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
❦

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 MillionsTen MillionsMillions Hundred ThousandsTen ThousandsThousands HundredsTensOnes
$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

NUMBER THEORY I
Prime Number Definition
$$p \in \mathbb{Z}^+, \ p > 1 \text{ with factors } \{1, p\}$$
Has exactly two distinct factors. Primes $< 30$: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
NUMBER THEORY II
Composite Number Definition
$$c > 1 \text{ having } \ge 3 \text{ positive factors}$$
Can be partitioned into rectangular arrays of size $> 1$: $4, 6, 8, 9, 10, 12 \dots$
NUMBER THEORY III
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
2Last digit is even: $\{0, 2, 4, 6, 8\}$$3,846$ (ends in $6$) ✓
3Sum of all digits is divisible by $3$$4,842 \to 4+8+4+2=18 \implies 18\div 3=6$ ✓
4Last two digits form a number divisible by $4$$75,324 \to 24\div 4=6$ ✓
5Last digit is $0$ or $5$$9,415$ (ends in $5$) ✓
6Divisible by BOTH $2$ and $3$$732$ (even and digits sum to $12$) ✓
9Sum of all digits is divisible by $9$$8,271 \to 8+2+7+1=18 \implies 18\div 9=2$ ✓
10Last digit is strictly $0$$42,190$ (ends in $0$) ✓
§ 4.5

Fraction Operations: Addition, Subtraction & Whole Scaling

FRACTION LAW I
Like Denominator Addition & Subtraction
$$\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$$
Operate strictly across numerators; keep the denominator identical: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.
FRACTION LAW II
Whole Number × Unit Fraction
$$n \times \frac{a}{b} = \frac{n \times a}{b}$$
Scaling a fraction by whole number: $4 \times \frac{2}{3} = \frac{8}{3} = 2 \frac{2}{3}$.
FRACTION LAW III
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

RECTANGLE AREA
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 $l = 72 \div 8 = 9\text{ cm}$.
RECTANGLE PERIMETER
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

  • $1 \text{ ft} = 12 \text{ in.} \quad \vert \quad 1 \text{ yd} = 3 \text{ ft}$
  • $1 \text{ lb} = 16 \text{ oz} \quad \vert \quad 1 \text{ Ton} = 2,000 \text{ lb}$
  • $1 \text{ gal} = 4 \text{ qt} = 8 \text{ pt} = 16 \text{ c}$
  • $1 \text{ m} = 100 \text{ cm} = 1,000 \text{ mm}$
  • $1 \text{ km} = 1,000 \text{ m} \quad \vert \quad 1 \text{ kg} = 1,000 \text{ g}$