Rosette Exercises - EscherMath. Instructor:Rosette Exercises Solutions. In Escher's Castrovalva there are no obvious mathematical symmetries, but Escher's eye for patterns is already evident. Find at least four areas of this print, which have repetition of pattern. Describe the area and the pattern that repeats. One of Escher's earliest works to feature symmetry was Paradise. Carefully examine and analyze the details of this print. Discuss the mathematical symmetry of his composition. If falling short of being mathematically symmetric, are there parts of the design that are at least balanced? High School Materials - EscherMath. K-12: Materials at high school level.

Course Materials The online materials are supplemented with worksheets from Key to Geometry by Hawley, Suppes, Gearhart, and Rasmussen (Key Curriculum Press) Product Information Other useful resources are: Read the introduction Geometry in Art and in the Real World. This will show you some examples of how geometry is used in the real world. Part I Points, Lines, Segments and Polygons In Key to geometry: Student Workbook 1, Lines and Segments the following pages cover the material: Part II Circles,Arcs, Congruence and Similarity In Key to geometry: Student Workbook 2, Circles the following pages cover the material: Part III Symmetry Resources: Part IV Tessellations Resources Part V Miscellaneous.

Tessellation - PowerPoint. Reflectional Symmetry Worksheet - EscherMath. Objective: Learn more about reflections. Figures with reflectional symmetry have a mirror line. Remember from your reading that a mirror line divides your figure exactly in half and one side of the figure will be the exact mirror image of the other half. Here are some examples: In these three examples the mirrolines are all vertical. That's a coincidence. Figures can have mirror lines in all direction. 1. It is also possible that figures have more than one mirror line. 2. 3. 4. It almost has a mirror line! A. What other parts of the drawing have a real mirror line? B. It doesn't quite work because they are different animals, but they are opposite one another and mirror each others behavior.

How many other pairs can you find that are almost mirror images? Semi-regular Tessellations. Regular tessellations use identical regular polygons to fill the plane. The vertices of each polygon must coincide with the vertices of other polygons. You can produce exactly three regular tessellations: Can you convince yourself that there are no more? Semi-regular tessellations (or Archimedean tessellations) have two properties:They are formed by two or more types of regular polygon, each with the same side lengthEach vertex has the same pattern of polygons around it. Here are two examples: In the first, triangle, triangle, triangle, square, square {3, 3, 3, 4, 4} meet at each point.

Can you find all the semi-regular tessellations? Can you show that you have found them all? To help you when you are working away from the computer, click below for multiple copies of the different polygons. Finding Unity in the Math Wars. I usually avoid current events, but recent skirmishes in the math world prompted me to chime in. To recap, there’ve been heated discussions about math education and the role of online resources like Khan Academy. As fun as a good math showdown may appear, there’s a bigger threat: Apathy. And Justin Bieber. Educators, online or not, don’t compete with each other. They struggle to be noticed in our math-phobic society, where we casually wonder “Should algebra be taught at all?”

Not “Can algebra be taught better?”. Entertainment is great; I love Starcraft. What do we need? I could be walking into a knife fight with an ice cream cone, but I’d like to approach each side with empathy and offer specific suggestions to bridge the gap. The Big Misunderstanding Superheroes need a misunderstanding before working together. Bad Teacher < Online Learning < Good teacher The problem is in considering each part separately. But, really, the ultimate solution is Online learning + Good Teachers. Why Do I Care? Math Avengers Wireframe - Google Drawings. Surprising Uses of the Pythagorean Theorem. The Pythagorean theorem is a celebrity: if an equation can make it into the Simpsons, I'd say it's well-known. But most of us think the formula only applies to triangles and geometry. Think again. The Pythagorean Theorem can be used with any shape and for any formula that squares a number.

Read on to see how this 2500-year-old idea can help us understand computer science, physics, even the value of Web 2.0 social networks. Understanding How Area Works I love seeing old topics in a new light and discovering the depth there. The area of any shape can be computed from any line segment squared. We can pick any line segment and figure out area from it: every line segment has an "area factor" in this universal equation: For example, look at the diagonal of a square ("d").

Now, use the entire perimeter ("p") as the line segment. Can we pick any line segment? You bet. Can we pick any shape? Sort of. Yes, every triangle follows the rule "area = 1/2 base * height". Cool, huh? Makes sense, right? Cochem germany frog. The Web's Most Popular Swim Shop! men's and women's swimwear, swim gear, swim store! 3Doodler Education | The 3Doodler. Silhouette America - Shop. A Taxonomy of Reflection: Critical Thinking For Students, Teachers, and Principals (Part 1) – Copy / Paste by Peter Pappas.

My approach to staff development (and teaching) borrows from the thinking of Donald Finkel who believed that teaching should be thought of as “providing experience, provoking reflection.” He goes on to write, … to reflectively experience is to make connections within the details of the work of the problem, to see it through the lens of abstraction or theory, to generate one’s own questions about it, to take more active and conscious control over understanding. ~ From Teaching With Your Mouth Shut Over the last few years I’ve led many teachers and administrators on classroom walkthroughs designed to foster a collegial conversation about teaching and learning. The walkthroughs served as roving Socratic seminars and a catalyst for reflection. But reflection can be a challenging endeavor. It’s not something that’s fostered in school – typically someone else tells you how you’re doing! 1. Take my Prezi tour of the Taxonomy A Taxonomy of Lower to Higher Order Reflection Like this: Like Loading...

Case Math - Improve your Mental Math Skills. Full Table of Contents. Geometry: Concepts and Applications. Illuminations. Cool Math - free online cool math lessons, cool math games & apps, fun math activities, pre-algebra, algebra, precalculus.

Habits of Mind | Mr G Online. Came across this infographic in my Scoop-It feed this morning and I couldn’t resist reflecting on its message. Schools are awash with opportunities for leadership building and modelling at 3 distinct levels – Peer Leadership (School Leadership teams/Curriculum leadership teams or individuals), Teacher-leading-Class and Student Leadership. It is telling to ponder the impact these 8 key leadership qualities can have in improving school environments and, as a result, performances. Peer Leadership: The keyword for me here is PROACTIVE. Tough decision making means knowing what may go wrong but being a believer in what can go right. Courageous leadership means not reacting to every bad standardised test result and looking for another solution because Plan A didn’t work straight away.

This just leads to one unfinished project after another that never really leads to sustained improvement and consistent achievement. Real change and improvement takes time. Student Leadership: Ditto. Integrating the 16 Habits of Mind. In outcomes-based learning environments, we generally see three elements in play: 1) learning objectives or targets are created from given standards; 2) instruction of some kind is given; and then 3) learning results are assessed. These assessments offer data to inform the revision of further planned instruction. Rinse and repeat. But lost in this clinical sequence are the Habits of Mind that (often predictably) lead to success or failure in the mastery of given standards. In fact, it is not in the standards or assessments, but rather these personal habits where success or failure -- in academic terms -- actually begin. Below are all 16 Habits of Mind, each with a tip, strategy or resource to understand and begin implementation in your classroom.

The habits themselves aren't new at all, and significant work has already been done in the areas of these "thinking habits. " And a renewed urgency for their integration. 1. 2. 3. 4. 5. Ask students to map out their own thinking process. 6. 7. 8. 10 ways to encourage student reflection… – What Ed Said. Split Screen Teaching Optimal learning occurs when students are active participants in their own learning, rather than passive recipients of teacher-delivered content. For this to be effective, students really need to think about their learning. I worked with a group of teachers recently who felt their young students were not capable of writing meaningful reflections for their end of semester reports. That might be true. But only if reflection and meta-cognition are not integral parts of the learning in their classes.

How do we encourage students to think about their learning? 1. Guy Claxton calls this ‘split screen teaching.’ 2. Stop thinking about how to teach the content. 3. Make sure you and your students know the purpose of every task and of how it will advance the learning. 4. Encourage students to plan how they will learn and to reflect on the learning process. 5. Make sure students have time to stop and think about why and how they learned, not just what. 6. 7. 8. 9. 10. Like this: Habits of Mind Home - Habits + Bloom's = Common Core. Differentiation is an Expectation: A School Leader’s Guide to Building a Culture of Differentiation. Admin - City Neighbors High School. Mathematical practices - Math Common Core resources.

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. —Common Core State Standards for Mathematics, page eight The eight Standards for Mathematical Practice are: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning What does all this mean?

And what do the Standards for Mathematical Practice look like in practice? Resources Think Math — CCSS Mathematical Practices This site is set up much like Wikipedia. Differences between, and connections between, Content and Practice Standards. Escher art gallery in Math Cats' Tessellation Town! Recognizable Tessellation Exercises - EscherMath. Look at all the sketches in Escher's regular division notebook. These are on pages 116-229 of Visions of Symmetry or at Regular Division of the Plane Drawings. Find ten sketches featuring four-legged mammals. (A four-legged mammal has four legs, and is a mammal - horse, dog, pegasus, lion, etc. etc. No people, fish, lizards.) Find the wallpaper symmetry group for each of the ten sketches. Instructor:Recognizable Tessellation Exercises Solutions.

The Strange Worlds of M C Escher. M C Escher at Work I try in my prints to testify that we live in a beautiful and orderly world, not in a chaos without norms, even though that is how it sometimes appears. M C Escher Maurits Cornelius Escher was born in June 1898, the youngest son of a prosperous Dutch government engineer. After a year in technical college, he attended the School of Architecture and Decorative Arts in Haarlem. He was diverted from a career in architecture by his teacher and mentor Jessurun Mesquita, who encouraged him to develop his drawing and printmaking skills. After visits to Spain and Italy, he was impressed by the dramatic landscapes, particularly mountainous and desert regions with olive trees and cacti. Eight Heads, woodcut stamped print, 1922 The print, Eight Heads, was made when Escher was still at art school, and is an example of a side-grained woodcut, where the white areas are cut away, leaving the raised areas to impress the ink.

Castrovalva, lithograph 1930 Reptiles, lithograph 1943. Pegasus (No. 105)