![]() ![]() We compare this to the Glencoe text - as this is the start of the second semester, the students out to be starting Chapter 7 (out of 12) - the first chapter of Volume 2. ![]() The class was working out of another MathLinks packet, 6-9, which is called Expressions and Equations 1. ![]() Today, as the second semester begins, I subbed in a sixth grade math classroom, for a teacher who is out for three days. Then when he tries to grab some of the oobleck, it pours through his fingers like a liquid once again. In a bowl it looks like a liquid, but when Kung punches the bowl, it feels solid on his fist. This causes oobleck to exhibit some of the properties of both solids in liquids. The Quick Conundrum for today involves something called "oobleck." Oobleck is just a mixture of corn starch and water, but the starch isn't completely dissolved in the water. My written description of this Rolling Spool Paradox won't do it justice, so let me provide a link where you can see what's going on: Basically, depending on which way you pull on the thread, it's possible to make the spool move forwards or backwards. His final paradox involves a spool of string. He adds that the idea that information takes time to transmit will show up in his later lectures. He says that it's as if information - namely, the fact that the Slinky has been cut - takes time for it to arrive at the bottom of the Slinky. The two opposing forces, gravity and the recoil, balance out exactly, so the bottom continues to float in the air until the top finishes recoiling as it falls - only then will the bottom of the Slinky fall to the ground. Well, when Kung cuts the Slinky, the bottom actually does neither. Then he asks, what would happen to the bottom of the Slinky if he were to detach the toy from the ceiling? It could be that gravity will cause the bottom to descend, but it could also be that the Slinky will recoil, thereby causing the bottom to ascend. He hangs a toy from the ceiling, which causes gravity to stretch it out a little. Next, Kung moves on to some paradoxes involving springs, but he demonstrates these by using a very large spring - also known as a Slinky. You simply take a large tank the same shape as the ship, pour in the gallon of water, and then add the ship. Kung begins the lecture by describing how to float a ship in one gallon of water. ![]() They are able to use technological tools to explore and deepen their understanding of concepts.Lecture 15 of David Kung's Mind-Bending Math is called "Enigmas of Everyday Objects." I've heard of many of these mathematical paradoxes before, but actually, most of the paradoxes Dave Kung describes in this lecture pertain more to physics and are new to me. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They detect possible errors by strategically using estimation and other mathematical knowledge. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Mathematically proficient students consider the available tools when solving a mathematical problem. Visit the video excerpts below to view multiple examples of these teachers. A teacher of adolescents and young adults might have established norms for accessing tools during the students' group "tinkering processes," allowing students to use paper strips, brass fasteners, and protractors to create and test quadrilateral "kite" models. A middle childhood teacher might have his students select different color tiles to show repetition in a patterning task. Teachers who are developing students' capacity to "use appropriate tools strategically" make clear to students why the use of manipulatives, rulers, compasses, protractors, and other tools will aid their problem solving processes. ![]()
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