Fractions, Pluto and the capital I got wrong at dinner

"Dad, what's three-fourths divided by one-half?"

It was 7:40 on a Tuesday. I was holding a dish towel and a pot I'd been "letting soak" since Sunday. Maddie, who's ten, had her pencil up like a conductor waiting for the orchestra. Sam, thirteen, was across the table with an essay due Thursday and the expression of a man who'd heard me say "easy" before.

I said "easy."

I always say easy. It's the word I use right before I'm not. Jen, who teaches middle school and could have answered from the next room, kept loading the dishwasher and said nothing at all. That's how you know she's enjoying it.

1. How do you divide fractions?

They think I know: Instantly, the way I know where the remote is.

What I actually know: There's a rhyme. Something about a flip. I tried "flip the first one" and got an answer smaller than both numbers, which even I found suspicious.

What you actually need to know: Keep the first fraction, change division to multiplication, flip the second one. So 3/4 ÷ 1/2 becomes 3/4 × 2/1, which is 6/4, or 1 1/2. Sanity check: you're asking how many halves fit into three-fourths. One and a half. If your answer doesn't pass that test, you flipped the wrong one.

2. Why can't I just carry the one?

They think I know: The method on the worksheet.

What I actually know: One method, learned decades ago, defended with the passion of a man protecting his grandmother's chili recipe.

What you actually need to know: Box models and "partial products" break numbers into tens and ones so kids see why multiplication works. 23 × 4 becomes 20 × 4 plus 3 × 4, which is 80 plus 12, which is 92. Same answer you'd get carrying the one. Ask your kid to teach you their way first. If they can explain it, they understand it. Don't spring your method on them the night before a test unless the teacher's fine with it.

3. What's 8 ÷ 2 × 4?

They think I know: PEMDAS cold.

What I actually know: PEMDAS, recited perfectly, applied wrong. I did the multiplication first because M comes before D. I got 1.

What you actually need to know: Multiplication and division are a tie. So are addition and subtraction. When there's a tie, you go left to right. 8 ÷ 2 is 4, then 4 × 4 is 16. Parentheses and exponents come first, but after that, the acronym is a ranking with two ties in it, not a strict line.

4. How do I solve for x?

They think I know: Algebra, the way I know the alphabet.

What I actually know: That x is a letter and that I used to be good at this.

What you actually need to know: Get x alone by undoing things in reverse order. For 3x + 5 = 20, subtract 5 from both sides to get 3x = 15, then divide both sides by 3. So x = 5. Now teach the habit that saves grades: plug it back in. Three times five plus five is twenty. If it checks, it's right. If it doesn't, the mistake is somewhere on the page, and it's findable.

5. What does photosynthesis actually do?

They think I know: Biology. I mow a lawn, after all.

What I actually know: Plants eat sunlight. Possibly they also drink it.

What you actually need to know: Plants take in carbon dioxide from the air and water through their roots, then use energy from sunlight to turn them into glucose, which is sugar the plant uses for food. Oxygen is the leftover, and it goes out into the air. The chemistry class version is 6CO2 + 6H2O → C6H12O6 + 6O2. Most of it happens in the leaves, in the chloroplasts, and chlorophyll is why they're green.

6. Is Pluto a planet?

They think I know: The solar system, in order.

What I actually know: A grudge.

What you actually need to know: In 2006 the International Astronomical Union reclassified Pluto as a dwarf planet because it hasn't cleared its orbit of other objects. The eight planets, in order from the sun: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune. Your kid's test will want eight. You can mourn on your own time.

7. Why is the sky blue?

They think I know: Science.

What I actually know: "It reflects the ocean." Sam looked at me the way you'd look at a man who just put diesel in a gas car.

What you actually need to know: Sunlight contains every color. When it hits the gases in our atmosphere, the shorter wavelengths get scattered around far more than the longer ones. Blue is short, so it bounces all over the sky and reaches your eyes from every direction. The name is Rayleigh scattering. At sunset, light travels through more air, the blue gets scattered out along the way, and you're left with the reds and oranges.

8. Affect or effect?

They think I know: Words. I write emails for a living.

What I actually know: I've been rewriting sentences for twenty years just to avoid choosing.

What you actually need to know: Most of the time, affect is the verb and effect is the noun. The rain affected the game. The rain had an effect on the game. A memory trick that works: A for action, E for end result. There are rare exceptions, but a seventh-grader who gets the main rule right is ahead of most adults. Including the one at my table.

9. What's a thesis statement?

They think I know: How to write an essay.

What I actually know: I wrote one in college the night before, by the light of a vending machine, and it was mostly a summary of the book.

What you actually need to know: A thesis is one sentence that makes a claim somebody could reasonably argue with. "The book is about a boy and a dog" is a summary. "The dog is the only character in the book who never lies" is a thesis, because now you have to prove it. Have your kid say their thesis out loud. If you can't say "I disagree" back, it isn't one yet. And let them write it. Your job is the questions, not the sentences.

The one I'm glad they asked

By the time we got to Sam's essay, I'd recovered enough confidence to mention, apropos of nothing, that Sydney is the capital of Australia.

"How do you know?" Sam asked.

And I didn't. I just knew it the way you know a song lyric you've been singing wrong since high school. Jen finally spoke up from the sink. "It's Canberra." She was right, of course. She usually is about anything with a map involved.

But "how do you know?" is the best question a kid can ask, and it's the one worth learning together. So we did it right there at the table, Sam on the laptop and me just watching. If you've got a kid old enough for the internet, try this tonight:

• Find two sources that don't depend on each other. A government site, an encyclopedia, a museum or a university. Not two blogs copying the same paragraph.
• Check who's behind the page. Scroll to the bottom. Click "About." A site selling something has a reason to tell you things.
• The textbook and the teacher win ties. If the internet disagrees with the assignment, bring the question to class. That isn't arguing. It's paying attention.
• Let the kid drive. They type, they read, they decide. You only ask, "How do you know that one's right?"

If you're worried about the laptop turning into an hour of videos, we've got screen rules that hold up on a Tuesday night. And if you're too tired for any of this, read what to do with your kids when you've got nothing left and do the five-minute version.

The answers on this page will be outdated the moment your kid hits calculus. The habit won't. Your kids will watch how you handle being wrong long after they've forgotten whether you got the fraction right, the same way your son is learning your work habits, not your work advice.

So when you get caught, say it out loud. "I was wrong. Let's look it up."

Sam wrote "Canberra" on a sticky note and put it on the fridge. It's still there. I didn't ask why.