Mathematical Facts, Constants & Formulas
Grade 1: Number Sense, Addition, Subtraction & Foundations
~13 min read
HESTEN ACADEMIC REFERENCE ARCHIVE
CALL NO: QA107.H47 2026 • DEWEY: 510.71 • VOL. I
Volume I • Primary Mathematics Compendium
First Grade Mathematics Reference Codex
The Complete Scholastic Guide to Counting to 120, Addition & Subtraction
Strategies, Place Value, Mensuration, Coins, Geometry, Mental Math Hacks & Synoptic Concordance
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An authoritative library reference handbook compiled for scholars, instructors, and independent students. Codifying fundamental numerical principles, arithmetic laws, spatial geometry, and mental calculation heuristics according to national mathematical standards.
§ 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)
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!
Associative Property (3 Addends)
$$(a + b) + c = a + (b + c)$$
When adding three numbers, group friend-of-ten pairs first: $(4 + 7) + 6
= (4 + 6) + 7 = 10 + 7 = 17$.
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
Greater Than ($>$)
$$a > b \iff a \text{ is larger than } b$$
Open mouth opens wide toward the bigger quantity: $45 > 38$.
Less Than ($<$)
$$a < b \iff a \text{ is smaller than } b$$
Small pointy vertex points directly at the smaller quantity: $29 <
51$.
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
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 $12 \implies 4:00$.
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
- $1 \text{ Hour} = 60 \text{ Minutes} \quad \vert \quad \text{Half-Hour} = 30 \text{ Min}$