Pre-Algebra · Grades 6–8 · Quick reference

Pre-Algebra(Arithmetic → Algebra)

Grades 6–86 pages

Place value and properties, divisibility and primes, fractions, decimals, integers, exponents and roots, ratios, percent, expressions and equations, the coordinate plane, geometry formulas, and data and probability: every rule and formula from the thirteen units, with worked examples and memory tricks.

1Number sense2Factors & primes3Fractions4Decimals5Integers6Exponents & roots7Ratios8Percent9Expressions & equations10Coordinate plane11Geometry12Data & probability

1Whole Numbers & Number SenseUnit 1 · place value, properties, order

Place value
Billions	Millions	Thousands	Ones
1,000,000,000	1,000,000	1,000	1
1 billion = 1,000 millions	1 million = 1,000 thousands		4,056 = 4 × 1,000 + 0 × 100 + 5 × 10 + 6 × 1

Read in groups of three: 4,056,210 is “four million, fifty-six thousand, two hundred ten.” Round: look one place right; 5 or more rounds up: 3,462 → 3,500. Estimate first: 48 × 21 ≈ 50 × 20 = 1,000.

Properties
Property	Addition	Multiplication
Commutative	a + b = b + a	a · b = b · a
Associative	(a + b) + c = a + (b + c)	(a · b) · c = a · (b · c)
Identity	a + 0 = a	a · 1 = a
Zero	—	a · 0 = 0
Distributive	—	a(b + c) = ab + ac

Mental math by distributing: 7 × 98 = 7(100 − 2) = 700 − 14 = 686; 6 × 43 = 6(40 + 3) = 240 + 18 = 258.

Order of operations

G grouping ( ) [ ] fraction bars · E exponents · MD multiply and divide, left → right · AS add and subtract, left → right

5² = 5 × 5 = 25 · 2³ = 2 × 2 × 2 = 8

Order of operations Worked

3 + 4 × 5 = 3 + 20 = 23multiply first
18 ÷ 3 × 2 = 6 × 2 = 12left → right
2[5 + (8 − 2)²] = 2[5 + 36] = 82inside out

Partners, not a line

M and D are partners, left to right; so are A and S. 18 ÷ 3 × 2 is 12, not 3.

2Factors, Multiples & PrimesUnit 2 · divisibility, GCF, LCM

Divisibility rules
By	Test	Example
2	last digit even	3,84
3	digit sum divisible by 3	5,13: 5+1+3 = 9
4	last two digits divisible by 4	7,316
5	ends in 0 or 5	2,35
6	divisible by 2 and 3	42: even, 4+2 = 6
9	digit sum divisible by 9	7,29: 7+2+9 = 18
10	ends in 0	90

Primes under 50

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. A prime has exactly two factors, 1 and itself; 1 is neither prime nor composite; 2 is the only even prime.

Prime factorization Worked

360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45divide by small primes

= 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5factor tree

GCF and LCM from factorizations Worked

48 = 2⁴ × 3   72 = 2³ × 3²
GCF = 2³ × 3 = 24lowest power of each shared prime
LCM = 2⁴ × 3² = 144highest power of every prime
GCF × LCM = 24 × 144 = 3,456 = 48 × 72check

Euclidean algorithm for GCF Worked

GCF(252, 105): 252 = 2 × 105 + 42larger = q × smaller + r
105 = 2 × 42 + 21GCF(larger, smaller) = GCF(smaller, r)
42 = 2 × 21 + 0 → GCF = 21last nonzero remainder

Words

factor: a number that divides evenly: factors of 12 are 1, 2, 3, 4, 6, 12
multiple: the times table: multiples of 4 are 4, 8, 12, …
GCF: greatest common factor, the largest shared factor
LCM: least common multiple, the smallest shared multiple
composite: has more than two factors

GCF goes down, LCM goes up

The GCF is never bigger than the smaller number; the LCM is never smaller than the larger number. If one number divides the other, that number is the GCF and the other is the LCM.

Traps

A number is not prime just because it is odd: 9 = 3 × 3, 51 = 3 × 17.
GCF(a, b) × LCM(a, b) = a × b works for two numbers only.

Pre-AlgebraPre-Algebra · Grades 6–8 · continued

3FractionsUnit 3 · the piece is named by its denominator

Rules

Equivalent: ab = a × kb × k; simplify by dividing top and bottom by their GCF: 1824 = 34.
Compare: ab > cd exactly when ad > bc. 58 vs 23: 15 < 16, so 58 < 23.
Add or subtract: same denominator first (the LCD = LCM of the denominators), then add the tops. ac ± bc = a ± bc.
Multiply: tops × tops, bottoms × bottoms; cancel first. ab × cd = acbd.
Divide: keep, change, flip. ab ÷ cd = ab × dc.
Mixed numbers: abc = ac + bc; 325 = 175.

Adding unlike fractions Worked

14 + 23 → LCD of 4 and 3 is 12
312 + 812 = 1112multiply top and bottom alike
Estimate: 14 + a bit over 12 ≈ just under 1 ✓

Multiply and divide Worked

× 89 = 11 × 23 = 23cancel 3 and 4 first
÷ 23 = 56 × 32 = 1512 = 114keep, change, flip
× 113 = 52 × 43 = 206 = 313improper first

Common LCDs by heart

2 and 3 → 6 · 3 and 4 → 12 · 4 and 6 → 12 · 6 and 8 → 24 · 3 and 5 → 15 · 8 and 12 → 24. Estimate against the benchmarks 0, ½ and 1 before you compute.

Traps

Never add denominators: 13 + 14 ≠ 27.
Change the top when you change the bottom.
Simplify every answer; a mixed number should have a proper fraction part.

Word problems

Recipes: scale factor = servings wantedservings the recipe makes; new amount = scale factor × original. “Of” means multiply: 23 of 24 = 16.

4DecimalsUnit 4 · fractions with 10, 100, 1,000 underneath

Rules

0.1 = 110, 0.01 = 1100, 0.001 = 11,000; 0.7 = 0.70 = 0.700. Read 3.25 as “three and twenty-five hundredths.”
Add or subtract: line up the decimal points; fill with zeros. 4.5 + 0.37 = 4.50 + 0.37 = 4.87.
Multiply: multiply as whole numbers, then count the decimal places in both factors and put that many in the product. 0.3 × 0.2 = 0.06.
Divide: move the point in the divisor until it is a whole number and move the dividend’s point the same way. 4.8 ÷ 0.6 = 48 ÷ 6 = 8.
× 10, 100, 1,000 moves the point 1, 2, 3 places right; ÷ moves it left.

Fraction ↔ decimal ↔ percent
Fraction	Decimal	Percent
12	0.5	50%
14	0.25	25%
34	0.75	75%
15	0.2	20%
18	0.125	12.5%
110	0.1	10%
13	0.3	33⅓%
23	0.6	66⅔%

ab = a ÷ b. In lowest terms, a fraction terminates only when its denominator is 2^m × 5ⁿ; otherwise it repeats.

Change and unit price Worked

Paid $20 for $13.45: 20.00 − 13.45 = $6.55line up points
12 oz for $2.88: 2.88 ÷ 12 = $0.24 per ozunit price = total ÷ units
0.25 × 1.2 = 25 × 12 = 300 → 0.300 = 0.33 decimal places

Points stay put

Adding and subtracting: points in a column. Multiplying: count places. Dividing: move both points the same number of places.

Pre-AlgebraPre-Algebra · Grades 6–8 · continued

5Integers & Rational NumbersUnit 5 · signs and the number line

Signed numbers

|a| is the distance from a to 0: |−7| = 7. Farther right on the number line is greater: −2 > −5.
Add, same signs: add the absolute values, keep the sign. −4 + (−7) = −11.
Add, different signs: subtract the absolute values, keep the sign of the larger. −9 + 4 = −5.
Subtract = add the opposite: a − b = a + (−b). 5 − (−3) = 5 + 3 = 8; −2 − 6 = −8.
Multiply or divide: same signs → +, different signs → −. (−6)(−4) = 24; −18 ÷ 3 = −6.
An even count of negative factors gives +, an odd count gives −.

Mixed signs Worked

−3 + 8 − (−2) = 5 + 2 = 7change − (−2) to + 2
(−2)(−3)(−4) = 6 × (−4) = −24three negatives: odd
−34 + 12 = −34 + 24 = −14rational numbers, same rules
(−5)² = 25 but −5² = −25the exponent touches only the 5

Change

change = new value − old value. A temperature from −4° to 7° changes by 7 − (−4) = +11°.

Same signs, happy sign

For × and ÷: signs alike give a positive; signs different give a negative. Adding is different: look at which absolute value is bigger.

Zero

0 ÷ a = 0 (a ≠ 0), but a ÷ 0 is undefined.
−ab = a−b = −ab: one negative sign may sit anywhere.

6Exponents & Square RootsUnit 6 · powers, scientific notation, roots

Laws of exponents
Law	Rule	Example
Product	a^m · aⁿ = a^m+n	2³ · 2⁴ = 2⁷
Quotient	a^maⁿ = a^m−n	5⁶ ÷ 5² = 5⁴
Power	(a^m)ⁿ = a^mn	(3²)⁴ = 3⁸
Zero	a⁰ = 1 (a ≠ 0)	7⁰ = 1
Negative	10⁻ⁿ = 110ⁿ	10⁻³ = 11,000 = 0.001

aⁿ means n factors of a: 2⁵ = 32. 10ⁿ is 1 followed by n zeros. (−3)² = 9 but −3² = −9.

Scientific notation

a × 10ⁿ with 1 ≤ a < 10. 4,500,000 = 4.5 × 10⁶; 0.0032 = 3.2 × 10⁻³. Multiply: (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹.

Square roots

√a = b means b × b = a and b ≥ 0. √49 = 7; √(a²) = a for a ≥ 0.
√ab = √a√b: √916 = 34.
Estimate: if a² < n < b², then a < √n < b. 7² = 49 < 55 < 64 = 8², so √55 ≈ 7.4.
√2, √3, π are irrational: their decimals never end or repeat.

The number family

counting ⊂ whole ⊂ integers ⊂ rational ⊂ real; the irrationals fill the rest of the real line.

Perfect squares to 20²
n	1	2	3	4	5	6	7	8	9	10
n²	1	4	9	16	25	36	49	64	81	100
n	11	12	13	14	15	16	17	18	19	20
n²	121	144	169	196	225	256	289	324	361	400

Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.

Estimating a root Worked

√40: 36 < 40 < 49 → 6 < √40 < 7between perfect squares

40 is 4 past 36 of a 13 gap → √40 ≈ 6.36.3² = 39.69 ✓

Pre-AlgebraPre-Algebra · Grades 6–8 · continued

7Ratios, Rates & ProportionsUnit 7 · comparing by division

Ratios and rates

a to b = a : b = ab; scale a ratio by multiplying both parts: 2:3 = 6:9.
Unit rate: first quantitysecond quantity with a second quantity of 1. 150 miles in 3 hours = 50 mi/h.
speed = distancetime; distance = speed × time; unit price = total pricenumber of units.
Proportion: ab = cd exactly when ad = bc (cross products).

Solving a proportion Worked

34 = x20 → 4x = 3 × 20 = 60cross multiply

x = 60 ÷ 4 = 15

Scale drawings

scale = drawing lengthactual length, both in the same unit. Map scale 1 in : 50 mi; 3.5 in on the map = 3.5 × 50 = 175 mi.

Unit conversion Worked

5 yd to inches: 5 yd × 3 ft1 yd × 12 in1 ft = 180 inmultiply by fractions equal to 1

2.5 km to m: 2.5 × 1,000 = 2,500 mmetric: move the point

Conversions
Length	Weight / Capacity
12 in = 1 ft · 3 ft = 1 yd · 5,280 ft = 1 mi	16 oz = 1 lb · 2,000 lb = 1 ton
1 km = 1,000 m · 1 m = 100 cm · 1 cm = 10 mm	8 fl oz = 1 c · 2 c = 1 pt · 2 pt = 1 qt · 4 qt = 1 gal
1 in = 2.54 cm · 1 mi ≈ 1.61 km	1 kg = 1,000 g · 1 L = 1,000 mL · 1 kg ≈ 2.2 lb

Set up each factor so the unit you are leaving cancels.

8PercentUnit 8 · per hundred

Rules

n% = n100. Percent → decimal: divide by 100 (35% = 0.35). Decimal → percent: multiply by 100 (0.08 = 8%).
n% of A = n100 × A. 10% of A = A ÷ 10; 1% = A ÷ 100; 5% is half of 10%; 25% = ¼; 50% = ½.
part = percent × whole; partwhole = percent100; percent = partwhole × 100%; whole = part ÷ percent.
Percent change = amount of changeoriginal amount × 100%.

Three questions, one triangle

Part = % × whole. Cover the one you want: part ÷ whole gives the percent; part ÷ percent gives the whole.

The three kinds Worked

What is 30% of 80? 0.30 × 80 = 24part
18 is what % of 24? 18 ÷ 24 = 0.75 = 75%percent
12 is 40% of what? 12 ÷ 0.40 = 30whole
Price 40 → 50: change 10; 10 ÷ 40 = 25% increaseoriginal on the bottom

Money

sale price = (100% − discount rate) × regular price: 20% off $45 = 0.80 × 45 = $36.
total with tax = (100% + tax rate) × price: 6% tax on $36 = 1.06 × 36 = $38.16.
tip: 15% of $40 = 10% + 5% = 4 + 2 = $6.
selling price = (100% + markup rate) × cost.
Simple interest: I = Prt; balance = P + I. $500 at 4% for 3 years: I = 500 × 0.04 × 3 = $60.
Compound interest adds r × balance each year, so the balance grows faster.

Pre-AlgebraPre-Algebra · Grades 6–8 · continued

9Variables, Expressions & EquationsUnits 9–10 · the language of algebra

Expressions

In 4x² − 7x + 9: 4 is a coefficient, 9 a constant, 4x², −7x and 9 are terms. x = 1x; −x = −1x.
Like terms have the same variable part: ax + bx = (a + b)x; 5y − 2y + 3 = 3y + 3.
Distribute: a(b + c) = ab + ac; −(b − c) = −b + c; 3(2x − 5) = 6x − 15. Factor backward: 4x + 12 = 4(x + 3).
Evaluate by substituting, then order of operations: 2l + 2w with l = 7, w = 3 is 14 + 6 = 20.

Words → symbols
Words	Symbols
b less than a	a − b
b more than a	a + b
k times the sum of a and b	k(a + b)
the quotient of a and b	ab
at least / at most	≥ / ≤
more than / fewer than	> / <
consecutive integers	n, n + 1, n + 2
consecutive even or odd	n, n + 2, n + 4

Solving equations

Whatever you do to one side, do to the other: if a = b then a + c = b + c, a − c = b − c, ac = bc, a ÷ c = b ÷ c (c ≠ 0). Undo + with −, × with ÷.

Simplify each side (distribute, combine like terms).
Get the variable terms on one side, constants on the other.
Undo + or −.
Undo × or ÷.
Check in the original equation.

Two-step and multi-step Worked

3x + 5 = 20 → 3x = 15 → x = 5undo +5, then ×3
x4 − 2 = 1 → x4 = 3 → x = 12undo −2, then ÷4
2(x − 3) = x + 4 → 2x − 6 = x + 4 → x = 10distribute, collect x
23x = 8 → x = 32 × 8 = 12multiply by the reciprocal

Inequalities

Solve like an equation, but flip the sign when you multiply or divide by a negative: −2x < 6 → x > −3.
Graph: open circle for < or >, closed circle for ≤ or ≥, shade toward the solutions.
“at least 12” means x ≥ 12; “fewer than 5” means x < 5.

Word problem Worked

Three consecutive integers sum to 48: n + (n+1) + (n+2) = 48
3n + 3 = 48 → 3n = 45 → n = 15 → 15, 16, 17check: 48 ✓
Perimeter 26, length 3 more than width: 2w + 2(w + 3) = 26 → w = 5

Golden rule of equations

Do unto one side as you do unto the other. Check every answer by putting it back.

10The Coordinate Plane & LinesUnit 11 · graphs, slope, y = mx + b

The plane

A point is (x, y): x right or left, then y up or down. Origin (0, 0).
Quadrants: I (+, +), II (−, +), III (−, −), IV (+, −).
Distance along a horizontal line: |x₂ − x₁|; along a vertical line: |y₂ − y₁|.
y = c is a horizontal line; x = c is a vertical line. A point is on a line when its coordinates make the equation true.

Tables and rules

input x → rule → output y. If y rises by m each time x rises by 1, the rule begins y = m·x + …. Plot the pairs; a linear rule makes a straight line.

Slope

(1, 3) and (4, 9): m = (9 − 3) ÷ (4 − 1) = 6 ÷ 3 = 2

m = y₂ − y₁x₂ − x₁ = riserun

Kinds of slope

uphill +, downhill −, horizontal 0, vertical undefined. Parallel lines share a slope.

Rise over run

Rise first (up is +, down is −), then run (right is +). Read a slope from a graph by walking from one lattice point to the next.

Slope-intercept form

m = slope, (0, b) = y-intercept. cost = fee + rate × hours.

y = mx + b

Graph y = 2x − 1 Worked

Start at (0, −1); slope 2 = rise 2, run 1 → (1, 1), (2, 3)plot, then draw the line

Check (3, 5): 2(3) − 1 = 5 ✓

Direct variation

y = kx with k = yx constant; the graph is a line through the origin. If y = 12 when x = 4, k = 3 and y = 3x.

Pre-AlgebraPre-Algebra · Grades 6–8 · continued

11GeometryUnit 12 · angles, polygons, area, volume, Pythagoras

Angles

complementary: ∠A + ∠B = 90°; supplementary: ∠A + ∠B = 180°; vertical angles are equal.
Parallel lines with a transversal: corresponding angles are equal, alternate interior angles are equal.
Triangle: ∠A + ∠B + ∠C = 180°. Polygon with n sides: (n − 2) × 180°; each angle of a regular n-gon: (n − 2) × 180° ÷ n. Hexagon: 720°, 120° each.

Pythagorean theorem

legs a, b; hypotenuse c (opposite the right angle). c = √(a² + b²); a = √(c² − b²). Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.

a² + b² = c²

Using it Worked

Legs 6 and 8: c = √(36 + 64) = √100 = 10
c = 13, b = 5: a = √(169 − 25) = √144 = 12
Distance (1, 2) to (4, 6): √(3² + 4²) = 5d = √((x₂−x₁)² + (y₂−y₁)²)
Sides 5, 7, 9: 25 + 49 ≠ 81 → not rightconverse

Perimeter and area
Shape	Area	Perimeter
Rectangle	A = lw	P = 2l + 2w
Square	A = s²	P = 4s
Parallelogram	A = bh	add the sides
Triangle	A = 12bh	add the sides
Trapezoid	A = 12(b₁ + b₂)h	add the sides
Circle	A = πr²	C = πd = 2πr

d = 2r; π ≈ 3.14 ≈ 22/7. Area is in square units, perimeter in plain units. Circle r = 5: C ≈ 31.4, A ≈ 78.5.

Volume and surface area
Solid	Volume	Surface area
Any prism or cylinder	V = Bh (B = base area)
Rectangular prism	V = lwh	SA = 2lw + 2lh + 2wh
Cylinder	V = πr²h	SA = 2πr² + 2πrh

Volume in cubic units; surface area in square units.

Area and volume Worked

Trapezoid b₁ = 6, b₂ = 10, h = 4: ½(16)(4) = 32
Box 3 × 4 × 5: V = 60; SA = 2(12) + 2(15) + 2(20) = 94
Can r = 2, h = 7: V = π(4)(7) ≈ 87.928π

Height stands up

In every area and volume formula, h is the perpendicular height, not a slanted side. The base B of a prism is a whole area, not a length.

Units

Convert before computing: 2 ft and 6 in → 24 in and 6 in.
Doubling every side multiplies area by 4 and volume by 8.

12Data & ProbabilityUnit 13 · averages and chance

Measures of center

mean = sum of the valuesnumber of values; sum = mean × count.
median = middle value when sorted (average the two middles for an even count).
mode = most frequent value; there may be none or several.
range = largest − smallest.
An outlier is far from the rest; it drags the mean, so report the median then.
Mean from a table: sum of (value × count)sum of the counts.

Find all four Worked

Data: 4, 7, 7, 10, 22
mean = 50 ÷ 5 = 10; median = 7; mode = 7; range = 18
Without the outlier 22: mean = 7, median = 7the median barely moves
Five tests average 84: need 84 × 6 = 504 for six; 504 − 420 = 84 on the nextsum = mean × count

Probability

P(event) = outcomes in the eventtotal equally likely outcomes, from 0 (impossible) to 1 (certain). P(not A) = 1 − P(A).
Experimental: times it happenednumber of trials. Expected count = P × trials.
Counting principle: m choices then n choices make m × n outcomes. 3 shirts × 4 pants = 12 outfits.
Independent events: P(A and B) = P(A) × P(B). Two heads: ½ × ½ = ¼.
One die: P(even) = 36 = 12; P(5) = 16; P(7) = 0.

n	1	2	3	4	5	6	7	8	9	10
n²	1	4	9	16	25	36	49	64	81	100
n	11	12	13	14	15	16	17	18	19	20
n²	121	144	169	196	225	256	289	324	361	400

n	1	2	3	4	5	6	7	8	9	10
n²	1	4	9	16	25	36	49	64	81	100
n	11	12	13	14	15	16	17	18	19	20
n²	121	144	169	196	225	256	289	324	361	400

n	1	2	3	4	5	6	7	8	9	10
n²	1	4	9	16	25	36	49	64	81	100
n	11	12	13	14	15	16	17	18	19	20
n²	121	144	169	196	225	256	289	324	361	400