Back to Natural Math® workshops

March 7th, 2003
Multiplication Adventures

Workshop participants and their "multiplication creatures"

This workshop is a part of the parent-children workshop series offered by Maria Droujkova at the wonderful recycle shop The Scrap Exchange in Durham, North Carolina. Here are some ideas for more games!

E-mail Maria with questions, comments, or to sign up for workshops.

"Family tree" activities:

Fractal and combinations models for multiplication

Creating "smiley families"

Draw or make a "child" (we used colorful plastic tubes to make smileys at the workshop). The child has two parents: attach two lines to the child, in the form of V, and attach "mom" and "dad" at the top of the lines. Continue with the next generation.

Child	Child and parents	Child, parents and grandparents	Child and three generations

Smiley family tree construction

Once the tree is built, it can be used for several activities. Start from common words, gradually moving to mathematical terms:

• How many grandparents are there? (point to the "grandparent" level on the tree). What about great-grandparents?
• We have one child, and we have two parents in the first generation from the child, and four grandparents in the second generation, and eight great-grandparents in the third generation... How many people are in the fourth generation? Fifth? How do you know?
• Mathematicians use the term "power" here. For example, we can say "grandparents" or "the second generation from the child" or "two to the second power." Two to the first power (parents) is two. Two to the second power (grandparents) is eight. What is two to the third power? There is a symbol for it:
  2³=8
  
• What generation has eight people? What power of two makes sixteen? Figuring out which generation each quantity means is a lot like logarithms. We can say, "What generation has sixteen people?" or we can write:
  log₂16=4
• Add up all generations up to a certain level, say, "grandparents". Compare to the number in the next level. What do you observe? Is it always the case?

Creating "smiley families"

After some work with the powers of two, introduce the base two system. In the decimal system, the one we usually use, all quantities are represented with powers of ten. In the base two system, everything depends on the powers of two.

• Put some ones and zeros (or any other markers for "count the level" and "do not count the level") next to levels of the tree. The result is "a base two number." What will it be in our (decimal) system? This is a program showing the family tree model of the binary system:
  Small screen    Medium screen    Large screen    
• What quantities can you show on the tree? 2, 4, 8 etc. are easy. What about 5 or 11? Can you show any quantity? How can you transform a (decimal) number into base two?

• Make a ternary (base three) or other trees and investigate them in a similar manner
• A tree is a kind of a fractal. Fractals are pictures with elements similar to the whole picture. This program has more ideas for fractal games:
  Small screen fractal game     Medium screen    Large screen
  
  
• Another type of fractal is a picture made out of smaller copies of itself. Here is a program that uses "picture within picture" fractal for work with binary (base two) system:
  Small screen    Medium screen    Large screen    
  
  
• Trees can be used to find all combinations of certain features. For example, three types of shirts and two types of skirts give 2*3=6 outfits:
  
  In the workshop, we used combinations of different colors of eyes, noses, etc. to make different kinds of smileys.

Powerful, connected, deep metaphors are the very basis of learning mathematics.

Metaphors in these activities:

• Two parents of each person=multiplication by two (How many parents of four grandparents are there? 4*2=8)
• Family tree=binary system
• From generations to quantities=exponents or powers (How many great-grandparents are there? 2³=8)
• From quantities to generations=logarithms (What generation has 16 people? log₂16=4)

"Egg tray" activities:

Array and combinatorics model for multiplication

Find a container with many holes in rows, such as an egg tray or a lab specimen box from The Scrap Exchange. For younger children, you can create a story about it, for example, holes can be "theater seats" or "nests."

The array model is probably the fastest way to represent multiplication visually. For example, to represent 2*3, fill a rectangle two columns wide and three rows long with counters:

Use interesting counters for added artistic dimension. Here is a program showing the array model in its simplest form:

Multiplication helper program

Array model is also used a lot, in many different forms, in human endeavors from bookkeeping to design, from sports to physics... For example, a table can be used to list results of testing pairs of drugs for compatibility. A table can help to choose a combination of a color and a pattern, as in this program:

Small screen Butterfly Times    Medium screen Butterfly Times    Large screen Butterfly Times    

Children may enjoy making their own tables and arrays out of numbers or objects or drawings. They can use tables to investigate multiplication questions such as, "How many designs can you make with two patterns and three colors?" or "How many chimeras can you make with three types of animal heads and four types of bodies?"

A combination table is often used to organize multiplication facts, as you can see on this page. Combination tables are very useful for computing probabilities; here is a page with a probability game about two dice.

• Rows and columns array=multiplication
• Combinations of features=multiplication

"Mirror book " activities:

Symmetry model for multiplication

Tape two mirror tiles together to make a "mirror book". Place some objects between the mirrors, open the mirrors by different angles and observe changes in reflections. This simple design produces fascinating results!

• How many copies of the object (counting the object) can you make with the mirror book? Can you make two? Three? Twenty?
• Open the mirror book at the right angle, using a corner of a standard sheet of paper for measuring. How many copies does it make this way?
• Fold the right angle of the paper in half. Open the mirror book that wide. How many copies are there?
• Open the mirror book to make, say, five copies of objects. Place two circles, three squares, four triangles, etc. inside the mirror book for "times five tables."
  

  Times five "tables"

• Here is a program that helps to explore different types of symmetry and tilings.

In connection with the mirror book, or as stand-alone games, snowflakes and other symmetrical pictures can be quite interesting. They can be used to explore multiplication, number of symmetry lines, angles, and fractions.

• It is easy to make snowflakes with one, two, four... lines of symmetry - why? Can you figure out how to fold paper to make a snowflake with three lines of symmetry? Here is the clue:
  
• Use a hole punch on paper folded in different ways. Can you guess how many holes will there be when the paper is opened? This can be a fun game for car trips and such.
• Here are a couple of "symmetry transformers." They can be used for multiplication games with younger children. Maybe children will come up with their own transformation ideas.
  
• Using some light materials and glue, folded paper snowflakes can be turned into "times tables" with applique.

  Multiplication by four with a "snowflake" design

  "Double snowflakes" - can be used for multiplying by 2, 4 or 8!

  Giant snowlakes!

  Multiplication by eight on a snowflake

Human body is very symmetrical. It may be fun to search for symmetry-based multiplication facts on your body, for example, "two times five" (fingers of two hands). Flowers, sea creatures, buildings, fabric designs, animals and cars can provide more ideas for symmetry games and crafts.

Fashionably symmetric eyewear doubles for "times two" tables

"Feet times" activities:

"Repeated addition " model for multiplication

Make a collection of creatures with different number of legs. The idea is to count in pairs. For multiplication by repeated addition, it is quite useful to have a separate word for the repeated unit, such as "pair." This difference in words, "pair" vs. "two" underlies the difference between the unit being repeated, and the number of times it is repeated.

Attaching four pairs of legs...

Just for the reference, octopuses and spiders have four pairs of legs, and lobsters, crabs and scorpions have five pairs (counting the pincers). The same activity can be done with wheels on different types of vehicles.

Here is the octopus!

After you collect a few animals with different numbers of legs, children can do several activities with them. Use some counters for "shoes." These activities can be fun for children as young as two.

• A ladybug lost its shoes. Some birds found the shoes. How many birds can get shod?
• Three dogs are going on a hike. Prepare enough boots for them. Older children can try to predict how many boots the dogs will need. They can also write down 3*4=12. Younger children can prepare the counters on a mat, doing the multiplication visually. Most often, they will make three groups of four counters. It is better to emphasize "three groups of four" much stronger than counting the total. We want to support multiplicative reasoning here.
• This is a game about lost and found shoes. Help groups of animals find shoes!
  Small screen Lost Shoes game    Medium screen Lost Shoes game    Large screen Lost Shoes game
• This is an interactive "animal legs times table":
  Small screen    Medium screen    Large screen    

Children may enjoy making collections of "intrinsic numbers" devoted to their favorite themes, and then doing math with these images. For example, 1-unicycle wheel, 2-bike wheels, 3-tricycle, 4-car, 5-office chair :-)... 18-big truck!