Unit 01
Integer Operations
⚡ Quick Memory Points
Say these out loud — they stick!
SAME signs → ADD, keep sign
DIFFERENT signs → SUBTRACT, keep bigger's sign
(−)(−) = +
(+)(−) = −
a − (−b) = a + b
Even # of negatives → Positive
Odd # of negatives → Negative
📖 Worked Example
Evaluate −8 − (−3)
−8 − (−3) = −8 + 3 = −5
Subtracting a negative flips to addition. Then different signs: |−8| > |3|, so result is negative.
What is −15 + (−4) ?
Both signs are NEGATIVE (same sign) → ADD, keep the negative sign.
15 + 4 = 19, so −15 + (−4) = −19.
Memory: SAME signs → ADD, keep sign.
15 + 4 = 19, so −15 + (−4) = −19.
Memory: SAME signs → ADD, keep sign.
Evaluate: −7 − (−12)
Subtracting a negative = adding a positive!
−7 − (−12) = −7 + 12
Different signs → subtract → 12 − 7 = 5. The bigger absolute value (12) is positive, so the answer is +5.
−7 − (−12) = −7 + 12
Different signs → subtract → 12 − 7 = 5. The bigger absolute value (12) is positive, so the answer is +5.
What is the product? (−4) × (−3) × (−2)
Count the negatives: 3 negatives = ODD → result is NEGATIVE.
4 × 3 × 2 = 24, odd number of negatives → answer = −24.
Rule: Even # of (−) = positive. Odd # of (−) = negative.
4 × 3 × 2 = 24, odd number of negatives → answer = −24.
Rule: Even # of (−) = positive. Odd # of (−) = negative.
Simplify: −36 ÷ (−9)
Negative ÷ Negative = Positive.
36 ÷ 9 = 4. Both are negative (same sign) → result = +4.
Same-sign division or multiplication always gives a positive result.
36 ÷ 9 = 4. Both are negative (same sign) → result = +4.
Same-sign division or multiplication always gives a positive result.
Unit 02
Rational Number Operations
⚡ Quick Memory Points
Fractions — the 3 golden rules
ADD / SUBTRACT → need Common Denominator (LCD)
MULTIPLY → straight across (num × num, den × den)
DIVIDE → Keep · Change · Flip (KCF)
Mixed number? Convert to improper first!
(−) × (−) fraction = positive
NEVER add denominators directly
📖 Worked Example — Division (KCF)
Calculate \(\dfrac{2}{3} \div \dfrac{4}{5}\)
Keep 2/3 · Change ÷ to × · Flip 4/5 → 5/4
= (2 × 5) / (3 × 4) = 10/12 = 5/6
What is \(\dfrac{3}{4} + \dfrac{1}{6}\) ?
LCD of 4 and 6 = 12. Rewrite both fractions.
3/4 = 9/12 | 1/6 = 2/12
9/12 + 2/12 = 11/12
Trap: Never just add 3+1=4 and 4+6=10. That's wrong!
3/4 = 9/12 | 1/6 = 2/12
9/12 + 2/12 = 11/12
Trap: Never just add 3+1=4 and 4+6=10. That's wrong!
Compute: \(\left(-\dfrac{2}{3}\right) \times \left(-\dfrac{9}{4}\right)\)
(−) × (−) = positive first, then multiply straight across.
(2 × 9) / (3 × 4) = 18/12 = 3/2. Sign = positive.
Answer: 3/2 = 1.5
(2 × 9) / (3 × 4) = 18/12 = 3/2. Sign = positive.
Answer: 3/2 = 1.5
Divide: \(\dfrac{5}{8} \div \left(-\dfrac{5}{4}\right)\)
KCF: Keep 5/8 · Change ÷ to × · Flip −5/4 → −4/5
= 5/8 × (−4/5) = −20/40 = −1/2
Positive ÷ Negative → answer is Negative.
= 5/8 × (−4/5) = −20/40 = −1/2
Positive ÷ Negative → answer is Negative.
Unit 03
Distributive Property
⚡ Quick Memory Points
Multiply EVERYTHING inside the parentheses
a(b + c) = ab + ac
a(b − c) = ab − ac
−a(b + c) = −ab − ac ← sign flips BOTH!
−a(b − c) = −ab + ac
Distribute BEFORE combining like terms
Like terms = same variable and exponent
🚨 Most Common Mistake
When the factor outside is negative, students forget to flip both signs inside the parentheses.❌ −3(x − 4) = −3x − 12 ✅ −3(x − 4) = −3x + 12
📖 Worked Example
Expand and simplify 2(3x − 5) + 4x
= 6x − 10 + 4x
= 10x − 10
Combine like terms: 6x + 4x = 10x, constants: −10.
Expand: 3(x + 5)
Distribute 3 to BOTH terms inside.
3 · x = 3x | 3 · 5 = 15
Answer: 3x + 15
Choice A is a classic trap: 3 must multiply 5 as well, not just x.
3 · x = 3x | 3 · 5 = 15
Answer: 3x + 15
Choice A is a classic trap: 3 must multiply 5 as well, not just x.
Expand: −4(2x − 7)
Negative outside flips BOTH signs inside!
(−4)(2x) = −8x | (−4)(−7) = +28
Answer: −8x + 28
Trap: Choice A writes −28 — forgetting that negative × negative = positive.
(−4)(2x) = −8x | (−4)(−7) = +28
Answer: −8x + 28
Trap: Choice A writes −28 — forgetting that negative × negative = positive.
Simplify: 5(2x − 3) − 3(x + 4)
Distribute each group first, then combine like terms.
5(2x−3) = 10x − 15
−3(x+4) = −3x − 12
10x − 15 − 3x − 12 = (10−3)x + (−15−12) = 7x − 27
5(2x−3) = 10x − 15
−3(x+4) = −3x − 12
10x − 15 − 3x − 12 = (10−3)x + (−15−12) = 7x − 27
Which expression equals −(3a − 2b + 5) ?
−1 distributes to every single term — ALL signs flip.
−1·(3a) = −3a | −1·(−2b) = +2b | −1·(5) = −5
Answer: −3a + 2b − 5
−1·(3a) = −3a | −1·(−2b) = +2b | −1·(5) = −5
Answer: −3a + 2b − 5
Simplify completely: 2(x + 3) + 3(2x − 1) − 4x
Distribute each group, then collect.
2(x+3) = 2x + 6
3(2x−1) = 6x − 3
Combine: 2x + 6 + 6x − 3 − 4x
x-terms: (2+6−4)x = 4x | constants: 6−3 = 3
Answer: 4x + 3
2(x+3) = 2x + 6
3(2x−1) = 6x − 3
Combine: 2x + 6 + 6x − 3 − 4x
x-terms: (2+6−4)x = 4x | constants: 6−3 = 3
Answer: 4x + 3
Unit 04
Factoring — Reverse Distribution
⚡ Quick Memory Points
Factoring = Undoing Distribution
Find the GCF first — always!
ab + ac = a(b + c)
Divide each term by GCF
Check: distribute back to verify ✓
GCF of numbers AND variables
Factor out (−) to make inside positive
🔑 GCF 4-Step Strategy
Step 1: List factors of each coefficient.Step 2: Find the largest one in common.
Step 3: Divide each term by the GCF.
Step 4: Always verify by distributing back!
📖 Worked Example
Factor: 12x² − 8x
GCF(12, 8) = 4 | GCF(x², x) = x → GCF = 4x
= 4x(3x − 2)
Verify: 4x · 3x = 12x², 4x · (−2) = −8x ✓
Factor out the GCF: 6x + 9
GCF(6, 9) = 3.
6x ÷ 3 = 2x | 9 ÷ 3 = 3
Answer: 3(2x + 3)
Verify: 3·2x + 3·3 = 6x + 9 ✓
6x ÷ 3 = 2x | 9 ÷ 3 = 3
Answer: 3(2x + 3)
Verify: 3·2x + 3·3 = 6x + 9 ✓
Factor completely: 15a² − 10a
GCF of coefficients: GCF(15,10) = 5. GCF of variables: GCF(a², a) = a. So GCF = 5a.
15a² ÷ 5a = 3a | 10a ÷ 5a = 2
Answer: 5a(3a − 2)
Choice A is not fully factored — 5(3a²−2a) still has a common 'a' inside.
15a² ÷ 5a = 3a | 10a ÷ 5a = 2
Answer: 5a(3a − 2)
Choice A is not fully factored — 5(3a²−2a) still has a common 'a' inside.
Which factored form equals −4x + 12 ?
Factor out −4 to keep the leading sign tidy.
−4x ÷ (−4) = x | 12 ÷ (−4) = −3
Answer: −4(x − 3)
Check: −4·x = −4x, −4·(−3) = +12 ✓
Trap: Choice D gives −4x − 12. Negative × negative = positive!
−4x ÷ (−4) = x | 12 ÷ (−4) = −3
Answer: −4(x − 3)
Check: −4·x = −4x, −4·(−3) = +12 ✓
Trap: Choice D gives −4x − 12. Negative × negative = positive!
Factor completely: 8x²y − 12xy²
Find GCF of coefficients AND variables separately.
GCF(8,12) = 4 | GCF(x²,x) = x | GCF(y,y²) = y → GCF = 4xy
8x²y ÷ 4xy = 2x | 12xy² ÷ 4xy = 3y
Answer: 4xy(2x − 3y)
Choice A is not complete — 4x−6y inside still has a GCF of 2.
GCF(8,12) = 4 | GCF(x²,x) = x | GCF(y,y²) = y → GCF = 4xy
8x²y ÷ 4xy = 2x | 12xy² ÷ 4xy = 3y
Answer: 4xy(2x − 3y)
Choice A is not complete — 4x−6y inside still has a GCF of 2.
Unit 05
Mixed Challenge Problems
⚡ Final Boss Tips
Watch for these traps on every problem
Distribute BEFORE combining like terms
Factor out the GREATEST (not just any) CF
Subtraction = adding the opposite
LCD before adding/subtracting fractions
Order: ( ) → × ÷ → + −
Use the distributive property to evaluate: 7 × 98
Hint: rewrite 98 = (100 − 2)
Hint: rewrite 98 = (100 − 2)
Mental math using distribution: 7(100 − 2) = 700 − 14 = 686
This is the distributive property applied to real arithmetic — it works for any number!
Answer: 686
This is the distributive property applied to real arithmetic — it works for any number!
Answer: 686
Simplify: \(\dfrac{-3}{4} - \dfrac{1}{2} + \dfrac{5}{8}\)
LCD of 4, 2, 8 = 8. Convert all.
−3/4 = −6/8 | 1/2 = 4/8
−6/8 − 4/8 + 5/8 = (−6 − 4 + 5)/8 = −5/8
Answer: −5/8
−3/4 = −6/8 | 1/2 = 4/8
−6/8 − 4/8 + 5/8 = (−6 − 4 + 5)/8 = −5/8
Answer: −5/8
If 6x + 18 = 6(x + ?), what is the missing number?
18 ÷ 6 = 3.
6(x + 3) = 6x + 18 ✓
This is the core idea of factoring — reversing the distribution to find what was inside.
6(x + 3) = 6x + 18 ✓
This is the core idea of factoring — reversing the distribution to find what was inside.
Simplify, then factor the result: 4(3x + 2) − 2(x − 5)
Step 1 — Distribute:
4(3x+2) = 12x + 8
−2(x−5) = −2x + 10 ← careful with the negative!
Step 2 — Combine: 12x + 8 − 2x + 10 = 10x + 18
Step 3 — Factor: GCF(10,18) = 2 → 2(5x + 9)
The most common error: writing −2(x−5) = −2x − 10. Remember: −2 × (−5) = +10!
4(3x+2) = 12x + 8
−2(x−5) = −2x + 10 ← careful with the negative!
Step 2 — Combine: 12x + 8 − 2x + 10 = 10x + 18
Step 3 — Factor: GCF(10,18) = 2 → 2(5x + 9)
The most common error: writing −2(x−5) = −2x − 10. Remember: −2 × (−5) = +10!