no squares.</p>
<p><img
alt="DIAGRAM WITH 3 FIGURES SHOWING STEPS 1 THROUGH 3. STEP 1 DOES NOT SHOW ANYTHING. STEP 2 HAS A RECTANGLE WITH 1 ROW AND 2 COLUMNS. 1 SQUARE IS ADDED TO THE TOP LEFT. STEP 3 HAS A RECTANGLE WITH 2 ROWS AND 3 COLUMNS. 2 SQUARES ARE ADDED TO THE TOP LEFT."
class="img-fluid atto_image_button_text-bottom" height="125"
src="https://k12.openstax.org/contents/raise/resources/8965a03887be419785389417c8ac9bb33490ead8" width="384"></p>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<ol class="os-raise-noindent">
<li> For the original step pattern in the statement, \(y=n^2+4\), write an equation to represent the relationship
between the step number, \(n\), and the perimeter, \(P\). <br>
<br>
<strong>Answer:</strong> \(P=4n+16\)
</li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> For the step pattern you created in Part 3 of the activity, \(y=n^2-1\), write an equation to represent the
relationship between the step number, \(n\), and the perimeter, \(P\). <br>
<br>
<strong>Answer:</strong> If the pattern was an \(n\)-by-\(n\) array of squares with a square missing from the
corner, the perimeter could be given by \(P=0\) when \(n=1\), and by \(P=4n\) when \(n>1\).
</li>
</ol>
<ol class="os-raise-noindent" start="3">
<li> Are these linear functions? <br>
<br>
<strong>Answer: </strong>Yes, in both cases, these are linear functions
</li>
</ol>
<h4>Anticipated Misconceptions</h4>
<p>Some students may wonder how to draw a pattern given the equation \(y=n^2-1\). Show them the warm up problems where
they subtracted 1 to remove the small square in Figure B and added 1 when there was an extra small square in Figure C.
Prompt them to describe subtracting 1 as removing one small square from each step. Some students may find it easier to
start drawing Step 2 or Step 3. They can work backward to draw Step 1, which would have 0 squares since \(1^2-1=0\).
Emphasize that making a table can help them figure out exactly how many small squares are needed in each step.</p>
<h4>Activity Synthesis</h4>
<p>Make sure students see the connection between the equation \(y=n^2+4\) and the composition of the squares in the
pattern: that regardless of what \(n\) is, the figure at Step \(n\) is composed of a square that is \(n\) by \(n\),
plus 4 small squares (1 at each corner).</p>
<p>Next, help students relate the work so far to the idea of functions. Discuss with students:</p>
<ul>
<li>“In each pattern we’ve seen, is the relationship between the step number and the number of squares a
function? How do you know?” (Yes. The step number is the input, and the number of squares is the output. For
every step number, there is a particular number of squares.) </li>
<li>“How can the relationship be expressed using function notation?” (If the function \(f\) gives the
number of squares at Step \(n\), we can define it as \(f(n)=n^2+4\).) </li>
</ul>
<p>Introduce quadratic function as a function that is defined by a quadratic expression. Like other functions, it can be
represented with an equation, a table of values, a graph, and a description.</p>
<h3>7.3.2: 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><img
alt="Three steps of a growing pattern. Step 1: Total of two squares, stacked one atop the other. Step 2: Total of five squares, two squares on the bottom row and the second row and one square on the top row. Step 3: Total of ten squares, three squares on the bottom row, row 2, and row 3, and one square on the top row."
height="155" src="https://k12.openstax.org/contents/raise/resources/86e2901e600899ad34039875f60c6cf062d3bb8d"
width="401"></p>
<p>Which equation represents the relationship between the step number \(n\) and the number of squares \(y\)?</p>
<table class="os-raise-textheavytable">
<caption></caption>
<thead>
<tr>
<th scope="col">
Answers
</th>
<th scope="col">
Feedback
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(y = n + 1\)
</td>
<td>
Incorrect. Let’s try again a different way: This only works for the first step. The answer is \(y=n^2+1\).
</td>
</tr>
<tr>
<td>
\(y = 2n+1\)
</td>
<td>
Incorrect. Let’s try again a different way: This only works for the second step. The answer is
\(y=n^2+1\).
</td>
</tr>
<tr>
<td>
\(y=n^2+1\)
</td>
<td>
That’s correct! Check yourself: When \(n\) is the number of steps, there are \(n\) rows and \(n\) columns,
or \(n\) times \(n\), then one additional block.
</td>
</tr>
<tr>
<td>
\(y = n(n+1)\)
</td>
<td>
Incorrect. Let’s try again a different way: The base of the area is always \(n\), but the height is not
always \(n+1\). The answer is \(y=n^2+1\).
</td>
</tr>
</tbody>
</table>
<br>
<h3> 7.3.2: Additional Resources</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>
<h4> Patterns Represented by Quadratics</h4>
<p>See Steps 1–3 of a pattern of squares below. Write an equation representing the relationship between the step
number \(n\) and the number of squares \(y\).</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="213" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/4ccf3dd0ae2703ffe11b06fc46c1a04927a29997" width="300"></p>
<p>Make a table for the number of columns and rows of each step and the number of squares:</p>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">
Step
</th>
<th scope="col">
# of columns
</th>
<th scope="col">
# of rows
</th>
<th scope="col">
# of squares
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>1</p>
</td>
<td>
<p>1</p>
</td>
<td>
<p>3</p>
</td>
<td>
<p>3</p>
</td>
</tr>
<tr>
<td>
<p>2</p>
</td>
<td>
<p>2</p>
</td>
<td>
<p>6</p>
</td>
<td>
<p>12</p>
</td>
</tr>
<tr>
<td>
<p>3</p>
</td>
<td>
<p>3</p>
</td>
<td>
<p>9</p>
</td>
<td>
<p>27</p>
</td>
</tr>
<tr>
<td>
<p>\(n\)</p>
</td>
<td>
<p>\(n\)</p>
</td>
<td>
<p>\(3n\)</p>
</td>
<td>
<p>\(3n^2\)</p>
</td>
</tr>
</tbody>
</table>
<br>
<p>First, count how many columns there are in each step (the length):</p>
<p>For each, there are \(n\) columns.</p>
<p>Next, count how many rows there are in each step (the height): </p>
<p>For each, there are \(3n\) rows.</p>
<p>The area of this figure is then \(l \times h\) or \(n \times 3n\), which becomes \(y=3n^2\).</p>
<h4>Try It: Patterns Represented by Quadratics</h4>
<p>See Steps 1–3 of a pattern of squares below. Write an equation representing the relationship between the step
number \(n\) and the number of squares \(y\).</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="90" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/7459295e7469e7c1d9d58452ad23659f223e851f" width="304"></p>
<p>Write down your answer, then select the<strong> solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to represent this pattern as a quadratic:</p>
<p>Make a table for the number of columns and rows of each step and the number of squares:</p>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">
Step
</th>
<th scope="col">
# of columns
</th>
<th scope="col">
# of rows
</th>
<th scope="col">
# of squares
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
1
</td>
<td>
0
</td>
<td>
0
</td>
<td>
0
</td>
</tr>
<tr>
<td>
2
</td>
<td>
2
</td>
<td>
2
</td>
<td>
3
</td>
</tr>
<tr>
<td>
3
</td>
<td>
3
</td>
<td>
3
</td>
<td>
8
</td>
</tr>
<tr>
<td>
\(n\)
</td>
<td>
\(n\)
</td>
<td>
\(n\)
</td>
<td>
\(n^2-1\)
</td>
</tr>
</tbody>
</table>
<br>
<p>For this pattern, notice that the area would be the number of columns times the number of rows, but then one block is
subtracted.</p>
<p>Note that for Step 1, there was a \(1 \times 1\) block there, but then it was subtracted with the equation given.</p>
<p>The equation, then, is \(y=n^2-1\).</p>