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Week 02 · Letters Stand for Numbers

Monday

The envelope
// An envelope hides a number, and a letter stands in for it
⏱ about 20 min

Monday: The Envelope

Rocket seals an envelope and sets it on the Puzzle Post bench. "There are beans in here. I will not say how many."

"Then we need a name for the number," says Raven. "Call it n. What do you notice about the beans next to it?"

Three loose beans sit beside the envelope. "So the bench holds n plus three beans," Rocket says. "Whatever n is."

Nova hovers over the envelope, her light tracing the flap. "Now say it with a letter," she says. "n + 3."

"And if I add a second envelope just like it?" Rocket asks, already reaching for one.

"Two n plus three," Raven says. "2n + 3. The letter does the hiding and the expression does the counting."

Rocket grins. "Puzzle two for the street: write what you see, then open the envelope."

Rocket seals an envelope at the garage bench while Raven writes n + 3 on a card and Nova hovers above.

A letter stands for a number

A variable is a letter that stands for a number. This week the number hides in an envelope, so we do not know it yet.

An expression with a letter still counts something. If the envelope holds n beans, then n + 3 counts the beans on the bench.

When the envelope opens and n turns out to be 8, the expression tells us: n + 3 = 11 beans.

From words to expressions

WordsExpressionWhy
3 more than nn + 3Adding 3 to the envelope.
Subtract y from 55 - yStart at 5, take y away. The order matters.
Twice n2n2 × n, written without the times sign.
3 more than twice n2n + 3Double first, then add 3.
Half of mm ÷ 2Split m into two equal parts.

In algebra the times sign is usually left out. 2n means 2 × n, and 3(n + 1) means 3 × (n + 1).

"Subtract y from 5" is 5 - y, not y - 5. Read the words slowly: what do you start with, and what do you take away?

WORDS TO EXPRESSIONS
  • Read the question.
  • Tap your answer.
Which expression says "subtract y from 5"?
Which expression says "3 more than twice n"?
Which expression says "4 less than m"?
Which expression says "the product of 6 and k"?
OPEN THE ENVELOPE
  • Read the question.
  • Tap your answer.
If n = 8, what is n + 3?
If n = 8, what is 2n + 3?
If y = 2, what is 5 - y?
If m = 14, what is m ÷ 2?
StatementTrue or false?
2n means 2 × n.?
"Subtract y from 5" is written y - 5.?
If n = 6, then n + 3 = 9.?
If n = 6, then 2n + 3 = 18.?
WHY THIS EXERCISEPutting 6 in for n gives 2 × 6 + 3 = 15. Adding first would give 2 × 9 = 18, which breaks the order.
The envelope holds n beans and 4 more sit beside it. If n = 9, how many beans in all? Type the number.
WHY THIS EXERCISEOnce the letter has a value, the expression names one number.
Try it
Put a secret number of beans in an envelope. Set two loose beans beside it and write the expression.
Have someone open it and check the total against your expression.
Safety first
Beans are counters, never food. They stay away from children under three and pets.
Seal envelopes with the flap or tape, not with licking.
Draw the bench with two envelopes marked n and three loose beans. Write the expression underneath.

A letter is hiding a number, and you can already count with it. Tomorrow you name the parts of an expression.