§ 1.1

Counting Principles, Cardinality & The 120-Chart

DEFINITION 1.1.1 Topic: Cardinality & One-to-One Correspondence

Cardinality — In an orderly enumeration of discrete objects, every element is matched to exactly one number tag ($1, 2, 3\dots$). The final number spoken represents the total magnitude (size) of the collection.

$$\text{Discrete Sequence: } \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \dots, 100, 110, 120\}$$

Student Guide: How to Navigate the 120-Chart like a Compass

The 120-Chart is organized into 10 columns and 12 rows. You can move across it without counting by ones:

RIGHT (→)

Add 1 ($+1$): Move one square right along the same row. Example: $34 \to 35$.

LEFT (←)

Subtract 1 ($-1$): Move one square left along the same row. Example: $34 \to 33$.

DOWN (↓)

Add 10 ($+10$): Drop down directly into the next row. The tens digit increases by $1$; the ones digit stays identical. Example: $34 \to 44$.

UP (↑)

Subtract 10 ($-10$): Jump straight up one row. The tens digit decreases by $1$; the ones digit stays identical. Example: $34 \to 24$.

Skip-Count Cadence	Cadence Pattern Sequence (First 10 Terms)	Visual / Physical Real-World Model
Skip Counting by 2s	$2, 4, 6, 8, 10, 12, 14, 16, 18, 20$ (All Even Numbers)	Pairs of shoes, socks, or hands ($2$ per person)
Skip Counting by 5s	$5, 10, 15, 20, 25, 30, 35, 40, 45, 50$ (Ends in $0$ or $5$)	Nickels ($5¢$) or tally mark bundles (gates)
Skip Counting by 10s	$10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120$ (Ends in $0$)	Dimes ($10¢$) or Base-10 Ten Rods

§ 1.2

Master Addition Strategies within 20 (How to Add)

ARITHMETIC LAW I

Commutative Property of Addition

$$a + b = b + a$$

Order never changes the sum. Example: $3 + 9 = 9 + 3 = 12$. Always begin with the larger addend!

ARITHMETIC LAW II

Associative Property (3 Addends)

$$(a + b) + c = a + (b + c)$$

When adding three numbers, group friend-of-ten pairs first: $(4 + 7) + 6 = (4 + 6) + 7 = 10 + 7 = 17$.

ARITHMETIC LAW III

Identity Property of Zero

$$a + 0 = a \quad \text{and} \quad a - 0 = a$$

Adding or taking away zero leaves the original quantity untouched: $14 + 0 = 14$.

Student Guide: The 4 Core Addition Strategies (Choose the Best Tool)

STRATEGY 1

Counting On (Best when adding $+1, +2, +3$): Lock the bigger number in your head, then count on. For $4 + 9$: start at $9$, then count on: 10, 11, 12, 13! ($4 + 9 = 13$).

STRATEGY 2

Make a 10 (Best when one number is close to 10: $8$ or $9$): Steal from the smaller number to make a full 10. For $8 + 5$: decompose $5$ into $2 + 3$. Give $2$ to $8$ to make $10$. Now $10 + 3 = 13$.

STRATEGY 3

Doubles & Near-Doubles (Best when numbers are contiguous: $6+7, 7+8$): Use known doubles: $6 + 6 = 12$. Since $7$ is $6 + 1$, the sum is $12 + 1 = 13$.

STRATEGY 4

Ten-Frames & Counters: Fill a top ten-frame ($5$ and $5$), then continue onto the second frame to easily see $10 + \text{ones}$.

EXEMPLUM 1.2 Associative Addition with 3 Addends

Problem: Calculate $3 + 8 + 7$ by finding friendly numbers first.

Step 1 (Scan for Friends of 10): Notice $3$ and $7$ are partners of $10$: $3 + 7 = 10$.

Step 2 (Regroup with Associative Law): $(3 + 7) + 8 = 10 + 8$.

Step 3 (Add the Remainder): $10 + 8 = 18$.

Result: $3 + 8 + 7 = \mathbf{18}$. ■ Q.E.D.

§ 1.3

Master Subtraction Strategies within 20 (How to Subtract)

SCHOLASTIC WARNING Subtraction is NOT Commutative

You can add in any order ($4 + 9 = 9 + 4$), but you CANNOT flip subtraction numbers! $$12 - 4 = 8 \quad \text{but} \quad 4 - 12 \neq 8$$ Always start with the total whole (minuend) when taking away.

Student Guide: The 3 Core Subtraction Strategies

STRATEGY 1

Counting Back (Best when subtracting $1, 2,$ or $3$): Start at the big number and hop backward on the number line. For $15 - 3$: start at $15$, count back: 14, 13, 12! ($15 - 3 = 12$).

STRATEGY 2

Counting Up / Finding the Distance (Best when numbers are close: $13 - 11$): Put the smaller number in your head and count up to the bigger number. From $11$ to $13$: 12, 13 (2 hops!). Difference is $2$.

STRATEGY 3

Think-Addition / Fact Family Bond: Turn subtraction into an addition question! For $14 - 9 = \square$, ask: "What plus 9 makes 14?" Since $9 + 5 = 14$, then $14 - 9 = 5$.

EXEMPLUM 1.3 Missing Addend / Part-Part-Whole Subtraction

Problem: Jordan has 16 marbles in a bag. 9 of them are blue and the rest are green. How many marbles are green?

Step 1 (Identify Whole & Parts): Whole $= 16$, Part 1 $= 9$, Part 2 $= \square$.

Step 2 (Formulate Equation): $9 + \square = 16 \iff 16 - 9 = \square$.

Step 3 (Bridge Through Ten): $16 - 6 = 10$, then $10 - 3 = 7$.

Result: Jordan has 7 green marbles ($16 - 9 = 7$). ■ Q.E.D.

§ 1.4

Word Problems Master Guide (Add To, Take From, Compare)

The 4-Step Word Problem Solving Framework

STEP 1: READ

Read the problem twice. Circle numbers. Underline what the question is asking you to discover.

STEP 2: DRAW

Draw a simple sketch, Part-Part-Whole box, or tape bar model to visualize the relationship.

STEP 3: WRITE

Write an equation with a box or question mark for the missing number ($\square$).

STEP 4: CHECK

Solve and check: Does your answer make logical sense? Use inverse math to verify!

Problem Type	Core Situation Archetype	Standard Equation Model	Key Clue Words
Add To (Result Unknown)	Start with an amount, someone gives you more.	$8 + 5 = \square \implies 13$	in all, altogether, joined, more
Take From (Change Unknown)	Start with total, some leave, tell how many left.	$15 - \square = 9 \implies 6$	gave away, lost, ate, left over
Put Together / Take Apart	Two parts combine into one total group.	$\text{Part} + \text{Part} = \text{Whole}$	combined, both, total
Compare (Difference Unknown)	Comparing two distinct amounts to find how many more or fewer.	$\text{Bigger} - \text{Smaller} = \text{Diff}$	how many more, how many fewer

§ 1.5

Place Value Foundations: Bundles of Tens & Ones

DEFINITION 1.5.1 Topic: Base-Ten Bundles

The Ten-Bundle Principle — $10$ individual ones are grouped together to form exactly $1$ bundle called a Ten (a Ten Rod). Any two-digit number represents a combination of tens and loose ones:

$$\text{Two-Digit Number } N = (T \times 10) + (O \times 1)$$ $$\text{Example: } 68 = 6 \text{ Tens } + 8 \text{ Ones } = 60 + 8$$

Number	Tens ($10$)	Ones ($1$)	Canonical Expanded Form	Physical Base-10 Model
17	1 ten (10)	7 ones (7)	$10 + 7$	1 Ten-Rod, 7 Loose Units
34	3 tens (30)	4 ones (4)	$30 + 4$	3 Ten-Rods, 4 Loose Units
50	5 tens (50)	0 ones (0)	$50 + 0$	5 Ten-Rods, 0 Loose Units
89	8 tens (80)	9 ones (9)	$80 + 9$	8 Ten-Rods, 9 Loose Units

Mental Math: How to Add and Subtract Multiples of 10

Because tens and ones live in separate houses, adding tens only changes the tens place!

2-Digit + 1-Digit (No Regrouping): Add ones to ones: $42 + 5 = 40 + (2 + 5) = 47$.
2-Digit + Multiple of 10: Add tens to tens: $35 + 40 = (30 + 40) + 5 = 75$.
Subtracting Multiples of 10: Think single-digit facts: $80 - 30 = \text{8 tens} - \text{3 tens} = \text{5 tens} = 50$.

§ 1.6

Comparing Numbers: Greater Than, Less Than & Equal

RELATIONAL AXIOM I

Greater Than ($>$)

$$a > b \iff a \text{ is larger than } b$$

Open mouth opens wide toward the bigger quantity: $45 > 38$.

RELATIONAL AXIOM II

Less Than ($<$)

$$a < b \iff a \text{ is smaller than } b$$

Small pointy vertex points directly at the smaller quantity: $29 < 51$.

RELATIONAL AXIOM III

Equal To ($=$)

$$a = b \iff \text{Both sides balance}$$

Both expressions represent the exact same value: $7 + 3 = 10$.

Student Guide: How to Compare Any Two 2-Digit Numbers

STEP 1

Look at the Tens place first! The number with more tens is ALWAYS larger, regardless of how big the ones are. Example: $51 > 49$ because $5 \text{ tens} > 4 \text{ tens}$.

STEP 2

If Tens are tied, compare the Ones place! For $67$ vs $63$: both have $6$ tens. Compare ones: $7 > 3$, so $67 > 63$.

STEP 3

If both Tens and Ones are identical, they are Equal! $84 = 84$.

§ 1.7

Linear Measurement & Length Comparison

DEFINITION 1.7.1 Topic: Non-Standard Unit Iteration

Linear Length Iteration — The length of an object is the number of same-size unit lengths (paper clips, cubes) that span it from end to end with no gaps and no overlaps.

$$\text{Length Rule: Align at baseline } \to \text{lay units end-to-end without gaps or overlaps}$$

Student Rules for Accurate Measurement

Baseline Alignment: Always align the zero end of your measuring tool or unit with the start of the object.
Indirect Comparison (Transitivity): If Pencil A is longer than Ribbon B, and Ribbon B is longer than Crayon C, then Pencil A is guaranteed longer than Crayon C!
Ordering 3 Objects: Order items strictly from shortest to longest or longest to shortest.

§ 1.8

Chronometry: Telling Time on Analog & Digital Clocks

CHRONOMETRY I

Telling Time to the Hour (:00)

$$\text{Long Hand on } 12 \implies \text{Hour:00 (O'Clock)}$$

Short hand points directly at the hour number. Example: Short hand on $4$, long hand on $12 \implies 4:00$.

CHRONOMETRY II

Telling Time to the Half-Hour (:30)

$$\text{Long Hand on } 6 \implies \text{Hour:30 (Half-Past)}$$

Short hand has traveled halfway between two hours. Always read the hour it has already passed!

Clock Feature	Short Hand (Hour Hand)	Long Hand (Minute Hand)
Movement Rate	Moves slowly: takes 1 whole hour to travel from one number to the next.	Moves faster: completes an entire 360° circle every 60 minutes.
Position at :00	Points directly at the active hour digit.	Points straight up at the 12 ($0$ minutes elapsed).
Position at :30	Points halfway between the past hour and the upcoming hour.	Points straight down at the 6 ($30$ minutes / half circle elapsed).

§ 1.9

Money Foundations: Identifying & Counting US Coins

Coin	Value	Physical Distinguishing Traits	Counting Skip-Cadence
Penny	$1¢$ ($0.01$)	Copper/bronze color, smooth edge, Abraham Lincoln	Count by 1s: $1¢, 2¢, 3¢, 4¢\dots$
Nickel	$5¢$ ($0.05$)	Silver color, thick, smooth edge, Thomas Jefferson	Count by 5s: $5¢, 10¢, 15¢, 20¢\dots$
Dime	$10¢$ ($0.10$)	Silver color, smallest & thinnest coin, grooved edge, FDR	Count by 10s: $10¢, 20¢, 30¢, 40¢\dots$
Quarter	$25¢$ ($0.25$)	Silver color, largest common coin, grooved edge, Washington	Count by 25s: $25¢, 50¢, 75¢, \$1.00$ ($4 \text{ quarters} = \$1$)

Student Guide: How to Count a Pocket of Mixed Coins

STEP 1

Sort coins into piles by denomination: Quarters first, then Dimes, Nickels, and Pennies last.

STEP 2

Start counting with the highest value coin. If you have $1$ Dime, $2$ Nickels, $3$ Pennies:

STEP 3

Count on continuously: Start at Dime ($10¢$), add Nickels by 5s ($15¢, 20¢$), add Pennies by 1s ($21¢, 22¢, 23¢$). Total $= \mathbf{23¢}$!

§ 1.10

Geometry: 2D Shapes, 3D Solids & Equal Share Fractions

Figure	Type	Sides / Faces	Corners / Vertices	Defining Attribute
Triangle	2D Plane	3 straight sides	3 vertices	Closed 3-sided rectilinear polygon
Rectangle	2D Plane	4 straight sides	4 square corners ($90^\circ$)	Opposite sides are congruent and parallel
Square	2D Plane	4 equal sides	4 square corners ($90^\circ$)	Special regular rectangle with all 4 sides equal
Trapezoid	2D Plane	4 straight sides	4 corners	Quadrilateral with at least 1 pair of parallel sides
Hexagon	2D Plane	6 straight sides	6 vertices	Closed 6-sided polygon (honeycomb shape)
Cube	3D Solid	6 square faces	8 vertices, 12 edges	Solid block where every face is an identical square
Cylinder	3D Solid	2 circular flat faces	0 vertices	Can roll on its curved surface or slide on flat circular ends
Sphere	3D Solid	0 flat faces	0 vertices, 0 edges	Perfect round ball; rolls in every direction

DEFINITION 1.10.1 Topic: Fraction Foundations (Equal Shares)

Equal Shares Rule — When a whole shape (circle or rectangle) is partitioned into equal shares:

Halves ($\frac{1}{2}$): Decomposes the whole into exactly $2$ equal-sized shares. Two halves make the whole ($2 \text{ halves} = 1 \text{ whole}$).
Fourths / Quarters ($\frac{1}{4}$): Decomposes the whole into exactly $4$ equal-sized shares. Four fourths make the whole ($4 \text{ fourths} = 1 \text{ whole}$).
Axiom of Inverse Share Size: Decomposing into more equal shares creates smaller shares! (A fourth is smaller than a half: $\frac{1}{4} < \frac{1}{2}$).

§ 1.11

Data Representation: Tally Marks, Picture Graphs & Bar Graphs

How to Organize & Interpret 1st Grade Data

TALLY MARKS

Draw vertical lines for each item: $|, ||, |||, ||||$. For the 5th item, draw a diagonal slash across the four lines: $\rlap{||||}/$ to make a closed gate of $5$. Count gates by 5s!

PICTURE GRAPHS

Each picture symbol represents $1$ object (e.g. ★ $= 1$ student favorite). Count symbols in each category row to compare.

COMPARISON QUESTIONS

To answer "How many more did A have than B?", find the difference: $\text{Category A} - \text{Category B} = \text{Difference}$.

§ 1.12

Scholia & Computational Mental Math "Cheat" Hacks

HACK I

The "Make a 10" Bridge Hack

$$8 + 6 = (8 + 2) + 4 = 10 + 4 = 14$$

Always look for pairs that make 10: $(9+1, 8+2, 7+3, 6+4, 5+5)$. Steal just enough from the other number to make 10, then add the rest!

HACK II

The "Plus-9 is Plus-10 Minus-1" Shortcut

$$9 + 7 = (10 + 7) - 1 = 17 - 1 = 16$$

Adding 9 is easy! Just add 10 to the number, then drop back 1. Example: $9 + 5 \to 15 - 1 = 14$.

HACK III

Doubles ± 1 (Near-Doubles Hack)

$$7 + 8 = (7 + 7) + 1 = 14 + 1 = 15$$

Memorize your doubles ($5+5=10, 6+6=12, 7+7=14, 8+8=16$). Then any neighbors are just 1 more or 1 less!

HACK IV

Turn Subtraction Into Missing Addition

$$15 - 8 = \square \iff 8 + \square = 15 \implies 7$$

Our brains add faster than they subtract. Never count backward from 15. Think: "What plus 8 equals 15?" Answer: 7!

§ 1.13

The Grand Synoptic Tables & Complete Grade 1 Student Cheat Sheet

Grade 1 Master Reference Concordance

Authorized curriculum reference concordance • Grade 1 Standards Complete

Friends of 10 Bonds

$1 + 9 = 10 \quad \vert \quad 9 + 1 = 10$
$2 + 8 = 10 \quad \vert \quad 8 + 2 = 10$
$3 + 7 = 10 \quad \vert \quad 7 + 3 = 10$
$4 + 6 = 10 \quad \vert \quad 6 + 4 = 10$
$5 + 5 = 10 \text{ (Symmetrical Double)}$
$10 + 0 = 10 \quad \vert \quad 0 + 10 = 10$

Doubles Roster (to 20)

$1 + 1 = 2 \quad \vert \quad 2 + 2 = 4$
$3 + 3 = 6 \quad \vert \quad 4 + 4 = 8$
$5 + 5 = 10 \quad \vert \quad 6 + 6 = 12$
$7 + 7 = 14 \quad \vert \quad 8 + 8 = 16$
$9 + 9 = 18 \quad \vert \quad 10 + 10 = 20$
$\text{Half of } 10 = 5 \quad \vert \quad \text{Half of } 20 = 10$

120-Chart Navigation Compass

Right (→): Add $1$ ($+1$)
Left (←): Subtract $1$ ($-1$)
Down (↓): Add $10$ ($+10$)
Up (↑): Subtract $10$ ($-10$)
$10 \text{ ones} = 1 \text{ ten rod}$
$100 = 10 \text{ tens} = 100 \text{ ones}$

Coins & Currency Equivalents

Penny: $1¢$ (Count by 1s)
Nickel: $5¢$ ($5 \text{ pennies}$)
Dime: $10¢$ ($2 \text{ nickels} = 10 \text{ pennies}$)
Quarter: $25¢$ ($2 \text{ dimes} + 1 \text{ nickel}$)
One Dollar (\$1): $100¢ = 4 \text{ quarters} = 10 \text{ dimes}$

Shapes & Partitioning

Triangle: 3 sides, 3 vertices
Rectangle: 4 sides, 4 square corners
Square: 4 equal sides, 4 square corners
Hexagon: 6 sides, 6 vertices
Halves ($\frac{1}{2}$): 2 equal shares
Fourths ($\frac{1}{4}$): 4 equal shares (smaller than halves!)

Relational & Time Axioms

$>$: Greater Than ($15 > 9$)
$<$: Less Than ($8 < 14$)
$=$: Balance / Equivalent ($6 + 4 = 10$)
Hour (:00): Long hand on 12
Half-Hour (:30): Long hand on 6
$1 \text{ Hour} = 60 \text{ Minutes} \quad \vert \quad \text{Half-Hour} = 30 \text{ Min}$