1<?xml version="1.0" encoding="UTF-8"?><rss version="2.0" 2 xmlns:content="http://purl.org/rss/1.0/modules/content/" 3 xmlns:wfw="http://wellformedweb.org/CommentAPI/" 4 xmlns:dc="http://purl.org/dc/elements/1.1/" 5 xmlns:atom="http://www.w3.org/2005/Atom" 6 xmlns:sy="http://purl.org/rss/1.0/modules/syndication/" 7 xmlns:slash="http://purl.org/rss/1.0/modules/slash/" 8 > 9 10<channel> 11 <title>Open Middle®</title> 12 <atom:link href="https://www.openmiddle.com/feed/" rel="self" type="application/rss+xml" /> 13 <link>https://www.openmiddle.com</link> 14 <description></description> 15 <lastBuildDate>Tue, 10 Mar 2026 01:16:23 +0000</lastBuildDate> 16 <language>en-US</language> 17 <sy:updatePeriod> 18 hourly </sy:updatePeriod> 19 <sy:updateFrequency> 20 1 </sy:updateFrequency> 21 22 <item> 23 <title>Constructing a Nondifferentiable Function</title> 24 <link>https://www.openmiddle.com/constructing-a-nondifferentiable-function/</link> 25 <comments>https://www.openmiddle.com/constructing-a-nondifferentiable-function/#comments</comments> 26 27 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 28 <pubDate>Mon, 09 Feb 2026 06:08:17 +0000</pubDate> 29 <category><![CDATA[Calculus]]></category> 30 <category><![CDATA[DOK 3: Strategic Thinking]]></category> 31 <category><![CDATA[Owen Kaplinsky]]></category> 32 <guid isPermaLink="false">https://www.openmiddle.com/?p=9103</guid> 33 34 <description><![CDATA[<p>Directions: Using the digits -9 to 9, at most two times each, place a digit in each box so that the function is nondifferentiable at the given x value. Extension: Can you make the function continuous but nondifferentiable? Source: Owen Kaplinsky</p> 35The post <a href="https://www.openmiddle.com/constructing-a-nondifferentiable-function/">Constructing a Nondifferentiable Function</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 36 <content:encoded><![CDATA[<p>Directions: Using the digits -9 to 9, at most two times each, place a digit in each box so that the function is nondifferentiable at the given x value.</p> 37<p>Extension: Can you make the function continuous but nondifferentiable?</p> 38<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/02/Constructing-a-Nondifferentiable-Function.png" data-caption="" title=""><img decoding="async" src="https://www.openmiddle.com/wp-content/uploads/2026/02/Constructing-a-Nondifferentiable-Function.png" alt="" /></a> 39 40 <div class="clear"></div> 41 <div class="toggle tie-sc-close"> 42 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 43 <div class="toggle-content">What things can make a function nondifferentiable? 44 </div> 45 </div> 46 47 48 <div class="clear"></div> 49 <div class="toggle tie-sc-close"> 50 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 51 <div class="toggle-content">There are many answers across the multiple types of nondifferentiability. Some examples are:<br /> 521. |4x+8|/|x-2| is nondifferentiable at -2 because of a sharp turn and at 2 because of a vertical asymptote.<br /> 532. |2x + -6|/|x-3| is nondifferentiable at 3 because it is undefined at that point.</p> 54<p>If you are using the extension, then the only type of nondifferentiability allowed is a sharp turn. 55 </div> 56 </div> 57 58<p>Source: Owen Kaplinsky</p>The post <a href="https://www.openmiddle.com/constructing-a-nondifferentiable-function/">Constructing a Nondifferentiable Function</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 59 60 <wfw:commentRss>https://www.openmiddle.com/constructing-a-nondifferentiable-function/feed/</wfw:commentRss> 61 <slash:comments>3</slash:comments> 62 63 64 </item> 65 <item> 66 <title>Piecewise Continuity</title> 67 <link>https://www.openmiddle.com/piecewise-continuity/</link> 68 <comments>https://www.openmiddle.com/piecewise-continuity/#respond</comments> 69 70 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 71 <pubDate>Mon, 09 Feb 2026 05:00:24 +0000</pubDate> 72 <category><![CDATA[Pre-Calculus]]></category> 73 <category><![CDATA[DOK 2: Skill / Concept]]></category> 74 <category><![CDATA[Owen Kaplinsky]]></category> 75 <guid isPermaLink="false">https://www.openmiddle.com/?p=9097</guid> 76 77 <description><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the function continuous/discontinuous at 3. Source: Owen Kaplinsky</p> 78The post <a href="https://www.openmiddle.com/piecewise-continuity/">Piecewise Continuity</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 79 <content:encoded><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the function continuous/discontinuous at 3.</p> 80<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/02/piecewisecontinuity.png" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/02/piecewisecontinuity.png" /></a> 81 82 <div class="clear"></div> 83 <div class="toggle tie-sc-close"> 84 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 85 <div class="toggle-content">What does it mean for a function to be discontinuous at a point?<br /> 86What happens when you plug 3 into the function? 87 </div> 88 </div> 89 90 91 <div class="clear"></div> 92 <div class="toggle tie-sc-close"> 93 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 94 <div class="toggle-content">There are many answers.<br /> 95For a continuous function, one example is 1x+2 and 3x-4.<br /> 96For a discontinuous function, one example is 2x+5 and 4x-3 97 </div> 98 </div> 99 100<p>Source: Owen Kaplinsky</p>The post <a href="https://www.openmiddle.com/piecewise-continuity/">Piecewise Continuity</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 101 102 <wfw:commentRss>https://www.openmiddle.com/piecewise-continuity/feed/</wfw:commentRss> 103 <slash:comments>0</slash:comments> 104 105 106 </item> 107 <item> 108 <title>A New Way to Use Open Middle: Interactive Problems</title> 109 <link>https://www.openmiddle.com/a-new-way-to-use-open-middle-interactive-problems/</link> 110 <comments>https://www.openmiddle.com/a-new-way-to-use-open-middle-interactive-problems/#respond</comments> 111 112 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 113 <pubDate>Sun, 08 Feb 2026 23:22:42 +0000</pubDate> 114 <category><![CDATA[Articles]]></category> 115 <guid isPermaLink="false">https://www.openmiddle.com/?p=9020</guid> 116 117 <description><![CDATA[<p>Learn how interactive Open Middle problems let students drag, try, and revise ideas directly on screen.</p> 118The post <a href="https://www.openmiddle.com/a-new-way-to-use-open-middle-interactive-problems/">A New Way to Use Open Middle: Interactive Problems</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 119 <content:encoded><![CDATA[<h4>Interactive Open Middle Problems</h4> 120<p>We uploaded our first Open Middle problem in 2014. Since then, weâve added over 700 problems to our collection. For most of that time, the way Open Middle problems were used stayed the same: teachers projected or printed problems, and students used paper or physical number tiles to work out the problem.</p> 121<h4>Embedded Problems</h4> 122<p>In 2025, we introduced <a href="https://www.openmiddle.com/about-embedding/">embedded problems</a> as the first step toward expanding how Open Middle problems could be used digitally. Embedded problems allow teachers to place Open Middle problems directly into platforms like Canvas, Google Classroom, Microsoft Teams, and websites, while keeping students focused on a single problem.</p> 123<p>Embedded problems give teachers control over what students see, including the title, hint, and answer, and remove site navigation so the problem itself remains the focus.</p> 124<h4>Introducing Interactive Open Middle Problems</h4> 125<p>Building on embedded problems, weâre now introducing interactive Open Middle problems.</p> 126<p>Interactive problems allow students to work directly with a problem on screen, reducing classroom setup time and removing the need for printed or physical materials.</p> 127<p>Interactive problems have the same workflow to use as embedded problems, with support for Canvas, Google Classroom, and other platforms.</p> 128<p><img fetchpriority="high" decoding="async" class="" src="https://www.openmiddle.com/wp-content/uploads/2026/02/draggingtiles.gif" width="370" height="370" /></p> 129<h4>Key Features</h4> 130<ul> 131<li><strong>Automatic answer checking</strong><br /> 132Interactive problems include built-in checking that verifies whether the current configuration satisfies the problemâs conditions.</li> 133<li><strong>Spam protection and intentional pacing</strong><br /> 134To prevent rapid guessing, answer checks include a short delay when conditions are not met. This encourages students to pause, reconsider, and revise rather than repeatedly clicking to check.</li> 135<li><strong>Optional hints on incorrect checks</strong><br /> 136When a configuration does not satisfy all conditions, problems can provide brief hints that point students back to the constraints without reveal
136ing a solution.</li> 137<li><strong>Annotations</strong><br /> 138Students can mark up the workspace directly, making it easier to keep track of ideas and test reasoning while working through the problem.</li> 139<li><strong>Teacher-controlled settings</strong><br /> 140Teachers can customize interactive problems by enabling or disabling hints, adjusting answer-check delays, and choosing how much feedback students receive.</li> 141<li><strong>Problem versions</strong><br /> 142Some problems include multiple versions, such as extensions or variations. Teachers can choose which version students see when embedding the problem.</li> 143</ul> 144<h4>Using Interactive Problems</h4> 145<p>Interactive problems use the same workflow as embedded problems.</p> 146<p>On any problem page, click the <strong>Use this problem</strong> button. In the embedding panel, set the <strong>Mode</strong> to <strong>Interactive</strong>.</p> 147<p>From there, you can embed the problem into Canvas, Google Classroom, Microsoft Teams, or copy the embed code for use in other platforms.</p> 148<p><img decoding="async" class="" src="https://www.openmiddle.com/wp-content/uploads/2026/02/mode.png" width="310" height="150" /></p> 149<h4>Problem Coverage</h4> 150<p>Not all Open Middle problems have been converted to interactive versions. Converting problems takes time, and we’re continuing to add interactive support where it makes sense. Some problems, such as language-heavy word problems, are not well suited to interactivity and will remain non-interactive. Interactive problems are currently available in English only.</p> 151<p>If a problem isnât available in Interactive mode, the Mode option will appear disabled. Hovering over it will explain that the problem isnât supported yet.</p> 152<p><img decoding="async" class="" src="https://www.openmiddle.com/wp-content/uploads/2026/02/notsupported.png" width="451" height="124" /></p> 153<p>We are actively converting more problems every day, focusing on those where interactivity adds clear value.</p> 154<h4>Whatâs Next</h4> 155<p>Interactive Open Middle problems are new, and weâre excited to learn from real classroom use. If you have suggestions, notice an issue, or encounter a bug with a specific problem, please let us know using the <a href="https://forms.gle/z16HWFKpP5fuX9dt7">Interactive Problems feedback form</a>.</p> 156<p>Thank you for continuing to explore Open Middle with us.</p>The post <a href="https://www.openmiddle.com/a-new-way-to-use-open-middle-interactive-problems/">A New Way to Use Open Middle: Interactive Problems</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 157 158 <wfw:commentRss>https://www.openmiddle.com/a-new-way-to-use-open-middle-interactive-problems/feed/</wfw:commentRss> 159 <slash:comments>0</slash:comments> 160 161 162 </item> 163 <item> 164 <title>Finding Equivalent Ratios 2</title> 165 <link>https://www.openmiddle.com/finding-equivalent-ratios-2/</link> 166 <comments>https://www.openmiddle.com/finding-equivalent-ratios-2/#comments</comments> 167 168 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 169 <pubDate>Sun, 01 Feb 2026 01:00:31 +0000</pubDate> 170 <category><![CDATA[Grade 6]]></category> 171 <category><![CDATA[Ratios & Proportional Relationships]]></category> 172 <category><![CDATA[6.RP.1]]></category> 173 <category><![CDATA[6.RP.2]]></category> 174 <category><![CDATA[DOK 2: Skill / Concept]]></category> 175 <category><![CDATA[Lorna McClory]]></category> 176 <guid isPermaLink="false">https://www.openmiddle.com/?p=8663</guid> 177 178 <description><![CDATA[<p>Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create equivalent ratios. Source: Lorna McClory</p> 179The post <a href="https://www.openmiddle.com/finding-equivalent-ratios-2/">Finding Equivalent Ratios 2</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 180 <content:encoded><![CDATA[<p>Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create equivalent ratios.</p> 181<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/01/equivalentratios2.png" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/01/equivalentratios2.png" /></a> 182 183 <div class="clear"></div> 184 <div class="toggle tie-sc-close"> 185 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 186 <div class="toggle-content">Have you tried using halves? 187 </div> 188 </div> 189 190 191 <div class="clear"></div> 192 <div class="toggle tie-sc-close"> 193 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 194 <div class="toggle-content">There are many possible solutions. Some examples include:<br /> 195{1:2, 35:70, 48:96}<br /> 196{1:2, 38:76, 45:90}<br /> 197{3:6, 07:14, 29:58}<br /> 198{3:6, 09:18, 27:54}<br /> 199{6:9, 18:27, 30:45} 200 </div> 201 </div> 202 203<p>Source: <a href="https://x.com/lorna_mcclory">Lorna McClory</a></p>The post <a href="https://www.openmiddle.com/finding-equivalent-ratios-2/">Finding Equivalent Ratios 2</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 204 205 <wfw:commentRss>https://www.openmiddle.com/finding-equivalent-ratios-2/feed/</wfw:commentRss> 206 <slash:comments>3</slash:comments> 207 208 209 </item> 210 <item> 211 <title>Quadratics and Number of Solutions</title> 212 <link>https://www.openmiddle.com/quadratics-and-number-of-solutions/</link> 213 <comments>https://www.openmiddle.com/quadratics-and-number-of-solutions/#comments</comments> 214 215 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 216 <pubDate>Sun, 01 Feb 2026 01:00:25 +0000</pubDate> 217 <category><![CDATA[High School: Algebra]]></category> 218 <category><![CDATA[High School: Functions]]></category> 219 <category><![CDATA[Interpreting Functions]]></category> 220 <category><![CDATA[Seeing Structure in Expressions]]></category> 221 <category><![CDATA[A-SSE.3]]></category> 222 <category><![CDATA[DOK 2: Skill / Concept]]></category> 223 <category><![CDATA[F-IF.8a]]></category> 224 <category><![CDATA[Ryan D. Fox]]></category> 225 <guid isPermaLink="false">https://www.openmiddle.com/?p=8664</guid> 226 227 <description><![CDATA[<p>Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create three quadratic equations: one with two imaginary solutions, one with one real solution, and one with two real (rational or irrational) solutions. Source: Ryan D. Fox</p> 228The post <a href="https://www.openmiddle.com/quadratics-and-number-of-solutions/">Quadratics and Number of Solutions</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 229 <content:encoded><![CDATA[<p>Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create three quadratic equations: one with two imaginary solutions, one with one real solution, and one with two real (rational or irrational) solutions.</p> 230<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/01/quadraticsandnumberofsolutions.png" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/01/quadraticsandnumberofsolutions.png" /></a> 231 232 <div class="clear"></div> 233 <div class="toggle tie-sc-close"> 234 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 235 <div class="toggle-content">How could you use the discriminant to help you identify each of the quadratics? 236 </div> 237 </div> 238 239 240 <div class="clear"></div> 241 <div class="toggle tie-sc-close"> 242 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 243 <div class="toggle-content">There are many possible solutions. One such set of answers could be: two imaginary solutions 0=2x^2 + 0x + 8, one real solution 0=1x^2 + -6x + 9, two real (irrational) solutions 0=5x^2 + 7x + -3. 244 </div> 245 </div> 246 247<p>Source: <a href="https://www.belmont.edu/profiles/ryan-fox/">Ryan D. Fox</a></p>The post <a href="https://www.openmiddle.com/quadratics-and-number-of-solutions/">Quadratics and Number of Solutions</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 248 249 <wfw:commentRss>https://www.openmiddle.com/quadratics-and-number-of-solutions/feed/</wfw:commentRss> 250 <slash:comments>3</slash:comments> 251 252 253 </item> 254 <item> 255 <title>Factoring using Greatest Common Factor</title> 256 <link>https://www.openmiddle.com/factoring-using-greatest-common-factor/</link> 257 <comments>https://www.openmiddle.com/factoring-using-greatest-common-factor/#comments</comments> 258 259 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 260 <pubDate>Sun, 01 Feb 2026 01:00:18 +0000</pubDate> 261 <category><![CDATA[Arithmetic w/ Polynomials & Rational Expressions]]></category> 262 <category><![CDATA[High School: Algebra]]></category> 263 <category><![CDATA[A-APR.1]]></category> 264 <category><![CDATA[David Groat]]></category> 265 <category><![CDATA[DOK 2: Skill / Concept]]></category> 266 <guid isPermaLink="false">https://www.openmiddle.com/?p=8665</guid> 267 268 <description><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the largest possible Greatest Common Factor for the polynomial. Then factor out the GCF. Extension: Repeat the activity with a two-digit GCF. Source: David Groat</p> 269The post <a href="https://www.openmiddle.com/factoring-using-greatest-common-factor/">Factoring using Greatest Common Factor</a>
269 first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 270 <content:encoded><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the largest possible Greatest Common Factor for the polynomial. Then factor out the GCF.</p> 271<p>Extension: Repeat the activity with a two-digit GCF.</p> 272<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/01/factorGCF.png" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/01/factorGCF.png" /></a> 273 274 <div class="clear"></div> 275 <div class="toggle tie-sc-close"> 276 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 277 <div class="toggle-content">Would it be more strategic to select the coefficients of the polynomial first or the GCF first?<br /> 278Given a GCF, how can you choose appropriate coefficients of the polynomial? 279 </div> 280 </div> 281 282 283 <div class="clear"></div> 284 <div class="toggle tie-sc-close"> 285 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 286 <div class="toggle-content">GCF = 9. Coefficients can be any combination of 18, 27, 36, or 45, 54, 63, 72, 81.<br /> 287For the extension problem: GCF = 18. Coefficients are 36, 54, and 72. 288 </div> 289 </div> 290 291<p>Source: David Groat</p>The post <a href="https://www.openmiddle.com/factoring-using-greatest-common-factor/">Factoring using Greatest Common Factor</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 292 293 <wfw:commentRss>https://www.openmiddle.com/factoring-using-greatest-common-factor/feed/</wfw:commentRss> 294 <slash:comments>1</slash:comments> 295 296 297 </item> 298 <item> 299 <title>Completing the Square</title> 300 <link>https://www.openmiddle.com/completing-the-square/</link> 301 <comments>https://www.openmiddle.com/completing-the-square/#respond</comments> 302 303 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 304 <pubDate>Sun, 01 Feb 2026 01:00:11 +0000</pubDate> 305 <category><![CDATA[High School: Algebra]]></category> 306 <category><![CDATA[Seeing Structure in Expressions]]></category> 307 <category><![CDATA[A-SSE.3]]></category> 308 <category><![CDATA[DOK 2: Skill / Concept]]></category> 309 <category><![CDATA[Kate Nerdypoo]]></category> 310 <guid isPermaLink="false">https://www.openmiddle.com/?p=8666</guid> 311 312 <description><![CDATA[<p>Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Kate Nerdypoo</p> 313The post <a href="https://www.openmiddle.com/completing-the-square/">Completing the Square</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 314 <content:encoded><![CDATA[<p>Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true.</p> 315<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/01/completingthesquare.png" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/01/completingthesquare.png" /></a> 316 317 <div class="clear"></div> 318 <div class="toggle tie-sc-close"> 319 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 320 <div class="toggle-content">What are the possible numbers of the coefficient of x?<br /> 321Therefore what needs to go in the binomial squared?<br /> 322What would normally be the last number in a perfect square trinomial? Therefore what is the last number here? 323 </div> 324 </div> 325 326 327 <div class="clear"></div> 328 <div class="toggle tie-sc-close"> 329 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 330 <div class="toggle-content">x^2+10x+28=(x+5)^2+3<br /> 331x^2+10x+29=(x+5)^2+4<br /> 332x^2+10x+34=(x+5)^2+9<br /> 333x^2+12x+43=(x+6)^2+7<br /> 334x^2+12x+45=(x+6)^2+9<br /> 335x^2+14x+52=(x+7)^2+3<br /> 336x^2+14x+58=(x+7)^2+9<br /> 337x^2+16x+73=(x+8)^2+9 338 </div> 339 </div> 340 341<p>Source: <a href="https://twitter.com/nerdypoo">Kate Nerdypoo</a></p>The post <a href="https://www.openmiddle.com/completing-the-square/">Completing the Square</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 342 343 <wfw:commentRss>https://www.openmiddle.com/completing-the-square/feed/</wfw:commentRss> 344 <slash:comments>0</slash:comments> 345 346 347 </item> 348 <item> 349 <title>Solving Quadratic Functions using the Square Root Method</title> 350 <link>https://www.openmiddle.com/solving-quadratic-functions-using-the-square-root-method/</link> 351 <comments>https://www.openmiddle.com/solving-quadratic-functions-using-the-square-root-method/#respond</comments> 352 353 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 354 <pubDate>Sun, 01 Feb 2026 01:00:02 +0000</pubDate> 355 <category><![CDATA[High School: Algebra]]></category> 356 <category><![CDATA[Reasoning with Equations and Inequalities]]></category> 357 <category><![CDATA[A-REI.B]]></category> 358 <category><![CDATA[DOK 2: Skill / Concept]]></category> 359 <guid isPermaLink="false">https://www.openmiddle.com/?p=8667</guid> 360 361 <description><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two separate quadratic equations. Source: Avanti Yamamoto</p> 362The post <a href="https://www.openmiddle.com/solving-quadratic-functions-using-the-square-root-method/">Solving Quadratic Functions using the Square Root Method</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 363 <content:encoded><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two separate quadratic equations.</p> 364<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/01/quadraticSRM.png" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/01/quadraticSRM.png" /></a> 365 366 <div class="clear"></div> 367 <div class="toggle tie-sc-close"> 368 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 369 <div class="toggle-content">What step would you take the square root in?<br /> 370What numbers could you put there? 371 </div> 372 </div> 373 374 375 <div class="clear"></div> 376 <div class="toggle tie-sc-close"> 377 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 378 <div class="toggle-content">4x^2+3=7, x=±1 (x-5)^2=9, x=8,2<br /> 3792x ^2+6x=8, x=+_1, (x-7)^2=4, x= 5,9 380 </div> 381 </div> 382 383<p>Source: Avanti Yamamoto</p>The post <a href="https://www.openmiddle.com/solving-quadratic-functions-using-the-square-root-method/">Solving Quadratic Functions using the Square Root Method</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 384 385 <wfw:commentRss>https://www.openmiddle.com/solving-quadratic-functions-using-the-square-root-method/feed/</wfw:commentRss> 386 <slash:comments>0</slash:comments> 387 388 389 </item> 390 <item> 391 <title>Composite Volume</title> 392 <link>https://www.openmiddle.com/composite-volume/</link> 393 <comments>https://www.openmiddle.com/composite-volume/#respond</comments> 394 395 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 396 <pubDate>Sun, 01 Feb 2026 00:59:46 +0000</pubDate> 397 <category><![CDATA[Geometric Measurement and Dimension]]></category> 398 <category><![CDATA[High School: Geometry]]></category> 399 <category><![CDATA[DOK 3: Strategic Thinking]]></category> 400 <category><![CDATA[G-GMD.3]]></category> 401 <guid isPermaLink="false">https://www.openmiddle.com/?p=8668</guid> 402 403 <description><![CDATA[<p>Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. The composite figure is made of a rectangular prism and a rectangular pyramid; the base for each is congruent. All dimensions must be greater than zero. Source: Jedidiah Butler</p> 404The post <a href="https://www.openmiddle.com/composite-volume/">Composite Volume</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 405 <content:encoded><![CDATA[<p>Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. The composite figure is made of a rectangular prism and a rectangular pyramid; the base for each is congruent. All dimensions must be greater than zero.</p> 406<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/01/compositevolume.jpg" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/01/compositevolume.jpg" /></a> 407 408 <div class="clear"></div> 409 <div class="toggle tie-sc-close"> 410 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 411 <div class="toggle-content">Think about what dimensions produce integer calculations. 412 </div> 413 </div> 414 415 416 <div class="clear"></div> 417 <div class="toggle tie-sc-close"> 418 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 419 <div class="toggle-content">There are many possible solutions. Some examples include:<br /> 420l=3 w=2 h1=5 h2=9 v=48<br /> 421l=3 w=7 h1=5 h2=9 v=168<br /> 422l=3 w=2 h1=10 h2=9 v=78 423 </div> 424 </div> 425 426<p>Source: Jedidiah Butler</p>The post <a href="https://www.openmiddle.com/composite-volume/">Composite Volume</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 427 428 <wfw:commentRss>https://www.openmiddle.com/composite-volume/feed/</wfw:commentRss> 429 <slash:comments>0</slash:comments> 430 431 432 </item> 433 <item> 434 <title>Equivalent Exponents</title> 435 <link>https://www.openmiddle.com/equivalent-exponents-2/</link> 436 <comments>https://www.openmiddle.com/equivalent-exponents-2/#respond</comments> 437 438 <dc:creator><![CDATA[Owen Kaplinsky]]></dc:creator> 439 <pubDate>Sun, 01 Feb 2026 00:59:39 +0000</pubDate> 440 <category><![CDATA[High School: Algebra]]></category> 441 <category><![CDATA[Seeing Structure in Expressions]]></category> 442 <category><![CDATA[A-SSE.3]]></category> 443 <category><![CDATA[DOK 2: Skill / Concept]]></category> 444 <guid isPermaLink="false">https://www.openmiddle.com/?p=8669</guid> 445 446 <description><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a true in equality and a true equation. An extension is to do so while leaving the greatest or smallest number unused. Source: John Joseph Vasko jr</p> 447The post <a href="https://www.openmiddle.com/equivalent-exponents-2/">Equivalent Exponents</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></description> 448 <content:encoded><![CDATA[<p>Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a true in equality and a true equation.</p> 449<p>An extension is to do so while leaving the greatest or smallest number unused.</p> 450<a class="lightbox-enabled" href="https://www.openmiddle.com/wp-content/uploads/2026/01/equivalentexponents.png" data-caption="" title=""><img decoding="async" alt="" src="https://www.openmiddle.com/wp-content/uploads/2026/01/equivalentexponents.png" /></a> 451 452 <div class="clear"></div> 453 <div class="toggle tie-sc-close"> 454 <h3 class="toggle-head">Hint <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 455 <div class="toggle-content">What does an exponent mean?<br /> 456What two exponents could be used for the equivalence?<br /> 457What bases are possible? 458 </div> 459 </div> 460 461 462 <div class="clear"></div> 463 <div class="toggle tie-sc-close"> 464 <h3 class="toggle-head">Answer <i class="fa fa-angle-down" aria-hidden="true"></i></h3> 465 <div class="toggle-content">8^2=4^3 or base 2 and 4 with exponents 6 and 3 or base 3 and 9 with exponents 2, 1 or 4, 2 or 8, 4… possibly other answers as well. 466 </div> 467 </div> 468 469<p>Source: John Joseph Vasko jr</p>The post <a href="https://www.openmiddle.com/equivalent-exponents-2/">Equivalent Exponents</a> first appeared on <a href="https://www.openmiddle.com">Open Middle®</a>.]]></content:encoded> 470 471 <wfw:commentRss>https://www.openmiddle.com/equivalent-exponents-2/feed/</wfw:commentRss> 472 <slash:comments>0</slash:comments> 473 474 475 </item> 476 </channel> 477</rss>
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