Ok, that’s a neat connection, and maybe we can prove it by drawing some right triangles (of course) and re-arranging terms.

But what does it mean?... Read article

]]>Ok, that’s a neat connection, and maybe we can prove it by drawing some right triangles (of course) and re-arranging terms.

But what does it mean?

Rather than the Law of Sines, think of the Law of Equal Perspectives:

**Each angle & side can independently find the circle that wraps up the whole triangle.** This connection lets us start with one angle and work out facts about the others.

I occasionally frighten the neighborhood children by unchaining the mutant gorilla in my front yard.

The kids run screaming, telling different stories of what they’ve seen:

“Alice claims the monster was 20 feet tall, but we all know she exaggerates by doubling. And Billy’s a bit of a crybaby, and said it was 30 feet tall. Charlie’s fairly no-nonsense and said the beast was exactly 10 feet high.”

If we know a kid’s “exaggeration factor” and the size they claim, we can deduce the true size of the monster. (Furious George has a name, you know.)

Even better, we can predict what *other* kids might have said: If Alice claimed it was 40 feet, what would Charlie have said?

What do kids running from monsters have to do with triangles? Well, every triangle is trapped inside its own Monster Circle:

Whatever triangle we draw, there’s *some* circle trying to gobble it up (technically, “circumscribe it”). Try this page to explore an example on your own.

Now here’s the magic: just knowing a single angle and its corresponding side, we can figure out the Monster Circle.

Here’s how. Let’s say we have a triangle like this:

We don’t know anything except the angle A (call it 30 degrees) and the length of side a (call it an inch).

First off: is this the correct drawing of the triangle? Probably not! We don’t know the other sides, so this is equally valid:

It still has the same angle (A = 30 degrees) and the size of the base hasn’t changed (still one inch).

What if we start drawing more possibilities?

Whoa. From A‘s point of view *all* the possible triangles that have “A=30 degrees, a=1 inch” are on this circle. Whatever B and C end up being, they need to pick an option from this circle.

Similarly, we can argue this from the other perspectives:

- We can lock down angle B and side b, and trace out a circle of possibilities
- We can lock down angle C and side c, and trace out a circle of possibilities

This is the meaning of the Law of Sines: each angle unknowingly generates the same circle as the others.

(How do we *prove*, not just see that the possibilities lie on a circle? That’s the Inscribed Angle Theorem, for another day.)

We’ve figured out that there *is* a Monster Circle, now let’s see how big it is. Um… how?

Remember, we can slide around the circle and keep A (30 degrees) and a (1 inch) the same. So let’s slide until we make a right triangle:

Ah! Now we can use sine. Remember that sine is the percentage height compared to the max possible. The max possible height is the full diameter (d) of the Monster Circle.

(Why is a 90-degree angle across from the full diameter? Draw a square inside the circle, touching the sides. It must be symmetric, the diagonals pass through the center along the diameter, and are opposite a 90-degree angle.)

With a little re-arranging, we get:

Using the same logic for the other sides, we get:

In a way, sin(A) is the “exaggeration factor” that converts the size the angle measured (a) to the full diameter (d). Each angle is a different kid, and some really misjudge the size of the full circle based on what they see. (90-degrees is right on target.)

In our example above, A is 30 degrees and a is 1 inch.

We can calculate the diameter pretty fast. First, we get the sine:

That means our length a is 50% of the max height, so the full diameter must be 2 inches.

This isn’t enough to figure out the triangle by itself. Let’s say angle B comes along and says it is 45 degrees. How long is b?

Well,

which means that b is .707 of the max diameter. Therefore,

Previously, I would plug numbers into the Law of Sines formula and chug away algebraically. Now I can think in terms of the Monster Circle: “Ok, I have the max diameter. I take the sine, and get the fraction of the max diameter for that side.”

Most books write the formula with sin(A) in the numerator. It might read better “Sine A over A” but it distorts the conclusion that frac(a)(sin(A)) is the size of the circle.

Put the concept in your own words. The “Law of Sines” is a generic description of what’s in the formula, but the “Law of Equal Perspectives” explains what it means:

- All parts of the triangle have a perspective on the whole
- Sine is the “exaggeration factor” that scales up an individual side to the full diameter. (Sine is the percentage of the max possible, and we divide by it.)

Happy math.

Technically, because B is over 90 degrees, we can’t ever spin it and have either A or C be a right angle (if we could, the triangle would have over 180 degrees).

What to do? Realize the 180-degree complement of B (call it B·) acts like a stand-in on the other side:

B· has the same sine as B, which should make sense: they both point upwards along the same trajectory. To help us sleep better at night, we start with B· in the right-angle setup:

and get to the same conclusion as before. Phew.

However, the fact that B and B· can be swapped can lead to problems.

If I have a triangle where I know A (30 degrees) and a (1 inch), and then say b is 1.5 inches, what can you deduce?

The max diameter is 2 inches as before, so

Unfortunately, there are two angles with that sine value: a calculator says sin^{-1}(.75) = 48 degrees, but 180 – 48 = 132 degrees would work too (more details).

Also, the triangle may not be possible given a hypothetical scenario. If I say b is 3 inches, you know something’s amiss. The max diameter was already calculated to be 2. Even a 90-degree angle, the best possible, could only have a side of 2 inches.

ADEPT Topic | Law of Sines |
---|---|

Analogy | Imagine kids describing the same monster with varying degrees of exaggeration. |

Diagram | |

Example | Suppose A=30 and a=1 inch. Since sin(A) = 0.5, the Monster Circle is 1 / 0.5 = 2 inches wide. Given another angle, I can figure out the length of its side. If B = 45 degrees, then side b takes up sin(45) = .707 of the diameter, and is 1.414 inches. |

Plain-English | Any angle + side can deduce the size of the wrapping circle. |

Technical |

**Make explanations ADEPT: Use an Analogy, Diagram, Example, Plain-English description, and then a Technical description.**

Here’s how to teach yourself a difficult idea, or explain one to others.... Read article

]]>**Make explanations ADEPT: Use an Analogy, Diagram, Example, Plain-English description, and then a Technical description.**

Here’s how to teach yourself a difficult idea, or explain one to others.

Most new concepts are variations, extensions, or combinations of what we already know. So start there!

In our decades of life, we’ve encountered thousands of objects and experiences. Surely *one* of them is vaguely similar to this new topic and can be the starting point.

Here’s an example: Imaginary numbers. Most lessons introduce them in a void, simply saying “negative numbers can have square roots too.”

Argh. How about this:

- Negative numbers were distrusted until the 1700s: How could you have
*less*than nothing? - We overcame this by realizing numbers could exist on a number line, allowing us to move forward or backward from zero.
- Imaginary numbers express the idea that we can move upwards and downwards, or
*rotate*around the number line.

Instead of just going East/West, we can go North/South too – or even spin around in a circle. Neat!

Analogies are fuzzy, not 100% accurate, and yet astoundingly useful. They’re a raft to get across the river, and leave behind once you’ve crossed.

We often think diagrams are a crutch if you aren’t macho enough to directly interpret the symbols. Guess what? Academic progress on imaginary numbers took off only *after* the diagrams were made!

Favor the easiest-to-absorb explanation, whether that comes from text, diagram, or interpretative dance. From there, we can work to untangle the symbols.

So, here’s a visualization:

Imaginary numbers let us rotate around the number line, not just move side-to-side.

Starting to get a visceral sense for what they can *do*, right?

Half our brain is dedicated to vision processing, so let’s use it. (And hey, maybe for this topic, twirling around in an interpretative dance would help.)

Oh, now’s our chance to hit the student with the fancy terminology, right?

Nope. Don’t tell someone the way things are: let them experience it. (How fun is hearing about the great dinner I had last night? The movie you didn’t get to see?)

But that’s what we do for math. “Someone smarter than you thought this through, found out all the cool connections, and labeled the pieces. Memorize what they discovered.”

That’s no fun: let people make progress themselves. Using the rotation analogy, what happens after 4 turns?

How about 2 turns? 4 turns clockwise?

If you genuinely experienced an idea, you should be excited to describe it:

- Imaginary numbers seem to point North, and we can get to them with a single clockwise turn.
- Oh! I guess they can point South too, by turning the other way.
- 4 turns gets us pointing in the positive direction again
- It seems like two turns points us backwards

These are all correct conclusions, just not yet written in the language of math. But you can still reason in plain English!

The final step is to convert our personal understanding to the formal notation. It’s like sharing a song you’ve made up: you can hum it to yourself, but need sheet music for other people to use.

Math is the sheet music we’ve agreed upon to share ideas. So, here’s the technical terminology:

- We say
*i*(lowercase) is 1.0 in the imaginary dimension Multiplying by

*i*is a 90-degree counter-clockwise turn, to face “up” (here’s why). Multiplying by*-i*points us SouthIt’s true that starting at 1.0 and taking 4 turns puts us at our starting point:

And two turns points us negative:

which simplifies to:

so

In other words, *i* is “halfway” to -1. (Square roots find the halfway point when using multiplication.)

Starting to get a feel for it? Just spitting out “i is the square root of -1″ isn’t helpful. It’s not explaining, it’s *telling*. Nothing was experienced, nothing was internalized.

Give people the chance to make an idea their own.

I used to be satisfied with a technical description and practice problem. Not anymore.

ADEPT is a checklist of what I need to feel comfortable with an idea. I don’t think I’ve actually learned a topic unless I have a metaphor that ties everything together. Here’s a few places to look:

- Analogy – ?
- Diagram – Google Images
- Example – Khan Academy for practice problems
- Plain-English – Forums like /r/math or Math Overflow
- Technical – Wikipedia or MathWorld

Unfortunately, there aren’t many resources focused on analogies, especially for math, so you have to make your own. (This site exists to share mine.)

It seems logical to assume we can present facts in order, like transmitting data to a computer. But who actually learns like that?

I prefer the blurry-to-sharp approach to teaching:

Start with a rough analogy and sharpen it until you’re covering the technical details.

Sometimes, you need to untangle a technical description on your own, so must work backwards to the analogy.

Starting with the technical details:

- Can you explain them in your own words?
- Can you solve an example problem, describing the steps in your own words?
- Can you create a diagram that represents how the concept fits together for you?
- Can you relate the concept to what you already know?

With this initial analogy, layer in new details and examples, and see if it holds up. (It doesn’t need to be perfect, but iterate.)

If we’re honest, we’ll admit that we forget 95% of what we learn in a class. What sticks? A scattered analogy or diagram. So, make them for yourself, to bootstrap the rest of the understanding as needed.

In a year, you probably won’t remember much about imaginary numbers. But the quick analogy of “rotation” or “spinning” might trigger a flurry of recognition.

I’m wary of making a contrived acronym, but ADEPT does capture what I need to internalize a new concept. Let’s stop being shy about thinking out loud: does a fact-only presentation really work for you? What other components do you need? I have a soft, squishy brain that needs the connecting glue, not just data.

Scott Young uses the Feynman Technique to explain concepts in everyday words and work backwards to an analogy and diagram. (Richard Feynman was a world-class expositor and physicist, and one of my teaching heroes.)

Prof. Barb Oakley runs an excellent, free course on Learning How To Learn. I was honored to do an interview with her for the class:

Click to watch the interview — I recommend the full course. The first session had over 180,000 students and was a great success.

Beyond any technique, raise your standards to find (or create) explanations that truly work for you. It’s the only way to have concepts stick.

Happy math.

“BE” is a nice prefix for the style to use when teaching:

Brevity is beautiful.

Empathy makes us human. Use your natural style, relate to common experience, and anticipate questions in your explanation.

I’ve yet to complain that a lesson respected my time too much, or related too well to how I thought.

ADEPT is like a nutrition label for an explanation: what are the key ingredients?

Concept | Euler’s Formula |
---|---|

Analogy | Imaginary numbers spin exponential growth into a circle. |

Diagram | |

Example | Let’s figure out the value of `3^i` . (It’s on the unit circle.) |

Plain-English | Raising an exponent to an imaginary power spins you on the unit circle. The same destination can be written with polar (distance and angle) or rectangular coordinates (real part and imaginary part). |

Technical |

Concept | Fourier Transform |
---|---|

Analogy | Like filtering a smoothie into ingredients, the Fourier Transform extracts the circular paths within a pattern. |

Diagram | Smoothie being filtered: |

Example | Split the sequence `(4 0 0 0)` into circular components: |

Plain-English / Technical |

Concept | Distributed Version Control |
---|---|

Analogy | Distributed Version Control is like sharing changes to a group shopping list with your friends. |

Diagram / Example | |

Plain-English | We check out, check in, branch, and share differences (“diffs”). |

Technical | `git checkout -b branchname` `git diff branchname` |

Combine ingredients with your own style. Steps might merge, but shouldn’t be skipped without a good reason (*“Zombies coming, no time for biochem, use this serum for the cure.”*). The site cheatsheet has a large collection of analogies.

What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?

I’d say it’s an eye for simplification, capturing the essence of an idea.

For example, let’s say we want to understand Ed O’Neill:

A literal-minded artist might portray him like this:

While the technical skill is impressive, does it really capture the essence of the man?... Read article

]]>What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?

I’d say it’s an eye for simplification, capturing the essence of an idea.

For example, let’s say we want to understand Ed O’Neill:

A literal-minded artist might portray him like this:

While the technical skill is impressive, does it really capture the essence of the man? Look at his eyes in particular.

Wow! The cartoonist recognizes:

The unique shape of his head. Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect.

The wide-eyed bewilderment. The whites of his eyes, the raised brows, the pursed lips – the cartoonist saw and amplified the emotion inside.

So, who really “gets it”? It seems the technical artist worries more about the shading of his eyes than the message they contain.

Think about the first numbers, the tally system:

I, II, III, IIII …

Those are… drawings! Cartoons! Caricatures of an idea!

They capture the essence of “existing” or “having something” without the specifics of what it represents.

Og the Cavemen Accountant might have tried drawing individual stick figures, buffalos, trees, and so on. Eventually he might realize a shortcut: draw one buffalo symbol to show the type, then a line for each item. This captures the essence of “something is there” and our imaginations do the rest.

Math is an ongoing process of simplifying ideas to their cartoon essence. Even the beloved equals sign (=) started as a drawing of two identical lines, and now we can write “3 + 5 = 8″ instead of “three plus five is equal to eight”. Much better, right?

So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come in *after* we’ve seen the essence.

We agree that multiplication makes things bigger, right?

Ok. Pick your favorite number. Now, multiply it by a random number. What happens?

- If that random number is negative, your number goes negative
- If that random number is between 0 and 1, your number is destroyed or gets smaller
- If that random number is greater than 1, your number will get larger

Hrm. It seems multiplication is more likely to *reduce* a number. Maybe we should teach kids “Multiplication generally reduces the original number.” It’ll save them from making mistakes later.

No! It’s a technically correct and real-life-ily horrible way to teach, and will confuse them more. If the technically correct behavior of *multiplication* is misleading, can you imagine what happens when we study the formal definitions of more advanced math?

There’s a fear that without every detail up front, people get the wrong impression. I’d argue people get the wrong impression *because* you provide every detail up front.

As George Box wrote, “All models are wrong, but some are useful.”

A knowingly-limited understanding (“Multiplication makes things bigger”) is the foothold to reach a more nuanced understanding. (“People generally multiply positive numbers greater than 1, so multiplication makes things larger. Let’s practice. Later, we’ll explore what happens if numbers are negative, or less than one.”)

I wrap my head around math concepts by reducing them to their simplified essence:

Imaginary numbers let us rotate numbers. Don’t start by defining

*i*as the square root of -1. Show how if negative numbers represent a 180-degree rotation, imaginary numbers represent a 90-degree one.The number e is a little machine that grows as fast as it can. Don’t start with some arcane technical definition based on limits. Show what happens when we compound interest with increasing frequency.

The Pythagorean Theorem explains how all shapes behave (not just triangles). Don’t whip out a geometric proof specific to triangles. See what circles, squares, and triangles have in common, and show that the idea works for any shape.

Euler’s Formula makes a circular path. Don’t start by analyzing sine and cosine. See how exponents and imaginary numbers create “continuous rotation”, i.e. a circle.

Avoid the trap of the guilty expert, pushed to describe every detail with photorealism. Be the cartoonist who seeks the exaggerated, oversimplified, and yet *accurate* truth of the idea.

Happy math.

PS. Here’s my cheatsheet full of “cartoonified” descriptions of math ideas.

]]>**Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)**

- Logarithms reveal the inputs that caused the growth
- Exponents find the final result of growth

**Are we talking about the grower’s perspective, or an observer’s?**... Read article

**Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)**

- Logarithms reveal the inputs that caused the growth
- Exponents find the final result of growth

**Are we talking about the grower’s perspective, or an observer’s?**

- e and the natural log are from the grower’s instant-by-instant perspective
- Base 10, Base 2, etc. are measurements convenient for a human observer

In my head, I put the options in a table:

and I have thoughts like *“I need the cause, from the grower’s perspective… that’s the natural log.”*. (Natural log is abbreviated with lowercase LN, from the high-falutin’ *logarithmus naturalis*.)

I was frustrated with classes that described the inner part of the table, the raw functions, without the captions that explained when to use them!

That won’t fly, let’s get direct practice thinking with logs and exponents.

Here’s a typical example of growth:

- From 2000 to 2010, the US GDP changed from 9.9 trillion to 14.4 trillion

Ok, sure, those numbers show change happened. But we probably want insight into the cause: What average annual growth rate would account for this change?

Immediately, my brain thinks “logarithms” because we’re working backwards from the growth to the rate that caused it. I start with a thought like this:

A good start, but let’s sharpen it up.

First, which logarithm should we use?

By default, I pick the natural logarithm. Most events end up being in terms of the grower (not observer), and I like “riding along” with the growing element to visualize what’s happening. (Radians are similar: they measure angles in terms of the mover.)

Next question: what change do we apply the logarithm to?

We’re really just interested in the ratio between start and finish: 9.9 trillion to 14.4 trillion in 10 years. This is the same growth *rate* as going from $9.90 to $14.40 in the same period.

We can sharpen our thought:

Ok, the cause was a rate of .374 or 37.4%. Are we done?

Not yet. Logarithms don’t know about how long a change took (we didn’t plug in 10 years, right?). They give us a rate as if all the change happened in a single time period.

The change could indeed be a single year of 37.4% continuous growth, or 2 years of 18.7% growth, or some other combination.

From the scenario, we know the change took 10 years, so the rate must have been:

From the viewpoint of instant, continuous growth, the US economy grew by 3.74% per year.

Are we done now? Not quite!

This continuous rate is from the grower’s perspective, as if we’re “riding along” with the economy as it changes. A banker probably cares about the human-friendly, year-over-year difference. We can figure this out by letting the continuous growth run for a year:

The year-over-year gain is 3.8%, slightly higher than the 3.74% instantaneous rate due to compounding. Here’s another way to put it:

- From an instant-by-instant basis, a given part of the economy is growing by 3.74%, modeled by e
^{.0374 · years} - On a year-by-year basis, with compounding effects worked out, the economy grows by 3.81%, modeled by 1.0381
^{years}

In finance, we may want the year-over-year change which can be compared nicely with other trends. In science and engineering, we prefer modeling behavior on an instantaneous basis.

I detest contrived examples like “Assume bacteria doubles every 24 hours, find its growth formula.”. Do bacteria colonies replicate on clean human intervals, and do we wait around for an exact doubling?

A better scenario: “Hey, I found some bacteria, waited an hour, and the lump grew from 2.3 grams to 2.32 grams. I’m going to lunch now. Figure out how much we’ll have when I’m back in 3 hours.”

Let’s model this. We’ll need a logarithm to find the growth rate, and then an exponent to project that growth forward. Like before, let’s keep everything in terms of the natural log to start.

The growth factor is:

That’s the rate for one hour, and the general model to project forward will be

If we start with 2.32 and grow for 3 hours we’ll have:

Just for fun, how long until the bacteria doubles? Imagine waiting for 1 to turn to 2:

We can mechanically take the natural log of both sides to “undo the exponent”, but let’s think intuitively.

If 2 is the final result, then ln(2) is the growth input that got us there (some rate × time). We know the rate was .0086, so the time to get to 2 would be:

The colony will double after ~80 hours. (Glad you didn’t stick around?)

Figuring out whether you want the input (cause of growth) or output (result of growth) is pretty straightforward. But how do you visualize the grower’s perspective?

Imagine we have little workers who are building the final growth pattern (see the article on exponents):

If our growth rate is 100%, we’re telling our initial worker (Mr. Blue) to work steadily and create a 100% copy of himself by the end of the year. If we follow him day-by-day, we see he does finish a 100% copy of himself (Mr. Green) at the end of the year.

But… that worker he was building (Mr. Green) starts working as well. If Mr. Green first appears at the 6-month mark, he has a half-year to work (same annual rate as Mr. Blue) and he builds Mr. Red. Of course, Mr. Red ends up being half done, since Mr. Green only has 6 months.

What if Mr. Green showed up after 4 months? A month? A day? A second? If workers begin growing immediately, we get the instant-by-instant curve defined by e^{x}:

The natural log gives a growth rate in terms of an individual worker’s perspective. We plug that rate into e^{x} to find the final result, with all compounding included.

Switching to another type of logarithm (base 10, base 2, etc.) means we’re looking for some pattern in the overall growth, not what the individual worker is doing.

Each logarithm asks a question when seeing a change:

- Log base e: What was the instantaneous rate followed by each worker?
- Log base 2: How many doublings were required?
- Log base 10: How many 10x-ings were required?

Here’s a scenario to analyze:

- Over 30 years, the transistor counts on typical chips went from 1000 to 1 billion

How would you analyze this?

- Microchips aren’t a single entity that grow smoothly over time. They’re separate editions, from competing companies, and indicate a general tech trend.
- Since we’re not “riding along” with an expanding microchip, let’s use a scale made for human convenience. Doubling is easier to think about than 10x-ing.

With these assumptions we get:

The “cause of growth” was 20 doublings, which we know occurred over 30 years. This averages 2/3 doublings per year, or 1.5 years per doubling — a nice rule of thumb.

From the grower’s perspective, we’d compute ln(text(1 billion)/1000) / text(30 years) = 46% continuous growth (a bit harder to relate to in this scenario).

We can summarize our analysis in a table:

Learning is about finding the hidden captions behind a concept. When is it used? What point view does it bring to the problem?

My current interpretation is that exponents ask about cause vs. effect and grower vs. observer. But we’re never done; part of the fun is seeing how we can recaption old concepts.

Happy math.

Here’s how to think about switching bases. Assuming a 100% continuous growth rate,

- ln(x) is the time to grow to x
- ln(2) is the time to grow to 2

Since we have the time to double, we can see how many would “fit” in the total time to grow to x:

For example, how many doublings occur from 1 to 64?

Well, ln(64) = 4.158. And ln(2) = .693. The number of doublings that fit is:

In the real world, calculators may lose precision, so use a direct log base 2 function if possible. And of course, we can have a fractional number: Getting from 1 to the square root of 2 is “half” a doubling, or log_{2}(1.414) = 0.5.

Changing to log base 10 means we’re counting the number of 10x-ings that fit:

Neat, right? Read Using Logarithms in the Real World for more examples.

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