Back to Natural Math® workshops

April 4 th, 2003
Natural Math for preschools: A teacher workshop

This teacher workshop was done in collaboration with The Scrap Exchange, a creative reuse shop in Durham, North Carolina. Teachers came up with many interesting activities and wonderful games that may be connected with learning mathematics.

"Family tree" activities:

Fractal and combinations models for multiplication

"Family reunion": a binary tree sculpture

Draw or make a "child." 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

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?

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:
  
  

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

Patterns on a grid...

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 also very useful for computing probabilities; here is a page with a probability game about two dice.

Buildings are often arranged in arrays, especially large skyscrapers. As children play with construction toys, they can connect their buildings with tables and multiplication.

A building with the "2*2=4" roof!

Grids can be used to make simple and complex patters. A pattern can be connected with music, where each part of a pattern corresponds to a sound, or to dance where each pattern corresponds to a move. Here is a program that connects visual and sound patterns, from the PBS's new math cartoon "Cyberchase."

Patterns in a building...

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

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."
  

  "Math cheerleader": a symmetric sculpture

• 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.

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.

Symmetric butterflies and multiplication. The same activity can be done with paints.

"Repeated addition" model for multiplication

Make a collection of creatures with different number of legs, or vehicles with different numbers of wheels, or any other "intrinsic numbers" that are a part of the image.

A ladybug: three pairs of legs makes the total of 6, or 2*3

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. As children already have an image of "bunny ears" or "car wheels," they can move from that image to the idea of a number.

Two ends of a telephone, two holes in "eyeglasses" - and eight legs of a spider!

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.

"Lobster times" - antennae, eyes...

After you collect a few objects of the same kind, children can do several activities with them. These activities can be fun for children as young as two.

"Can you prepare enough ears for these three bunnies?" 2*3=6, an Easter multiplication fact...

• 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    
• The idea of "so many per each..." can be carried into the notion of ratio and proportion. For example, children can solve simple puzzles such as "I am making dog sculptures. Here are four ears for my sculptures. How many legs will I need?"

Train proportions: 1car, 3 stripes, 4 wheels; 2 cars, 6 stripes, 8 wheels; 3 cars, 9 stripes, 12 wheels...

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!

Vehicle proportions. A bike: 1 stripe, 2 wheels, 1 person; a car: 2 stripes, 4 wheels, 2 people...

Multiplication can also be connected with familiar shapes, such as a triangle or a square.

4 sides of a square, 6 stripes to a side... 4*6=24

Made-up creatures such as aliens or dragons can be used to move from concrete images strongly connected to each number (e.g. 4 legs of a dog) to more abstract ideas of quantity, since an alien can have any number of legs...

A Chinese dragon in the making: two spikes for every purple stripe, three purple stripes... 2*3=6 But it could be a different length - here is the opportunity for the movement into more abstract quantities not directly related to an image.