Week 2 Task 3: Zoom and powers

From http://www.numbersleuth.org/universe/

Today your mission is…

Explore the Universe, from its largest to its tiniest objects. Scale analysis helps us to make sense of our complex universe in terms of easy models. It lets us compare magnitudes of essential quantities.

Ready, Set, Go

Start with the powers of two, because doubling and halving is easy even for toddlers.

Trace your hand or your child’s hand on paper, and cut out the shape. This is your unit, your beginning, your 1. (The One?) Go around with this unit and hunt for other objects that are approximately the same size. You can take photos as you go along.

Now double the size - this is your 2 - and find objects that big. You can cut a piece of yarn, or a strip of paper, as long as your 2. Keep doubling your units (4, 8, 16 and so on) and finding objects of about that size. You can arrange examples of objects in a row, with the measuring units next to them. You can also halve your 1 again and again for fractional units: ½, ¼, ⅛…

You can try doing the same with powers of 10, since it’s traditional in science. Do you run out of space quicker or slower with 10s, compared to 2s?

Your forum response

Many children’s stories, such as Alice in Wonderland, feature a hero who magically grows or shrinks. Other stories have characters that are much smaller or much larger than everyone else around them, such as Thumbelina, Hagrid, The Beanstalk Giant. Reading such a story or just looking through the illustrations can be a great lead-in to this activity.

Suggest a game of pretend where you will have a magic wand that can double or halve objects around you.

Toddlers

Your hands are always with you. Keep measuring everything with one or two hands wherever you are. Encourage your toddler to estimate which objects will fit one hand or two, before measuring. Trace or print hands on long strips of paper for your 2, 4, and 8 - rather than using “abstract” strips of paper or yarn.

Young kids

Invite children to multiply or divide units of measure by ten, as they zoom from a human to galaxy and then the other way, to atoms. Use interactives to zoom:

http://www.nikon.com/about/feelnikon/universcale/

http://www.numbersleuth.org/universe/

Older kids

To fit many, many powers of the universe into one picture, invite your kids to use logarithmic scales. For example, your dog is twice as tall as your cat, but you only draw (cartoon) it one unit taller. You are twice as tall as the dog, but you draw yourself one unit taller than the dog, and so on. Here are examples from XKCD: https://xkcd.com/482/ (zoom out to galaxies) and http://xkcd.com/485/ (zoom in to strings).

How is this multiplication?

Read Addition Is Useless, Multiplication Is King: Channeling Our Inner Logarithm article for a discussion of zoom and scales. There is strong evidence that zoom and scale reasoning is inborn, but our culture so far has been imposing counting and linear reasoning. To quote:

We need multiplication, not addition. We need multiplicative number lines, not linear number lines. If we start out with logarithmic intuitions — and the evidence is strong that we do — let’s not shed them, let’s amplify them!

Read Scale Analysis: What Scientific Concept Would Improve Everybody’s Cognitive Toolkit? A favorite quote:

There is a well-known saying: dividing the universe into things that are linear and those that are non-linear is very much like dividing the universe into things that are bananas and things that are not. Many things are not bananas.

Inspired by calculus

On a hot summer day, Californians say the temperature is in the triple digits (Fahrenheit). In basketball, a double is a two-digit score in one of the five category. A great player can get a triple double in a game, that is, two-digit scores in three categories. Scale analysis gives us quick snapshots about the key characteristics of the system.

Algebra = patterns of arithmetic; calculus = patterns of algebra. Let’s look at examples of questions about zoom.