The Fall Skills — Exam Prep — Architecture of Grace

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Study Cards

One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.

Helpful Hints

🚶 Integers: walk the line

Adding a positive = walk RIGHT. Adding a negative = walk LEFT. Subtracting = add the opposite, then walk. −3 − 5 becomes −3 + (−5) = −8.

Subtracting is just adding the opposite. Every time.

✖️ Multiplying & dividing: sign first, size second

Two separate questions. Sign: same signs → positive; different signs → negative. Size: just multiply or divide. −6 × −4: same signs → +; 6×4 = 24 → +24.

Same signs positive, different signs negative.

⚠️ The error gallery — traps with names

Right distance, wrong direction: −3 + 5 = −8? You walked the right distance the wrong way. Answer: 2.
The 0.34 trap: 3/4 is NOT 0.34. The bar means DIVIDE: 3 ÷ 4 = 0.75.
“5 less than n” is n − 5, never 5 − n. The order flips.
Forgetting the x: 3(x + 4) = 3x + 12 — the 3 multiplies BOTH terms.
|−7| = 7, never −7. Distance can’t be negative.
Left-to-right ties: 12 ÷ 3 × 2 = 8, not 2. × and ÷ are equals.

🧳 Equations: last on, first off

Solving 3x + 5 = 20: the x was multiplied FIRST, the 5 added LAST. Undo in reverse — subtract 5 first (3x = 15), then divide by 3 (x = 5). Same move to BOTH sides, always.

Last on, first off — like a suitcase.

🎯 How the test will ask

“Which step comes next?” — know the move, not just the answer.
Clue words: sum = +, difference/less than = −, product = ×, quotient = ÷, “is” = equals.
Properties ask what MOVED: order = commutative; grouping = associative; spread = distributive.
Check every equation answer by plugging it back in — ten free seconds of insurance.

✅ Before the test, can you…

Do −3 − 5 by adding the opposite?
Give the sign of −6 × −4 before the size?
Turn 3/4 into a decimal by dividing?
Distribute 3(x + 4) AND factor 6x + 9 back?
Write “5 less than a number” correctly?
Solve 3x + 5 = 20 and check it?

Term	Meaning in 10 words or less
Integer	Whole numbers and their negatives (…−2, −1, 0, 1, 2…).
Opposite	Same distance from zero, other side (5 and −5).
Absolute value	Distance from zero — always positive or zero.
Zero pair	A number plus its opposite: +3 and −3 make 0.
Commutative property	Order can travel: a + b = b + a.
Associative property	Grouping can change: (a+b)+c = a+(b+c).
Distributive property	Multiply through the parentheses: a(b+c) = ab + ac.
Like terms	Terms with the same variable part (3x and 5x).
Coefficient	The number multiplying the variable (the 3 in 3x).
Expression	Math phrase with no equals sign (2x + 1).
Equation	Two expressions set equal (2x + 1 = 7).
Inverse operation	The undo move: + / − undo each other; × / ÷ too.
Solution	The value that makes the equation true.

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Math — The Fall Skills