Math You Can See: Art, Nature, Patterns, Society, and More

SBAC OPT-OUT ASSIGNMENT: 

Do some reading from links below. Follow instructions on this form and fill it out for a site that interests you. If you were "invited" to join my SBAC Opt-Out class (by email to your PPS account), then when you join, you can click on the Reflections assignment, click on its form, fill in the form from there and submit it to me electronically! Cool!! It is also fine to save the form to your drive, fill it in, then share viewing privileges with me (jwright2@apps4pps.net or jwright2@pps.net), or to do a printed copy by hand and hand it in. Thanks -- I like reading about which sites interest you and what you learned!

GENERAL

Math Munch: A Weekly Digest of the Mathematical Internet: Something for everyone!

ART AND ARCHITECTURE

• Math Art Tools | Math Munch
  Fantastic online math art tools!
• Muslim rule and compass: the magic of Islamic geometric design | Alex Bellos, The Guardian
  Includes a step-by-step guide for an Islamic-style tiling pattern.
• Readers’ Gallery | Math Munch
  Fabric art and drawings from Math Munch readers that show math (especially geometry).
• {3,3,7} Honeycomb Meets Plane at Infinity | Visual Insight
  Gorgeous computer-generated image of "honeycomb in 3D hyperbolic space," etc. Just look at it.
• Henry Segerman's Math Art
  Mathematical art; 3-D sculptures & 2-D prints. Very cool colorful & printable dodecahedron on the "Autoglyphs" page!
• Bathsheba Sculpture on Shapeways | Bathsheba Grossman
• Impossible World: Art
  "Impossible art" by artists like MC Escher.
• Geometry of Iran's Mosques | Inhabitat

VISUAL PATTERNS AND VISUALIZATION OF MATH

• Visual Patterns Not just tiles! Dozens of visual patterns with drawings and some real objects like Legos.
• Factorization - Visual illustration of divisor pairs | GeoGebraTube
• The Harriss Spiral | Alex Bellos, The Guardian
  Pretty fractal curve made golden-ratio-style. This article has a great description of what is going on mathematically.
• Polyhedra from Cardstock (Slide-Togethers)
  George Hart's templates and demonstrations of polyhedral constructions that can be made from cardstock.
• Paper Clifford Torus | cutoutfoldup.com
• How To Teach Math With LEGOs | The Hub
  Fractions, area & perimeter, multiplication, statistics...

MUSIC

• The Saddest Thing I Know about the Integers | Evelyn Lamb
  Math shows that you can't get perfect octaves from a bunch of perfect fifths.
• What Pi sounds like | Michael Blake, YouTube
  Surprisingly beautiful.

SCIENCE, NATURE, AND FOOD

• Fibonacci Numbers in Nature
  Slides and photos.
• Huge 3200 Year Old Tree | Distractify
  BIIIIIIG.
• What 2000 Calories Looks Like - YouTube
• Incommensurate Cheez-Its | Christopher Danielson (Trianglemancsd)
  Are Big Cheez-Its REALLY "Twice the size!" of regular? Christopher and his kids are on the case.
• Gauss / Curvature & Strength (& Pizza) | WIRED

SOCIETY AND SOCIAL JUSTICE

• Are you reflected in the new Congress? | The Guardian
  Enter demographic info (sex, age, etc.) and find out how many members of Congress share it.
• Mapping Migration in the United States | New York Times
  Fascinating map showing where people living in each state in 1900, 1950, or 2012 were born.
• Parable of the Polygons | Vi Hart
  How small, harmless-seeming choices can make a harmful world: discrimination simulation.
• Math in the Real World | Get The Math
  Some "real world" math for various art/sports/jobs; some dubious connections, and some are better videos than others. The Math in Music one has some interesting proportions. 

What's the Angle? (But more interestingly: how did you solve it?)

So far I've filled out angles, drawn an isosceles triangle and redrawn the diagram. Thus ends my productive struggle. pic.twitter.com/Yo5gOWKCT4

— Michael Pershan (@mpershan) May 11, 2015

I saw this problem on Twitter. This high school geometry teacher wanted to know what solution paths people could come up with. (He did solve it in the end himself too!)

da Vinci Geometry students: what solution methods do you see? You've given me a new appreciation for Law of Sines, so I used that; I'm curious whether you see other paths you like. Let me know and I will add them to the online discussion! (I'll put a link to that, too, eventually, but I want to see what original stuff you come up with first.)

Geometry: Construction Project

The posters you see here (and in the da Vinci hallways) were created by my Geometry students as a way to learn, practice, and demonstrate their knowledge of constructing exact geometrical shapes, angles, and line segments using only a compass and unmarked straightedge.

Somewhere on one of the posters, each da Vinci Geometry student demonstrated how to do nine different constructions using a compass and an unmarked straightedge:

• Copy a line segment 
• Construct a circle 
• Construct a triangle that is congruent to another scalene triangle 
• Construct an angle bisector (essentially, split an angle into two evenly) 
• Construct a perpendicular bisector of a line segment 
• Construct a regular hexagon, regular octagon, or 30°-60°-90° triangle 
• Construct the perpendicular to a given line through an external point 
• Construct a line parallel to a given line and through an external point 
• Construct a non-equilateral isosceles triangle 

If you look closely, you can see some pencil marks from their construction process!

I got the idea and some materials for this project from a teacher in San Francisco who goes by the name cheesemonkeysf online. At her blog, she wrote: “Einstein was right — imagination IS more important than knowledge. […]I created this Constructions Castle project to give students plenty of practice doing constructions while also giving them a chance to develop their understanding of how shapes and angles fit together.”

Thanks to cheesemonkeysf for the great idea and to the Geometry class for doing such a fabulous job on this project. Your posters are much more beautiful than a pile of test papers!

Where Sixth Graders See PERCENTS

At the beginning of our percent mini-unit earlier this month, I asked my Math 6 classes to tell me about where they had encountered percents (or percentages) outside of math class. Here's what they told me about! (Period 4's answers are shown first, then Period 5's, then Period 6's.)

Decimal Multiplication and Division Practice

Here's an idea for creating your own practice problems for multiplication and division of decimal numbers.

1) Invent a decimal number with at least 4 non-zero digits. Use your favorite digits, or all of the same digits, or consecutive digits, or...  Examples: 12.34, .007777, 6544.56, etc.

2) Take a 1-digit or 2-digit number and make it into a decimal number. Examples: 0.000021, 0.6, 1.4, etc.

3) For division practice, make your number from step 1 the DIVIDEND and your number from step 2 the DIVISOR, then divide (remember the "ugly fraction" step). For multiplication practice, multiply your numbers from step 1 and step 2 together.

4) Check your answer(s) with a calculator. Did it work?? If you get REALLY stuck, take a picture or scan of your work and email it to me.

Pi Day!

Why are so many people excited about today, 3/14/15? It's "Pi Day," so called because the number pi rounded to the nearest ten-thousandth is 3.1415. If you want to round off to the nearest billionth, pi is 3.141592654, so if you trust your clock enough, you can get especially rapturous over Pi Day at 9:26 and 54 seconds! (I'm posting this after the morning 9:26, but you can still celebrate the evening one... the last 3/14/15 9:26 of the century.)

Long Division Practice Sites

Good long division practice sites! Click on the option to get help if you need it.

Easier (divisor has 1 digit)
Harder (divisor has 2 digits)

Math 6: Decimal Games & Practice

(If you're looking for Math 8 Pythagorean Theorem homework, see the post below this!)

Here's a list of places you can practice your skills with decimals. I will keep updating it as I find more sites.

For Math 6 homework due Wednesday 3/4/15, your job is to read this list till you find the instructions for the "crazy word" that will tell me you looked at it. (Don't tell your friends -- make them look themselves so they definitely know how to find my website!)

Fruit Splat/Place Value Decimals: This game is great practice for thinking about place value in decimals and for adding using mental math. It's designed to have several different levels, and you can play in timed mode or "relaxed" mode.
Flower Power: put decimals in order of size -- a nice complicated game (read the directions)
Balloon Pop Decimals Level 1 and Balloon Pop Decimals Level 2: Pop the balloons from smallest decimal to largest. If a balloon won't pop, it's because you haven't found the smallest. Score is based on time, but you can ignore it if you want and still get the practice.
Balloon Pop Decimal Patterns: Pop the balloons that continue a pattern.
Hungry Puppies: add decimals (quick mental math; fun speed challenge, but problems are not terribly complicated)
phschool.com: the textbook company's site (you need the codes from me; ask me or email me); includes online quizzes. Your crazy word is your name backwards! For fractions, decimals, and percents, try any of the Bits and Pieces books' links. Try doing the easier 4 or 5 problems in each quiz without a calculator, and use an online calculator for the others.
Sheppard Software decimal activities: Most of the activities on this list are mainly useful if you're having trouble understanding what decimals mean.
IXL (for-profit site which lets non-members practice a few minutes with standard problems) is SURE to have other decimal practice, but I haven't found the exact links yet. You can probably find something useful by exploring.

COMING SOON: PERCENTS:
Decention: finding equivalent fractions, decimals, and percentages

Fraction/Decimal/Percent Jeopardy: quiz yourself on converting between these. Use "0.3..." for 0.3 with a bar (repeating decimal)
Troy's Toys: prices and percents: find out amounts of discount from percent, or vice versa; you pick the level of difficulty by picking the toys
Balloon Invaders: a good challenge for finding percents FAST! only works if you are quicker with the keyboard than I am!

Informal Proofs of the Pythagorean Theorem (Math 8)

We've explored some right triangles in class and found that if you make squares off each side of a right triangle and find their areas, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the legs. (For the picture below, the blue area + the red area = the green area.) This relationship is one way of expressing the Pythagorean Theorem.



Author: Pythagorean.svg: en:User:Wapcaplet
Derivative work: CP QQY (talk)
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Therefore, the Pythagorean Theorem says that if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then 

a2 + b2 = c2

There are LOTS of different ways to prove the Pythagorean Theorem holds true for right triangles. You will explore two of them online and sketch your results in your math notebook.

1. Open Hotmath's Pythagorean Theorem activity. (If that link doesn't work, try clicking on the Pythagorean Theorem activity at hotmath.com's Geometry activities. Don't click on "Geometry" at the side.)
2. By default, the tab at the top for Dissection is chosen. In geometry, dissection is about cutting up a shape and rearranging it to make another shape. The square sticking off the longer leg is cut into four quadrilaterals. See if you can drag these pieces and the square sticking off the shorter leg to make the square sticking off the hypotenuse. If you get stuck, or just for a review, hit Animation.
3. Next, click on the Chinese tab at the top to explore another proof. Follow the directions; if you get stuck, or to review, hit Animation.
4. In your notebook, for both of these methods, sketch the "before" and "after" pictures that show a2 + b2 = c2.

(If you have technical difficulties, have a parent write a brief, signed note in your notebook explaining, and I will excuse you from the assignment and add a note to that effect.)