Mathematical Facts, Constants & Formulas
Grade 7: Signed Integers, Direct Proportions, Percentages & Two-Step Equations
~9 min read
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:
$$a + (-a) = 0 \quad \text{and} \quad a - (-b) = a + b$$ $$\text{Example: } 7 - (-4) = 7 + 4 = 11 \quad \vert \quad (-15) + 9 = -6$$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.
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 Increase / Decrease
$$\% \Delta = \frac{|\text{New} - \text{Original}|}{\text{Original}} \times 100\%$$
Always divide by the original starting baseline value!
Simple Interest Formula
$$I = P \cdot r \cdot t \quad \text{and} \quad A = P + I$$
$I = \text{Interest}$, $P = \text{Principal}$, $r = \text{Annual rate (decimal)}$, $t = \text{Time in years}$, $A = \text{Total Balance}$.
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 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!
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
Volume Formula
$$V = B \cdot h = \pi r^2 h$$
Base area of circle multiplied by vertical cylinder height.
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