#75 Glossary
Glossary
Algorithm
An algorithm is a step-by-step description of actions. You can conveniently reuse algorithms again and again in similar situations, and share them with others. Even very young children develop sophisticated algorithms. We usually describe them as the “going to bed routine” or the “playing house scenario”. Algorithms are challenging to some people because they have many steps (assembling a bike), or because they require guessing (long division), or because they demand special skills (like manual dexterity for tying shoelaces). It often helps to capture algorithms in step-by-step pictures or symbols.
Axis of Symmetry
The axis of symmetry (or line of symmetry) is an imaginary line your mind’s eye can learn to see in some shapes. Imagine folding the symmetric shape along this line. Both sides will match exactly, like two wings of a butterfly, or two clapping hands. Some shapes have no lines of symmetry within them, like one wing of most butterflies, or one hand.
Chirality
Raise your right hand in front of the mirror, wearing a glove or a mitten. Do you notice how your mirror image raises the other, left hand?! If you could reach through the mirror and give your right glove to your mirror image, it would not fit the raised, left hand. An object, such as your hand, is chiral if its mirror image is different from it. You can think of chirality as opposed to symmetry. A square, a butterfly, or any other object with line symmetry won’t be chiral (but one butterly wing is still chiral).
Commutative
Commuting means “moving around”, and commutative means “moving around without making any change.” For example, you can add the numbers two, five, and three in any order: the result will be ten all the same. How do you cook mashed potatoes? First, you boil them, then you mash them. Try changing the order of these operations. What is it you say? Mashing hard, raw potatoes does not work so well? That’s because “mashing” and “boiling” are not commutative. The type of joke called “Russian reversal” plays on noncommutativity: “In Soviet Russia, TV watches you!”
Composition of Functions
When you use the output of one function as the input of another, you are composing functions. Have you been on a field trip to any place that uses mass production? An assembly line for cars or computers may consists of hundreds of individual functions, all done in order. Making a bagel only consists of a dozen steps or so.
Covariance
Covariance, in this book, means that two or more things vary together. For example, bigger packages are also usually more expensive.
Cycle
Cycles are sequences of objects or series of operations that repeat. For example, “winter, spring, summer, autumn” is the cycle of seasons that repeats every year. There are cyclic stories and songs, like “99 bottles of beer.” Many household tasks are cyclic, such as washing dishes – “lather, rinse, repeat.” Cycles relate to recursion and iteration. Not all infinite sequences have cycles; for example, there are no cycles in the digits of Pi.
Domain of a Function
The domain is all allowed inputs. A coffee mug can go into a dishwasher and be expected to come out clean at the end of the cycle. An elephant or a car, on the other hand, cannot; you have to wash them by other methods. A coffee mug belongs to the dishwasher’s domain, while an elephant or a car does not. In related news, you can’t divide by zero (until you start working in calculus; stay tuned for our next book!) – so zero does not belong to the domain of the function that divides by inputs.
Fractal
Fractals blow our minds, because they consist of parts that are exactly the same as the whole thing, if you only zoom in. You can think of fractals as patterns made up of recursions. For example, a fern leaf consists of a stalk and leaflets. If you look closely, each leaflet consists of a stalk and sub-leaflets. And if you look even closer, each sub-leaflet… You get the picture!
Function
A function is a machine that converts values to other values, or finds correspondences between values. Function machines work by rules people make up. The starting values are called input. The converted or corresponding values are called output. The rule must find a single output for each input. Your stove is a function machine: it starts with the input of raw eggs, milk and spices, and makes the output of an omelet. The fantasy machine that starts with the same input and makes either omelets or live chickens is not a function.
Function Machine
The function machine is a traditional metaphor for exploring functions, created circa 1960s. Picture Dr. Seuss’ fancy machines!
Iconic Quantity
Iconic quantities are objects or actions (not symbols) that represent the quantity for you. Play a quick game of associations. Finish these sentences:
- There are four…
- There are seven…
- There are twenty-four…
What came to your mind? Say, four legs of a dog, or seasons; seven days in a week, or colors of a rainbow; twenty-four hours in a day, or Chopin’s preludes. There are many iconic representations for each quantity. But objects with variable or changing quantities, such as petals on a flower or pennies in jars, are not iconic.
Inverse Function
What a function does, its inverse will undo. Of course, some deeds cannot be undone. The function of “baby drops a cup of grapes on the floor” has an inverse, “parental unit crawls around, searching for grapes and putting them back into the cup”. On the other hand, the function of “baby drops a cup of orange juice on the floor” and the function “you break an egg for an omelet” do not have inverses.
Isometric Transformation
Isometric means the result of the transformation is the same exact size. Let’s say you are making stars using a cookie cutter. You are trying to make as many cookies as possible from the rolled out dough, so you turn the cutter this way and that. But no matter how you rotate the cutter, even if you turn it over completely, the size of the star and sizes of all its parts remain the same. But your transformations won’t be isometric if you stretch or shrink your star after it’s cut out to make the dough thinner or thicker.
Iteration
Iteration is applying the same operation repeatedly. For example, a child may break a slice of melon in half to share with her teddy bear, then break a piece of apple in half, break a piece of bread in half and so on, until all food is shared. In particular, recursion applies the same operation to previous results. For example, when you fold a piece of paper in half, then fold the already-folded paper in half again and so on, you use recursion.
Gradient
Take all your spoons – the tiniest dessert spoon, a dinner spoon, a ladle – and lay them out from the smallest to the largest. Turn the faucet on, slowly changing it from a trickle to Niagara Falls. Think of all the books you’ve read, and mentally arrange them from the most to the least boring. Those are all variations of ordering by a change in quantity: size, volume, or the number of boredom-induced yawns. Some of the most beautiful things in life are gradients, such as the change of light at sunrise, or the change of affection as you fall in love. Unless it’s love at the first sight, which is not a gradient but an instant jump in affection!
Operation
Operation is another name for function. It is used when we focus on the rule of the function, rather than inputs, outputs, domain, range and other aspects. In the “Stone Soup” story, new value (delicious soup) is produced from inputs of many different values (ingredients, including the stone itself). This is similar to the mathematical operation of addition, but better. A good soup, much like the eventually-cooperating village in the story, is more than the sum of its parts.
Pattern
Pattern means repetition or systematic change, by rules. You can see patterns: in arrangements of tiles on a bathroom wall; in polka dot prints on your child’s rain boots; in the rhythmic flow of nursery rhymes; or in hopscotch squares on a sidewalk. As always in math, you can make up your own patterns. Fork, plate, knife… Fork, plate, knife… Fork, plate, knife… Fork, plate, knife. Now we have a table set for four, but we’ve also created a pattern by arranging things following the rules. As with rules and functions, “whatever I want to come next” won’t create a pattern – the key is predictability.
Radial Symmetry
Objects with radial symmetry have three or more imaginary lines that radiate from one point, called the center of symmetry. The lines separate the objects into parts. Each two adjacent parts are either identical, or reflections of one another along the line that separates them. Starfish, tulips, daisies, snowflakes, and lace doilies all have such parts regularly arranged around centers.
Range of a Function
The range is all possible outputs. Once the dishwasher is done running, you can get clean dishes or, if your dishwasher isn’t very good, somewhat dirty dishes. But you will not get clean, or dirty, elephants or cars out of a dishwasher. Both clean and somewhat dirty dishes belong to the dishwasher’s range, while elephants or cars do not. Can you guess which machines have clean cars within their ranges?
Recursion
What do “This Is the House That Jack Built,” slicing pizza, and stacking up blocks have in common? They are all examples of recursion: applying the same operation to previous results. This produces patterns that change in a self-similar ways, much like fractals. You put the next block on top of all past blocks; you repeat – and an amazing block tower grows till the baby knocks it down. You slice each piece of your pizza in half; you repeat – until there are enough slices for everyone. You keep on adding verses – while keeping the entire previous rhyme, much like the block tower:
This is the dog that worried the cat
That killed the rat that ate the malt
That lay in the house that Jack built.
Reflection
Reflection is what a shape would see if it looked at itself in a mirror. Two parts of a shape divided by the axis of symmetry are reflections of one another. The exact copy is not a reflection, which you can see comparing mirror images with some webcam or cellphone images. When you raise your right hand, the reflection raises its left hand while the exact copy raises its right hand.
Rotation
Rotation happens when every point on an object moves along a circle, and all these circles center on the same axis or same point. Every time a little girl puts on a tutu and twirls, she performs a rotation. The tips of her fingers make larger circles than her elbows, but all the parts of the girl rotate around the same axis. Going down a slide is not a rotation: elbows and fingers and all other parts move along the slide (hopefully, together).
Rule
A rule is a description of how to perform actions. Mathematics uses systems of objects and symbols, and rules for operating on them. Of course, we and our children encounter rules in everyday lives. Let’s try making math rules! For example, the rule for + is adding quantities together. Try to invent the rule for @ and another one for #. Keep in mind that “whatever I want it to be next time” is not a rule, though kids try to use it as such!
Subitizing
Subitizing is knowing how many objects are in a group without having to count or calculate. Ever played a board game with dice? If so, did you have to count the number of dots on the dice, or did you recognize the number of dots without counting, just by looking? Babies can subitize to three or four, but not count, at birth. Grab a handful of beans or markers: usually, you can’t subitize quantities larger than six exactly, even if you can estimate them pretty closely.
Symmetry
Symmetry means that two or more parts of an object are the same or similar. In physical space, symmetry leads to balance; in art, it defines order and beauty; and among people, it signifies fairness, serenity, or harmony. Slight lack of symmetry adds interest; for example, human faces normally have slightly different sides.
Tessellation
Tessellation is a pattern of repeating shapes that can cover all space with no gaps or overlaps. You can see tessellations in tiled floors, chevron rugs, soccer balls, or pineapples.
Unitizing
Unitizing is a way of thinking where multiple objects or measures are considered together, as one whole thing. For example, you can think of a dozen eggs as a unit – a full carton. You need to unitize to understand the modern number system. For example, to read “253” you need to see hundreds and tens as things that you can count, just as you count ones: two hundreds, five tens, and three ones. More broadly, you need to unitize to understand very large and very small objects and quantities in the universe, such as the rather puny limits of human attention (1-2 objects at once), human instant memory (5-7 objects at once), and human magnitude perception (3-4 orders of magnitude at once).
