Today your mission is…

Make combination tables and trees, and play guessing games with them.

Ready, Set, Go

Once you build tables and trees, hide parts of them and play guessing games. Or make a puzzle: cut the whole table apart into cells, and put it back together again. In some cases, the same cells can form many different, correct tables!

Option 1. Animal table

Your two variables are heads and bodies. Draw a cat head (or your kid’s favorite animal) in each cell of the first column, and a cat body in each cell of the first row. The corner cell will have the whole cat. Ask the child to pick the next animal. Draw the head of that animal in each cell of the second column, and the body in each cell of the second row.

Option 2. Elements of art and elements of math tables

Make a row or a column with the same color, shape, or visual symbols such as antennae for robots. Print or draw clothes for a table dress-up.

Try systematic changes from cell to cell. For example, keep adding a side to your polygons, or another eye to your monsters.

Option 3. More variables: 3D tables and combination trees

Want to use more than two variables? For color-shape-size or any other three variables, build 3D tables. Sticks and styrofoam, or yarn suspended from a cardboard grid takes you to the third dimension. A faster, easier way is to draw combination trees. You might have seen them in sandwich or ice cream shops.

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Help kids keep the structure. Bring up famous chimeras and modular constructs from mythology or engineering. Kids may get bored drawing similar pictures in many cells, so help them finish grids quickly. Then play with hiding cells, taking grids apart and other puzzles. Find examples of using grids for combinations in everyday and scientific media.

Ask “How many?” questions about combinations, which lead to multiplication. For example, a grid for three heads and four bodies has twelve chimeras total, because 3x4=12.

Toddlers

Your toddler can place cut-out parts into cells. Some kids get upset if you cut pictures of whole animals into parts, so do that step by yourself, or draw parts separately. Babies who can’t talk yet can point at toys or pictures to help you select the content of the next row or column. Toddlers and young kids often like combinatorics character creation tools in computer games, such as Hero Machine, or combinatorics toys such as Potato Head.

Young children

Kids may want to draw by themselves, or have you draw with their hand in yours. Use

different colors for different rows and columns. Children can “drive” a toy truck along rows and columns, “delivering” parts to each cell. Be prepared: between rows and columns, young children not only change the value of their variables (e.g. red, green, blue for color), but also the type of their variables (e.g. red in the first column, a hat in the second column, triangular shape in the third column). This is fine; just help kids to stay consistent within the same row or column.

Older children

As you read fairy tales or science news, keep an eye on combinations. For example, luminosity and temperature defines different types of stars, such as a red dwarf or a blue giant. Try making number tables, also called structured variation tables. Label columns and rows 1, 2, 3… and come up with your own ways of combining the labels. If you just multiply row and column labels, you will get the familiar times table.

But what if you double the column label and add the row label? Or take the column label to the power of the row label? Or do something else, like the mystery table below? The possibilities for creativity and guessing games are endless! You can make these tables by hand, or program formulas in spreadsheet software.

How is this multiplication?

The number of combinations in a table is the number or rows multiplied by the number of columns. In 3D tables, or trees, you multiply the number of values for three or more variables to get the total number of combinations. These visual structures model multiplication even for the youngest kids. Yet we frequently meet adults who have never perceived links between multiplication and combining different pants and shirts, or selecting bread and fillings for sandwiches. It is delightful for kids and adults alike to make this new connection.

Inspired by algebra

Combination tables and trees introduce variables: what varies from row to row, or column to column. Tables prepare kids for coordinate planes. Two tasks, “Find the point with coordinates (2, 3)” and “Find the cell with red square” mean the same algebraic action. The more subtle algebraic idea is covariation, which requires kids to notice two or more variables at once. Whenever you pay attention to both rows and columns, you reason with covariation. Kids use covariation in guessing games with missing cells in tables, or to put a table back together again when it’s cut up into cells.

Frequently Asked Question

My child is simply not interested in multiplication. Whenever I try to bring it up, show him some examples, or explain it to him, he tries to change the subject, finds something else to do, or just doesn’t pay attention. I’ve tried various approaches - living math books, board games, just plain old flash cards, but nothing piques his interest. What can I do? What if he never becomes interested in learning multiplication?

Looks like it’s time for you to move on. Some kids simply dislike some topics, and won’t learn them directly. In such a case, the child will absorb the topic from its connections and applications - if only you keep doing other, interesting mathematics. Math is not linear (even if some curricula are), so it’s very easy to skip a topic for a long while and do something more productive.

Focus on algebraic patterns, explore geometry, get inspired by calculus - there are literally thousands of math topics out there. Print out small times tables with just the results, because the patterns stand out more this way. Put these tables where you do math, for easy reference. Don’t even mention the m-word for a couple of years. Notice how our lists of words that go with activities don’t have multiplication in them…

Words

Variable, combinations, combinatorics, table, tree, covariation

Scavenger hunt

Many cultures have myths about chimeras: creatures combined from animal or human parts, such as pegasus or sphynx. Production companies from car manufacturers to t-shirt makers offer combinations of features. Hunt for words that are made out of other words, such as microphone, telephone, microscope, and telescope. Online memes sometimes feature combination tables or trees, such as alignment charts.

Course links

• All course tasks
• Multiplication Lounge: our open forum for questions, comments, and ideas
• Course participants survey

Today your mission is…

Find words within words, make words out of words, and illustrate compound words.

Ready, Set, Go

Mark Twain said, “Some German words are so long, they have a perspective.” In this task, you add mathematical perspective to English, making a synthetic language of your own: MathLexicon!

Start with some nouns. As usual, select a few of your and your child’s favorite things: cat, truck, apple, love, your own name.

Now make a list of mathematical prefixes, that is, parts of words that attach before other parts. Try size prefixes like micro- and macro-, number prefixes like uni-, bi-, tri-, quadro-, penta-, spatial prefixes like omni- and under-, logic prefixes like anti-, and so on.

Attach your prefixes to your nouns. You’ll have microcat and pentalove, antitruck and omniapple. What are all the words in your new synthetic language? Make a table to find out! Can you illustrate your new MathLexicon?

For even more combinatorics fun, attach endings too. A few good math endings are -gon, meaning of angles, -plex, meaning of many, and -oid, meaning similar. How about a quadrocatgon - a quadruple cat angle, or maybe a cat with four angles? Use combination trees to keep track of all your prefixes and endings.

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?

• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Enter your child’s name in our online MathLexicon word maker and read the silly results! http://www.naturalmath.com/mathlexicon/new.html After you played for a while, try predicting numbers, for example, “How many different words can you make out of two nouns and three prefixes?” Some kids will be interested in counting combinations, some in table and tree structures, some in illustrating the new words, and some in telling stories explaining words. Follow your child’s interests: it’s all mathematics, of different kinds!

Toddlers

Make picture tables, with a few prefixes and nouns. Try using small toys or playdough for tactile play. For example, turn any toy animal into an octa-animal with eight playdough or tape tentacles.

By Anders Iden

Young children

Learn history and origins of words using etymology dictionaries. Hunt for parts of words systematically, for example, in names of polygons (pentagon, hexagon…) or in the metric system (megaton, gigaton…). Write your word parts on cards, attach them to magnets, and take MathLexicon to your fridge!

Older children

Mathematical linguistics is full of smart games that require systematic, logical reasoning. You can seek problems from linguistic Olympiads for children, or start at this excellent collection of puzzles and stories of exotic languages: http://lingclub.mycpanel.princeton.edu/challenge/puzzles.php

How is this multiplication?

The connections to multiplication are very similar to the Combinations task. Your variables are the numbers of prefixes, nouns, and endings. The resulting number of compound words is the number of combinations of these variables, that is, their product.

Inspired by algebra

Again, the connections to variables, tables, and trees are similar to the Combinations task. But MathLexicon has additional bridges to algebra, geometry, and other parts of mathematics, because words mean things. Each prefix or ending makes the rest of the world into an open, playful variable. For example, penta- injects fiveness into anything: pentacat, pentagon, pentawhatsoever!

Frequently Asked Question

If kids do not memorize their times tables and do math in an inquiry based way for years, does this hinder them when they enter more achievement-oriented math arenas later on?

We often ask this question of people who have had open and free choices in whether to memorize times tables, and if so, when and how. Several possible scenarios emerge. It seems that the bad or dangerous scenarios are largely due to social pressures, rather than inherently mathematical dangers. The good news is that there are many different happy ends! We do not know yet why one person may happily work as an engineer without ever memorizing times tables, while another person develops a crippling math anxiety. More research is needed. Meanwhile, here are the scenarios:

• As a kid, memorizing times tables never starts, or is quickly abandoned. As an adult, this person calculate facts as needed, sometimes using patterns. There are no issues in science and math courses or everyday life that the grown-up can see.

• A kid tries to memorize times tables, but it does not happen, or there is a lack of fluency. Math anxiety results, and lingers for a while, sometimes into the adulthood.

• A kid runs into times tables before encountering formal algebra and calculus, and memorizes them. These are usually kids who like number patterns, or have a strong sequential or auditory memory.

• Come formal algebra or calculus, a kid who never memorized before focuses on multiplication facts, to achieve more fluency with these higher subjects. Very few people become anxious over multiplication at this stage, but some become anxious over their lack of fluency in algebra. Then the road forks:

  • Some kids get most times tables via algebraic patterns.

  • Some kids memorize (it’s easier for older kids because of brain maturity and learning skills).

  • Some kids realize they can increase fluency by other means, such as printed-out times tables, or software.

• Come formal algebra or calculus, a kid who never memorized times tables on purpose notices that facts got absorbed by osmosis. This happens to those who spend a lot of time around people in STEM fields, do science and math for fun, or are otherwise immersed in a math-rich environment.

Words

Variable, combinations, combinatorics, table, tree, prefix, ending

Scavenger hunt

Find words within words as you read and talk. What do parts mean? Where does the meaning come from? Find examples where the same word part means different things, like bicycle and billion, or quadratic (power of two) and quadrilateral (four-sided). Find other examples where different word parts mean the same, like bicycle and dipole (both bi- and di- meaning 2). How would you redesign language to be more consistent?

Today your mission is to…

Browse a gallery of factorization diagrams below, and then make your own! Factorization diagrams show numbers in terms of their prime factors. This helps you to analyze the anatomy of numbers so you can get insights into their structure.

Ready, Set, Go

Search the web for factorization diagrams, or start with our overview of features. You have several math, art, and learning choices, for example: do you show all factors or just some? How do you tackle prime numbers? Do you always show the same factor with the same elements of art (shape, color, line, tone) or do you change the elements? Do you make your diagram straightforward or puzzling?

Count on Monsters by Richard Evan Schwartz (2003)

• Math: Geometric (triangle for 3) and abstract representations

• Art: Color, shape, and portraits of primes as unique creatures

Learning: Puzzling, perplexing, and unpredictable; abstract: the final quantity is not shown, just the factors

Primitives by Alec McEachran (2008)

• Math: Nested sets show factors of factors

• Art: Color, size, and composition of circles

Learning: Recognizable symmetric patterns, subitizing, systematic and orderly; pictures of numbers are fully predictable from smaller numbers

Diagrams by Brent Yorgey and friends (2012)

• Math: Composite shapes (circles, polygons, fractals)

• Art: Size, shape, compositions of small circles, color in some versions

Learning: Recognizable symmetric patterns, subitizing, systematic and orderly; color-coding has some perplexity; pictures of numbers are fully predictable from smaller numbers

Number and letter art by Mark Gonyea (2014)

• Math: A mix of symmetry, arrays, fractals, polygons, etc.

• Art: Exploring one element per poster, such as line, circle, or dot

Learning: A balance of order and randomness: pictures of numbers mostly look symmetric, but are not predictable from past examples

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Option 1: See then make

Look at the factorization diagrams together. Ask kids to see what they notice. Accept all answers, even if they are about superficial features of the art. Ask children to find similarities, differences, and patterns:

• Cover up a number on one of the predictable diagrams and try to guess what it should look like.

• Find numbers that look similar. Why did the artist choose to make them similar? It may be an artistic choice, or there may be math reasons.

• Reproduce a part of a diagram in a different medium, from Minecraft to found treasures. How does the medium influence what you do with math?

Are there numbers you would do differently? Remix some of the diagrams.

Option 2. Make then see

Invite kids to make portraits of quantities from 1 to 10, to 25, or to whatever number does not seem tedious. You may need to explain the difference between a symbol and a portrait, a number and a quantity. Portraits can be drawings or arrangements of objects, in physical or virtual worlds. Do that several times, together. Go for humour, pattern, beauty, order, and variety. Use LEGO, natural materials, found treasures, pencils, action figures, or whatever other medium your children enjoy. Take pictures as you go along. When children created some of their own diagrams and there is a pause, show them other people’s diagrams.

Toddlers

Use small numbers, up to 12 or so. Prepare beautiful stuff in just a few quantities at a time, on empty sheets of paper. Invite your toddler to arrange each group nicely, and play along yourself. You can trade object for object, but keep quantities. Your child may need to play with the stuff freely before doing that. Take pictures of arrangements.

Young children

Draw, use stickers or cutouts, use art software, make collages out of found objects. Invite kids to make connections between several numbers, for example, use the same star shape whenever the factor of five appears. Turn it into a puzzle: guess what the next number should look like from all the previous numbers. Turn it into a domino sorting game.

Older children

Explore ways to test divisibility ahead of time. For example, how do you find if a number is divisible by 3? By 5? An easier problem is to skip-count by a given number, so you find all the numbers divisible by it. Then you can use color, shape, or other elements of art to make these numbers look different, such as orange for multiples of 5 in Letters by the Number. Check out the Sieve of Erathosthenes animations. Use programming or modeling software to assist your explorations. Make the music analog of factorization diagrams: polyrhythms.

How is this multiplication?

Factorization diagrams show multiplication and division in one image. Now you see a number divided into factors; now you see factors coming together with the number.

Inspired by algebra

The most fundamental algebraic idea of factorization diagrams is reversibility. The diagrams promote the holistic understanding of multiplication and division as the reversible, connected, complementing actions.

Divisibility is the first step to building algebraic structures known as groups. Divisibility makes you focus on overarching patterns and algebraic “Whys”: numbers divisible by 4 are divisible by 2 twice over, numbers divisible by 10 always end in zero, numbers divisible by 3 have the sum of their digits also divisible by three, and so on.

Frequently Asked Question

In daily life, what are some of the very first comments we might make that can lead to more observations about multiplication?

When it comes to good chats with kids, go for:

• Your own sincere interest. Kids sense emotions. It is okay to share fear or frustration as your emotions, as long as you are open about your feelings, and have a bit of fascination to go with them, as Blake does in The Tyger poem below. Easy start: notice something cute, funny, or nice that also has multiplication, like a kitten, a flower, or your favorite spaceship.

• Details and particulars. These make for good stories. Easy start: make a cheat list of 3-5 particular terms that go with multiplication, for example: array, fractal, symmetry, scaling, combinations. Use these words to make your multiplication chat less generic and more detailed.

Play and adventure. Connect the conversation with children’s actions or favorite imaginary worlds. Easy start: find outdoor examples of multiplication large enough to climb, or use toys to pretend-play jumping, climbing, and otherwise adventuring in or on smaller objects.

By GapingVoid

Words

Factors, dividers, divisibility, factorization, group

Scavenger hunt

Find well-arranged quantities in nature or culture. Playing cards have groups of symbols. Windows and muffin tins come in arrays. Many folk dances, such as square dances, arrange and rearrange participants in groups that divide the total number. The children’s verse “The ants are marching one by one… two by two… three by three…” is a puzzle about different arrays made of the same ant army by different divisors.

By Traditional Music Co

Course links

• All course tasks
• Multiplication Lounge: our open forum for questions, comments, and ideas
• Course participants survey

Today your mission is…

Make geometric art based on skip-counting formulas.

Ready, Set, Go

There is little room for creativity in paint by number. In contrast, design by formula is creative, because you can tweak, remix, and change your formulas. Algebra has variables, and variables vary! In these activities, you will create beautiful designs, in a very meditative process.

Option 1. Waldorf stars

Draw a circle and mark its circumference with ten evenly spaced dots, numbered 0 to 9. Why? These are all the possible remainders when you divide by 10. Remember this fact for later.

Next, choose the number you will multiply by. Let’s use 3 as an example. Find dot 3. From there, go by the dots, skip-counting by 3 clockwise, or by the numbers, using multiplication tables. Your second dot is three dots over, or 3 * 2 = 6. Draw a line connecting 3 and 6. Your third dot is 3 * 3 = 9, so draw a line from 6 to 9.

By HowTheSunRose

Your fourth dot, three dots over from 9, is 2. So connect 9 to 2, even though 3 * 4 = 12 - because from now on, you will go by the remainders, throwing the tens of the number away. Keep going around the circle for a while. What do you notice about your path? Try with other numbers, to observe other paths. Experiment with color patterns.

Option 2. Spirolaterals

For this activity you will need graph paper. You can make graph paper with a ruler, print out custom graph paper from this site, or use square tiles and chalk, outdoors. The younger your kids, the larger you should make your grid cells.

Pick your favorite number; we’ll use 3 again. Write down the first few results from times 3 table. We’ll use 3 * 1 = 3, 3 * 2 = 6 and 3 * 3 = 9, so our numbers are 3, 6 and 9. On your graph paper, draw a line 3 cells long. Turn right, as if driving along the grid, and draw a line 6 cells long. Turn right again and draw a line 9 cells long. Turn right and draw a line 3 cells long, and so on. What do you notice about your path? Young kids can produce the same pattern visually, without knowing times tables or even counting beyond 3: go 3-step once, twice, thrice, then repeat.

By SharynIdeas

Can you get different shapes if you use a different times table or a different number of multiplication facts from it? Experiment and observe! The number of facts you use is called the order. So an order-4 spirolateral for the times 3 table uses numbers 3, 6, 9, 12. An order-5 spirolateral for the times 2 table uses numbers 4, 6, 8, 10, 12.

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?

• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

For visual activities, the best way to explain the rules is to show them. Count out loud as you make spirolaterals or stars. Use colors. With spirolaterals, kids get turned around and disoriented. You may need to help them stay oriented by turning paper for them, or announcing turns out loud. Try driving a pretend car along your grid paper, with a mark for its right-hand turn signal.

Toddlers

Use yarn with peg stars, and times tables with small numbers, like 2s and 3s. Make giant spirolaterals (painters’ tape is perfect for this) and invite your toddler to walk them, or to drive a toy car along them. Even if your kid can’t count yet, the geometry sinks in!

Young children

Experiment with changes in formulas, colors, or types of grids. Help kids implement their “What if?” ideas and test conjectures. What if we turn left rather than right? What if we use triangular or hexagonal grids? (Print them out here.) Do different numbers always give you different grids? Can you work backwards: draw a spirolateral first, and then figure out which formula goes with it? Or have a grown-up figure it out...

Older children

Try making stars and spirolaterals in different bases, such as binary or base five. For the stars, mark the circle from 0 to the number one less than your base, and use remainders from dividing by the base. Try to build in virtual worlds like Minecraft, or program in modeling software, such as GeoGebra or Scratch. Here is an example in Scratch.

How is this multiplication?

Finished stars and spirolaterals visualize times tables. The process of making them is all about thoughtful skip-counting with several inviting bridges to algebraic and geometric patterns.

Inspired by algebra

The stars build bridges to divisibility patterns and modulo operations. The original stars throw away the tens of the number, that is, operate in modulo 10. You can change that variable and work in different bases.

Spirolaterals have three variables: the angle of your turns, the times table, and the number of results from that times table (the order). You can play around with these variables, or change them systematically to catch patterns.

Frequently Asked Questions

If I do not tell my child that a particular activity is about multiplication, how will my child connect it to multiplication on her own? What if she enjoys the activity, but doesn’t realize how or why it is multiplication? How do I check for this understanding? And what do I do if the connection is missing?

There are always more connections than anyone realizes. We’ve been leading some of these activities for years. Yet new bridges to ideas keep coming up, from a parent on a forum, a kid in a math circle, or a researcher at a conference. There number of connections from any given activity to ideas or applications is literally infinite!

So, we are all in the same boat here: there are infinitely many connections, and we only notice some. First, ask children what are their favorite things about the activity. For example, your child may like unexpected results (such as the shapes of spirolaterals) or repeating patterns (such as skip-counting on stars going over the same star again and again). If you know of a neat connection to this favorite idea, you can share it. If not, you can research it together with the kid, or on your own time, and share later.

Connections only build over time. Put up a picture from the activity where you will see it often: a frame in a busy corridor, your bathroom mirror, or your fridge. As you seek more and more connections, add words and pictures reminding you of them. Your activity is a pioneer, pitching the first tent at the site of the future city of ideas.

Words

Variable, spirolateral, star, divisibility, modulo operations, base, order

Scavenger hunt

This time, create surprise mart for others to find and admire. Where can you leave your mathematical masterpieces? Busy places like parks and playgrounds work very well. You can create intricate designs with found objects. For the spirolaterals, collect sticks, pebbles, leaves, or pinecones that are about the same length. You can yarn bomb a fence with spirolaterals, or make chalk or tempera stars on manhole covers. Take a picture of your creation and share with us!

By Demakersvan

Course links

• All course tasks
• Multiplication Lounge: our open forum for questions, comments, and ideas
• Course participants survey

Today your mission is to…

Use rotational symmetry to create multiple copies of an object. And, since the number of copies can be changed from one to infinity, you will end up with a multiplication model AND a model of a variable.

Ready, Set, Go

Option 1: Mirror Book

Lay two plain rectangular mirrors on top of one another, face to face. Connect them at one edge with duct tape. You can usually find small mirrors in dollar stores and school supply or craft stores, and larger mirror tiles in home improvement stores. Put interesting things – like favorite toys or your face – inside the open mirror book. Change the angle between the mirrors and admire the greatest show in math!

Option 2: Snowflake

Fold a piece of paper through its center repeatedly, then sketch and cut out a snowflake. Real snowflakes always have 3, 6, or 12 sides, but abstract mathematical snowflakes can have any number, even 5 or 7, if you are clever about folding them. Learn different folds and make a snowflake multiplication table!

By Bloke School

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Children are immediately intrigued by mirror books. Let kids explore the books first. Younger children might bring favorite toys and pile them between the mirrors. Older children or those with more experience with the mirror book will build elaborate designs, act out stories, write coded messages and create puzzles.

Young children might need help with folding paper and cutting the designs. You can minimize frustration by asking the child to draw the design on a folded snowflake for you to cut out. It works especially well if you make a giant snowflake. Some children might not want to cut any designs at first. Instead, they might prefer to fold, then make one or two straight cuts. Go for it! It’s a perfect opportunity to admire how even very simple steps, repeated enough times and mirrored, can create complex patterns.

Toddlers

Make giant snowflakes out of newspapers or wrapping paper, for easier cutting. Your toddler can guide your hand with scissors, or you can wrap your hand around your child’s hand to help. Find full-body mirrors at an angle in dressing rooms, or make your own giant mirror books out of large door mirrors, using any door and its adjacent wall.

Young children

Combine two mirror books and let the kids explore infinity. If playing with paper, fold the paper once, cut or punch out a small circle, and ask the child to predict how many circles will there be once you unfold the snowflake. What if you fold the paper twice? Three times? Four times? Draw or write numbers, letters and words inside the mirror book, or cut them out of snowflakes. How many symmetries, hidden in these shapes and symbols, can your kids discover?

Older children

Become a puzzle maker. With a mirror book, ask if it is possible to make a square (or any other shape) with one toothpick. Or challenge the child to turn a slice of pizza into a whole pizza. How can we have our (whole) cake and eat it too? Fold paper for the snowflake and ask your child to cut it in such a way that when unfolded, there would be only one circle. Invite the child to create new puzzles! Model radial symmetry in software, such as GeoGebra or Scratch.

How is this multiplication?

Rotational symmetry connects counting and grouping, visual and sequential, algebra and geometry. That’s how the nature multiplies the same pattern (by the same genetic code) across all arms of an octopus, or all petals of a flower. You can add objects inside the mirror book, or cut to the folded snowflake, one by one. The reflections multiply that group you create into all the copies, all at once.

By Ernst Haeckel

Inspired by algebra

The mirror book has two variables: the number of objects and the number of reflections. But you can take a more continuous stance, if only you take the angle between mirrors as your variable. With some angles, you can have fractional reflections! With snowflakes, the number of sectors and the number of cuts are the two variables. But cuts on edges or the middle don’t get multiplied by the full number of folds, so the system is more complex than just multiplying cuts by sectors.

Frequently Asked Question

I showed a mirror book to my child and tried to explain what to do. He didn’t seem interested. Then he dumped a pile of small toys into it, looked for exactly one second and ran away. What did I do wrong?

If at all possible, we encourage you to make all your math manipulatives, including a mirror book, together with your child. There is a lot of hands-on mathematics in the process of making things. Besides, children are more eager to play with the toys they made themselves or helped making.

There are, of course, situations when making a math toy with a child is not possible or practical. In this case, make sure to give your child plenty of time to free play with the toy. Do not try to provide any directions, explanations, or rules at this point.

Leave the mirror book (or any other math play object) where a child can easily see and use it. Children will come back to this activity, if only for a few seconds. Observe and record (if possible) their play and its results. You will soon notice how their play, however short, becomes more complex and purposeful.

Finally, next time your child has a playdate, let children play with the mirror book. You will be amazed at the complexity of designs and the depth of discussions that accompany this activity.

Words

Radial symmetry, reflection, rotation, angle, variable

Scavenger hunt

Radial and reflection symmetry are beautiful! It is easy to find, but not that easy to capture. Look for examples of it in nature (hint: near bodies of water on a calm day), architecture (all that glass and polished highly reflective surfaces), and at home. See how many multiplication facts you can find or create.

Course links

• All course tasks
• Multiplication Lounge: our open forum for questions, comments, and ideas
• Course participants survey

Today your mission is…

Play the multiplication model bingo to see a variety of models wherever you go.

Ready, Set, Go

This is the same scavenger hunt you did during Week 1 task, What Is Multiplication. This time, invite your kids to hunt with you. Many people say they get stuck seeing the same model everywhere. Can you find all 12 models? You can print out this bingo sheet with spaces for your finds.

Your forum response

• What multiplication ideas do you see in this topic? What bridges connect this to your everyday life, sciences, and arts?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

This activity is sort of like a scavenger hunt, only you are allowed to create your own models.

You might end up being the first who notices a particular model. Point it out to your child and connect it to an activity you did in the previous weeks. If you are stuck looking for a particular model, go through the artwork and other artifacts you and the children created over the last couple of weeks.

Find different examples of the same model. Help kids point three tips of a leaf with three fingers, hold the puppy four paws up, put feet on the two levels of a climbing structure and otherwise show numbers with bodies, as you take photos. If your child really likes a find, reproduce it in LEGO, playdough, Minecraft, paper, etc.

Frequently Asked Question

When I do these activities with my child, I feel like I need to actively teach - talk about what goes on and why. I think I’m doing too much of it though, taking over my child’s math exploration. What should I do?

You can lecture occasionally - when your child asks for an explanation. Keep it under a couple of minutes. What to do during the rest of the time? Be active, yes, but in a variety of ways: as a coworker, a naturalist-researcher, and a guru. As your child’s coworker, play in parallel: do your own experiments and explorations of the same math your child is exploring, because it creates a smooth workflow and inspires ideas. As a researcher, observe the elusive natural behaviors of your kid, take notes and photos, gently ask non-leading questions, because it will help you preserve your child’s well-being, and nurture a healthy ecology of learning. As a guru, your goal is to help kids be heroes of their own adventures. Play Yoda to your kid’s Luke.

Bob Kaplan, a math circle leader and author, says the trick is to become invisible. Active, but invisible! Read more here - http://www.moebiusnoodles.com/2013/09/becoming-invisible/

Words

Array, area, number line, set, combinations, symmetry, fractal, scale

Today your mission is…

Fill times tables with personally meaningful examples. This task helps to develop fluency and applications of multiplication.

Ready, Set, Go

This activity scales up and down: you can spend five minutes, or five hours, or five years hunting for the elusive facts. Don’t worry about finishing it all in a day. Just as with any other collections, this one will grow and become more meaningful over time.

Go on a scavenger hunt for examples of multiplication in nature, designs, or stories. Prepare index cards or sticky notes, and a times table grid large enough to fit them. Every time you find an example, sketch or describe it, and put it in the right cell. Some combinations are easier to find than others, so you can have multiple cards in the same cell. Decide what to do about the symmetric times facts, like 2*5 and 5*2. Take photos of happy hunters with their finds and creations!

For an extra challenge, seek iconic examples, where objects always go with particular numbers. Hands and fingers always mean 2x5, and the chessboard is always 8x8.

If you are into arts, crafts, or engineering, make your collection beautiful. Go from a sketch to a full art piece. Compose your photos nicely. Design a complex Lego sculpture or Minecraft building. Turn a description of a situation into an engaging story. Make a poster, a collage, a gallery, or a diorama out of your collection.

By Sabdha C.

Your forum response

• What multiplication ideas do you see in this topic? What bridges connect this to your everyday life, sciences, and arts?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Where can you go hunting?

• On a field trip. Your yard, a park, a museum, a fire station, a zoo… Give each group of participants a camera to capture their finds. Invite them to point out multipliers they find with their hands or whole bodies for extra awesome photos.
• To your home. Find multiplication in toys, pets, decorations, books. Make mobiles or sculptures about multiplication.
• To a virtual world. Find and build multiplication in the universes people created, such as Minecraft, Star Wars, Dungeons and Dragons, and Harry Potter. Let kids pick destinations.
• To a box of construction toys and art materials. How many different ways can you show the same multiplication example with Legos? Can you paint it, sculpt it, fold it, cut it?
• Through history and ethnography. Check out what different times and cultures did for their multiplication, from playing dice to building abacuses.
• Into a trade. How do nurses, astronauts, musicians, or homesteaders use multiplication? You can interview a real tradesperson, or pretend-play about sci-fi, historical, or future professions, such as alchemists, Mars settlers, or time travelers.
• Into the imagination. Make up stories and art devoted to multiplication. Play “Imagine That!” by making silly changes to the world: a hand with fingers at the tips of fingers, a dog with eight legs, an alien planet where numbers only went up to three.

Frequently Asked Question

You frequently mention celebrating mistakes. But how to do it? And why? In life, making a mistake may lead to unfortunate or tragic consequences. Does celebrating mistakes mess up children’s respect for serious thinkers of the past?

You would not celebrate a mistake in building a bridge, or in getting the distance to Mars wrong and missing your landing window! However, there is a side of mathematics that is more whimsical and free. Mathematics can be not only about measuring this world where we live, but about making up other, fantastic worlds. In that, mathematics is closer to fairy tales, or science fiction, or fine arts where you can create rather freely. So when I talk about celebrating mistakes, I mean taking them from our world into one of those mathematical Wonderlands.

For example, my math circle kids kept confusing squares and cubes, or circles and spheres. But we played with it, and by now, they talk about three-dimensional squares - or even four- and five-dimensional! This road from mistakes to abstractions is a big part of the history of serious mathematics, from imaginary numbers to non-Euclidian geometries.

Words

Array, area, number line, set, combinations, symmetry, fractal, scale

Today your mission is…

Build decanomial squares, using the array model of multiplication. Array is the best model to explore algebraic properties of multiplication such as commutativity, and algebraic identities such as (a+b)^2=a^2+2ab+b^2

Ready, Set, Go

A decanomial square is a multiplication table made out of arrays. You can use different materials to build your squares.

Graph paper, by Somewhat in the Air. You can design and print graph paper online or make it by hand. Use larger cells with younger kids.

Homemade Montessori beads, by Our Homeschool Journey. This takes a long time to make by hand!

Felt, by Walk Beside Me. Note that unmarked pieces have more perplexity, forming visual, geometric puzzles. Bead pieces or graph paper pieces are easier to sort and push toward number patterns and algebra. You can make the same design out of construction paper.

Cuisenaire rods, by Simon Gregg. This combines visual aspects (rods don’t have unit marks) with counting (the number of rods).

Your forum response

• What multiplication ideas do you see in this topic? What bridges connect this to your everyday life, sciences, and arts?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Reach for your child’s favorite medium whether it’s clay, paper, beads or anything else. Start the pattern and ask the child for help. Play “what goes next” game. Make mistakes in the pattern to give your child an opportunity to correct you.

If you are working with younger children, keep the squares small, only two (binomial) or three (trinomial) numbers. The squares can grow with your child.

Photo of trinomial square and cube by Simanaitis Says

Frequently Asked Question

When it comes to “continue the pattern” activities, my child is quick to figure out the pattern, but starts making mistakes after just a couple of steps. Why? How can I gently correct my child’s mistakes? And should I do it in the first place?

For the reasons why kids break patterns, revisit our activity Your Child, the Divergent Thinking Genius. Consciously or not, children play around, change patterns, notice and immediately implement new possibilities. An adult is likely to hold a neat alternative for later, while carrying on the current pattern. Many kids don’t see any point in carrying a pattern beyond the moment they understood it.

Once a kid gets a bright new idea, it’s very hard to carry on the old pattern, even if the kid wishes to do so (which is rare). Children don’t have enough short-term memory or attention to hold two patterns in mind. So for children, the healthy way is to implement their bright ideas right away, lest they forget!

Don’t think of pattern-breaking as a mistake, because most often it isn’t. It’s play or creativity. Don’t: distract the child from brainstorming about patterns! Let the child play. Do:

1. Observe quietly for a while, to understand (or to make up) a pattern behind your child’s new idea.
2. Copy that new pattern yourself, making 4-5 steps of it. When your child pauses from play, ask if you got the idea right.
3. Sometimes children keep changing the rules again and again. Make all the patterns you notice.
4. Also make the old, original pattern. Admire, compare, and photograph all the patterns you made together.

Think of yourself as a catcher of the wild patterns your child generates. Gotta catch them all!

Words

Decanomial, commutativity, square, binomial, trinomial

Today your mission is…

Find and color-code patterns in the times table. If you look at the times table long enough, algebra will look back at you. Instead of 100 (or 144) disjointed facts, you’ll see a jigsaw of clear, orderly, beautiful patterns. The mastery of patterns is required to be a mathematician, a scientist, an artist, or an engineer.

Ready, Set, Go

Some kids like to memorize number patterns. Others do so because of requirements. Others keep a pocket times table with them, or use calculators. This activity is designed to help all these kids, in different ways.

Invite your kids to look at the times table and color any sequence or pattern they notice. It might make sense for adults to describe such a pattern as, “A group of facts you know without any need to memorize.” Younger kids may not understand what “need to memorize” means, but they can still seek patterns. For example, there is the counting sequence 1, 2, 3 in the table; the counting sequence with zeroes behind the numbers (times ten); and the doubles, which many kids find easy, especially after activities from the past weeks. There are more exotic and hidden patterns for those who like number play. Keep seeking!

Your forum response

• What multiplication ideas do you see in this topic? What bridges connect this to your everyday life, sciences, and arts?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

It is not necessary to reduce the table to something smaller if you work with young kids. After all, seeing more examples might help them notice the patterns. However, you do need larger cells, with numbers in large font. For toddlers and young kids who don’t yet know or are not fluent at recognizing numbers, this dot version will work. You can make your own posters using factorization diagrams.

Adapted from factorization diagrams at Math in Your Feet, Nike Naylor, and DataPointed.

It may help you to know, ahead of time, what types of patterns other people have found in the times tables so far. This does not mean you need to guide your kid to find all these patterns - let alone on the first day! This is to prepare you to expect the unexpected, because patterns come in many types. For each pattern, investigate why it works.

Patterns by the skip provide a sequence of skips that is easy on the eye.

• *10 all end in 0s: 10, 20, 30, 40…
• *2 are even numbers: 2, 4, 6, 8...
• *5 end in 5s and 0s by turn: 5, 10, 15, 20…
• *1 are counting numbers: 1, 2, 3, 4…
• *11 have repeating digits: 11, 22, 33, 44...

Personal patterns are sequences or individual facts you love and remember, for reasons of your own.

• People who like geometry love square numbers that sit on one diagonal of the times table: 1, 4, 9, 16…
• Poets love and remember rhyming or alliterating facts: six times six is thirty-six; five, six, seven, eight - fifty-six is seven times eight; five times five is twenty-five, etc.
• Psychologists note 7*8, 6*9, 7*9, and 6*8 as the facts that cause people the most confusion (see Dehaene, “The Number Sense” chapter “The multiplication table: An unnatural practice?”)
• Western musicians pay a lot of attention to *4, because many songs are in measures of four.

The symmetry pattern means that 2*5 is the same as 5*2. It makes the top right and bottom left corners of the table hold the same exact numbers, as if the diagonal were the mirror. Some people call this the commutativity pattern after the name of the property. Toddlers and young kids often like pretty math terms like this! For anyone who memorizes facts, this pattern reduces the workload almost in half.

In the nines pattern, there are many ways you can notice regularities.

• The first digit is increasing by one and the second digit is decreasing by one: 09, 18, 27, 36…
• Times nines are 1, 2, 3… away from the corresponding times tens: 09 and 10 (1 away), 18 and 20 (2 away), 27 and 30 (3 away) and so on.
• Spread your ten fingers and fold away the Nth from the left. You will see the answer to 9*N in the remaining fingers. Here’s a video from PBS: https://www.youtube.com/watch?v=Wu3JSnRaaV0
• The sum of the digits in the times nine results (up to 90) is always nine. The first digit is one less than the number you multiply by nine.

Calculation patterns are for people who love number crunching or number puzzles.

• Off-diagonal numbers run parallel to the square diagonal. They are facts like 4*6, 7*9, and in general (N-1)*(N+1). They are one less than square numbers.
• You can find a similar off-off-diagonal pattern for (N-2)(N+2), and so on.
• Times 4 is doubling, twice.
• Times 8 is doubling, thrice.
• A finger reckoning trick from 15th-century merchants, now a fun parlor trick, shows how to multiply numbers 6-9 on your hands. See details here: http://www.moebiusnoodles.com/2014/04/multiplication-a-parlor-finger-trick/

Frequently Asked Questions

Parent 1: My child believes that he is not good at math. In fact, his conceptual understanding is strong, but he’s a bit slow on calculations. Any time I suggest we play a math game or do a math activity, he just shuts down if he suspects it involves calculations. I don’t want to force him, but I do want him to be braver, and to learn how to calculate.

Parent 2: My child is overly confident, I think. Five minutes into a new activity he proclaims he’s good at it, so he’s done. He doesn’t stop to check if his answers make sense, such as 5000 mph car speed. I don’t want to lessen his confidence in himself, but I do want him to check his work, and not to be arrogant.

The two very different stories reflect the same key difference between kids and adults: metacognition, or knowing about knowing. Parent 1 and Parent 2, it’s frustrating that your kids don’t know what they know, but the good news is this is unlikely to be about character flaws of timidness and arrogance.

Use math tools to help kids grow their metacognitive skills, such as:

• Know what to check when you solve problems, where you are prone to mistakes (such as forgetting to check if 5000 mph is realistic for cars).
• Use your strengths to your advantage, pick tasks that match your strengths (Child 1 is beginning to get this awareness).
• Know your weaknesses and how to compensate for them; seek techniques, tools, learning, and helpers that can cover you (for example, using key terms from a text to seek images, if you don’t learn well from texts).
• Be aware of math values such as precision and rigor, and your personal math value system (for example, little engineers say you can’t really split a fractal into parts infinitely many times, because you have to stop at atom size, since they value realism).
• Know the size of your working memory; optimize your use of working, short-term, and long-term memory for your tasks (for example, use the partial quotient long division algorithm, which is less taxing on working memory).

Some people believe metacognitive skills can’t develop until adolescence, but we don’t think so. You can and should develop healthy math habits early on. This includes children appreciating some psychology of learning mathematics. For example, check answers after each and every problem, at all times, as a habit - but discuss how best to check for each problem. Play memory games and solve the same memory-intensive math exercises during different times of the day, so you can compare and discuss differences in how your mind works. Try to present the same problem as a picture, text, hands-on model, or whole body experience, and see which way makes sense. It will vary by problem, but some people do have favorite learning styles. In short, learn about mathematical metacognition yourself, and help your kids know what they know. Some resources to get you started:

• The Brilliant Blog
• Math Mind Hacks
• How to solve it (awareness in problem-solving)

Words

Pattern, formula, rule

- Why are waiters good at multiplication?

- Because they know their tables!

Today your mission is…

Learn the spaced repetition method for memorizing bodies of facts.

We memorize times tables to

1. Have an easy, reliable, quick access to each individual fact.
2. Notice and use more number patterns.

Notice that we say access, not recall. Most people don’t memorize facts like 2*2, 1*7, or 10*9, because they can calculate the answers very easily.

Strongly visual people often close their eyes to access facts (to imagine some pictures), auditory people mutter, and kinesthetic people move their fingers. These techniques aid either memory or calculation, and should be encouraged.

Quiz before you start

Which of the following methods of memorizing serve Goal 1, 2, or both? Answers with comments are at the end.

1. Memorizing by reciting whole times tables in order (*2, *3, *4…)
2. Memorizing by reciting times tables (*2, *3, *4…) plus training access to individual facts out of order.
3. Memorizing multiplication songs.
4. Using pictures and stories with mnemonic clues for the multiplication facts.
5. Memorizing by patterns.
6. Memorizing by patterns plus training access to individual facts out of a pattern.

Ready, Set, Go

This activity is for people who consider memorizing multiplication facts. We do not have enough data to tell you whether memorizing times tables is a good idea, mathematically speaking. But many people will like this as a mental exercise, just as people enjoy memorizing music and poetry. Many will find it useful for social reasons. We recommend memorizing by patterns plus training facts out of order.

First, do the triage of patterns from the activity Coloring the Table. Which patterns do you know well enough that you don’t need to memorize? Throw these facts out of your table! Cross them out. Different people make different decisions. We strongly recommend getting rid of ones, tens, and the symmetry (commutativity) pattern, leaving you with this:

You are down to 36 facts from 100, without doing any memorization! You may cross out even more patterns, for example, many people remove times two. Also cross out individual facts you happen to know. Now pick a pattern from the list of what is left, and use the spaced repetition method on these facts.

Option 1: Paper cards

Preparation

1. Make cards for each fact from one pattern. Write the multiplication (such as 3*9) on one side of the card and the answer (27) on the other side.
2. Cross these facts out in the table to mark your progress.
3. Make three boxes for the cards: Easy, Good, and Again. Place all your cards in the Again box for now, multiplication side up.

Work with your Again box two times a day

1. Take the stack from the Again box, shuffle it, and try to access each answer in your mind. Then turn the card over to check yourself.
2. If you accessed the fact as fast as you want, and correctly, move it to the Easy box. If you accessed the fact a bit too slowly but correctly, move it to the Good box.
3. If you could not access the fact, were very slow, or made a mistake, do a micro-exercise for your memory. Say the fact out loud: “Three times nine is twenty-seven” and/or imagine any visuals that go with that pattern, such as a three by ten array with the top three counters crossed out. Put the card back into the Again box.
4. If your Again box is empty, congratulation! You made great progress on that pattern. Refill the Again box with new cards, from the next pattern you want to tackle.

Work with your Good box once a day, and with your Easy box once a week. Occasionally, you may need to move a card back from Easy to Good, or from Good to Again - that’s fine, don’t worry. Eventually, all cards will migrate to Easy. Workouts should be under two minutes.

Option 2: Memory software

We highly recommend the free software Anki. You can make a deck of Anki cards for each pattern. The software will keep track of which card to use, and how often. In the long run, it is much easier to use and more efficient than paper cards. You can use the same tool for learning foreign words, scientific terms, flags of the world, and other bodies of facts.

As an example, you can download MariaD’s deck of some squares.

Your forum response

• What multiplication ideas do you see in this topic? What bridges connect this to your everyday life, sciences, and arts?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Help your kid make all the choices about the process. When during the day is it better to do this work? Which patterns to train, and which not to train? Which pattern comes next? If you work with paper cards, help to make the cards and their boxes beautiful. But don’t decorate cards with stickers or extra pictures, because that will mess up your memory. You can decorate the boxes.

Frequently Asked Question

I thought there were no pre-requisites for the activities in this course. Doesn’t my child need to know numbers first?

With children who do not yet know number symbols, use dots arranged in meaningful patterns, or objects that show quantities. Say the description out loud. Only use two or three cards or toys at one setting, unless your kid insists on more. Make it possible to answer with pointing or picking the right object, rather than words.

Here are some good patterns to use with toddlers:

• Dice and playing cards
• Montessori beads, Dienes blocks, algebra tiles
• Triangular, square, and other shape numbers
• Animal legs or other iconic quantities

Words

Recall, fluency, access, pattern

Quiz discussion

Goals:

1. Easy, reliable, quick access to each individual fact.
2. Noticing and using more number patterns.

Only two methods serve both goals. Of these, #2 (Memorizing by reciting times tables (*2, *3, *4…) plus training access to individual facts out of order) takes longer, is less engaging, and will serve the second goal less than #6 (Memorizing by patterns plus training access to individual facts out of pattern). We recommend that method, #6.

If you do #2 (Memorizing by reciting whole times tables in order (*2, *3, *4…)) or #5 (Memorizing by patterns), Goal 1 is at risk, because you have to access the whole list or pattern.

More dangerous are #3 (Memorizing songs) and #4 (Using pictures and stories with mnemonic clues for the multiplication facts), and other non-mathematical mnemonics. They can jeopardize you seeing mathematical patterns, thus your understanding of algebra. Though songs are less dangerous, since people are less likely to overuse them.

Hi I’m Rachel. I’m a high school student. I need your help with a book I’m co-authoring.

Imagine you are sitting in math class. The room is stiflingly hot. Your teacher is droning on and on about factoring, something you really couldn’t care less about. The kids next to you are passing notes, quietly murmuring. A stifled burst of laughter. Your eyes start to droop. Your desk is such a comfy pillow. You’ll just close your eyes for a few minutes, and then you’ll pay attention. Just a few minutes…

It’s a classic problem. Let’s face it, everyone gets bored sometimes. What should students do when they get totally bored in math class?

Lots of people have invented different coping strategies. When Vi Hart is bored, she doodles. When Trachtenberg was bored, he mentally manipulated numbers. The Happiness Project says to notice your surroundings, or to just plain keep trying. In internment camp, Pilates invented, well, Pilates. When Gauss was given boring math exercises to do, he invented an algorithm to make them go faster so he could do something interesting. What should you do when you’re bored in math class?

I’m collecting input for my book and I want your ideas. Please reply with them. Thank you so much!

Posting for a young friend - MariaD

Camp Logic is a book for teachers, parents, math circle leaders, and anyone who nurtures the intellectual development of children. It is not necessary to have any mathematical background at all to use these activities – only to have a willingness to dig in and work toward solving problems where there is no clear path to a solution.

You can help us publish Camp Logic! Let your friends, colleagues, and the world know you want children to enjoy the underlying structures of math: share the crowdfunding page on your sites, blogs, and social media. Join the crowdfunding event to contribute money for editing, printing, and distributing the book, and to field test the activities. The book will by published in 2014, under an open Creative Commons license.

-- Mark Saul and Sian Zelbo, authors

The campaign reached its first milestone, so we are opening the draft of the first chapter for everyone's preview and discussion. It is attached to this post. Answer to share your thoughts, feedback, and questions for Mark and Sian!

Camp Logic Day One PDF

If you are reading this, you should have received Chapter One of the Camp Logic book by email. Please answer some of these questions, leave other feedback about Chapter One, or ask questions. Any feedback will help Mark Saul and Sian Zelbo to improve the book. They will reply to everyone, and incorporate the suggestions posted before the end of June.

• How good was Chapter One overall? What caught your eye?
• Were there parts where you wished for more details?
• Were there confusing parts, problems you did not know how to start, or other roadblocks?
• Have you noticed anything we need to fix about spelling, grammar, and style? Any other comments to the editors?
• What do you think of illustrations?
• Do you feel you could run these activities with kids after reading? If not, what other support would you need?

If you are reading this, you should have received Chapter Two of the Camp Logic book by email. Please answer some of these questions, leave other feedback about Chapter Two, or ask questions. Any feedback will help Mark Saul and Sian Zelbo to improve the book. They will reply to everyone, and incorporate the suggestions posted before the end of June.

• How good was Chapter One overall? What caught your eye?
• Were there parts where you wished for more details?
• Were there confusing parts, problems you did not know how to start, or other roadblocks?
• Have you noticed anything we need to fix about spelling, grammar, and style? Any other comments to the editors?
• What do you think of illustrations?
• Do you feel you could run these activities with kids after reading? If not, what other support would you need

Do you know examples of being immersed in a math culture from the moment the baby was born? How can we make it happen? More details at http://www.moebiusnoodles.com/2014/08/math-like-music/

Today your mission is…

Make combination tables and trees, and play guessing games with them.

Ready, Set, Go

Once you build tables and trees, hide parts of them and play guessing games. Or make a puzzle: cut the whole table apart into cells, and put it back together again. In some cases, the same cells can form many different, correct tables!

Option 1. Animal table

Your two variables are heads and bodies. Draw a cat head (or your kid’s favorite animal) in each cell of the first column, and a cat body in each cell of the first row. The corner cell will have the whole cat. Ask the child to pick the next animal. Draw the head of that animal in each cell of the second column, and the body in each cell of the second row.

Option 2. Elements of art and elements of math tables

Make a row or a column with the same color, shape, or visual symbols such as antennae for robots. Print or draw clothes for a table dress-up.

Try systematic changes from cell to cell. For example, keep adding a side to your polygons, or another eye to your monsters.

Option 3. More variables: 3D tables and combination trees

Want to use more than two variables? For color-shape-size or any other three variables, build 3D tables. Sticks and styrofoam, or yarn suspended from a cardboard grid takes you to the third dimension. A faster, easier way is to draw combination trees. You might have seen them in sandwich or ice cream shops.

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Help kids keep the structure. Bring up famous chimeras and modular constructs from mythology or engineering. Kids may get bored drawing similar pictures in many cells, so help them finish grids quickly. Then play with hiding cells, taking grids apart and other puzzles. Find examples of using grids for combinations in everyday and scientific media.

Ask “How many?” questions about combinations, which lead to multiplication. For example, a grid for three heads and four bodies has twelve chimeras total, because 3x4=12.

Toddlers

Your toddler can place cut-out parts into cells. Some kids get upset if you cut pictures of whole animals into parts, so do that step by yourself, or draw parts separately. Babies who can’t talk yet can point at toys or pictures to help you select the content of the next row or column. Toddlers and young kids often like combinatorics character creation tools in computer games, such as Hero Machine, or combinatorics toys such as Potato Head.

Young children

Kids may want to draw by themselves, or have you draw with their hand in yours. Use

different colors for different rows and columns. Children can “drive” a toy truck along rows and columns, “delivering” parts to each cell. Be prepared: between rows and columns, young children not only change the value of their variables (e.g. red, green, blue for color), but also the type of their variables (e.g. red in the first column, a hat in the second column, triangular shape in the third column). This is fine; just help kids to stay consistent within the same row or column.

Older children

As you read fairy tales or science news, keep an eye on combinations. For example, luminosity and temperature defines different types of stars, such as a red dwarf or a blue giant. Try making number tables, also called structured variation tables. Label columns and rows 1, 2, 3… and come up with your own ways of combining the labels. If you just multiply row and column labels, you will get the familiar times table.

But what if you double the column label and add the row label? Or take the column label to the power of the row label? Or do something else, like the mystery table below? The possibilities for creativity and guessing games are endless! You can make these tables by hand, or program formulas in spreadsheet software.

How is this multiplication?

The number of combinations in a table is the number or rows multiplied by the number of columns. In 3D tables, or trees, you multiply the number of values for three or more variables to get the total number of combinations. These visual structures model multiplication even for the youngest kids. Yet we frequently meet adults who have never perceived links between multiplication and combining different pants and shirts, or selecting bread and fillings for sandwiches. It is delightful for kids and adults alike to make this new connection.

Inspired by algebra

Combination tables and trees introduce variables: what varies from row to row, or column to column. Tables prepare kids for coordinate planes. Two tasks, “Find the point with coordinates (2, 3)” and “Find the cell with red square” mean the same algebraic action. The more subtle algebraic idea is covariation, which requires kids to notice two or more variables at once. Whenever you pay attention to both rows and columns, you reason with covariation. Kids use covariation in guessing games with missing cells in tables, or to put a table back together again when it’s cut up into cells.

Frequently Asked Question

My child is simply not interested in multiplication. Whenever I try to bring it up, show him some examples, or explain it to him, he tries to change the subject, finds something else to do, or just doesn’t pay attention. I’ve tried various approaches - living math books, board games, just plain old flash cards, but nothing piques his interest. What can I do? What if he never becomes interested in learning multiplication?

Looks like it’s time for you to move on. Some kids simply dislike some topics, and won’t learn them directly. In such a case, the child will absorb the topic from its connections and applications - if only you keep doing other, interesting mathematics. Math is not linear (even if some curricula are), so it’s very easy to skip a topic for a long while and do something more productive.

Focus on algebraic patterns, explore geometry, get inspired by calculus - there are literally thousands of math topics out there. Print out small times tables with just the results, because the patterns stand out more this way. Put these tables where you do math, for easy reference. Don’t even mention the m-word for a couple of years. Notice how our lists of words that go with activities don’t have multiplication in them…

Words

Variable, combinations, combinatorics, table, tree, covariation

Scavenger hunt

Many cultures have myths about chimeras: creatures combined from animal or human parts, such as pegasus or sphynx. Production companies from car manufacturers to t-shirt makers offer combinations of features. Hunt for words that are made out of other words, such as microphone, telephone, microscope, and telescope. Online memes sometimes feature combination tables or trees, such as alignment charts.

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Today your mission is…

Find words within words, make words out of words, and illustrate compound words.

Ready, Set, Go

Mark Twain said, “Some German words are so long, they have a perspective.” In this task, you add mathematical perspective to English, making a synthetic language of your own: MathLexicon!

Start with some nouns. As usual, select a few of your and your child’s favorite things: cat, truck, apple, love, your own name.

Now make a list of mathematical prefixes, that is, parts of words that attach before other parts. Try size prefixes like micro- and macro-, number prefixes like uni-, bi-, tri-, quadro-, penta-, spatial prefixes like omni- and under-, logic prefixes like anti-, and so on.

Attach your prefixes to your nouns. You’ll have microcat and pentalove, antitruck and omniapple. What are all the words in your new synthetic language? Make a table to find out! Can you illustrate your new MathLexicon?

For even more combinatorics fun, attach endings too. A few good math endings are -gon, meaning of angles, -plex, meaning of many, and -oid, meaning similar. How about a quadrocatgon - a quadruple cat angle, or maybe a cat with four angles? Use combination trees to keep track of all your prefixes and endings.

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?

• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Enter your child’s name in our online MathLexicon word maker and read the silly results! http://www.naturalmath.com/mathlexicon/new.html After you played for a while, try predicting numbers, for example, “How many different words can you make out of two nouns and three prefixes?” Some kids will be interested in counting combinations, some in table and tree structures, some in illustrating the new words, and some in telling stories explaining words. Follow your child’s interests: it’s all mathematics, of different kinds!

Toddlers

Make picture tables, with a few prefixes and nouns. Try using small toys or playdough for tactile play. For example, turn any toy animal into an octa-animal with eight playdough or tape tentacles.

By Anders Iden

Young children

Learn history and origins of words using etymology dictionaries. Hunt for parts of words systematically, for example, in names of polygons (pentagon, hexagon…) or in the metric system (megaton, gigaton…). Write your word parts on cards, attach them to magnets, and take MathLexicon to your fridge!

Older children

Mathematical linguistics is full of smart games that require systematic, logical reasoning. You can seek problems from linguistic Olympiads for children, or start at this excellent collection of puzzles and stories of exotic languages: http://lingclub.mycpanel.princeton.edu/challenge/puzzles.php

How is this multiplication?

The connections to multiplication are very similar to the Combinations task. Your variables are the numbers of prefixes, nouns, and endings. The resulting number of compound words is the number of combinations of these variables, that is, their product.

Inspired by algebra

Again, the connections to variables, tables, and trees are similar to the Combinations task. But MathLexicon has additional bridges to algebra, geometry, and other parts of mathematics, because words mean things. Each prefix or ending makes the rest of the world into an open, playful variable. For example, penta- injects fiveness into anything: pentacat, pentagon, pentawhatsoever!

Frequently Asked Question

If kids do not memorize their times tables and do math in an inquiry based way for years, does this hinder them when they enter more achievement-oriented math arenas later on?

We often ask this question of people who have had open and free choices in whether to memorize times tables, and if so, when and how. Several possible scenarios emerge. It seems that the bad or dangerous scenarios are largely due to social pressures, rather than inherently mathematical dangers. The good news is that there are many different happy ends! We do not know yet why one person may happily work as an engineer without ever memorizing times tables, while another person develops a crippling math anxiety. More research is needed. Meanwhile, here are the scenarios:

• As a kid, memorizing times tables never starts, or is quickly abandoned. As an adult, this person calculate facts as needed, sometimes using patterns. There are no issues in science and math courses or everyday life that the grown-up can see.

• A kid tries to memorize times tables, but it does not happen, or there is a lack of fluency. Math anxiety results, and lingers for a while, sometimes into the adulthood.

• A kid runs into times tables before encountering formal algebra and calculus, and memorizes them. These are usually kids who like number patterns, or have a strong sequential or auditory memory.

• Come formal algebra or calculus, a kid who never memorized before focuses on multiplication facts, to achieve more fluency with these higher subjects. Very few people become anxious over multiplication at this stage, but some become anxious over their lack of fluency in algebra. Then the road forks:

  • Some kids get most times tables via algebraic patterns.

  • Some kids memorize (it’s easier for older kids because of brain maturity and learning skills).

  • Some kids realize they can increase fluency by other means, such as printed-out times tables, or software.

• Come formal algebra or calculus, a kid who never memorized times tables on purpose notices that facts got absorbed by osmosis. This happens to those who spend a lot of time around people in STEM fields, do science and math for fun, or are otherwise immersed in a math-rich environment.

Words

Variable, combinations, combinatorics, table, tree, prefix, ending

Scavenger hunt

Find words within words as you read and talk. What do parts mean? Where does the meaning come from? Find examples where the same word part means different things, like bicycle and billion, or quadratic (power of two) and quadrilateral (four-sided). Find other examples where different word parts mean the same, like bicycle and dipole (both bi- and di- meaning 2). How would you redesign language to be more consistent?

Today your mission is to…

Browse a gallery of factorization diagrams below, and then make your own! Factorization diagrams show numbers in terms of their prime factors. This helps you to analyze the anatomy of numbers so you can get insights into their structure.

Ready, Set, Go

Search the web for factorization diagrams, or start with our overview of features. You have several math, art, and learning choices, for example: do you show all factors or just some? How do you tackle prime numbers? Do you always show the same factor with the same elements of art (shape, color, line, tone) or do you change the elements? Do you make your diagram straightforward or puzzling?

Count on Monsters by Richard Evan Schwartz (2003)

• Math: Geometric (triangle for 3) and abstract representations

• Art: Color, shape, and portraits of primes as unique creatures

Learning: Puzzling, perplexing, and unpredictable; abstract: the final quantity is not shown, just the factors

Primitives by Alec McEachran (2008)

• Math: Nested sets show factors of factors

• Art: Color, size, and composition of circles

Learning: Recognizable symmetric patterns, subitizing, systematic and orderly; pictures of numbers are fully predictable from smaller numbers

Diagrams by Brent Yorgey and friends (2012)

• Math: Composite shapes (circles, polygons, fractals)

• Art: Size, shape, compositions of small circles, color in some versions

Learning: Recognizable symmetric patterns, subitizing, systematic and orderly; color-coding has some perplexity; pictures of numbers are fully predictable from smaller numbers

Number and letter art by Mark Gonyea (2014)

• Math: A mix of symmetry, arrays, fractals, polygons, etc.

• Art: Exploring one element per poster, such as line, circle, or dot

Learning: A balance of order and randomness: pictures of numbers mostly look symmetric, but are not predictable from past examples

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Option 1: See then make

Look at the factorization diagrams together. Ask kids to see what they notice. Accept all answers, even if they are about superficial features of the art. Ask children to find similarities, differences, and patterns:

• Cover up a number on one of the predictable diagrams and try to guess what it should look like.

• Find numbers that look similar. Why did the artist choose to make them similar? It may be an artistic choice, or there may be math reasons.

• Reproduce a part of a diagram in a different medium, from Minecraft to found treasures. How does the medium influence what you do with math?

Are there numbers you would do differently? Remix some of the diagrams.

Option 2. Make then see

Invite kids to make portraits of quantities from 1 to 10, to 25, or to whatever number does not seem tedious. You may need to explain the difference between a symbol and a portrait, a number and a quantity. Portraits can be drawings or arrangements of objects, in physical or virtual worlds. Do that several times, together. Go for humour, pattern, beauty, order, and variety. Use LEGO, natural materials, found treasures, pencils, action figures, or whatever other medium your children enjoy. Take pictures as you go along. When children created some of their own diagrams and there is a pause, show them other people’s diagrams.

Toddlers

Use small numbers, up to 12 or so. Prepare beautiful stuff in just a few quantities at a time, on empty sheets of paper. Invite your toddler to arrange each group nicely, and play along yourself. You can trade object for object, but keep quantities. Your child may need to play with the stuff freely before doing that. Take pictures of arrangements.

Young children

Draw, use stickers or cutouts, use art software, make collages out of found objects. Invite kids to make connections between several numbers, for example, use the same star shape whenever the factor of five appears. Turn it into a puzzle: guess what the next number should look like from all the previous numbers. Turn it into a domino sorting game.

Older children

Explore ways to test divisibility ahead of time. For example, how do you find if a number is divisible by 3? By 5? An easier problem is to skip-count by a given number, so you find all the numbers divisible by it. Then you can use color, shape, or other elements of art to make these numbers look different, such as orange for multiples of 5 in Letters by the Number. Check out the Sieve of Erathosthenes animations. Use programming or modeling software to assist your explorations. Make the music analog of factorization diagrams: polyrhythms.

How is this multiplication?

Factorization diagrams show multiplication and division in one image. Now you see a number divided into factors; now you see factors coming together with the number.

Inspired by algebra

The most fundamental algebraic idea of factorization diagrams is reversibility. The diagrams promote the holistic understanding of multiplication and division as the reversible, connected, complementing actions.

Divisibility is the first step to building algebraic structures known as groups. Divisibility makes you focus on overarching patterns and algebraic “Whys”: numbers divisible by 4 are divisible by 2 twice over, numbers divisible by 10 always end in zero, numbers divisible by 3 have the sum of their digits also divisible by three, and so on.

Frequently Asked Question

In daily life, what are some of the very first comments we might make that can lead to more observations about multiplication?

When it comes to good chats with kids, go for:

• Your own sincere interest. Kids sense emotions. It is okay to share fear or frustration as your emotions, as long as you are open about your feelings, and have a bit of fascination to go with them, as Blake does in The Tyger poem below. Easy start: notice something cute, funny, or nice that also has multiplication, like a kitten, a flower, or your favorite spaceship.

• Details and particulars. These make for good stories. Easy start: make a cheat list of 3-5 particular terms that go with multiplication, for example: array, fractal, symmetry, scaling, combinations. Use these words to make your multiplication chat less generic and more detailed.

Play and adventure. Connect the conversation with children’s actions or favorite imaginary worlds. Easy start: find outdoor examples of multiplication large enough to climb, or use toys to pretend-play jumping, climbing, and otherwise adventuring in or on smaller objects.

By GapingVoid

Words

Factors, dividers, divisibility, factorization, group

Scavenger hunt

Find well-arranged quantities in nature or culture. Playing cards have groups of symbols. Windows and muffin tins come in arrays. Many folk dances, such as square dances, arrange and rearrange participants in groups that divide the total number. The children’s verse “The ants are marching one by one… two by two… three by three…” is a puzzle about different arrays made of the same ant army by different divisors.

By Traditional Music Co

Course links

• All course tasks
• Multiplication Lounge: our open forum for questions, comments, and ideas
• Course participants survey

Today your mission is…

Make geometric art based on skip-counting formulas.

Ready, Set, Go

There is little room for creativity in paint by number. In contrast, design by formula is creative, because you can tweak, remix, and change your formulas. Algebra has variables, and variables vary! In these activities, you will create beautiful designs, in a very meditative process.

Option 1. Waldorf stars

Draw a circle and mark its circumference with ten evenly spaced dots, numbered 0 to 9. Why? These are all the possible remainders when you divide by 10. Remember this fact for later.

Next, choose the number you will multiply by. Let’s use 3 as an example. Find dot 3. From there, go by the dots, skip-counting by 3 clockwise, or by the numbers, using multiplication tables. Your second dot is three dots over, or 3 * 2 = 6. Draw a line connecting 3 and 6. Your third dot is 3 * 3 = 9, so draw a line from 6 to 9.

By HowTheSunRose

Your fourth dot, three dots over from 9, is 2. So connect 9 to 2, even though 3 * 4 = 12 - because from now on, you will go by the remainders, throwing the tens of the number away. Keep going around the circle for a while. What do you notice about your path? Try with other numbers, to observe other paths. Experiment with color patterns.

Option 2. Spirolaterals

For this activity you will need graph paper. You can make graph paper with a ruler, print out custom graph paper from this site, or use square tiles and chalk, outdoors. The younger your kids, the larger you should make your grid cells.

Pick your favorite number; we’ll use 3 again. Write down the first few results from times 3 table. We’ll use 3 * 1 = 3, 3 * 2 = 6 and 3 * 3 = 9, so our numbers are 3, 6 and 9. On your graph paper, draw a line 3 cells long. Turn right, as if driving along the grid, and draw a line 6 cells long. Turn right again and draw a line 9 cells long. Turn right and draw a line 3 cells long, and so on. What do you notice about your path? Young kids can produce the same pattern visually, without knowing times tables or even counting beyond 3: go 3-step once, twice, thrice, then repeat.

By SharynIdeas

Can you get different shapes if you use a different times table or a different number of multiplication facts from it? Experiment and observe! The number of facts you use is called the order. So an order-4 spirolateral for the times 3 table uses numbers 3, 6, 9, 12. An order-5 spirolateral for the times 2 table uses numbers 4, 6, 8, 10, 12.

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?

• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

For visual activities, the best way to explain the rules is to show them. Count out loud as you make spirolaterals or stars. Use colors. With spirolaterals, kids get turned around and disoriented. You may need to help them stay oriented by turning paper for them, or announcing turns out loud. Try driving a pretend car along your grid paper, with a mark for its right-hand turn signal.

Toddlers

Use yarn with peg stars, and times tables with small numbers, like 2s and 3s. Make giant spirolaterals (painters’ tape is perfect for this) and invite your toddler to walk them, or to drive a toy car along them. Even if your kid can’t count yet, the geometry sinks in!

Young children

Experiment with changes in formulas, colors, or types of grids. Help kids implement their “What if?” ideas and test conjectures. What if we turn left rather than right? What if we use triangular or hexagonal grids? (Print them out here.) Do different numbers always give you different grids? Can you work backwards: draw a spirolateral first, and then figure out which formula goes with it? Or have a grown-up figure it out...

Older children

Try making stars and spirolaterals in different bases, such as binary or base five. For the stars, mark the circle from 0 to the number one less than your base, and use remainders from dividing by the base. Try to build in virtual worlds like Minecraft, or program in modeling software, such as GeoGebra or Scratch. Here is an example in Scratch.

How is this multiplication?

Finished stars and spirolaterals visualize times tables. The process of making them is all about thoughtful skip-counting with several inviting bridges to algebraic and geometric patterns.

Inspired by algebra

The stars build bridges to divisibility patterns and modulo operations. The original stars throw away the tens of the number, that is, operate in modulo 10. You can change that variable and work in different bases.

Spirolaterals have three variables: the angle of your turns, the times table, and the number of results from that times table (the order). You can play around with these variables, or change them systematically to catch patterns.

Frequently Asked Questions

If I do not tell my child that a particular activity is about multiplication, how will my child connect it to multiplication on her own? What if she enjoys the activity, but doesn’t realize how or why it is multiplication? How do I check for this understanding? And what do I do if the connection is missing?

There are always more connections than anyone realizes. We’ve been leading some of these activities for years. Yet new bridges to ideas keep coming up, from a parent on a forum, a kid in a math circle, or a researcher at a conference. There number of connections from any given activity to ideas or applications is literally infinite!

So, we are all in the same boat here: there are infinitely many connections, and we only notice some. First, ask children what are their favorite things about the activity. For example, your child may like unexpected results (such as the shapes of spirolaterals) or repeating patterns (such as skip-counting on stars going over the same star again and again). If you know of a neat connection to this favorite idea, you can share it. If not, you can research it together with the kid, or on your own time, and share later.

Connections only build over time. Put up a picture from the activity where you will see it often: a frame in a busy corridor, your bathroom mirror, or your fridge. As you seek more and more connections, add words and pictures reminding you of them. Your activity is a pioneer, pitching the first tent at the site of the future city of ideas.

Words

Variable, spirolateral, star, divisibility, modulo operations, base, order

Scavenger hunt

This time, create surprise mart for others to find and admire. Where can you leave your mathematical masterpieces? Busy places like parks and playgrounds work very well. You can create intricate designs with found objects. For the spirolaterals, collect sticks, pebbles, leaves, or pinecones that are about the same length. You can yarn bomb a fence with spirolaterals, or make chalk or tempera stars on manhole covers. Take a picture of your creation and share with us!

By Demakersvan

Course links

• All course tasks
• Multiplication Lounge: our open forum for questions, comments, and ideas
• Course participants survey

Today your mission is to…

Use rotational symmetry to create multiple copies of an object. And, since the number of copies can be changed from one to infinity, you will end up with a multiplication model AND a model of a variable.

Ready, Set, Go

Option 1: Mirror Book

Lay two plain rectangular mirrors on top of one another, face to face. Connect them at one edge with duct tape. You can usually find small mirrors in dollar stores and school supply or craft stores, and larger mirror tiles in home improvement stores. Put interesting things – like favorite toys or your face – inside the open mirror book. Change the angle between the mirrors and admire the greatest show in math!

Option 2: Snowflake

Fold a piece of paper through its center repeatedly, then sketch and cut out a snowflake. Real snowflakes always have 3, 6, or 12 sides, but abstract mathematical snowflakes can have any number, even 5 or 7, if you are clever about folding them. Learn different folds and make a snowflake multiplication table!

By Bloke School

Your forum response

• What multiplication ideas do you see in this topic? How about ideas inspired by algebra?
• Did you use this with kids or students? How did it go? What did they say and do? What questions did they ask?

How to help your child to get started

Children are immediately intrigued by mirror books. Let kids explore the books first. Younger children might bring favorite toys and pile them between the mirrors. Older children or those with more experience with the mirror book will build elaborate designs, act out stories, write coded messages and create puzzles.

Young children might need help with folding paper and cutting the designs. You can minimize frustration by asking the child to draw the design on a folded snowflake for you to cut out. It works especially well if you make a giant snowflake. Some children might not want to cut any designs at first. Instead, they might prefer to fold, then make one or two straight cuts. Go for it! It’s a perfect opportunity to admire how even very simple steps, repeated enough times and mirrored, can create complex patterns.

Toddlers

Make giant snowflakes out of newspapers or wrapping paper, for easier cutting. Your toddler can guide your hand with scissors, or you can wrap your hand around your child’s hand to help. Find full-body mirrors at an angle in dressing rooms, or make your own giant mirror books out of large door mirrors, using any door and its adjacent wall.

Young children

Combine two mirror books and let the kids explore infinity. If playing with paper, fold the paper once, cut or punch out a small circle, and ask the child to predict how many circles will there be once you unfold the snowflake. What if you fold the paper twice? Three times? Four times? Draw or write numbers, letters and words inside the mirror book, or cut them out of snowflakes. How many symmetries, hidden in these shapes and symbols, can your kids discover?

Older children

Become a puzzle maker. With a mirror book, ask if it is possible to make a square (or any other shape) with one toothpick. Or challenge the child to turn a slice of pizza into a whole pizza. How can we have our (whole) cake and eat it too? Fold paper for the snowflake and ask your child to cut it in such a way that when unfolded, there would be only one circle. Invite the child to create new puzzles! Model radial symmetry in software, such as GeoGebra or Scratch.

How is this multiplication?

Rotational symmetry connects counting and grouping, visual and sequential, algebra and geometry. That’s how the nature multiplies the same pattern (by the same genetic code) across all arms of an octopus, or all petals of a flower. You can add objects inside the mirror book, or cut to the folded snowflake, one by one. The reflections multiply that group you create into all the copies, all at once.

By Ernst Haeckel

Inspired by algebra

The mirror book has two variables: the number of objects and the number of reflections. But you can take a more continuous stance, if only you take the angle between mirrors as your variable. With some angles, you can have fractional reflections! With snowflakes, the number of sectors and the number of cuts are the two variables. But cuts on edges or the middle don’t get multiplied by the full number of folds, so the system is more complex than just multiplying cuts by sectors.

Frequently Asked Question

I showed a mirror book to my child and tried to explain what to do. He didn’t seem interested. Then he dumped a pile of small toys into it, looked for exactly one second and ran away. What did I do wrong?

If at all possible, we encourage you to make all your math manipulatives, including a mirror book, together with your child. There is a lot of hands-on mathematics in the process of making things. Besides, children are more eager to play with the toys they made themselves or helped making.

There are, of course, situations when making a math toy with a child is not possible or practical. In this case, make sure to give your child plenty of time to free play with the toy. Do not try to provide any directions, explanations, or rules at this point.

Leave the mirror book (or any other math play object) where a child can easily see and use it. Children will come back to this activity, if only for a few seconds. Observe and record (if possible) their play and its results. You will soon notice how their play, however short, becomes more complex and purposeful.

Finally, next time your child has a playdate, let children play with the mirror book. You will be amazed at the complexity of designs and the depth of discussions that accompany this activity.

Words

Radial symmetry, reflection, rotation, angle, variable

Scavenger hunt

Radial and reflection symmetry are beautiful! It is easy to find, but not that easy to capture. Look for examples of it in nature (hint: near bodies of water on a calm day), architecture (all that glass and polished highly reflective surfaces), and at home. See how many multiplication facts you can find or create.

Course links

• All course tasks
• Multiplication Lounge: our open forum for questions, comments, and ideas
• Course participants survey