HESTEN ACADEMIC REFERENCE ARCHIVE CALL NO: QA107.H47 2026 • DEWEY: 510.71 • VOL. VII

Volume VII • Pre-Algebraic Compendium

Seventh Grade Mathematics Reference Codex

The Complete Scholastic Guide to Signed Rational Numbers, Constant of Proportionality ($y=kx$), Multi-Step Percents, Two-Step Equations & Inequalities, Circle Geometry ($\pi$), Probability & Synoptic Concordance

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An authoritative academic handbook for seventh-grade scholars, secondary educators, and pre-algebra specialists. Formulating signed rational arithmetic, direct linear variations through the origin, fiscal percent algorithms and simple interest, inequality sign reversals, Euclidean circle mensuration, and theoretical probability distributions.

§ 7.1

Signed Rational Number Arithmetic

AXIOM 7.1.1 Topic: Additive Inverses & Double Negatives

Additive Inverse Law — For every rational number $a$, there exists an opposite $-a$ such that:

$$a + (-a) = 0 \quad \text{and} \quad a - (-b) = a + b$$ $$\text{Example: } 7 - (-4) = 7 + 4 = 11 \quad \vert \quad (-15) + 9 = -6$$

ADDITION RULES

Same vs. Different Signs

$$\begin{aligned} (-a) + (-b) &= -(a + b) \\ (-a) + b &= b - a \end{aligned}$$

Same signs: Add absolute values and keep common sign. Different signs: Subtract smaller absolute value from larger, take larger's sign.

MULTIPLICATION & DIVISION

Product & Quotient Signs

$$\begin{aligned} (+) \times (+) &= + \quad & (-) \times (-) &= + \\ (+) \times (-) &= - \quad & (-) \times (+) &= - \end{aligned}$$

An even count of negative factors yields POSITIVE; an odd count yields NEGATIVE.

§ 7.2

Direct Proportions & Constant of Proportionality ($k$)

THEOREM 7.2.1 Topic: Direct Variation Axioms

Proportional Relationship — Two variable quantities $x$ and $y$ are directly proportional if and only if their ratio is strictly invariant:

$$y = kx \iff k = \frac{y}{x} \quad (k = \text{Constant of Proportionality / Unit Rate})$$ $$\text{Two Universal Tests: } \begin{cases} \text{1. Numerical: The ratio } \frac{y}{x} \text{ is identical for all data pairs.} \\ \text{2. Graphical: The line is strictly straight AND passes through origin } (0, 0). \end{cases}$$

§ 7.3

Multi-Step Percents & Simple Interest ($I = Prt$)

PERCENT CHANGE

Percent Increase / Decrease

$$\% \Delta = \frac{|\text{New} - \text{Original}|}{\text{Original}} \times 100\%$$

Always divide by the original starting baseline value!

FINANCIAL ALGEBRA

Simple Interest Formula

$$I = P \cdot r \cdot t \quad \text{and} \quad A = P + I$$

$I = \text{Interest}$, $P = \text{Principal}$, $r = \text{Annual rate (decimal)}$, $t = \text{Time in years}$, $A = \text{Total Balance}$.

SINGLE MULTIPLIER

Efficient Tax & Discount

$$\begin{aligned} \text{7% Tax: } &\text{Total} = \text{Price} \times 1.07 \\ \text{25% Off: } &\text{Sale} = \text{Price} \times 0.75 \end{aligned}$$

Calculate directly in 1 step without separate subtraction.

EXEMPLUM 7.3 Simple Interest Loan Calculation

Problem: A student deposits $\$1,200$ in a savings bond earning $4.5\%$ annual simple interest for $3\text{ years}$. Find the interest earned and total final balance.

Step 1 (Convert Rate to Decimal): $r = 4.5\% = 0.045$.

Step 2 (Apply $I = Prt$): $I = 1,200 \times 0.045 \times 3 = 1,200 \times 0.135 = \$162$.

Step 3 (Calculate Total Balance): $A = P + I = \$1,200 + \$162 = \$1,362$.

Result: Interest $= \mathbf{\$162}$; Total Balance $= \mathbf{\$1,362}$. ■ Q.E.D.

§ 7.4

Solving Two-Step Algebraic Equations ($ax + b = c$)

Reverse PEMDAS (SADMEP) Strategy

STEP 1: UNDO ADD / SUB

Use inverse operations to cancel the constant term $b$: $ax = c - b$.

STEP 2: UNDO MULT / DIV

Divide or multiply by the coefficient $a$ to isolate $x$: $x = \frac{c - b}{a}$.

STEP 3: CHECK

Substitute $x$ back into the original equation to ensure both sides balance!

§ 7.5

Linear Inequalities & The Golden Sign-Flip Rule

MANDATORY INEQUALITY INVARIANT Multiplying or Dividing by a Negative Reverses the Direction!

Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must flip direction: $$-4x < 20 \implies \text{Divide by } -4 \implies x \mathbf{>} -5$$ $$-2x + 7 \ge 15 \implies -2x \ge 8 \implies x \mathbf{\le} -4$$

§ 7.6

Scale Drawings & Geometric Angle Pairs

Angle Pair Classification	Geometric Relationship	Defining Equation	Visual Identifier
Complementary	Two angles summing to a right corner	$\angle 1 + \angle 2 = 90^\circ$	Forms an $L$-corner ($90^\circ$)
Supplementary	Two angles summing to a straight line	$\angle 1 + \angle 2 = 180^\circ$	Linear pair on a straight line
Vertical Angles	Opposite angles across intersecting lines	$\angle 1 = \angle 2$ (Congruent)	Form an $X$-intersection
Adjacent Angles	Share a common vertex and common ray	Side-by-side neighbors	No overlapping interiors

§ 7.7

Circle Geometry: Circumference & Area Formulas

CIRCUMFERENCE (PERIMETER)

Circumference Formula

$$C = 2\pi r = \pi d \quad (\pi \approx 3.14159 \approx \frac{22}{7})$$

Continuous boundary path around circle. Ratio $\frac{C}{d} = \pi$ for ALL circles!

ENCLOSED 2D SURFACE

Area of a Circle

$$A = \pi r^2 = \pi \times r \times r$$

Remember: Square the radius $r$ first before multiplying by $\pi$! ($r = \frac{d}{2}$).

§ 7.8

3D Solid Cross-Sections & Right Circular Cylinders

CYLINDER VOLUME

Volume Formula

$$V = B \cdot h = \pi r^2 h$$

Base area of circle multiplied by vertical cylinder height.

CYLINDER SURFACE AREA

Total Surface Area

$$SA = 2\pi r^2 + 2\pi r h$$

Sum of top/bottom circular bases ($2\pi r^2$) and unrolled rectangular lateral wrapper ($2\pi rh$).

§ 7.9

Theoretical Probability, Compound Events & Sampling

THEOREM 7.9.1 Topic: Classical Probability Axiom

The theoretical probability of event $E$ in a sample space of equally likely outcomes is: $$P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \quad \text{where } 0 \le P(E) \le 1$$ $$\text{Independent Events: } P(A \text{ and } B) = P(A) \times P(B)$$

§ 7.10

Scholia & Computational Mental Math Speed Hacks

HACK I

The 10% + 5% Restaurant Tip Hack

$$15\% \text{ of } \$60 \implies 10\% = \$6.00, \ 5\% = \$3.00 \implies \$9.00$$

Shift decimal 1 spot left for 10%, take half of that for 5%, sum them! Calculate in 2 seconds.

HACK II

One-Step Discount Multiplier

$$\text{30% Off } \implies \times 0.70 \quad \vert \quad \text{8.5% Tax } \implies \times 1.085$$

Never calculate discount and subtract in separate steps. Multiply price directly by $(1 - \text{rate})$!

HACK III

Fraction Clearing in Equations

$$\frac{2}{3}x + \frac{1}{4} = 5 \implies \times 12 \implies 8x + 3 = 60$$

Multiply entire equation across by LCD to eliminate all fractions instantly!

HACK IV

$\frac{22}{7}$ Fractional $\pi$ Hack

$$\text{For radius multiple of 7: } C = 2 \times \frac{22}{7} \times 14 = 2 \times 22 \times 2 = 88$$

When radius or diameter divides by 7, use $\frac{22}{7}$ instead of $3.14$ for effortless integer cancellation!

§ 7.11

The Grand Synoptic Tables & Complete Grade 7 Student Cheat Sheet

Grade 7 Master Reference Concordance

Authorized curriculum reference concordance • Grade 7 Standards Complete

Signed Integers

$a - (-b) = a + b$
$(-) \times (-) = (+)$
$(+) \times (-) = (-)$
$(-) \div (-) = (+)$
Even number of negatives $= +$

Proportions & Percents

$y = kx \iff k = \frac{y}{x}$
Graph passes through origin $(0, 0)$
$I = Prt$ (Simple Interest)
$\% \Delta = \frac{|\text{New}-\text{Old}|}{\text{Old}} \times 100\%$
$15\% \text{ tip} = 10\% + 5\%$

Equations & Inequalities

$ax + b = c \implies x = \frac{c - b}{a}$
$\times / \div$ by negative $\implies$ Flip sign!
$-2x < 8 \implies x > -4$
Multiply by LCD to clear fractions
Open circle: $<, >$ • Closed: $\le, \ge$

Circles & Cylinders

$C = 2\pi r = \pi d$
$A = \pi r^2$
$\pi \approx 3.14159 \approx \frac{22}{7}$
Cylinder Volume: $V = \pi r^2 h$
Cylinder SA: $2\pi r^2 + 2\pi rh$

Angle Pairs

Complementary: Sum $= 90^\circ$
Supplementary: Sum $= 180^\circ$
Vertical Angles: Congruent ($=$)
Scale Factor: $\frac{\text{Drawing}}{\text{Actual}}$
Area Scale Factor $= k^2$

Probability

$P(E) = \frac{\text{Favorable}}{\text{Total Possibilities}}$
$0 \le P(E) \le 1$
Independent: $P(A \cap B) = P(A) \cdot P(B)$
Complement: $P(\text{not } E) = 1 - P(E)$
Representative random sample required