Week 03 · Equivalent Expressions

Thursday

Rocket's cards and Raven's cards

// Two expressions that always give the same number

⏱ about 20 min

Thursday: Rocket's Cards and Raven's Cards

Rocket counts his puzzle cards for the round. "Two per house on the street, plus three for the board. 2n + 3, if n is houses."

"Mine is three per house, minus the one I keep," says Raven. "3n - 1. What do you notice if we add our piles together?"

"Five per house," Rocket says slowly. "And three minus one is two. 5n + 2."

Nova hovers between the two piles, her light passing over each. "Like terms across two expressions," she says. "Same rule."

"And how many more do I have than you?" Raven asks.

Rocket frowns, then grins. "Subtract mine from yours. 3n minus 2n is n. Minus one minus three is minus four. n - 4."

Nova hums. "Check it with a real number of houses before you trust it."

Adding and subtracting expressions

To add two expressions, combine like terms across both: letter terms together, plain numbers together.

(2n + 3) + (3n - 1) = 2n + 3n + 3 - 1 = 5n + 2.

To subtract, subtract every term of the second expression. Keep track of each sign.

(3n - 1) - (2n + 3) = 3n - 2n - 1 - 3 = n - 4. The result is negative for small n, and that is fine.

Raven's card log

The crew's own card counts for three practice rounds, where n is the number of houses on the route.

Houses n	Rocket: 2n + 3	Raven: 3n - 1	Total: 5n + 2	Difference: n - 4
2	7	5	12	-2
4	11	11	22	0
7	17	20	37	3

With n = 2, Rocket has 7 cards and Raven has 5. The total is 12, and 5n + 2 with n = 2 gives 12. It checks.

The difference n - 4 is -2 when n = 2: Raven has fewer. By n = 7 it is 3: Raven has more.

Fraction coefficients

Coefficients can be fractions or decimals. The rules do not change.

1/2n + 1/2n = n, two halves of an envelope make one envelope. And 0.5n + 3 + 1.5n = 2n + 3.

Factoring works too: 1/2n + 2 = 1/2(n + 4). Half of n plus half of 4 is half of n plus 2.

ADD AND SUBTRACT THE PILES

Read the question.
Tap your answer.

Which expression is equivalent to (2n + 3) + (3n - 1)?

Which expression is equivalent to (3n - 1) - (2n + 3)?

Which expression is equivalent to (4n + 5) + (n - 2)?

Which expression is equivalent to (6n + 1) - (2n + 4)?

FRACTIONS AND DECIMALS

Read the question.
Tap your answer.

Which expression is equivalent to 1/2n + 1/2n?

Which expression is equivalent to 0.5n + 3 + 1.5n?

Which expression is equivalent to 1/2n + 2?

Which expression is equivalent to 3/4x + 1/4x - 2?

With n = 4 houses, how many cards do Rocket and Raven have together? Type the number.

With n = 7 houses, how many more cards does Raven have than Rocket? Type the number.

From the card log	True or false?
(2n + 3) + (3n - 1) and 5n + 2 are equivalent.	?
(3n - 1) - (2n + 3) and n + 2 are equivalent.	?
11 + 11 = 22	?
Subtracting an expression means subtracting every one of its terms.	?

WHY THIS EXERCISE(3n - 1) - (2n + 3) is n - 4, not n + 2. Forgetting to subtract the 3 flips the sign of the result.

Try it

Write two expressions for cards of your own, like 4n + 1 and n + 5. Add them, then subtract them.
Check both answers with n = 3 by working out the original expressions.

Draw Rocket's pile and Raven's pile as envelopes and beans for n = 2, then the combined pile.

Puzzle Notebook: add (2n + 3) and (3n - 1) your own way, then check with n = 10.Rocket got n - 4 for the difference. Write how the signs worked when he subtracted the 3.Write an expression with a fraction coefficient and combine it with another one.

Two piles, one expression. Tomorrow you find equivalent expressions in everyday places and review the week.

Grown-up checked my answers

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