Mathematical Facts, Constants & Formulas
Grade 4: Multi-Digit Algorithms, Factors, Fraction Operations & Angles
~10 min read
HESTEN ACADEMIC REFERENCE ARCHIVE
CALL NO: QA107.H47 2026 • DEWEY: 510.71 • VOL. IV
Volume IV • Intermediate Grade Mathematics Compendium
Fourth Grade Mathematics Reference Codex
The Complete Scholastic Guide to Place Value to 1,000,000, Multi-Digit Algorithms, Division with Remainders, Primes & Factors, Fraction Operations, Decimals, Angles, Symmetry & Synoptic Concordance
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An exhaustive reference codex designed for fourth-grade scholars, educators, and academic practitioners. Formulating multi-digit base-10 arithmetic, the Euclidean division algorithm, prime sieve decomposition, rational fraction arithmetic with whole-number scaling, decimal positional equivalence, and protractor angle geometry according to national curricula standards.
§ 4.1
Place Value to 1,000,000, Periods & Rounding
DEFINITION 4.1.1
Topic: The 10-to-1 Metric Power Principle
Positional Base-10 Law — In any multi-digit numeral, a digit in one place represents exactly $10$ times as much as it represents in the place immediately to its right:
$$\text{Value}(\text{Position } k) = 10 \times \text{Value}(\text{Position } k-1)$$ $$\text{In } 44,444: \text{The } 4 \text{ in the thousands place } (4,000) = 10 \times \text{the } 4 \text{ in the hundreds place } (400).$$| Millions Period | Thousands Period | Units (Ones) Period | ||||||
|---|---|---|---|---|---|---|---|---|
| Hundred Millions | Ten Millions | Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
| $100,000,000$ | $10,000,000$ | $1,000,000$ | $100,000$ | $10,000$ | $1,000$ | $100$ | $10$ | $1$ |
Student Guide: The 4-Step Rounding Protocol
STEP 1: UNDERLINE
Underline the target place value digit you are rounding to. (Example: Round $47,682$ to nearest thousand $\to 4\underline{7},682$).
STEP 2: CIRCLE NEIGHBOR
Look directly at the neighbor digit to its right: $6$.
STEP 3: BENCHMARK 5
If neighbor is $0-4$, underline stays same. If neighbor is $5-9$, add $+1$ to underlined digit ($7+1=8$).
STEP 4: ZERO OUT
Turn all digits to the right into zeros: $\mathbf{48,000}$!
§ 4.2
Multi-Digit Multiplication: Partial Products & Standard Algorithm
ALGORITHM 4.2.1
Topic: Area Model (Partial Products)
To multiply $24 \times 35$, decompose both numbers by expanded place value:
$$24 \times 35 = (20 + 4) \times (30 + 5)$$
$$= (20 \times 30) + (20 \times 5) + (4 \times 30) + (4 \times 5)$$
$$= 600 + 100 + 120 + 20 = \mathbf{840}$$
EXEMPLUM 4.2
Standard 2-Digit by 2-Digit Multiplication
Problem: Calculate $63 \times 47$ using the standard vertical algorithm.
Step 1 (Multiply by Ones): $7 \times 63$: $7 \times 3 = 21$ (write $1$, carry $2$). $7 \times 6 = 42 + 2 = 44$. First partial product: $441$.
Step 2 (Place Zero & Multiply Tens): Write $0$ in ones column. $4 \times 63$: $4 \times 3 = 12$ (write $2$, carry $1$). $4 \times 6 = 24 + 1 = 25$. Second partial product: $2,520$.
Step 3 (Add Partial Products): $441 + 2,520 = 2,961$.
Result: $63 \times 47 = \mathbf{2,961}$.
■ Q.E.D.
§ 4.3
Long Division Algorithm & The Division Theorem
THEOREM 4.3.1
Topic: The Division with Remainder Theorem
For any integer dividend $a$ and non-zero divisor $b$, there exist unique integers quotient $q$ and remainder $r$ such that:
$$a = (b \times q) + r \quad \text{where } 0 \le r < b$$
$$\text{Example: } 257 \div 4 \implies 257 = (4 \times 64) + 1 \implies \mathbf{64 \text{ R } 1}$$
The DMSB Division Cycle (Does McDonald's Serve Burgers?)
D: DIVIDE
How many times does the divisor fit into the current working digits without exceeding? Write digit in quotient.
M: MULTIPLY
Multiply quotient digit by divisor. Write product underneath.
S: SUBTRACT
Subtract product from working digits. Result must strictly be smaller than divisor!
B: BRING DOWN
Bring down the next digit from the dividend. Repeat the cycle until all digits are processed.
§ 4.4
Factors, Multiples, Prime & Composite Numbers
Prime Number Definition
$$p \in \mathbb{Z}^+, \ p > 1 \text{ with factors } \{1, p\}$$
Has exactly two distinct factors. Primes $< 30$: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
Composite Number Definition
$$c > 1 \text{ having } \ge 3 \text{ positive factors}$$
Can be partitioned into rectangular arrays of size $> 1$: $4, 6, 8, 9, 10, 12 \dots$
The Special Case of 1
$$1 \text{ is NEITHER prime NOR composite}$$
1 has only one single factor ($1$). $2$ is the ONLY even prime in existence!
| Divisor | Divisibility Invariant Test | Exemplary Verification |
|---|---|---|
| 2 | Last digit is even: $\{0, 2, 4, 6, 8\}$ | $3,846$ (ends in $6$) ✓ |
| 3 | Sum of all digits is divisible by $3$ | $4,842 \to 4+8+4+2=18 \implies 18\div 3=6$ ✓ |
| 4 | Last two digits form a number divisible by $4$ | $75,324 \to 24\div 4=6$ ✓ |
| 5 | Last digit is $0$ or $5$ | $9,415$ (ends in $5$) ✓ |
| 6 | Divisible by BOTH $2$ and $3$ | $732$ (even and digits sum to $12$) ✓ |
| 9 | Sum of all digits is divisible by $9$ | $8,271 \to 8+2+7+1=18 \implies 18\div 9=2$ ✓ |
| 10 | Last digit is strictly $0$ | $42,190$ (ends in $0$) ✓ |
§ 4.5
Fraction Operations: Addition, Subtraction & Whole Scaling
Like Denominator Addition & Subtraction
$$\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$$
Operate strictly across numerators; keep the denominator identical: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.
Whole Number × Unit Fraction
$$n \times \frac{a}{b} = \frac{n \times a}{b}$$
Scaling a fraction by whole number: $4 \times \frac{2}{3} = \frac{8}{3} = 2 \frac{2}{3}$.
Mixed Number Deconversion
$$W \frac{a}{b} = \frac{(W \times b) + a}{b}$$
Example: $3 \frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{15}{4}$.
§ 4.6
Decimals: Tenths, Hundredths & Currency Equivalence
| Fraction Notation | Decimal Notation | Positional Base-10 Meaning | US Currency Equivalence |
|---|---|---|---|
| $\frac{1}{10}$ | $0.1 = 0.10$ | $1 \text{ tenth}$ | $1 \text{ Dime } (\$0.10)$ |
| $\frac{1}{4} = \frac{25}{100}$ | $0.25$ | $2 \text{ tenths} + 5 \text{ hundredths}$ | $1 \text{ Quarter } (\$0.25)$ |
| $\frac{1}{2} = \frac{50}{100}$ | $0.5 = 0.50$ | $5 \text{ tenths}$ | $1 \text{ Half-Dollar } (\$0.50)$ |
| $\frac{3}{4} = \frac{75}{100}$ | $0.75$ | $7 \text{ tenths} + 5 \text{ hundredths}$ | $3 \text{ Quarters } (\$0.75)$ |
| $\frac{100}{100} = 1$ | $1.00$ | $1 \text{ whole unit}$ | $1 \text{ Dollar Bill } (\$1.00)$ |
§ 4.7
Geometric Mensuration: Area & Perimeter Formulas
Area Formula
$$A = l \times w \iff l = \frac{A}{w}$$
Find missing dimensions: If $A = 72\text{ cm}^2$ and $w = 8\text{ cm}$, then $l = 72 \div 8 = 9\text{ cm}$.
Perimeter Formula
$$P = 2l + 2w = 2(l + w)$$
Continuous perimeter boundary distance around the quadrilateral.
§ 4.8
Geometry: Angles, Protractor Measurement & Symmetry
| Angle Classification | Angular Measure ($\theta$) | Geometric Defining Trait |
|---|---|---|
| Acute Angle | $0^\circ < \theta < 90^\circ$ | Sharp, smaller than a right corner |
| Right Angle | $\theta = 90^\circ$ | Forms perpendicular lines ($\perp$); exactly $\frac{1}{4}$ of a full rotation |
| Obtuse Angle | $90^\circ < \theta < 180^\circ$ | Wide, larger than a right corner |
| Straight Angle | $\theta = 180^\circ$ | Forms a continuous straight line; exactly $\frac{1}{2}$ of a circle |
| Full Rotation | $\theta = 360^\circ$ | Full complete circular revolution |
Lines & Symmetry Axioms
- Parallel Lines ($\parallel$): Lines in a plane that never intersect, no matter how far extended. Always equidistant.
- Perpendicular Lines ($\perp$): Two intersecting lines that form exact $90^\circ$ right angles.
- Line of Symmetry: A fold line that divides a plane figure into two identical mirror-image halves that match exactly.
§ 4.9
Measurement Systems & Unit Conversion Ratios
| Dimension | US Customary Equivalences | Metric System Equivalences |
|---|---|---|
| Length | $1\text{ ft} = 12\text{ in.}$ • $1\text{ yd} = 3\text{ ft} = 36\text{ in.}$ • $1\text{ mi} = 5,280\text{ ft}$ | $1\text{ m} = 100\text{ cm} = 1,000\text{ mm}$ • $1\text{ km} = 1,000\text{ m}$ |
| Weight / Mass | $1\text{ lb} = 16\text{ oz}$ • $1\text{ Ton} = 2,000\text{ lb}$ | $1\text{ kg} = 1,000\text{ g}$ |
| Capacity | $1\text{ gal} = 4\text{ qt} = 8\text{ pt} = 16\text{ cups} = 128\text{ fl oz}$ | $1\text{ L} = 1,000\text{ mL}$ |
| Time | $1\text{ min} = 60\text{ s}$ • $1\text{ hr} = 60\text{ min}$ • $1\text{ day} = 24\text{ hr}$ • $1\text{ yr} = 365\text{ days} = 52\text{ weeks}$ | |
§ 4.10
Scholia & Computational Mental Math Speed Hacks
HACK I
Multiplying by 11 Shortcut
$$35 \times 11 \implies 3 \ [3+5] \ 5 = \mathbf{385}$$
Spread the two digits and place their sum in the middle! If sum $\ge 10$, carry $1$ to the left digit ($48 \times 11 = 528$).
HACK II
Quarter Scaling (×25)
$$48 \times 25 = (48 \div 4) \times 100 = 12 \times 100 = \mathbf{1,200}$$
Multiplying by 25 is dividing by 4 and appending two zeros! Think of counting quarters into dollar bills.
HACK III
Divisibility by 3 and 9 Sum Test
$$5,814 \implies 5+8+1+4 = 18 \implies \text{Divisible by 9!}$$
Sum all digits. If sum divides by 9, the entire number divides by 9. If sum divides by 3, it divides by 3.
HACK IV
Division Inverse Sanity Check
$$(\text{Quotient} \times \text{Divisor}) + \text{Remainder} = \text{Dividend}$$
Always perform the inverse algorithm check before submitting division results. Guarantees 100% computational accuracy.
§ 4.11
The Grand Synoptic Tables & Complete Grade 4 Student Cheat Sheet
Grade 4 Master Reference Concordance
Authorized curriculum reference concordance • Grade 4 Standards Complete
Multi-Digit Operations
- Place value grows $\times 10$ each left step
- Partial products: $(20+4)(30+5)$
- $a = (b \times q) + r \quad (r < b)$
- Rounding: 5 or more, raise the score
- $35 \times 11 = 385$ • $48 \times 25 = 1,200$
Primes & Factors
- Primes: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$
- $2$ is the only even prime number
- $1$ is neither prime nor composite
- Divisible by $3/9$: Sum of digits test
- Divisible by $4$: Last 2 digits test
Fractions & Decimals
- $\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$
- $4 \times \frac{2}{3} = \frac{8}{3} = 2 \frac{2}{3}$
- $\frac{1}{10} = 0.1 \quad \vert \quad \frac{1}{100} = 0.01$
- $\frac{1}{4} = 0.25 \quad \vert \quad \frac{1}{2} = 0.50$
- $\frac{3}{4} = 0.75 \quad \vert \quad \frac{100}{100} = 1.00$
Angles & Lines
- Acute: $0^\circ < \theta < 90^\circ$
- Right: $\theta = 90^\circ$ ($\perp$ Perpendicular)
- Obtuse: $90^\circ < \theta < 180^\circ$
- Straight: $\theta = 180^\circ$
- Full Turn: $\theta = 360^\circ$
- Parallel ($\parallel$): Never intersect
Area & Perimeter
- $A = l \times w$ (Square units)
- $P = 2l + 2w = 2(l + w)$ (Linear)
- Missing side: $l = A \div w$
- Square Area: $A = s^2$ • $P = 4s$
- Symmetry: Identical fold reflection
Measurement Units
- $1 \text{ ft} = 12 \text{ in.} \quad \vert \quad 1 \text{ yd} = 3 \text{ ft}$
- $1 \text{ lb} = 16 \text{ oz} \quad \vert \quad 1 \text{ Ton} = 2,000 \text{ lb}$
- $1 \text{ gal} = 4 \text{ qt} = 8 \text{ pt} = 16 \text{ c}$
- $1 \text{ m} = 100 \text{ cm} = 1,000 \text{ mm}$
- $1 \text{ km} = 1,000 \text{ m} \quad \vert \quad 1 \text{ kg} = 1,000 \text{ g}$