§ 2.1

3-Digit Place Value: Hundreds, Tens & Ones

DEFINITION 2.1.1 Topic: Base-10 Positional Notation

Positional Base-10 Decomposition — Any 3-digit natural number $N$ represents a linear combination of Hundreds ($H$), Tens ($T$), and Ones ($O$):

$$N = (H \times 100) + (T \times 10) + (O \times 1)$$ $$\text{Example: } 742 = (7 \times 100) + (4 \times 10) + (2 \times 1) = 700 + 40 + 2$$

NOTATION TYPE I

Standard Form

$$583$$

Written directly using compact base-10 Hindu-Arabic numeral digits.

NOTATION TYPE II

Expanded Form

$$500 + 80 + 3$$

Decomposes each digit into its explicit face value scaled by positional power.

NOTATION TYPE III

Word Form (Orthographic)

\text{"Five hundred eighty-three"}

Formal linguistic representation without using colloquial "and".

Student Guide: How to Read & Build Any 3-Digit Number

STEP 1: HUNDREDS

Look at the leftmost digit. Count flats ($100\text{s}$). For $648$, $6$ flats equal $600$.

STEP 2: TENS

Look at the middle digit. Count rods ($10\text{s}$). For $648$, $4$ rods equal $40$.

STEP 3: ONES

Look at the rightmost digit. Count unit cubes ($1\text{s}$). For $648$, $8$ unit cubes equal $8$. Combine: $600 + 40 + 8 = 648$.

Unit Level	Base-10 Physical Manipulative	Numerical Value	Regrouping Equivalence
Unit Cube	Single $1 \times 1$ cube	$1$	$10 \text{ unit cubes} = 1 \text{ Ten Rod}$
Ten Rod (Long)	Column of 10 connected cubes	$10$	$10 \text{ Ten Rods} = 1 \text{ Hundred Flat}$
Hundred Flat	$10 \times 10$ square grid (100 cubes)	$100$	$10 \text{ Hundred Flats} = 1 \text{ Thousand Cube}$
Thousand Cube	$10 \times 10 \times 10$ 3D block (1,000 cubes)	$1,000$	$1,000 \text{ ones} = 100 \text{ tens} = 10 \text{ hundreds}$

§ 2.2

2-Digit & 3-Digit Addition with Regrouping (Carrying)

AXIOM 2.2.1 Topic: The Column Carrying Principle

Regrouping Axiom for Addition — Because each place value column can hold at most a single digit ($0$ through $9$), whenever the column sum $S \ge 10$:

$$\text{Write } (S \bmod 10) \text{ in the column, and carry } \left\lfloor \frac{S}{10} \right\rfloor \text{ to the next place value on the left.}$$ $$10 \text{ Ones} \longrightarrow 1 \text{ Ten}, \quad 10 \text{ Tens} \longrightarrow 1 \text{ Hundred}$$

Student Guide: The 3 Golden Rules of Column Addition

RULE 1: ALIGN

Always line up digits vertically by their place value columns (Ones directly over Ones, Tens over Tens, Hundreds over Hundreds).

RULE 2: RIGHT-TO-LEFT

Always begin adding at the rightmost column (Ones). Never start adding from the left when using the standard algorithm!

RULE 3: NEVER FORGET CARRIES

Write your carried digit neatly on the roof of the next column so you remember to add it into that column's total.

EXEMPLUM 2.2 3-Digit Addition with Double Regrouping

Problem: Calculate $368 + 275$ using the standard vertical algorithm.

Step 1 (Add Ones Column): $8 + 5 = 13 \text{ ones}$. Write down $3$ in the ones place, and carry $1$ ten above the tens column.

Step 2 (Add Tens Column): $1 \text{ (carried)} + 6 + 7 = 14 \text{ tens}$. Write down $4$ in the tens place, and carry $1$ hundred above the hundreds column.

Step 3 (Add Hundreds Column): $1 \text{ (carried)} + 3 + 2 = 6 \text{ hundreds}$. Write down $6$ in the hundreds place.

Result: $368 + 275 = \mathbf{643}$. ■ Q.E.D.

§ 2.3

Multi-Digit Subtraction with Regrouping (Decomposition)

SCHOLASTIC WARNING More on the Floor? Go Next Door and Get 10 More!

If the top digit in a column is smaller than the bottom digit ($A_{\text{top}} < B_{\text{bottom}}$), you cannot just flip the digits! You must decompose $1$ unit from the column to the left into $10$ units for the current column. $$53 - 28 \neq 35 \quad \text{(Error: subtracting } 8 - 3 \text{ is invalid!)}$$ $$\text{Correct: Borrow } 1 \text{ ten from } 5 \implies (4 \text{ tens}, 13 \text{ ones}) - 28 = 25$$

Student Guide: How to Borrow Across Zeros ($500 - 247$)

STEP 1: INSPECT

You need to subtract $0 - 7$ in ones, but the tens place is also $0$! You cannot borrow from zero.

STEP 2: GO TO HUNDREDS

Borrow $1$ from the $5$ hundreds. $5$ becomes $4$. The tens place receives $10$ tens.

STEP 3: PASS TO ONES

Now borrow $1$ ten from the $10$ tens. The tens place becomes $9$, and the ones place receives $10$ ones.

STEP 4: SUBTRACT

Ones: $10 - 7 = 3$. Tens: $9 - 4 = 5$. Hundreds: $4 - 2 = 2$. Total: $\mathbf{253}$!

EXEMPLUM 2.3 Standard Decomposition Across Columns

Problem: Calculate $724 - 386$.

Step 1 (Ones): $4 < 6$. Borrow $1$ ten from $2$ (tens becomes $1$). Ones becomes $14$. $14 - 6 = 8$.

Step 2 (Tens): $1 < 8$. Borrow $1$ hundred from $7$ (hundreds becomes $6$). Tens becomes $11$. $11 - 8 = 3$.

Step 3 (Hundreds): $6 - 3 = 3$.

Step 4 (Inverse Verification): Check by adding: $338 + 386 = 724$ ✓.

Result: $724 - 386 = \mathbf{338}$. ■ Q.E.D.

§ 2.4

Even vs. Odd Parity & Rectangular Arrays

DEFINITION 2.4.1 Topic: Parity Laws of Whole Numbers

Even Integer — An integer divisible by $2$ with zero remainder. Can always be paired into two equal teams. Ends in $\{0, 2, 4, 6, 8\}$.
Odd Integer — An integer with a remainder of $1$ when paired into groups of $2$. Ends in $\{1, 3, 5, 7, 9\}$.

$$\text{Parity Laws: } \text{Even} + \text{Even} = \text{Even}, \quad \text{Odd} + \text{Odd} = \text{Even}, \quad \text{Even} + \text{Odd} = \text{Odd}$$

ALGEBRAIC FOUNDATION I

Rectangular Array Formula

$$\text{Total Items} = \text{Rows } (R) \times \text{Columns } (C)$$

An orderly grid arranged in horizontal rows and vertical columns. $4 \text{ rows of } 5 = 20$.

ALGEBRAIC FOUNDATION II

Repeated Addition Law

$$3 \text{ rows of } 6 = 6 + 6 + 6 = 18$$

Multiplication represents rapid repeated addition of equal-sized groups.

Student Guide: How to Model and Write Equations for Arrays

STEP 1: COUNT ROWS

Count how many horizontal rows go across (side to side). Let rows $= 3$.

STEP 2: COUNT COLUMNS

Count how many items sit in each row (vertical columns). Let columns $= 4$.

STEP 3: WRITE TWO EQUATIONS

By rows: $4 + 4 + 4 = 12$. By columns: $3 + 3 + 3 + 3 = 12$. Both equal $12$!

§ 2.5

Money & US Coin Currency Arithmetic

Denomination	Cents Value	Decimal Notation	Equivalence to \$1.00	Physical Distinguishing Marker
Penny	$1\text{¢}$	$\$0.01$	$100 \text{ pennies} = \$1.00$	Copper/bronze zinc, smooth rim, Lincoln
Nickel	$5\text{¢}$	$\$0.05$	$20 \text{ nickels} = \$1.00$	Thick silver color, smooth rim, Jefferson
Dime	$10\text{¢}$	$\$0.10$	$10 \text{ dimes} = \$1.00$	Smallest and thinnest coin, reeded rim, FDR
Quarter	$25\text{¢}$	$\$0.25$	$4 \text{ quarters} = \$1.00$	Large silver color, reeded rim, Washington
Half Dollar	$50\text{¢}$	$\$0.50$	$2 \text{ half dollars} = \$1.00$	Very large silver coin, Kennedy
Dollar Bill (\$1)	$100\text{¢}$	$\$1.00$	$1 \text{ whole dollar}$	Green paper currency, George Washington

Student Guide: How to Solve Money Word Problems

Always convert dollars and cents into total cents ($100\text{¢} = \$1$) before adding or subtracting:

COUNTING CHANGE

Group like coins: 3 quarters ($75\text{¢}$) + 2 dimes ($20\text{¢}$) + 4 pennies ($4\text{¢}$) $= 75 + 20 + 4 = 99\text{¢} = \$0.99$.

MAKING CHANGE FROM \$1.00

Subtract cost from 100: If an apple costs $68\text{¢}$, change is $100 - 68 = 32\text{¢}$ (1 quarter, 1 nickel, 2 pennies).

§ 2.6

Chronometry: Telling Time to 5 Minutes & A.M./P.M.

CHRONOMETRY RULE I

Analog Clock 5-Minute Skip Rule

$$\text{Minutes Elapsed} = \text{Clock Number} \times 5$$

Every big number on the dial equals 5 minutes. If long hand points to $7$: $7 \times 5 = 35\text{ minutes}$.

CHRONOMETRY RULE II

A.M. vs. P.M. Meridiem Cycles

$$\text{A.M. (Ante)}: 12\text{ Midnight} \to 11:59\text{ AM}$$ $$\text{P.M. (Post)}: 12\text{ Noon} \to 11:59\text{ PM}$$

A full day contains exactly $24 \text{ hours} = 2 \times 12\text{-hour cycles}$.

Student Guide: The 2-Step Protocol for Reading Any Analog Clock

STEP 1: SHORT HAND

Read the short hour hand first. Look at the number it has passed. If it is between $4$ and $5$, the hour is strictly $4$!

STEP 2: LONG HAND

Read the long minute hand. Start at 12 ($:00$) and skip count by 5s around the clock: $5, 10, 15, 20, 25\dots$ until reaching the hand.

§ 2.7

Standard Linear Measurement, Units & Line Plots

System	Standard Unit	Abbreviation	Standard Equivalence	Benchmark Object
US Customary	Inch	$\text{in. or } ''$	$1\text{ in.}$	Length of a paperclip or thumb knuckle
US Customary	Foot	$\text{ft. or } '$	$1\text{ ft} = 12\text{ inches}$	Standard 12-inch classroom ruler
US Customary	Yard	$\text{yd.}$	$1\text{ yd} = 3\text{ ft} = 36\text{ in.}$	Width of a standard classroom door
Metric System	Centimeter	$\text{cm}$	$1\text{ cm} = 10\text{ mm}$	Width of a pinky fingernail
Metric System	Meter	$\text{m}$	$1\text{ m} = 100\text{ cm}$	Height of a kitchen countertop

Measuring & Plotting on a Line Plot

Baseline Zero: Always align the zero tick mark ($\mathbf{0}$) of the ruler with the left edge of the object, not the physical edge of the wooden stick!
Unit Invariance Rule: Measuring with a smaller unit (cm) yields a larger numerical count than measuring with a larger unit (inches).
Line Plots: Each measured object receives an $\mathbf{X}$ or dot stacked vertically above its measured value on a continuous horizontal number line.

§ 2.8

Geometry: 2D Polygons, 3D Solids & Equal Share Partitioning

Geometric Shape	Sides / Faces	Angles / Vertices	Defining Scholastic Classification
Triangle	3 straight sides	3 angles / 3 vertices	Closed 3-sided polygon ($\Sigma \text{ angles} = 180^\circ$)
Quadrilateral	4 straight sides	4 angles / 4 vertices	Includes rectangles, squares, rhombuses, trapezoids
Pentagon	5 straight sides	5 angles / 5 vertices	Closed 5-sided polygon
Hexagon	6 straight sides	6 angles / 6 vertices	Closed 6-sided polygon (honeycomb cell)
Cube	6 identical square faces	8 vertices, 12 equal edges	Regular hexahedron; every edge is congruent

DEFINITION 2.8.1 Topic: Fraction Partitioning & Conservation of Area

Equal Share Partitioning — Partitioning circles or rectangles into equal shares creates fractional shares:

Halves ($\frac{1}{2}$): Two equal parts. Whole $= 2 \text{ halves}$.
Thirds ($\frac{1}{3}$): Three equal parts. Whole $= 3 \text{ thirds}$.
Fourths ($\frac{1}{4}$): Four equal parts (quarters). Whole $= 4 \text{ fourths}$.
Conservation of Area: Equal shares of identical wholes do not need to have the same shape to have equal area!

§ 2.9

Data Representation: Picture Graphs & Scaled Bar Graphs

Student Guide: How to Read Scaled Graphs

CHECK THE KEY / SCALE

Always check the legend key! If $\odot = 2\text{ students}$, then $4$ symbols represent $4 \times 2 = 8\text{ students}$. If a symbol is cut in half, it equals $1$.

READ BAR HEIGHT / LENGTH

Trace the top of the bar across to the vertical scale (or right edge to horizontal scale). Check if the grid counts by $1\text{s}, 2\text{s}, 5\text{s},$ or $10\text{s}$.

HOW MANY MORE / FEWER

Subtract to find the difference between categories: $\text{Category A} - \text{Category B} = \text{Difference}$.

§ 2.10

Scholia & Computational Mental Math Speed Hacks

HACK I

Left-to-Right Mental Addition

$$47 + 36 = (40 + 30) + (7 + 6) = 70 + 13 = 83$$

Add the tens first in your head, then add the ones. This eliminates working memory carrying strain!

HACK II

Compensation Hack (Friendly Tens)

$$58 + 27 = (58 + 2) + 25 = 60 + 25 = 85$$

When an addend ends in 8 or 9, steal from the neighbor to round up to a clean ten. $58+2=60$, then add $25$.

HACK III

Cashier's "Add-Up" Subtraction

$$72 - 58 \implies 58 + \mathbf{2} = 60, \ 60 + \mathbf{12} = 72 \implies 2 + 12 = 14$$

Never borrow across columns in your head. Hop up from the smaller number to the bigger number via friendly multiples of 10.

HACK IV

Quarter Counting Rhythm

$$\mathbf{25\text{¢}}, \mathbf{50\text{¢}}, \mathbf{75\text{¢}}, \mathbf{\$1.00}, \mathbf{\$1.25}, \mathbf{\$1.50}, \mathbf{\$1.75}, \mathbf{\$2.00}$$

Chant the 25 cadence aloud like music. You will count change and tell time (:15, :30, :45) with lightning speed.

§ 2.11

The Grand Synoptic Tables & Complete Grade 2 Student Cheat Sheet

Grade 2 Master Reference Concordance

Authorized curriculum reference concordance • Grade 2 Standards Complete

Place Value to 1,000

$10 \text{ ones} = 1 \text{ ten rod}$
$10 \text{ tens} = 1 \text{ hundred flat}$
$10 \text{ hundreds} = 1 \text{ thousand cube}$
$742 = 700 + 40 + 2$
Even ends in: $\{0, 2, 4, 6, 8\}$
Odd ends in: $\{1, 3, 5, 7, 9\}$

Regrouping Axioms

Addition Carrying: If sum $\ge 10$, write ones, carry tens.
Subtraction Borrowing: Top digit smaller? Borrow $10$ from left column.
$10 \text{ ones} \leftrightarrow 1 \text{ ten}$
$10 \text{ tens} \leftrightarrow 1 \text{ hundred}$
Verify subtraction with inverse addition.

Arrays & Foundations

$\text{Total} = \text{Rows} \times \text{Columns}$
$3 \text{ rows of } 4 = 4 + 4 + 4 = 12$
$4 \text{ cols of } 3 = 3 + 3 + 3 + 3 = 12$
$\text{Even} + \text{Even} = \text{Even}$
$\text{Odd} + \text{Odd} = \text{Even}$
$\text{Even} + \text{Odd} = \text{Odd}$

Money & Currency

Penny: $1\text{¢} = \$0.01$ ($100 = \$1$)
Nickel: $5\text{¢} = \$0.05$ ($20 = \$1$)
Dime: $10\text{¢} = \$0.10$ ($10 = \$1$)
Quarter: $25\text{¢} = \$0.25$ ($4 = \$1$)
$1\text{ Dollar} = 100\text{¢} = \$1.00$
$50\text{¢} = 2 \text{ quarters} = 5 \text{ dimes}$

Time & Measurement

$1 \text{ Hour} = 60 \text{ Minutes}$
Minutes on clock: Number $\times 5$
$12 \text{ inches} = 1 \text{ foot}$
$3 \text{ feet} = 1 \text{ yard} = 36 \text{ in.}$
$100 \text{ cm} = 1 \text{ meter}$
A.M. = Midnight to Noon

Geometry & Fractions

Triangle: 3 sides, 3 vertices
Quadrilateral: 4 sides, 4 angles
Pentagon: 5 sides • Hexagon: 6
Cube: 6 square faces, 8 vertices, 12 edges
Halves ($\frac{1}{2}$): 2 equal shares
Thirds ($\frac{1}{3}$): 3 equal shares
Fourths ($\frac{1}{4}$): 4 equal shares