tag:blogger.com,1999:blog-71144213106344373992024-03-18T05:00:34.309-07:00Thoughts on Teaching Math with technologyLee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.comBlogger3076125tag:blogger.com,1999:blog-7114421310634437399.post-53579549452938839462024-03-18T05:00:00.000-07:002024-03-18T05:00:00.243-07:00A New Verification For Robots<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; margin: 1.25em 0px; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhL0EsGM09YtaeRvOcRvIuPZoHTmyPraqDkNYUYab25LHxug636VRhBjmf3Je08N7YjxhmiDMFzbfMt6ToCnIi8uiaTXyeZX2lDq6Fq_RLt-cKUbrY3UDl-X6ziqkivP2mEES_1Pq1CNfy8iS6M398vrKGtiaZa56MgWd2waw8OB-8j67ltPFxOzlyOqy4/s640/robot-2740075_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="640" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhL0EsGM09YtaeRvOcRvIuPZoHTmyPraqDkNYUYab25LHxug636VRhBjmf3Je08N7YjxhmiDMFzbfMt6ToCnIi8uiaTXyeZX2lDq6Fq_RLt-cKUbrY3UDl-X6ziqkivP2mEES_1Pq1CNfy8iS6M398vrKGtiaZa56MgWd2waw8OB-8j67ltPFxOzlyOqy4/s320/robot-2740075_640.jpg" width="320" /></a></div><span style="font-size: medium;">In the field of robotics, ensuring safety is paramount, especially when it comes to robot motion in shared spaces with humans. A new safety-check technique has emerged, promising to prove with 100 percent accuracy that a planned robot motion will not result in a collision. This groundbreaking development has the potential to revolutionize the field of robotics and make human-robot interactions safer than ever before.</span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Traditional safety-check techniques rely on probabilistic methods, which can sometimes lead to false positives or negatives, posing a risk to safety. However, the new technique, known as "formal verification," takes a different approach. By using mathematical algorithms and logic, formal verification can rigorously prove that a planned robot motion will not result in a collision, eliminating the possibility of errors.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">One of the key advantages of formal verification is its ability to handle complex robotic systems with multiple moving parts. Traditional methods struggle to cope with the complexity of these systems, often leading to incomplete or inaccurate safety checks. Formal verification, on the other hand, can analyze intricate robotic motions and provide a definite answer regarding safety.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">The implications of this new safety-check technique are far-reaching. In industries such as manufacturing, healthcare, and logistics, where robots often work alongside humans, it improves safety which is crucial. Using this technique, companies can have confidence that their robotic systems will operate safely, reducing the risk of accidents and injuries.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Moreover, formal verification can also accelerate the development and deployment of robotic systems. By providing a fast and reliable method for safety checking, developers can streamline the testing process and bring their robots to market faster.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">While this improved technique represents a significant advancement in robotics safety, it is not without its challenges. Implementing formal verification requires specialized knowledge and expertise, and the technique may not be suitable for all robotic applications. However, with further research and development, formal verification has the potential to become a standard practice in ensuring the safety of robotic systems.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;">This new safety-check technique of formal verification promises to revolutionize the field of robotics by providing a rigorous and accurate method for ensuring collision-free robot motion. With its potential to enhance safety and accelerate development, formal verification represents a major step forward in the advancement of robotics technology. Let me know what you think, I'd love to hear. Have a great week.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-21327944410007971872024-03-17T05:00:00.000-07:002024-03-17T05:00:00.139-07:00Happy St. Patricks Day<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiY5FEtpyZ4vDkDY1KP2mrhbG4A_mXPxbijsLMLzlXGNQ2nsdd4aQvGFeylMgDjabi3p1Qf-1eXJnFrTTxPXVBdDmsG2aCX5YIFL-XQuEo8-NqWzIygjlRhbgILYkwxcTpQqJZ0zP0u0e6Ca3VYseWqt5CZWwVtqDE0GNMPbMsaobRCPUPmokqQd7ZMaYw/s640/happy-st-patricks-day-2654423_640.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="300" data-original-width="640" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiY5FEtpyZ4vDkDY1KP2mrhbG4A_mXPxbijsLMLzlXGNQ2nsdd4aQvGFeylMgDjabi3p1Qf-1eXJnFrTTxPXVBdDmsG2aCX5YIFL-XQuEo8-NqWzIygjlRhbgILYkwxcTpQqJZ0zP0u0e6Ca3VYseWqt5CZWwVtqDE0GNMPbMsaobRCPUPmokqQd7ZMaYw/w640-h300/happy-st-patricks-day-2654423_640.jpg" width="640" /></a></div><br /> <p></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-47765795755723485352024-03-15T05:00:00.000-07:002024-03-15T05:00:00.144-07:00How To Establish Mathematical Goals. <p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; margin: 1.25em 0px; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSZj9Y9bweDzXFFlVv5JGihM5UqEYct1NX1wbpigHqlggVsG53UA0i7dI7C_MzGLWC4mWZuN-8151MCQPTnPS8Pb2THtvZHRN3iwE36tLwQCuExvy4o9JRizByOtFxIJRwBgRvT3VFyMOXfdjfpCArmF5HZAj9vkbf8cA1iP18XPH60Yc3dcXa_zP8GfE/s640/goal-2045924_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="376" data-original-width="640" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSZj9Y9bweDzXFFlVv5JGihM5UqEYct1NX1wbpigHqlggVsG53UA0i7dI7C_MzGLWC4mWZuN-8151MCQPTnPS8Pb2THtvZHRN3iwE36tLwQCuExvy4o9JRizByOtFxIJRwBgRvT3VFyMOXfdjfpCArmF5HZAj9vkbf8cA1iP18XPH60Yc3dcXa_zP8GfE/s320/goal-2045924_640.jpg" width="320" /></a></div><span style="font-size: medium;">One of the seven effective mathematical teaching practices is to establish mathematical goals as a way of focusing learning. It is important to set clear and achievable mathematical goals since it is essential for guiding learning and ensuring academic success for students in grades K-12. These goals not only provide direction but also help educators tailor instruction to meet the diverse needs of students. Today, we'll explore the importance of establishing mathematical goals while outlining strategies that work across different grade levels.</span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Why is it important to establish mathematical goals? Setting mathematical goals help students and educators stay focused on what needs to be achieved, providing a clear path for learning progression. In addition goals provide a basis for assessing student progress and evaluating the effectiveness of instructional strategies. By establishing goals, educators can differentiate instruction to meet the individual needs of students, ensuring that all learners are appropriately challenged. Furthermore, clear goals can motivate students by providing a sense of achievement and progress as they work towards mastering mathematical concepts.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Setting goals will be different in the early grades from those used in high school. In the kindergarten to second grades, goals should focus on building foundational skills such as number recognition, counting, and basic operations. Goals may include mastering addition and subtraction within 20, understanding place value, and developing spatial reasoning skills.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Whereas the goals for grades 3 to 5 should expand to include more complex operations, such as multiplication and division, fractions, and basic geometry. Students should also develop problem-solving skills and the ability to apply mathematical concepts to real-world situations.</p><ol style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; counter-reset: list-number 0; display: flex; flex-direction: column; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; list-style: none; margin: 1.25em 0px; padding: 0px; white-space: pre-wrap;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;">Middle school goals should focus on deepening understanding of mathematical concepts, including algebra, geometry, and statistics. Students should also develop critical thinking skills and the ability to analyze and interpret data.</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><br /></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;">By the time we set goals for high school students, we need to set goals to prepare students for college and career readiness, focusing on advanced topics such as calculus, trigonometry, and advanced algebra. Students should also develop the ability to use mathematical models to solve real-world problems. </p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><br /></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;">In regard to actually writing goals, the goals should align with state and national standards to ensure that students are meeting grade-level expectations. One way is to use student data, such as assessment results and observations, to help with goal-setting and tracking student progress over time. Don't forget to involve students, parents, and other stakeholders in the goal-setting process to ensure that goals are meaningful and achievable. Finally, monitor student progress towards goals and adjust instruction as needed to ensure that all students are on track to meet their goals.</p></li></ol><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;">Remember that establishing clear and achievable mathematical goals is crucial for focusing learning and ensuring academic success for students in grades K-12. By setting appropriate goals and using data to monitor progress, educators can help all students achieve mathematical proficiency and develop the skills they need for future success. Let me know what you think, I'd love to hear.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-92170459589837570752024-03-14T05:00:00.000-07:002024-03-14T05:00:00.136-07:00Happy Pi Day<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEMvBMBTJOxakvgCpkjXAdEsCC78SjypZhk3jOo4dXjDFFXf_DhkIp_3R5A7a5bTMbzOFgfmO0s-VQ_p3MhYk9RBp3RbYfh3qYRj63xrdKU7Kq1KKndWMtXZOLKaly9KBtzQiWNOSP5ndFIJ6fTOXzffPhVE1tl2YV9Gv5tr1dUoxhpRqYCHrLeIisuP0/s640/tiramisu-6977659_640.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="429" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEMvBMBTJOxakvgCpkjXAdEsCC78SjypZhk3jOo4dXjDFFXf_DhkIp_3R5A7a5bTMbzOFgfmO0s-VQ_p3MhYk9RBp3RbYfh3qYRj63xrdKU7Kq1KKndWMtXZOLKaly9KBtzQiWNOSP5ndFIJ6fTOXzffPhVE1tl2YV9Gv5tr1dUoxhpRqYCHrLeIisuP0/w430-h640/tiramisu-6977659_640.jpg" width="430" /></a></div><h2 style="text-align: center;">Happy Pi Day to all</h2> <p></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-15492018666226063602024-03-13T05:00:00.000-07:002024-03-13T05:00:00.129-07:00Finding Solutions Through Visualization<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; margin: 1.25em 0px; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBw3Yd38VFDr_THumIEN6fPkWY6-Q8Jmf2CYjxP_P7NSyk0ZBuKxJsK1lhd-Je2KDZTYn00IlMmifTwBc5KJds2_EuZRbDMygt_3mtsHrxwNJxvAA2-Kxbhm6DyVyQpDHp3S9pJEEeZ-8SwkN6hetnNMz3dU4whYc7k-xy5NtsQsIlgcMvWOAYN5XX958/s640/business-4241792_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBw3Yd38VFDr_THumIEN6fPkWY6-Q8Jmf2CYjxP_P7NSyk0ZBuKxJsK1lhd-Je2KDZTYn00IlMmifTwBc5KJds2_EuZRbDMygt_3mtsHrxwNJxvAA2-Kxbhm6DyVyQpDHp3S9pJEEeZ-8SwkN6hetnNMz3dU4whYc7k-xy5NtsQsIlgcMvWOAYN5XX958/s320/business-4241792_640.jpg" width="320" /></a></div><span style="font-size: medium;">Mathematics is often seen as a subject of numbers and equations, but it can also be a visually creative endeavor. One of the ways to solve problems is to create some sort of drawing or visualization since drawing representations of mathematical problems not only helps in understanding complex concepts but also in predicting their resolutions. Today, we explore the art of drawing mathematical problems and how it can lead to insights into their solutions.</span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Drawing mathematical problems involves creating diagrams, graphs, or illustrations that represent the problem at hand and help visualize each problem. This visualization can provide valuable insights into the problem's structure, relationships between variables, and potential solutions. For example, drawing a graph of a function can help in understanding its behavior and identifying key points such as intercepts, maxima, and minima.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">In addition drawing mathematical problems can help in predicting possible solutions by allowing us to see patterns, relationships, and symmetries that may not be apparent from the equation alone. For instance, drawing a geometric figure can reveal hidden congruence or similarity relationships that can be used to solve a problem. Similarly, drawing a diagram of a trigonometric function can help in visualizing its periodic nature and predicting its behavior over a certain interval since you "see" all aspects of it.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">One famous example of how drawing can find solutions in mathematics is the Four Color Theorem. This theorem states that any map can be colored using only four colors in such a way that no two adjacent regions have the same color. While the proof of this theorem is complex, it was initially conjectured based on the observation that maps could be drawn in such a way that only four colors were needed, leading mathematicians to search for a proof of this conjecture.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">There are numerous benefits by creating visual representations. By drawing mathematical problems, people are not only able to find solutions but also has several other benefits. Drawing can aid people in understanding complex concepts, exploring mathematical ideas, and communicating solutions to others. Furthermore, drawing can enhance creativity, critical thinking, and problem-solving skills, thus making it a valuable tool in mathematical education.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;">So creating drawings of mathematical problems is an art that can lead to insights into possible solutions. By visualizing problems, we can see patterns and relationships that may not be apparent from equations alone, consequently helping us find solutions and deepen our understanding of mathematical concepts. So, it is important to teach students that the next time they encounter a mathematical problem, try picking up a pencil and sketching it out to see possible answers. Let me know what you think, I'd love to hear.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-83115531408166314992024-03-11T05:00:00.000-07:002024-03-11T05:00:00.133-07:00Pi Day Is Coming Up On Thursday.<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ZHS4gXvVm7ghhcEWFBIvOIcXlkFLt64E_zvw8NhQhT3btXbD1pIhKrABRwrgD-rUBmcR9N54JYHrb7C9jQEiUG_EZoUjDRv_be_AZibzNA_qhXWUQ9UttBRVcxodYTz33KBeK4Ipgk38Vh1RPLZclpaoUMfch617HhLkyzrd4MANePGkHu3Rlum050E/s640/tiramisu-6977659_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="429" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ZHS4gXvVm7ghhcEWFBIvOIcXlkFLt64E_zvw8NhQhT3btXbD1pIhKrABRwrgD-rUBmcR9N54JYHrb7C9jQEiUG_EZoUjDRv_be_AZibzNA_qhXWUQ9UttBRVcxodYTz33KBeK4Ipgk38Vh1RPLZclpaoUMfch617HhLkyzrd4MANePGkHu3Rlum050E/s320/tiramisu-6977659_640.jpg" width="215" /></a></div><span style="caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; white-space: pre-wrap;"><span style="font-size: medium;">On Thursday, we celebrate pi day which is a beautiful look at a wonderfully helpful irrational number. Every year on March 14th, mathematicians, scientists, and enthusiasts around the world celebrate Pi Day, a day dedicated to the mathematical constant π (pi). Pi, often approximated as 3.14, represents the ratio of a circle's circumference to its diameter and is a fundamental constant in mathematics, with an infinite number of decimal places that never repeat.</span></span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Pi Day was first celebrated in 1988 by physicist Larry Shaw at the San Francisco Exploratorium. Shaw, known as the "Prince of Pi," organized a march around the museum's circular spaces and concluded the event with a pie-eating celebration, honoring both the mathematical constant and the delicious dessert.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Today, Pi Day is celebrated worldwide with various activities, including pi recitation contests, baking and eating pies, and exploring the significance of pi in mathematics and science. Many educational institutions and organizations host events to raise awareness about the importance of mathematics and inspire interest in STEM (science, technology, engineering, and mathematics) fields.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Pi Day can be celebrated in so many different ways. One of the most popular ways to celebrate this spectacular day is to bake pies with the pi symbol on top. One place I worked provided pieces of pie for everyone. At that same place, I had a pi trivia search through the building. One of the facts included a British air squadron that used it for their symbol. Other possibilities include holding a contest to see who can recite the most digits in pi, or pi based art or music, host a pi run, or learn more about pi.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">In recent years, NASA has joined the celebration by hosting the NASA Pi Day Challenge, an educational activity that encourages students and the public to solve a series of math problems related to space exploration. The challenges are designed to showcase how pi is used in real-world scientific calculations, such as calculating the size of craters on Mars or the volume of propellant needed for a rocket launch. Do a quick check on the internet to find out more about these activities. In addition, NASA has previous years activities available should you want to look at some of those.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;">Remember Pi Day is not only a celebration of a fascinating mathematical constant but also a reminder of the importance of mathematics in understanding the world around us. Whether you're solving complex equations or simply enjoying a slice of pie, Pi Day is a time to appreciate the beauty and significance of mathematics in our lives.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-69472921352974489112024-03-08T05:00:00.000-08:002024-03-08T05:00:00.136-08:00I've Learned To Use Manipulatives To Clarify Misunderstandings.<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguRpn7s_0JEJHYFFCWp1F5n4reFuFFRT722VCCS4Lm0_EhqfXNNXlFsF3cqLPZYAW9PCOgxjG7ocHgTbDtA8oOdZuRkY0HwTl6-8tvLRnpWIuGvvY9v9FAzrIODANsYtMCwrLWzAIRB5r0T0sbFA5ifP01409GLNvOYoGJ5ctqDE1s-vvvlzJOa9_NA44/s640/color-4503279_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguRpn7s_0JEJHYFFCWp1F5n4reFuFFRT722VCCS4Lm0_EhqfXNNXlFsF3cqLPZYAW9PCOgxjG7ocHgTbDtA8oOdZuRkY0HwTl6-8tvLRnpWIuGvvY9v9FAzrIODANsYtMCwrLWzAIRB5r0T0sbFA5ifP01409GLNvOYoGJ5ctqDE1s-vvvlzJOa9_NA44/s320/color-4503279_640.jpg" width="320" /></a></div><br />Over this past year, I've discovered how useful manipulative are in helping to clarify missing information in a students knowledge base. As you know, I am currently teaching grades 6 to 12 in a two room school house (yes they still exist in Alaska) and I've resorted to manipulative to help clarify student misunderstanding.<p></p><p>I had study hall this past weekend and one of my Algebra I students was having difficulty distinguishing between tens and tenths. Apparently, she thought they were the same so I used a place value chart (borrowed from the elementary classroom) to help clarify this topic. I showed her the ones place and then pointed to the column on either side after which I identified the tens and tenths, emphasizing the whole numbers end only in s while the decimal value ends in ths. I then went to the hundreds and hundredths followed by thousands and thousandths. She said this is the first time she understood the difference.</p><p>Since the elementary teacher is out on medical leave, the sub in there just turned 21 and doesn't have a strong math background so she sends students to me. I had one who pretty much understood when fractions were different as to which was larger but equivalent fractions he struggled with so I pulled out those fraction strips and used them to show him. He could see using fractions strips better on how certain equivalents were the same rather than just coloring in sections of printed squares or rectangles.</p><p>In addition, he could see why you would need the same denominator to compare different fractions. He was also able to connect to why you multiply the denominator and numerator by the same number from "playing" with these strips.</p><p>For my 7th graders, we hit multiplying and dividing decimal numbers, so out come the base 10 manipulative so that we could use as we worked through questions. I found that using these for division was easier for them then for multiplication. They did understand that if the number was 3.4 divided by 6, they had to change some of the larger pieces into equivalent smaller ones. Out of my 3 students students in that class, one cannot multiply anything but one digit by one digit numbers so he is struggling to show his work. </p><p>Then there is the Algebra I group who had trouble adding like terms so again, I pulled out the base 10 pieces because I can use the ones as ones, the tens as x, the 100's as x^2 and the 1000 block as x^3. I used these pieces to represent the terms in the equation so they could see what could be combined. It worked so well and helped them see by x^3 doesn't get combined with x^2.</p><p>In the meantime, I look for other possibilities of using the limited supply of manipulative to clarify mathematical concepts and understanding. Let me know what you think, I'd love to hear. Have a great weekend.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-60887732545423315652024-03-06T05:00:00.000-08:002024-03-06T05:00:00.148-08:00Who Uses Math More When They Play Tetris.<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; margin: 0px 0px 1.25em; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis3t76QDDRXR9IIF3LcHcDqnSaqb8nwvoFV83np2zKs9AdCWj4xC476dhB5aWoVJaLMjJUbrM_m_8Lgz376YysKPX5z1ZBRzPthS-2tAqtPT5moHfpSVMoG6ADOSYKbgXAdxcR4ZWkG2Bh_YLYgW9SBI83Ad1hfWsabBQCSC9gJ8FpTbHqMoCNtF4rqB4/s640/tetris-8154682_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis3t76QDDRXR9IIF3LcHcDqnSaqb8nwvoFV83np2zKs9AdCWj4xC476dhB5aWoVJaLMjJUbrM_m_8Lgz376YysKPX5z1ZBRzPthS-2tAqtPT5moHfpSVMoG6ADOSYKbgXAdxcR4ZWkG2Bh_YLYgW9SBI83Ad1hfWsabBQCSC9gJ8FpTbHqMoCNtF4rqB4/s320/tetris-8154682_640.jpg" width="320" /></a></div><br /><span style="font-size: medium;">Today, we are looking at why certain players are more likely to use the mathematical under pinnings of Tetris because I saw a cool article on the topic. Tetris is a classic puzzle video game that has captured the hearts of players around the world for decades. Its simple yet addictive gameplay appeals to people of all ages and backgrounds, making it a game that anyone can pick up and enjoy. However, while anyone can play Tetris, certain professions, such as architects, animators, and engineers, are more likely to use the math underlying the game in their work.</span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">One of the reasons why architects, animators, and engineers are more likely to use the math underlying Tetris is because of the game's focus on spatial reasoning. In Tetris, players are tasked with fitting different shaped blocks together to create complete lines, requiring them to think about how the shapes will fit together in a limited space. This spatial reasoning is a valuable skill in professions such as architecture, animation, and engineering, where professionals often need to design and create objects or structures that fit together in a specific way.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Additionally, the math underlying Tetris can also be useful in these professions for calculating dimensions, angles, and proportions. For example, architects may use their spatial reasoning skills to design buildings that maximize space and efficiency, while animators may use math to create realistic movement and proportions in their animations. Engineers, on the other hand, may use math to calculate the structural integrity of a design or to determine the best way to assemble a complex machine.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Furthermore, playing Tetris can also help improve these professionals' problem-solving skills, as the game presents players with increasingly challenging puzzles that require quick thinking and strategic planning to solve. This ability to think quickly and solve problems efficiently is invaluable in professions where professionals are often faced with complex challenges that require creative solutions.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;">In conclusion, while anyone can play Tetris and enjoy its addictive gameplay, however architects, animators, and engineers are more likely to use the math within the game in their work. The spatial reasoning, problem-solving, and math skills developed through playing Tetris can be valuable assets in these professions, helping professionals to design, create, and problem-solve more effectively. Let me know what you think.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-84349014575943685832024-03-04T05:00:00.000-08:002024-03-04T05:00:00.142-08:00The Mathematical Underpinnings Of Tetris<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; margin: 0px 0px 1.25em; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY2OW3R8AT4sVP4L1cj47FOuvaA9xMIoTtGwpqFEy0oUCnFh-yOXLpSFbfdoTgdwv7ZF_mBoxq1OzWSxVaPCwr6Iov75gmIktXC79mYOUnKTModyMf2tP6MQ6Ktm-R50WqwJ7FQ1rhbXEC7TUOvSwpIxGN1wvT-hj7aHyPtjL-uNmzCQ6i1A1oiG1ponw/s640/nostalgia-1305079_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY2OW3R8AT4sVP4L1cj47FOuvaA9xMIoTtGwpqFEy0oUCnFh-yOXLpSFbfdoTgdwv7ZF_mBoxq1OzWSxVaPCwr6Iov75gmIktXC79mYOUnKTModyMf2tP6MQ6Ktm-R50WqwJ7FQ1rhbXEC7TUOvSwpIxGN1wvT-hj7aHyPtjL-uNmzCQ6i1A1oiG1ponw/s320/nostalgia-1305079_640.jpg" width="320" /></a></div><br /><span style="font-size: medium;">Today's column is the first in a series of two on Tetris. We'll look at the mathematical underpinnings of the game Tetris today and tomorrow we'll see which type of players are more likely to use the mathematics of the game when they play. </span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;">As you know, tetris, the iconic puzzle game was created by Russian designer Alexey Pajitnov in 1984. It is not just a test of quick reflexes and spatial awareness but it also has deep mathematical roots. At its core, Tetris revolves around the manipulation of geometric shapes, requiring players to fit them together to form complete lines. This simple yet challenging gameplay is supported by several mathematical concepts that contribute to its addictiveness and enduring appeal.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">One of the key mathematical principles used in Tetris, is the concept of polyominoes. Polyominoes are shapes made up of squares connected along their edges. In Tetris, the seven different tetrominoes (tetris pieces) are examples of polyominoes, ranging from the straight "I" shape to the square "O" shape and the various "L" and "T" shapes. The challenge in Tetris comes from arranging these tetrominoes in such a way that they form complete lines, which are then cleared from the playing field.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Another important mathematical concept in Tetris involves combinatorics, specifically permutations and combinations. In Tetris, players must consider all the possible ways in which a tetromino can be rotated and placed within the playfield. This requires an understanding of the different permutations and combinations of tetrominoes, as well as the ability to quickly analyze and choose the best placement for each piece.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Additionally, Tetris involves elements of probability theory. Since the order in which tetrominoes appear is random, players must make decisions based on the likelihood of certain pieces appearing. This requires an understanding of probability and the ability to make informed decisions based on the current game state and the potential future outcomes.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;">Furthermore, the scoring system in Tetris is based on mathematical principles. Points are awarded for clearing lines, with more points given for clearing multiple lines simultaneously (referred to as a "Tetris"). This scoring system incentivizes players to strategize and plan their moves to maximize their score, adding a layer of mathematical complexity to the game.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;">In conclusion, Tetris is not just a game of shapes and patterns; it is also a game rooted in mathematical principles. The concepts of polyominoes, combinatorics, probability, and scoring all contribute to the mathematical underpinnings of Tetris, making it a game that challenges players' mathematical skills as well as their gaming prowess. Let me know what you think, I'd love to hear.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-86817585367612880192024-03-01T05:00:00.000-08:002024-03-01T05:00:00.134-08:00The Complex Mathematics of Forests<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; margin: 0px 0px 1.25em; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-td6wxRI9Nv8XP8o80gkRxM-fgRK5tZdtE7vC63AT_hP80iECfxeU0-AnUdWBhsblnTwiWELvRTZUS3cI7xF-nk8XpJe9daXOz5lpAvZkT8csEwxqc192b-4Ml6u0K76X-eiVt5sSJkI9H6cl3_55btsHZEXWBTPQ0DLOc-ALjSdUFcCZuIHpuaPBaPg/s640/trees-3294681_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="360" data-original-width="640" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-td6wxRI9Nv8XP8o80gkRxM-fgRK5tZdtE7vC63AT_hP80iECfxeU0-AnUdWBhsblnTwiWELvRTZUS3cI7xF-nk8XpJe9daXOz5lpAvZkT8csEwxqc192b-4Ml6u0K76X-eiVt5sSJkI9H6cl3_55btsHZEXWBTPQ0DLOc-ALjSdUFcCZuIHpuaPBaPg/s320/trees-3294681_640.jpg" width="320" /></a></div><span style="font-family: georgia; font-size: medium;">Forests are often seen as lush expanses of trees and wildlife, however they are proving to be far more mathematically complex than previously understood. Recent research has revealed intricate patterns and structures within forests that challenge traditional mathematical models and expand our understanding of their ecological dynamics.</span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: georgia;">One of the biggest insights comes from studying fractals, which are geometric shapes that exhibit self-similarity at different scales. Trees and vegetation in forests often exhibit fractal patterns, with branches and leaves repeating similar shapes and structures as you zoom in or out. This self-similarity is not just a visual phenomenon; it reflects underlying mathematical principles that govern the growth and development of forest ecosystems.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: georgia;">Another aspect of forests lies in their network structures. Trees communicate and interact with each other through underground fungal networks called mycorrhizal networks. These networks facilitate the exchange of nutrients, water, and chemical signals between trees, allowing them to cooperate and support each other. The mathematics of these networks is highly complex, involving principles of graph theory and network science.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: georgia;">Furthermore, the spatial distribution of trees in a forest is not random but follows intricate patterns. Research has shown that trees tend to exhibit spatial patterns such as clustering, where trees of similar species are grouped together, and regularity, where trees are evenly spaced. These patterns are not just aesthetically pleasing but also serve important ecological functions, influencing factors like competition for resources and biodiversity.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: georgia;">Understanding the mathematical complexity of forests has significant implications for ecology, conservation, and sustainable forest management. By incorporating mathematical models that account for this complexity, scientists can better predict how forests will respond to environmental changes such as climate change or deforestation. This knowledge can inform conservation efforts and help us preserve these vital ecosystems for future generations.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;"><span style="font-family: georgia;">In conclusion, forests are far more mathematically complex than previously thought, with fractal patterns, network structures, and spatial distributions that challenge traditional mathematical models. Embracing this complexity not only enhances our understanding of forests but also underscores the importance of preserving these ecosystems for their ecological, aesthetic, and mathematical value. Let me know what you think, I'd love to hear.</span></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-12413586187721106502024-02-28T05:00:00.000-08:002024-02-28T05:00:00.135-08:00How Do They Predict How Long People Live?<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;"></span></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><span style="font-family: helvetica;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhyphenhyphenCnXsuW_gNSxqlmjR5mbtL5nPbAMY1R1SK_ex66RfEVhdOHFZj6UOdnuAXoRUYAWzQRglXO4GvN5m4u9QhD3H2oai76BnuLsunUaXfoOd1HypOWsvcCKFn9trp5SpihUzufxSvMwoX3ahFWEXq1XvfQSw3v3_vPL4YCDxIARaSmBhPLQ773oyu4nEXQ/s640/senior-3336451_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="480" data-original-width="640" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhyphenhyphenCnXsuW_gNSxqlmjR5mbtL5nPbAMY1R1SK_ex66RfEVhdOHFZj6UOdnuAXoRUYAWzQRglXO4GvN5m4u9QhD3H2oai76BnuLsunUaXfoOd1HypOWsvcCKFn9trp5SpihUzufxSvMwoX3ahFWEXq1XvfQSw3v3_vPL4YCDxIARaSmBhPLQ773oyu4nEXQ/s320/senior-3336451_640.jpg" width="320" /></a></span></div><span style="font-family: helvetica;"><span style="font-size: medium;">Today's topic came from thoughts of my father. He passed away a couple years ago, four months after my mother died. I know he missed her. Yesterday would have been is 100th birthday if he'd survived. I wondered how math was used to create actuarial tables used in insurance and other industries, so today we'll learn more about it.</span></span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;">Calculating how long a person will live involves some complex mathematical models that take into account various factors such as age, gender, health status, lifestyle choices, and genetic predispositions. While predicting an individual's lifespan with absolute certainty is impossible, actuarial science and life expectancy calculations provide valuable insights into average lifespans and mortality risks.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">Actuarial tables are a fundamental tool used in life expectancy calculations. These tables are based on large sets of population data and provide statistical probabilities of survival and mortality at different ages. Actuaries use these tables to estimate life expectancies for different demographic groups and to calculate insurance premiums and pension benefits.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">One of the key mathematical concepts in life expectancy calculations is the probability distribution function, which describes the likelihood of different outcomes. In the context of life expectancy, this function is used to model the distribution of ages at death within a population. By analyzing this distribution, actuaries can estimate the average lifespan and the probability of living to a certain age.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">Another important mathematical concept is the concept of conditional probability. This concept is used to calculate the probability of an event occurring given that another event has already occurred. In the context of life expectancy, conditional probability is used to calculate the probability of surviving to a certain age given that a person has already reached a certain age.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">Additionally, mathematical models such as the Gompertz law and the Lee-Carter model are used to analyze mortality trends and project future life expectancies. These models take into account factors such as historical mortality data, age-specific mortality rates, and cohort effects to make predictions about future mortality rates and life expectancies.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">In conclusion, calculating how long a person will live involves complex mathematical models that take into account various factors such as age, gender, health status, lifestyle choices, and genetic predispositions. While these models cannot predict an individual's lifespan with certainty, they provide valuable insights into average lifespans and mortality risks, which are essential for insurance, pension planning, and public health policy. Let me know what you think about this, I'd love to hear. Have a great day.</span></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-14353251749545202482024-02-26T05:00:00.000-08:002024-02-26T05:00:00.138-08:00Making Direct Instruction Better<p><span style="background-color: white;"><span face="Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji" style="color: #0d0d0d;"><span style="white-space: pre-wrap;"></span></span></span></p><div class="separator" style="clear: both; text-align: center;"><span face="Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji" style="color: #0d0d0d; font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDeEK_N7v68lRNhmqgBsPoGJ2_s1-T5fd2WqYxoMhU-0QhHw3k8cKd96JHmD8LZ62-W2HqdtU7eUaVW53cjwDTt9TvuHXgRk8sYcnn-E5w8Ps4OhXfUNHjzJals4rGGisDNJlGz_Uiqr4XNqQmmgaKVqaX3nURFQ7ZimGx2aL4aAScnJ35kBuubvtQnSg/s640/teacher-4784916_640.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDeEK_N7v68lRNhmqgBsPoGJ2_s1-T5fd2WqYxoMhU-0QhHw3k8cKd96JHmD8LZ62-W2HqdtU7eUaVW53cjwDTt9TvuHXgRk8sYcnn-E5w8Ps4OhXfUNHjzJals4rGGisDNJlGz_Uiqr4XNqQmmgaKVqaX3nURFQ7ZimGx2aL4aAScnJ35kBuubvtQnSg/s320/teacher-4784916_640.jpg" width="320" /></a></span></div><span style="font-family: trebuchet;"><span style="font-size: medium;"><span face="Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji" style="color: #0d0d0d;">No matter how you arrange your lesson, there is usually a span dedicated to direct instruction since direct instruction</span><span face="Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"" style="background-color: white; caret-color: rgb(13, 13, 13); color: #0d0d0d; white-space: pre-wrap;"> plays an important part in the math classroom.</span></span><span face="Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"" style="background-color: white; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; white-space: pre-wrap;"> It helps students understand complex concepts, develops problem-solving skills, and builds a solid foundation for future learning. However, determining the best time for direct instruction can be challenging, as it depends on a variety of factors such as student age, attention span, and the complexity of the material. We'll look at some of those factors in a bit more </span><span face="Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji" style="color: #0d0d0d;"><span style="caret-color: rgb(13, 13, 13); white-space: pre-wrap;">detail.</span></span></span><p></p><p><span style="font-family: trebuchet;"><span face="Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji" style="color: #0d0d0d;"><span style="caret-color: rgb(13, 13, 13); white-space: pre-wrap;"><span style="font-size: medium;">First, one needs to look at the ability of students to pay attention.</span> </span></span><span face="Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"" style="caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; white-space: pre-wrap;">Younger students generally have shorter attention spans, so direct instruction sessions should be shorter and more focused. For elementary school students, direct instruction sessions of 10-15 minutes are often ideal, with frequent breaks or transitions to keep them engaged. In addition, many students who game may have shorter attention spans. </span></span></p><ol style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; counter-reset: list-number 0; display: flex; flex-direction: column; font-size: 16px; list-style: none; margin: 1.25em 0px; padding: 0px; white-space: pre-wrap;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;">Then one needs to look at the complexity of the material being taught as it also influences the length of direct instruction. For more complex topics in classes such as advanced algebra or calculus, longer direct instruction sessions may be necessary to ensure students grasp the concepts fully. However, it is important to break down these longer sessions into smaller, more manageable segments to avoid overwhelming students.</span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;"><br /></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;">Furthermore, it is important to monitor student engagement since that is the key to effective direct instruction. Teachers should be mindful of the signs of student disengagement, such as fidgeting or inattentiveness, and adjust the length and pace of direct instruction accordingly. Interactive activities, hands-on learning experiences, and multimedia resources can also help maintain student engagement during direct instruction.</span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;"><br /></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;">Another area is the classroom environment as it can impact the effectiveness of direct instruction. A comfortable, well-organized classroom with minimal distractions can help students stay focused and engaged during direct instruction sessions.</span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;"><br /></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;">In addition, direct instruction should be followed by opportunities for students to practice, receive feedback and reflect on their learning. This can be done through group discussions, individual reflection exercises, or formative assessments.</span></p></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="font-family: trebuchet;"><br /></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;"><span style="background-color: white; font-family: trebuchet;">In conclusion, the best time for direct instruction in the math classroom depends on a variety of factors, including student age, attention span, the complexity of the material, student engagement, and the classroom environment. By considering these factors and adjusting direct instruction accordingly, teachers can ensure that students receive the support and guidance they need to succeed in math. Let me know what you think, I'd love to hear. Have a great day.</span></p></li></ol>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-68610909771790539652024-02-23T05:00:00.000-08:002024-02-23T05:00:00.248-08:00Creating Guided Notes To Go With Videos Used In The Math Classroom.<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; margin: 0px 0px 1.25em; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMjfpQLtX595sF3IfO9-MVDj3XZZISni5_l59wGmB75Fk3rjbHusPe9tedmC6-CPOCJbXG7y_2BCt8qKNh6f5kE3oI97rFlsVphG7uHlMncGa6T9RYup203nPLhPmFV4ZwpV1nvTCSKFjw9B6qd4lK-CduY_w4xkvUfJ6FoPb910yLdGK4SC9NOg0L9bg/s640/write-4491416_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="427" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMjfpQLtX595sF3IfO9-MVDj3XZZISni5_l59wGmB75Fk3rjbHusPe9tedmC6-CPOCJbXG7y_2BCt8qKNh6f5kE3oI97rFlsVphG7uHlMncGa6T9RYup203nPLhPmFV4ZwpV1nvTCSKFjw9B6qd4lK-CduY_w4xkvUfJ6FoPb910yLdGK4SC9NOg0L9bg/s320/write-4491416_640.jpg" width="214" /></a></div><span style="font-size: medium;">Guided notes are an effective tool for enhancing student learning during video presentations in math class. These notes provide a structured format for students to follow along with the video, focus on key concepts, and actively engage with the material. Here’s how you can create guided notes to accompany videos shown in math class:</span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;">The first step is to identify any key concepts covered in the video before you begin creating guided notes. These concepts should align with your learning objectives and the content of the video.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;">Next, create an outline for the guided notes based on the key concepts. Organize the notes in a logical sequence that follows the flow of the video. Don't forget to include prompts and questions. So in addition to listing key concepts include prompts and questions that encourage students to think critically about the material. These can be fill-in-the-blank statements, multiple-choice questions, or short-answer questions.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;">Furthermore leave space for students to write their responses to the prompts and questions. This allows them to actively engage with the material and helps them organize their thoughts. Take this one step further and incorporate visual aids such as diagrams, graphs, or equations into the guided notes. These visual representations can help students better understand the concepts being presented in the video.</p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;">In addition, provide cues to turn a passive experience into an active one. When you include cues in the guided notes, they prompt students to pay attention to specific parts of the video. For example, you can instruct students to underline key terms or circle important information. Once the guided notes are created, review them to ensure that they are clear, concise, and aligned with the content of the video. Revise as needed to improve clarity and effectiveness.</p><ol style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; counter-reset: list-number 0; display: flex; flex-direction: column; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; list-style: none; margin: 1.25em 0px; padding: 0px; white-space: pre-wrap;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(227, 227, 227); box-sizing: border-box; margin: 0px;">The final step is to decide whether to distribute the guided notes before or after showing the video. Distributing them before can help students focus on key points during the video, while distributing them after can serve as a review and reinforcement of the material.</p></li></ol><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;">In conclusion, creating guided notes for videos shown in math class can enhance student learning by providing a structured framework for understanding key concepts, encouraging active engagement with the material, and facilitating comprehension and retention of the content.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-44173648516976224992024-02-21T05:00:00.000-08:002024-02-21T05:00:00.248-08:00Scaffolding Direct Instruction With Videos.<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;"></span></p><div class="separator" style="clear: both; font-size: 16px; text-align: center;"><span style="font-family: helvetica;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXKWxjeBP3iXluL1dRVZ7E-L3dfQXamIoOB8JvWKFc6f2gqz6sNyIBbLQg84QYIM6cP5XMfIRHkxGdjAplvTkru9Mz3l357tC0XzA3B-2dDvK1jjTtLnCleDH-GZoqmWSf-eWzM7RW4CskPMWCxJ4WuuxoSluwuhXDr2gQsSrWmYmUn0xkQjHeO8RtAr0/s640/internet-315799_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="424" data-original-width="640" height="212" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXKWxjeBP3iXluL1dRVZ7E-L3dfQXamIoOB8JvWKFc6f2gqz6sNyIBbLQg84QYIM6cP5XMfIRHkxGdjAplvTkru9Mz3l357tC0XzA3B-2dDvK1jjTtLnCleDH-GZoqmWSf-eWzM7RW4CskPMWCxJ4WuuxoSluwuhXDr2gQsSrWmYmUn0xkQjHeO8RtAr0/s320/internet-315799_640.jpg" width="320" /></a></span></div><span style="font-family: helvetica;"><span style="font-size: medium;">Videos can be powerful tools in the math classroom, especially when used to scaffold direct instruction. By carefully selecting and incorporating videos into your lessons, you can enhance student understanding, engagement, and retention of mathematical concepts. Here’s how you can effectively use videos to scaffold direct instruction in your math class:</span></span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;">First, choose relevant and engaging videos. Look for videos that directly align with the concepts you are teaching. The videos should be age-appropriate, clear, and engaging to maintain student interest. Consider using a variety of video formats such as animations, real-world examples, and instructional videos.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;">Always preview the videos before you assign it. Ensure that the content is accurate, clear, and at an appropriate level for your students. Pay attention to the pacing, as videos should not be too fast or too slow for students to follow.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;">Take time to provide context by introducing the video, explaining its relevance to the lesson and how it connects to the concepts students are learning. This helps students understand why they are watching the video and what they should pay attention to.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;">Furthermore, use the videos as a pre-teaching tool. Videos can be used to introduce new concepts or as a review before a lesson. This can help students build background knowledge and prepare them for the upcoming instruction.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;">If you are playing it as part of the lesson, encourage active viewing by pausing the video at key points to ask questions or discuss concepts. This helps students process the information and clarify any misunderstandings. Always provide guided notes or worksheets to complete while watching the video. This keeps them focused and helps them actively engage with the content. Once the video is done, provide students with follow-up activities such as discussions, problem-solving tasks, or hands-on activities to reinforce the concepts learned.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: helvetica;">In addition, assess student understanding.Use the video as a formative assessment tool by asking questions or giving quizzes to check for understanding. This helps you identify any misconceptions that need to be addressed.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">By incorporating carefully selected videos into your math instruction, you can scaffold learning, enhance understanding, and make math more accessible and engaging for your students. Let me know what you think, I'd love to hear.</span></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-43110395532180663862024-02-19T05:00:00.000-08:002024-02-19T05:00:00.139-08:00Gerrymandering And Ham Sandwich Theorem.<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; margin: 1.25em 0px; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_vENU6dgOspDMEyR3-MFjpq7E5wWAsXsBn2ECjXHf6jH7hRXGowhD6NvqnJFJUsrudbj-6fEPZuBVCK4jb7dJ9K_ZXqJjTLfCNIawaJkXOCWmYpVOx_QQtuah3jm3txcjZppz4C7q7K4OIK-NEGa2QBnFLhqeXopa52A0UObDPe7iZTRdAHTNTHE4RCA/s640/ballot-box-2586557_640.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="601" data-original-width="640" height="301" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_vENU6dgOspDMEyR3-MFjpq7E5wWAsXsBn2ECjXHf6jH7hRXGowhD6NvqnJFJUsrudbj-6fEPZuBVCK4jb7dJ9K_ZXqJjTLfCNIawaJkXOCWmYpVOx_QQtuah3jm3txcjZppz4C7q7K4OIK-NEGa2QBnFLhqeXopa52A0UObDPe7iZTRdAHTNTHE4RCA/s320/ballot-box-2586557_640.png" width="320" /></a></div><span style="font-family: helvetica; font-size: medium;">I saw an article on the topic of gerrymandering and the ham sandwich theorem and my mind went huh? So I had to read it to see how they relate. I love how math explains so much. </span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">Gerrymandering, the practice of manipulating the boundaries of electoral districts to favor a certain political party, is a hotly debated topic in modern politics. There are court cases galore on this topic. While the concept of gerrymandering is rooted in political strategy, its implications can be understood through the lens of mathematics, particularly the Ham Sandwich Theorem.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">The Ham Sandwich Theorem, a fundamental principle in geometric measure theory, states that given any three objects in n-dimensional space (such as three shapes in a plane or three volumes in three-dimensional space), it is possible to divide them equally with a single cut, much like slicing a ham sandwich into two equal halves with a single slice. This theorem has interesting implications when applied to the concept of gerrymandering.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">In the context of gerrymandering, imagine the objects as representing different groups of voters, and the cut as representing the boundary lines of electoral districts. The Ham Sandwich Theorem suggests that it is theoretically possible to draw district boundaries in such a way that the political influence of each group is evenly balanced, ensuring fair representation for all.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">However, the practical application of the Ham Sandwich Theorem to gerrymandering is challenging due to the complexity of real-world political boundaries and the need to consider various factors such as population distribution, community interests, and legal requirements. In practice, gerrymandering often involves intricate boundary-drawing techniques that aim to maximize the political advantage of one party over another, rather than achieving true equality in representation.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;"><span style="font-family: helvetica;">Despite its limitations in addressing gerrymandering directly, the Ham Sandwich Theorem serves as a reminder of the importance of fairness and equality in the design of electoral systems. By understanding the mathematical principles behind gerrymandering, we can better appreciate the need for transparent and equitable practices in redistricting and electoral reform. Let me know what you think.</span></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-56529127291324469972024-02-16T05:00:00.000-08:002024-02-16T05:00:00.129-08:00The Importance Of Bell Ringers And Exit Tickets<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; margin: 0px 0px 1.25em; white-space: pre-wrap;"></p><div class="separator" style="clear: both; font-family: times; font-size: 16px; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgakgTOMRbsOoRbEXkXpnSvU3hcAFDL3piG5BeFDSdmPyjQ8UpThfkpzSsaN1ueRc15L38vQ_lJOkYaZLgkkjAXLyUVo71CrwSNrwVB9LIj6OQA9olQ6Lmi_bAm5Etwv9SVYJGTOqu52MLKhPTxX-pE8EunRX9nHZuQdxfGFYMajJYRRbhzZqDlsqDDSmA/s640/tact-3036245_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="473" data-original-width="640" height="237" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgakgTOMRbsOoRbEXkXpnSvU3hcAFDL3piG5BeFDSdmPyjQ8UpThfkpzSsaN1ueRc15L38vQ_lJOkYaZLgkkjAXLyUVo71CrwSNrwVB9LIj6OQA9olQ6Lmi_bAm5Etwv9SVYJGTOqu52MLKhPTxX-pE8EunRX9nHZuQdxfGFYMajJYRRbhzZqDlsqDDSmA/s320/tact-3036245_640.jpg" width="320" /></a></div><span style="font-family: times;"><br /><span style="font-size: medium;">In the world of teaching mathematics, bell ringers (or warm-ups) and exit tickets still serve as invaluable tools for improving student learning and engagement in the classroom. These brief activities, typically done at the beginning and end of a class session, offer numerous benefits that contribute to a more effective learning environment.</span></span><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: times;">First of all, bell ringers set the tone for the math lesson by activating students' prior knowledge and priming their minds for learning. By presenting students with a thought-provoking problem or question related to the day's topic, bell ringers stimulate curiosity and encourage students to mentally prepare for the upcoming lesson. This initial engagement helps capture students' attention from the outset, fostering a positive and focused learning atmosphere.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: times;">Next, bell ringers provide an opportunity for teachers to assess students' understanding of key concepts while identifying any misconceptions or gaps in their knowledge. By observing students' responses to the bell ringer activity, teachers can gauge the readiness of the class to work on the topic and tailor their instruction accordingly. This way, the teacher addresses any areas of misunderstanding or confusion before teaching the main lesson content. This aspect of bell ringers allows teachers to carry out formative assessment that allows teachers to differentiate instruction and provide targeted support to individual students as needed.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: times;">Similarly, exit tickets also serve as a valuable tool for assessing student learning and comprehension at the conclusion of a lesson. When a teacher poses a brief question or prompt related to the day's lesson objectives, it allows students to demonstrate their understanding and reflect on their learning experiences. This formative assessment feedback informs teachers so they can instructional decisions and helps them gauge the effectiveness of their teaching strategies, enabling them to adjust their approach as necessary to meet students' needs.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: times;">In addition to their assessment function, exit tickets also promote metacognitive awareness and reflective thinking among students. When prompting students to articulate what they have learned, what questions they still have, or how they plan to apply their learning, the exit tickets encourage students to engage in critical thinking and self-assessment. This reflective process not only reinforces learning but also fosters greater ownership and accountability for one's learning journey.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;"><span style="font-family: times;">Overall, the strategic use of bell ringers and exit tickets in the math classroom enhances student engagement, assesses understanding, and promotes reflective thinking, providing teachers with ongoing assessment. Teachers who incorporate these brief yet powerful instructional strategies into their teaching practice, are able to cultivate a dynamic and supportive learning environment that empowers students to succeed in mathematics and beyond. Let me know what you think, I'd love to hear.</span></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-51955486137279521462024-02-14T05:00:00.001-08:002024-02-14T05:00:00.135-08:00How Does Math Relate To Valentines Day?<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span color="var(--tw-prose-body)" style="font-size: 1rem;"><span style="font-family: trebuchet;"></span></span></p><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><span style="text-align: left;"><div class="separator" style="clear: both; font-family: trebuchet; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIO5KUE9xcP9R3R72KC2LVxuRKl3z8h6aQ_VHjkEe2tQcxICXQDYxqkgcIV5SdJLNghurfJV7FBDR8Z-mK0kAD-YqsMEN73KwmGmCRnUvOyXigFn_vpVI471F8FX9ZegJBe0T4S9MTUc62W7RDOhUYk_exsmhW6erq-hpxs9EAepEUEecXvWyebxVOiBM/s640/candles-1645551_640.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="417" data-original-width="640" height="209" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIO5KUE9xcP9R3R72KC2LVxuRKl3z8h6aQ_VHjkEe2tQcxICXQDYxqkgcIV5SdJLNghurfJV7FBDR8Z-mK0kAD-YqsMEN73KwmGmCRnUvOyXigFn_vpVI471F8FX9ZegJBe0T4S9MTUc62W7RDOhUYk_exsmhW6erq-hpxs9EAepEUEecXvWyebxVOiBM/s320/candles-1645551_640.jpg" width="320" /></a></div><span style="font-family: times; font-size: medium;">Valentine's Day is often celebrated when people give flowers, chocolates, and heartfelt messages to others. Although this might not seem like a holiday closely associated with mathematics, a world of mathematical concepts and principles lie beneath the surface of romantic gestures. These concepts and principles add depth and intrigue to the celebration of love.</span></span></div><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Furthermore, the exchange of flowers on Valentine's Day provides another opportunity to explore mathematical concepts. Florists meticulously arrange bouquets as they consider factors such as color, size, and shape to create visually stunning arrangements. The Fibonacci sequence, a famous mathematical pattern found in nature, often serves as a guide for arranging flowers in a visually pleasing manner. The spiral patterns observed in flowers such as sunflowers and roses closely follow the Fibonacci sequence, reflecting mathematical beauty in nature's design.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">In addition, gift-giving on Valentine's Day involves mathematical considerations, especially when it comes to budgeting. Individuals often set budgets for Valentine's Day gifts, balancing the desire to express affection with financial limitations. If individuals consider mathematical optimization techniques so they maximize the value of their gifts within their budgetary constraints while considering factors such as the preferences of their loved ones and the available options in the market.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Moreover, the celebration of Valentine's Day provides an opportunity to explore mathematical concepts related to symmetry and geometry. Heart-shaped chocolates, cards, and decorations abound on Valentine's Day, all reflecting the universal symbol of love and mathematicians study the properties of geometric shapes such as hearts, examining their symmetry, curvature, and mathematical representations.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">In conclusion, Valentine's Day may be a holiday centered on love and romance, but mathematics plays a hidden yet significant role in its celebration. From probability and combinatorics to geometry and optimization, mathematical concepts add depth and complexity to the rituals and traditions associated with Valentine's Day, reminding us that love and mathematics are intertwined in unexpected ways. Let me know what you think, I'd love to hear. Have a great day.</span></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-55823685091064068392024-02-12T05:00:00.000-08:002024-02-12T05:00:00.135-08:00Swarming Cicadas, Stock Traders, And Crowds.<p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: trebuchet;"></span></p><span style="font-family: trebuchet;"><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: -webkit-standard; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="font-family: trebuchet;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: trebuchet;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfhI5SUnEbOp7BVrtSTMdjdORL5kPRgBHAzh9H84YYL72YgiYfzKkpUjke2W5oUdWP9K9JdWI8Gc5CTvWSLY_1KKl-N5XZA-rBTLsFPdt2lz55i-bqG7km1y3GOttXehOJW4iPpWVJ1LbzEahOo-jlktejKDXVN3cHCOmFau4RhMEjuMP4wf2KK8n_9o4/s640/cicada-936145_640.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="396" data-original-width="640" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfhI5SUnEbOp7BVrtSTMdjdORL5kPRgBHAzh9H84YYL72YgiYfzKkpUjke2W5oUdWP9K9JdWI8Gc5CTvWSLY_1KKl-N5XZA-rBTLsFPdt2lz55i-bqG7km1y3GOttXehOJW4iPpWVJ1LbzEahOo-jlktejKDXVN3cHCOmFau4RhMEjuMP4wf2KK8n_9o4/s320/cicada-936145_640.jpg" width="320" /></a></span></div><p></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-size: 16px; margin: 0px 0px 1.25em; white-space: pre-wrap;"><span style="background-color: white;">Swarming cicadas, stock brokers, and the combined knowledge of a crowd might seem like disparate topics, but they all intersect in the fascinating realm of mathematics. </span><span style="font-family: trebuchet;">These seemingly unrelated phenomena actually share underlying principles that mathematicians and scientists use to understand collective behavior, decision-making processes, and patterns in nature and society.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: -webkit-standard; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Cicadas, insects known for their synchronized emergence in large numbers at certain intervals, exhibit a behavior known as swarming. This phenomenon, observed in various species of cicadas, is driven by mathematical principles related to prime numbers and survival strategies. Cicadas have evolved to emerge in large numbers at prime number intervals, such as 13 or 17 years, which reduces the likelihood of predators setting their breeding cycles with the cicadas' appearance, thus increasing their chance of survival.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: -webkit-standard; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Similarly, in the world of finance, stock brokers and investors rely on mathematical models and the combined knowledge of crowds to make informed decisions in the stock market. The wisdom of the crowd refers to the collective intelligence of a group of individuals, whose aggregated opinions or predictions tend to be more accurate than those of any single member. This concept is leveraged in various mathematical models, such as the efficient market hypothesis and the random walk theory, which posit that stock prices reflect all available information and follow a random pattern.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: -webkit-standard; font-size: 16px; margin: 1.25em 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">In addition, the mathematical principles underpinning swarming behavior and the wisdom of the crowd have applications beyond cicadas and stock markets. They are also relevant in fields such as artificial intelligence, where algorithms are designed to mimic the collective behavior of swarms or crowds to solve complex problems, and in decision-making processes in business, politics, and social sciences.</span></p><p style="border: 0px solid rgb(227, 227, 227); box-sizing: border-box; caret-color: rgb(13, 13, 13); color: #0d0d0d; font-family: -webkit-standard; font-size: 16px; margin: 1.25em 0px 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">In essence, the study of swarming cicadas, stock brokers, and the wisdom of the crowd exemplifies the interdisciplinary nature of mathematics and its relevance in understanding complex phenomena in nature, society, and beyond. By applying mathematical principles to analyze patterns, behaviors, and interactions, researchers can uncover hidden insights and develop strategies to address real-world challenges, from predicting cicada emergences to navigating financial markets and harnessing collective intelligence for problem-solving. Let me know what you think, I'd love to hear. Have a great day.</span></p></span><p></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-73413835878652409812024-02-09T05:00:00.000-08:002024-02-09T05:00:00.165-08:00How Short Can A Mobius Strip Be?<p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWBOyZy9ZAlKgHolUBXX0vuYj1DCerOnIM2yIeQUsRBJXMURBOwhcZbxVnPNuV8dEdnd0SNeVPMPCk97p4KHBYj90zBRDMoOVDzNRSqMutFA9BX1-sF5RxbQ7jSUMF0Ms6f-MhPMHQaqk9VMDnB-11HZM4213qjLjxH2lNbsAz9OanmCuJEAmnIwJyPZE/s640/cut-174860_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="480" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWBOyZy9ZAlKgHolUBXX0vuYj1DCerOnIM2yIeQUsRBJXMURBOwhcZbxVnPNuV8dEdnd0SNeVPMPCk97p4KHBYj90zBRDMoOVDzNRSqMutFA9BX1-sF5RxbQ7jSUMF0Ms6f-MhPMHQaqk9VMDnB-11HZM4213qjLjxH2lNbsAz9OanmCuJEAmnIwJyPZE/s320/cut-174860_640.jpg" width="240" /></a></div><span style="font-family: trebuchet;">The Möbius strip is a fascinating mathematical concept that continues to captivate minds with its seemingly impossible and paradoxical nature. One of the intriguing questions that arise from playing with a Möbius strip is just how short can one be? Today, we'll explore that question and learn more about this topic since many teachers have had students create one in class.</span><p></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="font-family: trebuchet;">Let's start with a bit of background. <span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;">The Möbius strip, named after the German mathematician August Ferdinand Möbius, the person who discovered it in 1858. It is a non-orientable surface with only one side and one boundary. This odd creation is formed by taking a strip of paper, giving it a half twist, and then connecting the ends. The result is a single-sided, continuous loop that challenges conventional notions of geometry. </span></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;"><span style="font-family: trebuchet;"> The Möbius strip's most remarkable property is that it has only one edge and one surface. If you trace your finger along the surface, you would find yourself on both sides without ever lifting your finger. This inh built paradoxical nature makes the Möbius strip a favorite subject for mathematical explorations and artistic creations.</span></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="font-family: trebuchet;"><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;">The question of how short this creation could be snagged the imagination of Richard Evan Swartz. He explored the </span><span style="color: #374151;"><span style="caret-color: rgb(55, 65, 81); white-space: pre-wrap;">topic using a computer program but due to a mistake in the program, he almost missed finding the answer. However, he kept playing with </span></span><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;">Möbius strips and that lead him to the answer.</span></span></p><ol style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-reset: list-number 0; display: flex; flex-direction: column; list-style: none; margin: 1.25em 0px; padding: 0px;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; counter-increment: list-number 1; display: block; font-size: 16px; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em; white-space: pre-wrap;"><span style="font-family: trebuchet;">The usual method of constructing a Möbius strip involves taking a rectangular strip of paper, twisting one end by 180 degrees, and then connecting the ends. The resulting Möbius strip has a length twice that of the original strip. Experience showed that a long thin strip is easier to make than a short, fat one. </span></li></ol><div><span style="color: #374151; font-family: trebuchet;"><span style="white-space: pre-wrap;">Back in 1977, several mathematicians theorized that a triangular shaped Möbius strip is as small as the strip can get but no one could prove it . They said the ratio between the length and width would be more than about 1.57 times or pi/2. It took another 50 years before someone one was able to come up with proof.</span></span></div><div><span style="color: #374151; font-family: trebuchet;"><span style="white-space: pre-wrap;"><br /></span></span></div><div><span style="color: #374151; font-family: trebuchet;"><span style="white-space: pre-wrap;">In order to address this question Swartz focused on the properties of a </span><span style="caret-color: rgb(55, 65, 81); white-space: pre-wrap;">Möbius strip. At every point, there is a direction that a line travels edge to edge with no curvature. It is completely flat. Swartz recognized that there are places where the two lines cross forming 90 degree angles forming an T shaped intersection.</span></span></div><div><span style="color: #374151; font-family: trebuchet;"><span style="caret-color: rgb(55, 65, 81); white-space: pre-wrap;"><br /></span></span></div><div><span style="color: #374151; font-family: trebuchet;"><span style="white-space: pre-wrap;">Swartz used these contortions to find a new length to width ration of 1.69. He moved on to other projects but still thought about this. One day, he realized that he'd made a basic error when he cut open a Möbius strip, realizing it was trapezoidal rather than a parallelogram. This lead to the understanding that he'd made a basic error in the computer program he'd been using to explore the topic.</span></span></div><div><span style="color: #374151; font-family: trebuchet;"><span style="white-space: pre-wrap;"><br /></span></span></div><div><span style="color: #374151; font-family: trebuchet;"><span style="white-space: pre-wrap;">This small change in understanding lead to the discovery that the ratio is the sqrt 3 or about 1.73 length times its width. In addition, the strip is so short, it ends up flattening into an equilateral triangle. Let me know what you think, I'd love to hear. Have a great day.</span></span></div><div><span style="caret-color: rgb(55, 65, 81); color: #374151; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; white-space: pre-wrap;"><br /></span></div><div><br /></div><div><span style="color: #374151; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="caret-color: rgb(55, 65, 81); white-space: pre-wrap;"><br /></span></span></div><ol style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-reset: list-number 0; display: flex; flex-direction: column; list-style: none; margin: 1.25em 0px; padding: 0px;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; counter-increment: list-number 1; display: block; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em; white-space: pre-wrap;"><br /></li><br class="Apple-interchange-newline" /></ol>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-79517504804210641712024-02-07T05:00:00.000-08:002024-02-07T05:00:00.152-08:00Looking At Math Based Riddles<p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiK6dzFgbpcKaj5NHlA6Kjxfx2h026w6PKDD9V55-lTS6HVtGaGfVGNRy5cf1lkEYY9ZBuhA-i5-n4zDruD4l3csJ3KuRv-juYiX4RvgmxymgXUswG47ylalovMtsaI4nGSjSpg7G2NCl18fqp-Q0rOHBEvyHDXyshPki0JOOfmk3vu3FdJsQsD_OqnU8I/s640/woman-687560_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="480" data-original-width="640" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiK6dzFgbpcKaj5NHlA6Kjxfx2h026w6PKDD9V55-lTS6HVtGaGfVGNRy5cf1lkEYY9ZBuhA-i5-n4zDruD4l3csJ3KuRv-juYiX4RvgmxymgXUswG47ylalovMtsaI4nGSjSpg7G2NCl18fqp-Q0rOHBEvyHDXyshPki0JOOfmk3vu3FdJsQsD_OqnU8I/s320/woman-687560_640.jpg" width="320" /></a></div><span style="font-family: trebuchet;">Mathematics often brings up images of complex equations and rigorous problem-solving, but hidden in the world of numbers lies the engaging world of math riddles. These brain-teasers not only offer a break from traditional learning but also provide an entertaining way to sharpen mathematical skills. In this article, we'll delve into the charm of math riddles and explore a few mind-bending examples to tickle your brain.</span><p></p><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Math riddles combine the fun of jokes with numbers. They challenge our problem-solving abilities, encourage creative thinking, and add an element of playfulness to mathematical concepts. Whether you're a teacher looking for a fun way to reinforce lessons or looking a mental workout, math riddles offer an accessible and enjoyable solution.</span></span><div><span style="color: #374151; font-family: trebuchet;"><span style="caret-color: rgb(55, 65, 81); white-space: pre-wrap;"><br /></span></span></div><div><span style="color: #374151; font-family: trebuchet;"><span style="caret-color: rgb(55, 65, 81); white-space: pre-wrap;"><br /></span></span><div><span style="color: #374151; font-family: trebuchet;"><span style="caret-color: rgb(55, 65, 81); white-space: pre-wrap;"><br /></span></span></div><div><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Let's look a classic math riddle:</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><em style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span style="font-family: trebuchet;">I am taken from a mine, and shut up in a wooden case, from which I am never released, and yet I am used by almost every person. What am I?</span></em></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><em style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span style="font-family: trebuchet;"><br /></span></em></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Answer: A pencil lead.</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">This riddle cleverly disguises a common item, highlighting the ability of math riddles to weave everyday objects into enigmatic puzzles.</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Here is another one referred to as the <b>Farmers Challenge</b>.</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Imagine you are a farmer with a sack of grain, a chicken, and a fox. You need to transport them across a river using a small boat, but the boat can only carry you and one other item at a time. The catch: if you leave the fox alone with the chicken, the fox will eat it; if you leave the chicken alone with the grain, the chicken will eat it. How do you get all three across the river safely?</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Answer: Take the fox across first, then go back and take the chicken across. Leave the fox on the other side and take the grain across. Finally, go back alone to get the fox. No harm done!</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Then there is the <b>Missing Dollar </b>riddle -</span></p><ol style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-reset: list-number 0; display: flex; flex-direction: column; list-style: none; margin: 1.25em 0px; padding: 0px;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; counter-increment: list-number 1; display: block; font-size: 16px; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em; white-space: pre-wrap;"><span style="font-family: trebuchet;">Three people check into a hotel room that costs $30. They each contribute $10, handing $30 to the hotel clerk. Later, the hotel clerk realizes there was a mistake, and the room should only cost $25. The clerk gives $5 to the bellboy and asks him to return it to the guests. On his way to the guests' room, the bellboy decides to keep $2 for himself and gives $1 back to each guest. Now, each guest has paid $9 (a total of $27), and the bellboy has kept $2, making a total of $29. What happened to the missing dollar?</span><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"></p></li></ol><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><em style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span style="font-family: trebuchet;"><br /></span></em></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Answer: There is no missing dollar. The $27 paid by the guests includes the $2 kept by the bellboy. The guests paid $25 for the room, and the bellboy kept $2.</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Check out the <b>Age Puzzle.</b></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><b><span style="font-family: trebuchet;"><br /></span></b></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">A puzzle to exercise your math and deductive reasoning:</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><em style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span style="font-family: trebuchet;">A father is 3 times as old as his son. In 20 years, he will be just twice as old as his son. How old are they now?</span></em></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Answer: Let the son's current age be x. The father's age is then 3x. In 20 years, the son's age will be x + 20, and the father's age will be 3x + 20. Since the father will be twice as old, the equation is 3x + 20 = 2(x + 20). Solving this, the son is currently 20 years old, and the father is 60.</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;">We are finishing off with my favorite. <b>St. Ives </b>one</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; margin: 0px; white-space: pre-wrap;"><span style="font-family: trebuchet;"><br /></span></p><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">As I was going to St. Ives,</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">I met a man with seven wives,</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">Each wife had seven sacks,</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">Each sack had seven cats,</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">Each cat had seven kits:</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">Kits, cats, sacks, and wives,</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">How many were there going to St. Ives?</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;"><br /></span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;">The answer is one.</span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-family: trebuchet;"><br /></span></dd><dd style="caret-color: rgb(32, 33, 34); color: #202122; font-size: 14px; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px; text-align: left;"><span style="caret-color: rgb(55, 65, 81); color: #374151; font-family: trebuchet; font-size: 16px; white-space: pre-wrap;">Math riddles inject an element of fun into the world of numbers, making learning an enjoyable and interactive experience. These can be used to start or end a class. Puzzles such as these </span><span style="caret-color: rgb(55, 65, 81); color: #374151; font-family: trebuchet; font-size: 16px; white-space: pre-wrap;">enhance problem-solving skills, promote creative thinking, and demonstrate that mathematics can be an exciting journey of discovery. So, have fun and think about building a library of math riddles to sprinkle through the year. Let me know what you think, I'd love to hear.</span></dd></div></div>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-71429440927407643372024-02-05T05:00:00.000-08:002024-02-05T05:00:00.365-08:00Make Instruction More Effective Part 2.<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiajm3W1wOrwdsY6ZSSYGgRHrWsMGALgwi-xj69DL3vunMGa6Pgk2zJwzkrzg-b5gMMC6-j8HycanOHXt3hMH69P7mjytxfs_GJLvBOM_cDVM14zOw3bfhj4q7CPOuOZR6ql965Z2u2BQVhzIuRbS9CpjDDYD9esok3pjrr0C-xCS6H1xuJs7xyFFCI-dI/s640/mistake-1966448_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="410" data-original-width="640" height="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiajm3W1wOrwdsY6ZSSYGgRHrWsMGALgwi-xj69DL3vunMGa6Pgk2zJwzkrzg-b5gMMC6-j8HycanOHXt3hMH69P7mjytxfs_GJLvBOM_cDVM14zOw3bfhj4q7CPOuOZR6ql965Z2u2BQVhzIuRbS9CpjDDYD9esok3pjrr0C-xCS6H1xuJs7xyFFCI-dI/s320/mistake-1966448_640.jpg" width="320" /></a></div>Last Friday, we started looking at ways that help students learn more in math class. Today, we'll finish the suggestions since I didn't want to overload brains. The last part of Friday's column was discussing ways to use mistakes in class to help improve student knowledge and we'll continue that today.<p></p><p>One important step is to have students slow down and really think about what they are doing. Create four incorrectly solved problems and place them around the room. Have the students rotate around the room, stopping at each problem where they must discuss the error and write down using complete sentences, why the error occurred and how to correct it. As they rotate around the room, students are engaged in conversations on what other students have written and they write down whether they support the answers or if they disagree with the thoughts, they need to write down their evidence.</p><p>Furthermore, use math journals where students can record thoughts on concepts that they are having difficulty with, They can talk about where they are getting tripped up, work the problem from start to finish, reflect on different strategies they used when they tried to solve it, or talk about other approaches they could have used. This type of journaling helps synthesis learning while looking at unanswered questions.</p><p>We want students to develop methods of solving complex problems rather than just mimicking the teachers method. One way to nudge students in this direction, is to give students a thinking task and ask them to work in small random groups. Thinking tasks are problem solving activities or mental puzzles designed to challenge thinking. Have students stand while they work on these problems and do calculations, writing, etc on surfaces like white boards, blackboards, or windows where they can easily erase ideas that are not working out. Teachers circulate but do not answer questions such as "Is this right?" so as not to stop their thinking processes.</p><p>The idea behind this is to encourage perseverance, collaboration, and explore their curiosity. Rather than assessing them on being right or wrong, assess them on their perseverance, collaboration, and thoughts on how to solve problems. Take it a step further by letting students know where they are and where they are going in their learning through the use of frequent checks, observations, and unmarked quizzes. </p><p>Furthermore, summative assessments should focus less on answers and more on the process of learning. Let me know what you think, I'd love to hear. Have a great day.</p><p><br /></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-9155083428792368302024-02-02T05:00:00.000-08:002024-02-02T05:00:00.156-08:00Make Instruction More Effective Part 1.<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9qRh9XL0-Fr0srwixdEcaiLxpgNUAb-_6Q0_tlh3TTbNenbhn4SOw9TeMQauqVewfQrjPITEwdh7kJLJ5qce3z6wFkuH8W1btrtsexbeBmpImkZpZikzh3bt3fw8a_LQcOyA3whOzFS6yogrJRPiwI0wtKCLge050-wJx_aaue9b4lHiih1piHOTPsh4/s640/teacher-4784916_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9qRh9XL0-Fr0srwixdEcaiLxpgNUAb-_6Q0_tlh3TTbNenbhn4SOw9TeMQauqVewfQrjPITEwdh7kJLJ5qce3z6wFkuH8W1btrtsexbeBmpImkZpZikzh3bt3fw8a_LQcOyA3whOzFS6yogrJRPiwI0wtKCLge050-wJx_aaue9b4lHiih1piHOTPsh4/s320/teacher-4784916_640.jpg" width="320" /></a></div>At anytime of the year, we as teachers often need to adjust the testing routine, work on eliminating math anxiety, creating an environment that encourages mistakes, and develop critical thinking skills. Although we often want to make changes in time for the new school year, changes can be made at the beginning of a semester, quarter, or after a break.<p></p><p>One of the big things teachers face are students who believe they cannot do math or are not born with the math gene. One way to check their beliefs and help to change them is to ask students to write a math autobiography where they answer questions such as "How do you feel about math?" or "How did your relationship with math change overtime?" and then asking them to share answers with each other. Furthermore, it is best if teachers eliminate comments about something being easy because it can demotivate them and they are less likely to ask questions to clarify their understanding.</p><p>Next, think about engaging students as soon as they arrive in the classroom. To do this, have a warm-up, bell ringer, math riddles, or challenging brain teaser. The brain teaser might be having students begin with a cross and then ask them to draw two more lines to intersect or cut the cross to form the most number of pieces. On the other hand, a math riddle might be something like "You have two coins that equal 30 cents but one of the coins is not a nickel so what are the coins?". Both brain teasers and riddles require students to problem solve and think critically. </p><p>Then think about ways that students can show what they learn in ways other than by taking a test especially as they are tested quite regularly though the school year. Instead of testing, give frequent, short assessments that are made up of current and past topics so they have to retrieve information. Give this type of assessment every two weeks just as a way of checking on students but they don't need to be graded. However, the teacher should note mistakes and change instruction accordingly.</p><p>If you do have to give tests, allow students to discuss material on the test before beginning it. One way to keep them from starting the test is to have students place their pencils on the floor, and then spend about 5 minutes to talk about the problems they see. </p><p>As far as mistakes go, it is best to let them make mistakes while giving them the opportunity to discuss mistakes with others can better place the information in their brains. Use mistakes as part of a class activity by dividing the class into small groups, then identify and reflect on common mistakes. One does this by dividing the class into small groups, they are asked to generate a problem and solve it incorrectly. Next the groups rotate and they must identify the mistake made by the other group and solve the problem correctly. Then students rotate again and they have to identify the mistake of the first group and how the second group corrected the problem.</p><p>In addition, try the which one is more right by presenting two incorrect problems - one conceptually incorrect and the other is incorrectly calculated - and ask students to determine which problem is more right. By doing this activity, they have to think about nuances in problem solving.</p><p>I think this is enough for today. Come Monday, I'll provide more suggestions. Let me know what you think, I'd love to hear. </p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-54415551587819919782024-01-31T05:00:00.000-08:002024-01-31T05:00:00.152-08:00Harvard Psychologist On Teaching Math.<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJRXlLRmxYIJvv1f81sRNAR8EpLXxZOcnAUucAb264NIba7p8Ngqdw0OufxGhDfzoEKS4pF4LfJnjjaQSebCyvUR8-DjB16ExSN3SvU2eTU0I7apJJ-_UXGQsHNNkqVdVyxuED7lkZC6dcWJMbJCdpKpkMTX648EAq8dUUPPJdMwHDKc9qx65Mg23C08M/s640/teacher-3765909_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJRXlLRmxYIJvv1f81sRNAR8EpLXxZOcnAUucAb264NIba7p8Ngqdw0OufxGhDfzoEKS4pF4LfJnjjaQSebCyvUR8-DjB16ExSN3SvU2eTU0I7apJJ-_UXGQsHNNkqVdVyxuED7lkZC6dcWJMbJCdpKpkMTX648EAq8dUUPPJdMwHDKc9qx65Mg23C08M/s320/teacher-3765909_640.jpg" width="320" /></a></div>As teachers, many of us are looking around for the best way to teach mathematics. One person, Jon R Star, a distinguished psychologist at Harvard University, has spent years researching this topic with professors from Vanderbilt university. Over time, he has determined better ways to teach math. We shall look at what he has to say.<p></p><p>It is well known in the field of psychology that the actual process of learning requires people to reach into their brains to find a certain piece of knowledge, pull it out, think about it, and them putting it back into the brain. This process is referred to as elaborative coding. The more a person does this process, the more they will have learned, remembered, and understood. </p><p>In math, he stated that it is important to make sense of the material being taught. Learning is more than just listening to what the teacher says. Instead, he emphasizes how important it is for students to develop conceptual knowing in mathematics, so it is important for teachers to present the information in a way that students make sense rather than memorizing or internalizing the material.</p><p>In fact, Jon recommends that teachers compare two ways of solving a problem by writing each process side by side and then lead a discussion designed to help students understand the difference between the two methods. It is the discussion because the teacher has asked students why a strategy works and students must dig into their heads to share what they understand. In addition, listening to other student's reasoning helps reinforce the process of learning since they have to think about it.</p><p>It is thought that when students learn multiple ways to solve problems, it deepens their understanding of content. In addition to providing a wonderful benefit to learning, it helps students see math differently. Instead of seeing math as needing to memorize methods with one method to solve each problem, they are shown that each problem may have multiple ways to solve.</p><p>Furthermore, this type of instruction can make class more interesting. It simply is best to show students the compare and contrast of two methods but it involves a lot of thinking for both the teacher and student. There is a concern that both teachers and students might experience information overload or that the teacher might want to take it further by comparing three or four different methods. </p><p>Although many curriculums already show the various methods for solving problems, they don't often focus on the comparing and contrasting step. In addition, this strategy often requires teachers spend time preparing the visuals so they can teach but it does help students learn more. Let me know what you think, I'd love to hear. Have a great day.</p><p><br /></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-30211060979441924222024-01-29T05:00:00.000-08:002024-01-29T05:00:00.205-08:00How Much Fruit Can Be Removed Before The Display Falls Down!<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvDmdD6rD_U6H8lh6LCpWnoTHN554uqqlyInYpPOVd4JJBxnBSPl5-iPtWscjPwPk5w-CoohVdJnZNDfJxf2Bmyd39WKQKoCdEPZac3BuVAjFHUC7yavJ3NF9_zzhov92rDsdxLBfGHUq3MbVEIN1wgysapg96x_LJDBZ-ZoXW9CCNmPnKKc0Nmg_dGLE/s640/grocery-1232944_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvDmdD6rD_U6H8lh6LCpWnoTHN554uqqlyInYpPOVd4JJBxnBSPl5-iPtWscjPwPk5w-CoohVdJnZNDfJxf2Bmyd39WKQKoCdEPZac3BuVAjFHUC7yavJ3NF9_zzhov92rDsdxLBfGHUq3MbVEIN1wgysapg96x_LJDBZ-ZoXW9CCNmPnKKc0Nmg_dGLE/s320/grocery-1232944_640.jpg" width="320" /></a></div>It's a standard scene in the movies. There is a chase seen in the store where someone touches a piece of fruit, or a kid grabs something and the whole pile falls off and rolls all over the floor. This scene lead to a person wondering how much fruit can be removed from the display before the whole thing comes tumbling down.<p></p><p>You might wonder why anyone would want to know the answer to this question but apparently the answer to this question is extremely important since it uses the same dynamics that cause avalanches and landslides.</p><p>Unfortunately, it is too difficult to figure out how much dirt or snow can be removed before it call comes crashing down as an avalanche or a landslide but the fruit actually offers the opportunity to see how removing objects results in movement. This study explored the physics of tumbling produce which is the same action that causes avalanches.Since most fruits are about the same size and shape and end up spread out in a nonrandom, crystal like form. This makes it easier to look at the impact of removing one item at a time from the overall structure. </p><p> So the researchers created a computer program that modeled a different number of spheres stacked in a variety of angles. Then they removed spheres one at a time to determine at which angle the display would fall instantly, not at all, or somewhere in-between. They took it one step further to determine how many objects had to be removed to make the display collapse if it didn't fall on its own.</p><p>Furthermore, they concluded the steeper the angle of the display, the fewer the objects removed before it starts falling and the gentler the angle, the more objects one has to remove. If the slope is gentle enough, the display won't collapse. They concluded that one could remove about 10 percent of the fruit in the display before it collapses. So if you have 300 fruit stacked in the display and 29 people grab a piece, it will fall when the 30th person takes one. </p><p>Researchers are hoping to take this a bit further by exploring how many objects can be removed when the objects are of different sizes and are randomly arranged. This is more involved with avalanches and landslides and the results could provide additional knowledge and open up additional avenues of investigation. Let me know what you think, I'd love to hear.</p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0tag:blogger.com,1999:blog-7114421310634437399.post-6858042018687806132024-01-26T05:00:00.000-08:002024-01-26T05:00:00.143-08:00Using Doodle Notes In Math part 2<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXEDe1viOqXjjhTcFNPz1Xf4ixjJSC6jf2ulruvxBzuabZPTW6at9lpxyjAExT_8xLSmwC0zCH9Gp8w7QkT0vXq_WisxvfL392SH0YPF6E5CVUSsMNGhPKFIBy5camTtXPV4Xpwu_Kj4ijsA1lV3wIeXURi-E5gFwp5TOUEhbq8rSABZqpHVHGEVAFrY0/s640/notebook-2178656_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: trebuchet;"><img border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXEDe1viOqXjjhTcFNPz1Xf4ixjJSC6jf2ulruvxBzuabZPTW6at9lpxyjAExT_8xLSmwC0zCH9Gp8w7QkT0vXq_WisxvfL392SH0YPF6E5CVUSsMNGhPKFIBy5camTtXPV4Xpwu_Kj4ijsA1lV3wIeXURi-E5gFwp5TOUEhbq8rSABZqpHVHGEVAFrY0/s320/notebook-2178656_640.jpg" width="320" /></span></a></div><span style="font-family: trebuchet;">Doodle notes are a great way to make note taking more interesting but most students, even if they have already learned to use them, don't know how to use them in math. So today, we'll look at helping students learn to use them effectively in the math classroom. </span><p></p><p><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Mathematics can be an intimidating subject for many students, but by incorporating doodle notes into the learning process, it can make class more engaging and accessible. Teaching students to create doodle notes in math class not only enhances their understanding but also taps into their creativity. In this article, we'll explore effective strategies for educators to teach students the art of doodle notes in the math classroom.</span></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="font-family: trebuchet;">Start by introducing the idea of doodle notes and explaining the benefits students will experience. Focus on explaining how doodling can enhance memory retention, promote active engagement, and make complex mathematical concepts more visually accessible. Emphasize that doodle notes are not just about drawing but it's also a great approach to learning. Take time to demonstrate doodling techniques to provide students with a visual model since students need examples. Show how to represent mathematical symbols, concepts, and equations through simple drawings and symbols. Model the integration of colors, lines, and shapes to create a visually appealing and organized set of doodle notes.</span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="font-family: trebuchet;"><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;">Offer visual templates that guide students in organizing their doodle notes and make sure templates possible include sections for definitions, examples, and problem-solving steps.When you provide a structured framework, it helps students get started and ensures that their doodle notes are comprehensive and well-organized. Then e</span><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;">mphasize the importance of creative expression in doodle notes. Encourage students to add their personal touch to the doodles, whether it's through unique symbols, colors, or visual metaphors. Also reinforce the idea that doodle notes are a form of self-expression that can make learning more enjoyable.</span></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="font-family: trebuchet;"><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;">Choose doodle-friendly examples that lend themselves well to visual representation. Select mathematical concepts that can be easily illustrated through doodles, such as geometric shapes, algebraic equations, or trigonometric functions. In addition, walk students through the process of creating doodle notes for these examples. </span><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;">Ensure that students have access to doodle-friendly materials. Provide colored pens, markers, drawing paper, and other materials that facilitate doodle note-taking. A variety of tools allows students to experiment with different visual elements and find the style that works best for them.</span></span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="font-family: trebuchet;">Furthermore, integrate doodle notes into various class activities to reinforce the learning process. Look at the upcoming concepts and assign specific topics for doodling during group discussions, create doodle-based assignments, or use doodle notes as a form of assessment. By integrating doodle notes into class activities, it reinforces their value as a learning tool. </span></p><p style="border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span style="caret-color: rgb(55, 65, 81); color: #374151; font-size: 16px; white-space: pre-wrap;"><span style="font-family: trebuchet;">Teaching students the art of doodle notes in math class opens up new avenues for engagement and understanding. By explaining the benefits, modeling doodling techniques, providing visual templates, using doodle-friendly examples, encouraging creative expression, incorporating real-world applications, facilitating peer collaboration, and providing doodle-friendly materials, educators can empower students to use doodle notes as a powerful learning tool. As students embrace the creative side of math through doodle notes, they not only enhance their understanding of mathematical concepts but also discover a more enjoyable and personalized approach to learning.</span></span></p>Lee MacArthurhttp://www.blogger.com/profile/06119112850683374383noreply@blogger.com0