## Exploding Dots

### 4.2 Piles and Holes; Dots and Antidots

Lesson materials located below the video overview.

See how Goldfish & Robin, “When Young Minds Collide,” explain this lesson: Kids Explain Math for Kids.

My disbelief in subtraction comes from another story that isn’t true. Briefly, it goes as follows.

As a young child I used to regularly play in a sandbox. And there I discovered the positive counting numbers as piles of sand: one pile, two piles, and so on. And I also discovered the addition of positive numbers simply by lining up piles. For example, I saw that two plus three equals five simply by lining up piles like this.

I had hours of fun counting and lining up piles to explore addition.

But then one day I had an astounding flash on insight! Instead of making piles of sand, I realized I could also make holes. And I saw right away that a hole is the opposite of a pile: place a pile and a hole together and they cancel each other out. Whoa!

Later in school I was taught to call a hole “$$-1$$”, and two holes “$$-2$$,” and so on and was told to do this thing called “subtraction.” But I never really believed in subtraction. My colleagues would read $$5-2$$, say, as ”five take away two,” but I was thinking of five piles and the addition of two holes. A picture shows that the answer is three piles.

Yes. This gives the same answer as my peers, of course: the two holes “took away” two of the piles. But I had an advantage. For example, my colleagues would say that $$7-10$$ has no answer. I saw that it did.

$$7-10 =$$  seven piles and ten holes $$=$$ three holes $$= -3$$.

Easy!

Subtraction is just the addition of the opposite.

(By the way, I will happily write $$7-10$$ as “$$7 + \; -10$$.” This makes the thinking more obvious.)

Let’s go back now to our dots-and-boxes machines, the $$1 \leftarrow 10$$ machine in particular.

There we work with dots, which I’ve been drawing as solid dots.

We now need the notion of the opposite of dot, like a hole is the opposite of a pile. I’ll draw the opposite of a dot as a hollow circle and call it an antidot.

Like matter and antimatter, and like $$1$$ and $$-1$$  in piles and holes, which each annihilate one another when brought together, a dot and an antidot should also annihilate too – POOF! – when brought together to leave nothing behind.

And we can conduct basic arithmetic with dots and antidots, just like we did with piles and holes.

Aside: By the way, some students prefer to call the opposite of a dot a tod. Can you guess what made them think of that name?

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