AMZB: Private Tutor

(Never fear, my dear readers: In the following post, names and identifying details have been changed to protect privacy.)

Arnold is ten years old, bright eyed, a bit chubby. He is staring at me in consternation.

"I don't know how to do this question. I don't get what it means."

"Which one?"

"This one. What does the triangle sign mean?"

"Oh, yes. Don't worry, you're not supposed to know what it means in advance. This is a trick the SSAT likes to play on you. They try to throw you off by using strange symbols. But see here? They've defined it for you."

I point at the top of the page where the strange operator is defined.

"Look at this equation. It tells you just what to do. 'A triangle B' just means 'multiply A by 3, then subtract B.' So you just have to plug numbers into this formula. Does that make sense?"

He nods.

"Okay. So suppose I have 5 triangle 6. What do we do with 5 and 6?"

"Do we multiply them?"

"Not quite. Okay. So what we're doing is we're taking the number before the triangle, and we're plugging it in wherever we see an A in this equation here" — I point to it — "and then we're taking the number after the triangle and plugging it in wherever we see a B. By 'plugging in,' I mean that we're replacing A and B with these numbers. We're switching out A and B for whatever numbers they give us."

"Oh, okay."

"So when we try to find 5 triangle 6, first, we replace A with 5, and then we replace B with 6. That means that everything that happens to A in this formula now happens to 5, and everything that happens to B now happens to 6. So we get 5 times 3 minus 6. Which is what?"

"Nine."

"Good! Now you try. Let's say we have 4 triangle 1. What, for starters, do we do with the number 4?"

"Do we multiply it by A?"

"No, see, we're going to replace A with 4 instead. See how A is being multiplied by 3? Now 4 is coming along and shoving A out of the way, and taking over the place where A used to be. That means we're multiplying 4 by 3 now. Does that make sense?"

"Yeah, I think so."

"So then if we had 6 triangle 8, what do we do with the 6?"

"Do we multiply it by 4?"

"Okay. Let's think of this another way. The letter A here is just a placeholder. It's like ... have you ever played that game madlibs? Where you get a paragraph of text but there are blanks where some of the words should be, and you choose new words without knowing what the paragraph says, and then you get silly sentences at the end?"

He brightens. "Yeah, we play that at camp!"

"Okay. So let's rewrite this equation as a madlib." I write out the equation with blank lines where there were variables. Under each line I write either "A" or "B" in the style of a madlib.

"Now we have a general madlib and we can stick any numbers we like in it to get a different equation each time. Let's put in 6 and 8 like before." I write them in. "Then what's 6 triangle 8?"

He squints for a minute. "Ten?"

"Yes, exactly! Great! Okay. So now what's 9 triangle 2? How do we figure that out?"

"Do we multiply 9 and 2?"