MariaD’s blog

Maggie F.:
I’ve got an idea that I’m hoping someone else can execute for the group. I’ve seen a project (I think Carol C. used this) where children used a template to draw the periodic chart onto pillowcases. .

I was thinking it would be great if we could do the same thing with the multiplication table chart, while I guess we could freehand it, it would be nicer if we had a template to trace from. I did an internet search and did not find an already existing template for such a project and I’m not computer design savvy enough to create on myself.

Jill:
One suggestion, Maggie, is to use the multiplication pie rather than table.

Or you would you could use your completed ‘Color the Monster Table‘ as your
template for the pillow case. Just blow it up?

Cindy S.:
I love this idea and I think my dd would love to make one to put on a tshirt!

MariaD:
You can sketch on fabric with regular pencils, and it comes off in laundry. I find rectangle grids easier to make with rulers, rather than tracing patterns. First, make a rectangle in the size of your table, then mark cell sizes on all sides and connect the dots using the ruler. It’s a good problem to figure with kids.

Here is a printable grid in that size. You will need to trace it a few times, next to each other, to make the whole table. I think figuring out how to make grids is a useful skill. It definitely involves a lot of multiplicative reasoning! Making your own patterns for arts and crafts is one of my favorite soapboxes, in general.

Spirolaterals are beautiful, and making them feels like a meditation. You can start on square grid graph paper, and then experiment with other paper types. It is a good idea to do most of your math work on graph paper, and to keep a good supply at home, but here is a good printables site for custom graph paper. Use paper with larger cells with younger kids! Some spirolateral artists, such as Robert Krawczuk, use complex grids and even three dimensions for their spirolateral creations.

Square grid spirolaterals, from Wolfram’s Math World

3d, 45-degree turn spirolateral on a cone, by Robert Krawczuk

It is hard to believe, but I could not find any spirolateral movies on Youtube. I think we will make it this week’s project to produce one.

Spirolaterals are based on patterns and turns. Multiplication is a pattern, so a spirolateral can depict multiplication tables. Times two tables are a good easy start. An order three spirolateral needs three numbers, so you can use 2, 4, 6. Trace a line two units long, turn right, trace four units, turn right, trace six units, turn right, trace two units… and so on. It is a bit harder than it sounds. I turn my maps driving, and I have to physically turn my paper drawing spirolaterals, or go very slowly. But drawing them feels so good! It probably does something between brain hemispheres, because counting is sequential and turning is visual-spatial.

Keep going until you see a pattern. Enjoy it for a while, maybe color it. And now for a little surprise! Take any other three consequtive numbers from any times table, say, 6, 9, 12. Make an order three spirolateral out of them. What do you see now? Why?! (Hint: this part has to do with proportions. You will notice it, visually, when you draw it.)

You can also make spirolaterals of order four, five, six, or even infinite if you use an infinite pattern for your steps instead of repeating a cycle. You can use patterns other than multiplication tables for your spirolateral art. For example, try powers, doubling and doubling again: 2, 4, 8… Another thing to change is the angle of your turn. On square paper, you always turn in right angles. Experiment with hex or triangle paper to turn by different angles. Share your pictures!

Function machines: a classic

You can create:

• Function machines!

How?
Imagine a machine that takes some numbers in, does something to them, then gives you the results. This is, quite possibly, the most popular metaphor in all mathematics: function as a machine! Working with function machines is simple. One person, the machine operator, makes up a machine. Other people offer numbers (or whatever input the machine takes). The operator gives machine’s outputs back. Repeat, until you can guess what the machine does.

Here is some of the complexity in the activity:

• What if the machine’s creator says the machine adds the number to itself, but your guess is the machine multiplies numbers by two? When are machines the same?
• Can you work your machines backwards? (Hint: you can do it with some of them). This is called "inverse functions."
• Can you connect several machines together? This is called "composition of functions."
• Can you attach the output exit of your machine to its input entrance and cycle it again and again? This is called "iterations."
• Come up with some non-number function machines. How about a machine that turns baby animals into adult animals (chicken-hen, foal-horse, kitten-cat…) or a machine that builds a traditional house for a person from each culture (tipi - North American Indians, igloo - Eskimos, round-windowed holes - hobbits…), or a machine that…
• Come up with funky ways your machine actually works. Does the machine that triples make clones, or is it based on reflections and mirrors?

You can do this activity in the car or on a walk. However, for more complex function machines, you will want to write down input-output tables and look for patterns to be able to guess. You can also plot inputs and outputs on coordinate plane, using graph paper. Excel spreadsheets with hidden formulas provide a way to program your function machines on computers.

There are many function machine applets available on the web. Example 1, Example 2, Example 3… Natural Math is working on an applet that will allow people to create their own function machines. Of course, most computer algebra systems do it, but they require at least as much programming as Excel. It would be nice to have something more kid-friendly.

Why?
Because this activity is empowering, since people can make up functions at their own level. Also, multiple representations and the metaphor help people with asynchronous development.

As you go

• Notice patterns in your machines
• Support and celebrate mathematical arguments among activity participants
• If people want to spend a lot of time drawing detailed machines, instead of coming up with fancy math rules, support them
• Record your machines and share with the family multiplication study group

Higher and deeper

• Functions are omnipresent in all branches of mathematics. Algebra, as a study of number patterns, has especially many functions.

Coloring the monster table

You can create:

• A poster with your own collection of multiplication table patterns
• Your own unique plan for memorizing times tables
• Conjectures about algebraic rules related to times tables

How?
If you are planning on memorizing times table, this activity is the first step. Print out or create a standard times table. Then stare at it for a while. Do you see any groups of facts that are very easy for you? Color each group’s cells with its own colored pencil, so you will still see the facts. Or you can draw little symbols in the cells - blue stars for times ten, red cats for doubles and so on. Symbols are better, because you can put several of them in each cell.

Each person will notice different things in times tables. I have done this activity with adult mathematicians and with four year olds, and everybody notices some patterns - different ones! Here are just a few examples:

• Times one and times ten are very easy.
• Times nine is a pattern - there is also a finger trick for it.
• Doubles (times two) are easy for most people and you can color them
• Times five have a pattern - some people think of them as half of times ten
• Off-diagonal numbers are one off complete squares (for example, 5*5=25, but 4*6=6*4=24)
• Actually, most multiplication facts come in pairs - like 3*7 and 7*3 - and you only need to memorize one fact out of each pair

The facts you color, those you think are easy for you, will be the facts you won’t have to memorize. It is a bit scary to look at the monster table with its 100 facts (some memorize facts 12, though it’s more rare now). As you notice patterns and color or mark facts, you will see the parts you still need to memorize shrinking. It is a great feeling of relief.

You can even color the facts you remember from silly jokes or rhymes, like "56=7*8" (counting five, six, seven, eight). A few of those are fine. I am scared of systems that use extrinsic mnemonics for many multiplication facts. There is a psychological danger there, called "runaway imagery" - the mnemonics can later block algebraic understanding.

Of course, the most interesting part, as far as math goes, is to figure out why each pattern works. For example, why are off-diagonal numbers one less than complete squares (numbers multiplied by themselves) on the diagonal? Figuring this out by making charts, experimenting with counters or doing algebra can lead to many investigations.

Why?
Because this activity helps you see many patterns in times tables, and patterns are algebra. Also, the activity makes you feel better about memorization, and organizes facts mathematically, by their properties and relationships. The activity is interesting for mathematicians who find more complicated patterns. And, last but not least, you can make a beautiful poster out of your decorated, colored and marked times table.

Higher and deeper

• Describing your patterns in general terms lead to algebra or number theory
• You can create times tables in different bases and see how patterns apply. Which facts are easy in base five?

You can create:

• Symmetric works of art
• Windows into infinity
• All times tables
• Conjectures and theories about symmetry, transformation, multiplication and more

How?
Two plain rectangular mirrors, laid facing each other and connected with duct tape - this is the simple recipe for a whole lot of beauty and fun. Here are a few mirror book activities, invented by club members:

• Doodle inside your mirror book! Everything is incredibly beautiful when reflected multiple times. Or make some sculptures and place them inside. Many contemporary artists and decorators use mirrors in their work.
• Put two mirror books next to each other to form a mirror square. Look inside from the top, to find infinity.
• Draw a line across the opening of the mirror book and start closing the book. Your line will form polygons with more and more sides, eventually turning into "circles" - too many sides to see angles. 
• Put your fingers inside the mirror book and wiggle for a very creepy effect.
• Light a candle inside the mirror book. Beautiful!
• Find all multiplication tables hidden in the mirror book. Open it 90 degrees for times four, 72 degrees for times five, 60 degrees for times six, or just count reflections - you will find those angles quite intuitively, even if you are a three year old.
• Hold the mirror horizontally right next to your face for some pretty alien visual effects. Four eyes?
• Hold the mirror over your head and stand in front of another mirror to see the back of your head.
• Get a pizza slice and magically turn it into the whole pizza.
• Move things around for easy animations.
• Write letters and words to explore symmetries hidden in the alphabet.
• Make a pentagon that has ten segments - how?!
• While you are at it, explore reflections and rotations - 2d transformations!
• Stuff a multitude of colorful objects inside the mirror book and see a gazillion reflections.

You can buy mirror tiles at any large home improvement store, such as Home Depot. I also found small rectangular mirrors at dollar stores and craft stores, for example, as mirror candle holders. School supply stores sell plastic mirrors, which may be a good idea if you are doing this with very young kids.

Why?
Because this activity is incredibly beautiful. It’s fun for babies, but also deep enough for adults. It helps with math anxieties, because there is nothing stressful in it, yet a lot of math possibilities.

As you go

• Seek multiplication, and ye shall find it in multiple places within your mirror book!

Higher and deeper

• You can explore a lot of symmetry algebra and geometric transformations in connection to mirrors.
• You can explore angles and other geometric topics, and prove theorems using your mirror book.

MathLexicon: More multiplicative words

You can create:

• New English words
• Silly pictures based on your words - visual representations of math
• Your own wildcard definitions for the English language extender MathLexicon

How?
First, head to MathLexicon software page, enter a word you like (your name may be a good idea) and see what math words the program makes out of it! Then you can really start playing. Search for collections of math word parts, such as decimal prefixes or Greek prefixes traditionally used for polygon names. Add your own words and art to MathLexicon gallery. Draw self-portraits with math parts attached, like Kelley did in the picture above.

Polygon prefixes have to do with counting. You can make and illustrate your own counting book using them. "Tricat, quadridog, pentafish"  if you pick pets as your theme, for example. Decimal prefixes or metric prefixes have to do with powers. Here is a nice video using them:

I found this online etymology dictionary to be quite helpful in searching for word part meanings and history.

Why?
Because this activity connects love for words, art and mathematics. Also, you can be silly, you can do it on many levels, in a group or by yourself, for one or two word parts or for whole collections.

As you go

• How can your draw math words to really show their meanings? For example, "milli-" means one-thousandth. How can you show that? This is where proportional and other multiplicative thinking kicks in.
• Find real words with math prefixes and suffixes, like "tricycle" or "millimeter" - though made-up words, like "tridog" or "millimaria" can be a lot of fun, too.

Higher and deeper

• Etymology is a fascinating branch of history and linguistics dealing with word origins.
• Combinatorics (in this case, making words out of parts) is a formal, scientific or mathematical way of organizing the world. Interestingly, both French and Russian revolutions heavily used combinatorics. In case of French, it was the metric system most of the world uses now. In case of Russians, it was a proliferation of made-up words, now mostly forgotten. Newspeak from Orwell’s "1984" is built this way.
• Many computer and internet entities are named by smushing words together, for example, Photoshop, Youtube, or blog (for "web log").

Youtube has many videos showing polyrhythms. I liked these two, for maracas and piano.

You can create:

• Polyrhythmic beat patterns
• Graphs of rhythms
• Conjectures and theories about musical ratios and multiples

How?
There are two different ways to play polyrhythms. Let’s look at them on graphs, using 3:2 polyrhythm often found in jazz.

Method 1 - the piano dude from the video. Imagine a metronome going. Your left hand plays on every second beat, and your right hand plays on every third beat. Practice it with clapping or any instrument. It’s hard, but doable. Here is a graph, called "time unit box system" in music:

Do you see when your left and right hand meet each other? Why does it work like that? The math term for "meeting beats" is "least common multiple."

Method 2 - the maracas dude from the video. The left hand goes "one-two" while the right hand, within the same interval, goes "one-two-three." Here is a graph of it, for percussion:

But if you play an instrument like a piano, your notes will have lengths, and your picture will be different. Here is how this method may look like:

Do you see the relationship between the lengths of your left hand and right hand notes here? This isn’t something you usually meet in European music! But you can experiment with fractional ratios of note lengths. In this case, the ratio is 3:2

You can try polyrythms based on different ratios, with both methods. Make graphs of your rhythms, and compose music based on them. Share your music with the multiplication study group! Here is a little applet that can help you experiment with the first method.

Why?
Because this activity is for music lovers and everybody who wants to be a Renaissance person, connecting math with everything else!

As you go

• Note which polyrhythms are easier for us to learn. 2:4 is easy, 3:5 is hard - why?
• Read on polyrhythms in music of different cultures
• Find polyrhythms in your favorite musical pieces

Higher and deeper

• Music composition theory is rich in mathematical topics. Multiples and ratios are just the tip of the iceberg.
• "Godel, Escher, Bach" is a more advanced book about music, math and art.

You can create:

• A game idea about multiplication - if it’s intrinsic and interesting, it may get programmed!
• A collection of multiplicative situations for your game

How?
Whatever you do, please do not have drill exercises like 2*2=? appear in your game. Even if you write your exercise on the tummy of a monster, or offer players to shoot balloons with right answers, it’s still a drill exercise. There are plenty of games with those, none of them very imaginative or interesting, in my opinion.

The real challenge is to create a game where multiplication is, somehow, a part of the story. Whether your story is about making chimeras out of spare parts (combinations=multiplication) or about sharing pirate booty in a certain proportion among the crew, the multiplication activities have to be intrinsic to the scenario.

A good method is to find some real-life or fiction situations that involve multiplication, and then build game mechanics on them. Game mechanics include guessing, collecting, capturing or eliminating, tiles and covering, and many more. Wikipedia has a decent article on the subject, if you need some inspiration. Also, try to find some multiplication in the computer games you already like. The picture above, for example, comes from Zoombinis - you can play their demo if you don’t know the game yet.

Why?
Because this activity is great for lovers of computer games. It is empowering and creative, as most activities that use design as the main component.

As you go

• Think of what makes good games.
• Reflect on intrinsic vs. extrinsic elements of games and life.
• Find multiplicative situations everywhere.

Higher and deeper

• "Serious games" use game mechanics to help people accomplish work tasks. Game mechanics research is heavy in psychology.
• Some researchers claim "learning game" is a contradiction in terms. There are certainly precious few good ones, among the sea of drill software somehow being called "games." As you design your multiplication game, try to have artistic or working in-game goals, rather than learning goals. Learning, like happiness, should probably not be pursued directly - at least in games.
• If you end up designing a multiplication game, though, your understanding of multiplication will become much deeper than before. This is a general feature of design tasks.

You can create:

• Multiplication wheels
• Colorful patterns on your wheels
• Conjectures and theories about times tables patterns

How?

This activity is inspired by Robinsunne’s "multiplication clock." The idea is quite simple: you create concentric circles, separate them into as many segments as you want for your tables, and fill them with multiplication results. Figuring out how to draw concentric circles and separate them evenly is a good engineering and geometry problem, though the site has detailed directions.

The most interesting thing, of course, is how to color your multiplication wheels. For example, you can make all numerals for even numbers one color and all numerals for odd numbers another color. What pattern will you see? Why? Try it with multiples of different numbers, say three or five, for more pattern fun. You can highlight full squares (number times itself) for a very special spiral. Any pattern you pick will produce interesting results.

If you want to play with many patterns, it may be easier to copy and then print your concentric circles. However, drawing circles is usually a fun tactile experience. You can use multiplication wheels to decorate your walls, at least for the duration of the study. I usually attach them with a nail or a pushpin in the middle, so they rotate. Remember to share your creations!

Why?
Because this activity gives a fresh representation on times tables; it has interesting potential for "text design" (or rather, numerals design). People who like to play with numerals will probably like it.

As you go

• Look at the wheel for any patterns in numbers. Color your patterns.
• Think of other wheels and round models people use, such as the color wheel, the seasons wheel, the clock - all of them are multiplicative in some way!

Higher and deeper

• Counting on wheels (or in cycles) is the basis of modulo arithmetic, a branch of number theory.
• Circle is a powerful metaphor of time and life cycles, widely used in art and psychology.
• Many coding systems use wheels; mathematics of wheel and cyphers is quite rich.

You can create:

• A collection of multiplicative situations relevant to your life and your interests
• A collaborative visual times table collage
• Conjectures and theories on uses, differences, good and bad features of different multiplication models

How?
Take a notebook, a camera or just your imagination and go on an expedition to find multiplication. You can search your backyard or the web, you can draw or photograph what you find, you can do it alone or with a group. Some multiplication instances are easy to see. Some are complex and hidden. Start by finding any and all. Then you can search for new kinds of multiplication.

You can hang a big piece of paper on the wall and keep adding your multiplication finds to it over some days or weeks. This way, there will be some reminder of multiplication in your daily life.

This gallery lists a dozen situations where you are likely to meet multiplication. I bet you there are several of them in the room where you are reading this description. You can store your finds in the gallery, too.

For a more difficult quest, try finding multiplication examples that always have some particular numbers and no other numbers. Say, a chess board is always 8*8, and our fingers are always 2*5. This is very hard to do! For a few examples, head to this "intrinsic multiplication" gallery. It’s somewhat empty, because intrinsic multiplication examples are hard to find.

Why?
Because this activity can be done at many levels, in many media from text to video, in long stretches of time or one-minute snippets. It connects each person’s interests and life to multiplication. It is a good fit both for organized learning (when you sort and categorize models) and for eclectic unschooling projects.

As you go

• See what kinds of numbers could fit into each multiplication situation. Check fractions, negative numbers, zero. Can area be a fraction, or a negative?
• Grab your favorite multiplication fact and find it in as many situations as possible, in as many multiplication models as you can.
• This activity answers the question, "What’s multiplication for?" - hopefully, before you even start doing multiplication formally.

Higher and deeper

• If you try to search for particular multiplication pictures on the web, you will see severe limitations of current search engines. Picture content search is a cutting edge topic of artificial intelligence.
• Crystals are never based on pentagons (can you figure out why?) - so to find the number five for your multiplication, look at either living nature, or culture. Before that first organic matter, Earth had no fives anywhere.