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!

Geometry: Exploration of Triangle Similarity

Conditions for triangle similarity: Try the SAS tab first, then select SSA at the top and experiment with that.

If two triangles have a proportional pair of side lengths and the angles between those sides are congruent, must they be similar?

If two triangles have a proportional pair of side lengths and the angles between one of those sides and the other side are congruent, must they be similar?

Unit Test Correction Policy

Tests are a big part of my students' grades. I want all students to have a chance to go back and relearn material they have trouble with on a test, and I want their grade to reflect those improvements. I also want all students to take classwork, homework, test review, and tests seriously as they happen, not wait till disaster strikes and then try to catch up. My policy on test corrections tries to balance these goals.

If you score below 70 on a unit test, you really need to do corrections. I don’t want anyone to have below a C and I know you can catch up if you work at it!

How To Get Higher Test Scores 

The best way to get a high score on a test is to do it on test day. If you didn't take test review seriously in class or for homework, rethink that for next time and you'll probably find it helps you get a better score on the next test.

You can raise your score significantly by doing corrections, though. Corrected problems will earn you half the missing points back, OR full corrections will give you an 80% (B-), whichever is better. Another way to think of this is that your post-correction test score is the average of your old test and your corrected test (or 80%, whichever is better). 

Examples: with full corrections, a 90 would become a 95, an 86 would become a 93, an 80 would become a 90, a 72 would become an 86, and anything 60 or below would become an 80.

(Some exceptions for higher correction or retake credit may be made in the case of excused absences.)

How To Do Corrections

Your corrections must be easily identifiable (I don't want to have to reread your whole test). You can do them on the original test and mark them clearly (highlighters are great for this), do them on a new copy of the test, or do them on a separate piece of notebook paper.

WRITE YOUR NAME on any new pieces of paper with corrections. Staple them to the original test for easy reference.

Each corrected problem must include a written explanation of what was wrong and how you fixed it (for instance, explain you mixed up factors and multiples, then do the problem correctly). The explanation can be brief. I just don't want to see anyone simply copying down the right final answers. Show work!!

Don't work on extra credit problems, if there are any. Those are a one-time opportunity.

There will sometimes be two due dates for corrections: one for drafts, and one for final corrections. Basically, you only get points for corrections that are actually correct, so if you want a second chance, you need to get me your corrections before the final due date so I can help you catch any errors.

Where To Do Corrections, and Who Can Help

Work on corrections at home, or at school outside of class time (especially after school). Students often find that just working in a calmer environment helps them remember some things they felt confused about on the original test.

You can get help from ANYONE with corrections: me, your friend, your parent, another family member, Khan Academy or elsewhere on the web, ... You can use a calculator and, of course, your class notes.

Ask me for help after school if you need it!! I am generally available on Mondays, Wednesdays, and for a shorter time on Thursdays and (sometimes) Fridays. If possible, let me know you’re coming so I’m sure to be in my room. I can also arrange to work with you at lunch with a day's warning. If those times don't work, contact me and we can try to work something else out.

Where to Put Corrections

When you finish corrections, put them in your period's in-box in Room 203.

Did I leave anything out? Please let me know if something was unclear.

Middle School Geometry Under Common Core Standards

The Common Core State Standards adopted by Oregon and most other states in the US describe what math students should understand and be able to do at various grades and in various "domains," or topics, such as Expressions and Equations (basically Algebra) or Statistics & Probability. The standards at each grade level are meant to build on the standards of the year before.

Here is my summary of what all middle school students are expected to learn about Geometry by the end of eighth grade, based on the Common Core state standards in math for sixth, seventh, and eighth grades. (Note: "e.g." means "for example".)

I. Area, perimeter, circumference, surface area, and volume

1. Know and use formulas for area of triangles, rectangles, parallelograms, and two-dimensional shapes made from these (6.G.1), and circle area and circumference (7.G.4).
2. Know and use formulas for the volumes of right rectangular prisms (boxes) with fractional edge lengths (6.G.2), three-dimensional objects composed of cubes and right prisms (7.G.6), and cones, cylinders, and spheres (8.G.9).
3. Find the surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.6)
4. Represent 3-D figures with nets, and use nets to find surface area. (6.G.4)
5. Describe what 2-D shapes you would get by slicing 3-D objects like boxes, cylinders or pyramids. (7.G.3)

II. Scale and Similarity

1. Interpret scale drawings, and reproduce scale drawings with a different scale. (7.G.1)
2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions (e.g. triangles with certain angle measures and side lengths). (7.G.2)
3. Recognize what conditions (e.g. combinations of side lengths or angles) determine a unique triangle, more than one triangle, or no triangle. (7.G.2)
4. Use informal arguments to establish facts about the angle-angle criterion for similarity of triangles. (8.G.5)

III. Angles

1. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (7.G.5)
2. Use informal arguments to establish facts about the angle sum and exterior angle of triangles. (8.G.5)
3. Use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal. (8.G.5)

IV. Transformations

1. Rotations, reflections, and translations: know side lengths & angle measures are preserved, and parallel lines are still parallel. (8.G.1)
2. Describe how to get one congruent or similar figure from another with transformation(s). (8.G.2 and 8.G.4)
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (8.G.3) 

V. The Pythagorean Theorem

1. Explain a proof of the Pythagorean Theorem and its converse. (8.G.6)
2. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in two and three dimensions. (8.G.7)
3. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (8.G.8)

Geometry Exploration: Pythagorean Theorem and Pantographs

Explored in class today:

Pythagorean Theorem (click on the Desmos link):

Pantograph

Fun Puzzle Websites

This is nowhere close to a complete list! There's so much fun stuff out there!

Oregonian's Puzzle Kingdom: Lots of logic puzzles of all sorts, including Battleships, Sudoku, Pic-a-Pix, Kakuro, and Hashi.
Maths Resources (the British call it maths): Dozens of fun puzzles and games, including classic card and board games as well as newer online games like 2048.
KenKen: Includes various difficulty levels. Great practice for thinking about numbers (especially factoring) and logic!
Numbrix: another neat logic puzzle. Try the easier levels first to get the hang of it.
Brain Teasers from the National Council of Teachers of Mathematics' Illuminations website
Calculation Nation online math games (also from NCTM)
Hotmath math games are at various levels; pick one that is appropriate for you (the cockroach one is pretty funny)
NRICH Enriching Mathematics: Lower secondary is probably the most appropriate level here
Vi Hart has a lot of amazing videos on YouTube. I haven't watched all of these, and some rely on high school or college math.
Lure of the Labyrinth is a computer game designed for pre-algebra middle schoolers. It has a storyline in which you are rescuing a lost pet from monsters in a labyrinth by solving complicated math puzzles. You can set up a free account to try it. I have not investigated it much yet. If you try it, let me know what you think of it!
Lewis Carroll Puzzles: How can you go wrong? I also strongly recommend reading Alice in Wonderland and Through the Looking Glass if you have not already!

Math Websites with Creative or Complicated Games

Here are some math websites I recommend for middle school students. This page is a work in progress; I am reviewing, sorting, and updating math links I originally collected on my former website, so you may want to look there too.

Calculation Nation: All free, no ads. Click on "GUEST PASS" or create a login at home with your family. I recommend READING DIRECTIONS before playing any game. Any are fine, but some are more fun or better for math than others. My favorites are:

• Square Off: Capture spaceships with rectangles with certain perimeters. Math concept level: high. Strategy level: high. Time pressure: medium.
• Factor Dazzle: Get points by finding factors, and keep your opponent's score low by giving them numbers without many factors. Can you figure out ahead of time how many points you will get from a certain move? Very similar to NCTM Illuminations' Factor Game but has a little extra fun. Math concept level: high. Strategy level: medium. Time pressure: low.
• Drop Zone: "Drop" your fractions on other fractions to add up to 1. More fun than it sounds! Math concept level: high. Strategy level: medium. Time pressure: low.
• Fraction Feud: Make a fraction (less than 1) smaller or larger than the one you’re “jousting” against. Consider using the Fraction Bar Chart. Try to figure out which cards are generally best to use or keep for later. Math concept level: high. Strategy level: high. Time pressure: low.
• Times Square: Tic-tac-toe with times tables, basically. Math concept level: medium. Strategy level: medium. Time pressure: low.
• Flip-n-Slide: Capture ladybugs with a triangle by translating, rotating, and reflecting it. Complicated; could turn out fun after several sessions. Math concept level: medium to high. Strategy level: high. Time pressure: low.
• neXtu: I haven't really played this, but it basically looks like a fun board game. Math concept level: low. Strategy level: high. Time pressure: low.

Fraction Game (NCTM Illuminations): Click on '+' symbols for information. Use equivalent fractions and estimation of fraction sizes to "play" fraction cards on fraction number lines. Play several times. How few cards can you use? What are good strategies to reduce the number of cards you use?

Troy's Toys: Percent discount game. In Level 1, find a discounted price from the original price and the percent discount. In Level 2, find the mystery discount percent from the price and discount. Not super creative, but fairly realistic and thorough.