Mathematical Facts, Constants & Formulas
Grade 6: Ratios, Unit Rates, Negative Integers & Pre-Algebra
~9 min read
HESTEN ACADEMIC REFERENCE ARCHIVE
CALL NO: QA107.H47 2026 • DEWEY: 510.71 • VOL. VI
Volume VI • Middle School Mathematics Compendium
Sixth Grade Mathematics Reference Codex
The Complete Scholastic Guide to Ratios, Rates, Percentages, Rational Numbers, One-Step Equations, Statistical Measures of Center & Spread, Geometric Nets & Synoptic Concordance
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An authoritative academic handbook for sixth-grade scholars, secondary educators, and pre-algebra students. Formulating proportional relationships, fraction division algorithms, the signed integer domain, four-quadrant coordinate grids, statistical dispersion, spatial surface area nets, and computational mental arithmetic.
§ 6.1
Ratios, Rates & Unit Rate Calculations
DEFINITION 6.1.1
Topic: Multiplicative Comparison
Ratio — A mathematical comparison of two quantities $a$ and $b$ ($b \neq 0$), expressed as $a : b$, $\frac{a}{b}$, or "$a$ to $b$".
Unit Rate — A rate simplified so the denominator equals exactly $1$ unit:
$$\text{Unit Rate } r = \frac{\text{Quantity } A}{\text{Quantity } B} \implies \frac{\$18}{3 \text{ books}} = \$6.00 \text{ per book}$$ Student Guide: Solving Best Buy Unit Rates
Scenario: Brand A sells $16\text{ oz}$ of cereal for $\$3.20$. Brand B sells $24\text{ oz}$ for $\$4.32$. Which is the better buy?
BRAND A UNIT PRICE
$\frac{\$3.20}{16\text{ oz}} = \$0.20 \text{ per ounce}$.
BRAND B UNIT PRICE
$\frac{\$4.32}{24\text{ oz}} = \$0.18 \text{ per ounce}$.
COMPARE
Brand B costs $\$0.02$ less per ounce. Brand B is the superior buy!
§ 6.2
Proportions & Percentages as Rates per 100
Cross-Product Property
$$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$$
$\frac{3}{5} = \frac{x}{20} \implies 5x = 60 \implies x = 12$.
Part-Whole Percent Proportion
$$\frac{\text{Part } (\text{is})}{\text{Whole } (\text{of})} = \frac{\%}{100}$$
What is $35\%$ of $80$? $\frac{x}{80} = \frac{35}{100} \implies x = 28$.
§ 6.3
Division of Fractions by Fractions: Multiplicative Inverses
THEOREM 6.3.1
Topic: Reciprocal Division Theorem
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c} \quad (b, c, d \neq 0)$$
EXEMPLUM 6.3
Fraction Division in Context
Problem: A recipe calls for $\frac{3}{4}$ cup of flour. You only have a $\frac{1}{8}$ cup scoop. How many scoops are needed?
Step 1 (Formulate Equation): $\frac{3}{4} \div \frac{1}{8} = \square$.
Step 2 (Apply KCF): Keep $\frac{3}{4}$, Change $\div \to \times$, Flip $\frac{1}{8} \to \frac{8}{1}$.
Step 3 (Cross-Cancel & Multiply): $\frac{3}{\cancel{4}^1} \times \frac{\cancel{8}^2}{1} = 3 \times 2 = 6$.
Result: Exactly $\mathbf{6\text{ scoops}}$ are required.
■ Q.E.D.
§ 6.4
The Integer System ($\mathbb{Z}$), Opposites & Absolute Value
The Set of Integers ($\mathbb{Z}$)
$$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$$
Closed under addition, subtraction, and multiplication.
Absolute Value ($|x|$)
$$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$
Geometric distance from zero on number line. Always $|x| \ge 0$. $|-14| = 14$.
| Quadrant | $x$-coordinate Sign | $y$-coordinate Sign | Sample Ordered Pair | Geometric Region |
|---|---|---|---|---|
| Quadrant I | Positive ($+$) | Positive ($+$) | $(+3, +4)$ | Top-Right |
| Quadrant II | Negative ($-$) | Positive ($+$) | $(-3, +4)$ | Top-Left |
| Quadrant III | Negative ($-$) | Negative ($-$) | $(-3, -4)$ | Bottom-Left |
| Quadrant IV | Positive ($+$) | Negative ($-$) | $(+3, -4)$ | Bottom-Right |
§ 6.5
Algebraic Expressions & Distributive Factoring
Distributive Law
$$a(bx + c) = abx + ac$$
$4(3x + 5) = 12x + 20$.
Factoring Out GCF
$$18x + 24 = 6(3x + 4)$$
Divide each term by greatest common factor: $\gcd(18, 24) = 6$.
§ 6.6
One-Step Algebraic Equations & Inequalities
Solving One-Step Equations via Inverse Operations
- Addition Inverse: $x + 15 = 42 \implies x = 42 - 15 = \mathbf{27}$.
- Subtraction Inverse: $x - 9 = 24 \implies x = 24 + 9 = \mathbf{33}$.
- Multiplication Inverse: $6x = 48 \implies x = \frac{48}{6} = \mathbf{8}$.
- Division Inverse: $\frac{x}{7} = 9 \implies x = 9 \times 7 = \mathbf{63}$.
§ 6.7
2D Polygon Mensuration: Triangle, Parallelogram & Trapezoid
Area of a Triangle
$$A = \frac{1}{2} b h = \frac{b \times h}{2}$$
Half the area of a rectangle with identical base and perpendicular height.
Area of a Parallelogram
$$A = b \times h$$
Base $b$ multiplied by vertical perpendicular altitude $h$.
Area of a Trapezoid
$$A = \frac{b_1 + b_2}{2} \times h$$
Average of parallel bases times perpendicular height.
§ 6.8
3D Surface Area & Geometric Nets
FORMULA 6.8.1
Topic: Total Surface Area of Rectangular Prisms
The total surface area ($SA$) is the sum of the areas of all six 2D face polygons unfolding from the 3D net:
$$SA = 2(l \cdot w) + 2(l \cdot h) + 2(w \cdot h) = 2(lw + lh + wh)$$
§ 6.9
Statistics: Measures of Center & Spread
| Statistical Measure | Domain | Mathematical Definition & Procedure |
|---|---|---|
| Mean ($\bar{x}$) | Center | $\bar{x} = \frac{\sum x_i}{n} = \frac{\text{Sum of all values}}{\text{Number of observations}}$ |
| Median | Center | Middle value of ordered dataset (average of two middle values if $n$ is even) |
| Mode | Center | Most frequently appearing value(s) in the dataset |
| Range | Spread | $\text{Range} = \text{Maximum} - \text{Minimum}$ |
| Interquartile Range (IQR) | Spread | $\text{IQR} = Q_3 (\text{75th percentile}) - Q_1 (\text{25th percentile})$ |
| Mean Absolute Deviation (MAD) | Spread | $\text{MAD} = \frac{\sum |x_i - \bar{x}|}{n}$ (Average distance from the mean) |
§ 6.10
Scholia & Computational Mental Math Speed Hacks
HACK I
Cross-Multiply Ratio Resolver
$$\frac{a}{b} = \frac{x}{d} \implies x = \frac{a \times d}{b}$$
Multiply the complete diagonal pair, then divide by the lone partner! Solves any proportion in 1 step.
HACK II
10% Benchmark Percentage Hack
$$10\% \text{ of } 75 = 7.5 \implies 5\% = 3.75, \ 20\% = 15$$
Find 10% by shifting decimal left 1 spot. Then double for 20% or halve for 5%!
HACK III
Distance as Absolute Difference
$$\text{Distance} = |a - b|$$
Distance between $-7$ and $5$: $|-7 - 5| = |-12| = 12\text{ units}$. Never get tripped up by negative coordinates.
HACK IV
Integer Multiplication Signs Rule
$$(+) \times (+) = + \quad \vert \quad (-) \times (-) = + \quad \vert \quad (+) \times (-) = -$$
Same signs produce POSITIVE. Different signs produce NEGATIVE. True for multiplication and division.
§ 6.11
The Grand Synoptic Tables & Complete Grade 6 Student Cheat Sheet
Grade 6 Master Reference Concordance
Authorized curriculum reference concordance • Grade 6 Standards Complete
Ratios & Rates
- Ratio: $a:b = \frac{a}{b}$
- Unit Rate: Per 1 unit ($\frac{\text{Miles}}{\text{Hour}}$)
- Proportion: $\frac{a}{b} = \frac{c}{d} \iff ad = bc$
- Percent: $\frac{\text{Part}}{\text{Whole}} = \frac{\%}{100}$
- $10\% \implies$ Slide decimal left 1 spot
Fraction Division
- $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$
- Keep, Change, Flip
- Reciprocal: $\frac{4}{7} \leftrightarrow \frac{7}{4}$
- $n \times \text{reciprocal} = 1$
- Cross-cancel before multiplying!
Integers & $|x|$
- $|x| \ge 0$ unconditionally
- $|-8| = 8 \quad \vert \quad |8| = 8$
- Distance: $|a - b|$
- Quadrant I: $(+, +)$ • II: $(-, +)$
- Quadrant III: $(-, -)$ • IV: $(+, -)$
Pre-Algebra
- $x + a = b \implies x = b - a$
- $x - a = b \implies x = b + a$
- $ax = b \implies x = \frac{b}{a}$
- $\frac{x}{a} = b \implies x = ab$
- GCF Factoring: $12x + 18 = 6(2x + 3)$
Area & Nets
- Triangle: $A = \frac{1}{2}bh$
- Parallelogram: $A = bh$
- Trapezoid: $A = \frac{b_1+b_2}{2}h$
- Prism SA: $2(lw + lh + wh)$
- Volume: $V = lwh$
Statistics
- Mean: $\bar{x} = \frac{\sum x}{n}$
- Median: Middle value
- Mode: Most frequent
- $\text{Range} = \text{Max} - \text{Min}$
- $\text{IQR} = Q_3 - Q_1$
- $\text{MAD} = \frac{\sum |x - \bar{x}|}{n}$