🧪 STEM & K-12 Mathematics Complete 25,000+ Word Guide

H3: Understanding Counting

Counting is the foundation of all mathematics. It helps us understand how many things we have and is the first math skill children learn. When we count, we say numbers in order while pointing to objects. The last number we say tells us how many objects there are. This is called the cardinality principle—the final number represents the total quantity.

One-to-one correspondence is the concept that each object being counted must be assigned one and only one number. Children develop this understanding by touching each object as they count. Rote counting (saying numbers in order) comes before rational counting (understanding that each number represents a quantity).

Counting forward from 1 to 20 is the first milestone. Practice counting objects around you: toys, snacks, fingers. Say each number clearly while touching each object. The order is: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. Notice that after 10, we have teen numbers (13-19) that start with the ones digit first: thir-teen (3 and 10), four-teen (4 and 10), etc.

H3: Counting to 100

After mastering 1-20, practice counting to 100. Notice the patterns: 20, 21, 22... 30, 31, 32... The tens are 20, 30, 40, 50, 60, 70, 80, 90, 100. Practice counting by tens: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Then practice counting by ones from any starting number.

Skip counting by 2s, 5s, and 10s builds the foundation for multiplication. Practice counting by 2s: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. By 5s: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. By 10s: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.

Place value is a critical component of number sense. In a two-digit number, the first digit tells how many tens, and the second digit tells how many ones. For example, in 34: 3 tens and 4 ones. That means 34 = 3 groups of ten plus 4 ones. Using base-ten blocks helps visualize this: rods represent tens, cubes represent ones.

H3: What is Addition?

Addition is putting things together to find how many there are in total. The plus sign (+) means we are adding. The equal sign (=) means "the same as." For example, 2 + 3 = 5 means "2 plus 3 equals 5" or "2 and 3 together make 5."

When starting addition, pictures help us see what's happening. Count all the objects to find the total. This is called counting all. Later, we learn to count on from the first number. For 3 + 2, start at 3 and count up 2 more: 4, 5. So 3 + 2 = 5.

H3: Addition Facts to 10

Memorizing basic addition facts makes adding faster. Practice with flashcards or games until you can answer quickly. Start with +0 (any number plus 0 equals itself), then +1, then doubles (1+1, 2+2, 3+3, 4+4, 5+5).

Making 10 strategy: For harder facts, use the making 10 strategy. Break one number apart to make a ten with the other number. This is a powerful mental math strategy.

Doubles facts: 1+1=2, 2+2=4, 3+3=6, 4+4=8, 5+5=10, 6+6=12, 7+7=14, 8+8=16, 9+9=18, 10+10=20. These are easy to remember and help with near-doubles.

H3: What is Subtraction?

Subtraction is taking away or finding the difference between two numbers. The minus sign (−) means we are subtracting. For example, 5 − 2 = 3 means "5 minus 2 equals 3" or "5 take away 2 leaves 3."

When starting subtraction, pictures help us see what's happening. Cross out the ones being taken away, then count what's left.

H3: Subtraction Facts

Practice basic subtraction facts until they're automatic. Start with subtracting 0 (any number minus 0 equals itself), subtracting 1 (count back one), and related facts to addition.

Relating addition and subtraction: Addition and subtraction are opposites. If 3 + 4 = 7, then 7 − 3 = 4 and 7 − 4 = 3. This is called a fact family. Knowing fact families helps solve subtraction problems quickly.

Count back strategy: For subtracting small numbers, count back from the larger number. 7 − 3: count back from 7: 6, 5, 4. So 7 − 3 = 4.

H3: What is Multiplication?

Multiplication is repeated addition. Instead of adding 3 + 3 + 3 + 3 (four times), we can write 4 × 3 = 12. The × sign means "times" or "groups of." 4 × 3 means 4 groups of 3.

Visualizing multiplication helps understand the concept. Arrays show rows and columns. A 3 × 4 array has 3 rows and 4 columns, total 12 objects.

H3: Multiplication Tables 0-10

Mastering multiplication facts is essential for future math success. Start with the easiest facts:

×0: Always equals 0 (4 × 0 = 0)

×1: Equals the number itself (7 × 1 = 7)

×2: Doubling (8 × 2 = 16)

×5: Patterns (ends in 0 or 5)

×9: Patterns (9 × 3 = 27, digits sum to 9)

×10: Add a zero (6 × 10 = 60)

H3: What is Division?

Division is splitting into equal groups. 12 ÷ 3 means "12 split into 3 equal groups" or "how many in each group?" It also means "how many groups of 3 in 12?"

Division is the inverse of multiplication. If you know 4 × 3 = 12, then you know 12 ÷ 4 = 3 and 12 ÷ 3 = 4.

H3: Division with Remainders

Sometimes numbers don't divide evenly. The leftover is called the remainder. 14 ÷ 3 = 4 remainder 2, because 3 × 4 = 12, with 2 left over.

Fact families: For 3, 4, 12: 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4, 12 ÷ 4 = 3.

Long division: For larger numbers, use long division: divide, multiply, subtract, bring down.

H3: Understanding Fractions

A fraction represents part of a whole. The top number (numerator) tells how many parts we have. The bottom number (denominator) tells how many equal parts in the whole. 3/4 means 3 parts out of 4 equal parts.

Fractions can be shown visually with shapes. A circle divided into 4 equal parts with 3 shaded shows 3/4. A number line from 0 to 1 divided into 4 equal parts shows 1/4, 2/4, 3/4, 4/4 = 1.

H3: Equivalent Fractions

Equivalent fractions name the same amount. 1/2 = 2/4 = 3/6. Multiply or divide numerator and denominator by the same number to find equivalents.

Comparing fractions: With the same denominator, larger numerator = larger fraction: 3/8 > 2/8. With the same numerator, smaller denominator = larger fraction: 2/3 > 2/5.

Adding fractions: When denominators are the same, add the numerators: 1/5 + 2/5 = 3/5.

H3: What are Integers?

Integers are whole numbers and their opposites: ...-3, -2, -1, 0, 1, 2, 3... Positive numbers are above zero, negative numbers below zero. Integers extend the number line in both directions from zero.

Real-world examples of negative numbers include temperature below zero, debt, elevation below sea level (Death Valley is -282 feet), and在地下停车场 levels (-1, -2). Understanding integers is essential for algebra and beyond.

Absolute value is distance from zero, always positive. |5| = 5, |−5| = 5. Distance can't be negative. Absolute value is shown by vertical bars.

H3: Adding & Subtracting Integers

Adding same signs: Add and keep the sign. 5 + 3 = 8, −5 + (−3) = −8.

Adding different signs: Subtract and keep the sign of the larger absolute value. 5 + (−3) = 2, −5 + 3 = −2.

Subtracting integers: Add the opposite. 5 − 3 = 5 + (−3) = 2. 5 − (−3) = 5 + 3 = 8. −5 − 3 = −5 + (−3) = −8.

H3: Multiplying & Dividing Integers

Multiplying same signs = positive: 5 × 3 = 15, −5 × −3 = 15

Multiplying different signs = negative: 5 × −3 = −15, −5 × 3 = −15

Dividing same rules: 15 ÷ 3 = 5, −15 ÷ −3 = 5, 15 ÷ −3 = −5, −15 ÷ 3 = −5

H3: Integer Word Problems

Temperature: The temperature was 5°C and dropped 8 degrees. What is the new temperature? 5 − 8 = −3°C

Elevation: A submarine is at −200 feet (below sea level) and dives another 150 feet. −200 − 150 = −350 feet

Debt: You owe $50 and earn $30. What's your net worth? −50 + 30 = −$20 (still in debt)

H3: Understanding Ratios

A ratio compares two quantities. 3:2 can be written 3 to 2 or 3/2. If a class has 3 boys for every 2 girls, the ratio is 3:2. Ratios can be part-to-part (boys to girls) or part-to-whole (boys to total students).

Equivalent ratios multiply or divide both parts by the same number. 3:2 = 6:4 = 9:6. These are equivalent ratios. Finding equivalent ratios is like finding equivalent fractions.

H3: Unit Rates

A unit rate compares to 1 unit. 60 miles in 2 hours = 30 miles per hour (unit rate). Find by dividing: 60 ÷ 2 = 30. Unit rates help compare prices and efficiency.

Unit price: $4.50 for 3 pounds = $1.50 per pound. Compare prices by finding unit rates.

Speed: 150 miles in 3 hours = 50 mph. Distance divided by time.

H3: Solving Proportions

A proportion says two ratios are equal. 3/4 = 6/8 is a proportion. Cross-multiply to check: 3 × 8 = 4 × 6 (24 = 24).

To solve a proportion, use cross multiplication: x/5 = 3/4 → 4x = 15 → x = 15/4 = 3.75

H3: Scale Drawings

Scale factors relate drawing to real object. 1 inch = 10 feet means scale factor 1:120 (since 10 feet = 120 inches). Scale drawings preserve proportions.

If a map scale is 1 inch = 5 miles, and two cities are 3 inches apart on the map, the actual distance is 3 × 5 = 15 miles.

H3: Understanding Percent

Percent means "per hundred." 25% means 25 out of 100, or 25/100 = 1/4. Percentages are fractions with denominator 100.

Converting: Percent to decimal: divide by 100 (move decimal left 2 places). 45% = 0.45. Decimal to percent: multiply by 100. 0.375 = 37.5%.

Fraction to percent: Convert to decimal first, then multiply by 100. 3/4 = 0.75 = 75%.

H3: Finding Percentages

Finding a percent of a number: Multiply the number by the percent (as decimal). 30% of 80 = 0.3 × 80 = 24.

Finding the whole: If 15 is 20% of what number? 15 = 0.2 × whole → whole = 15 ÷ 0.2 = 75.

Finding what percent one number is of another: Divide part by whole, multiply by 100. What percent is 12 of 60? 12 ÷ 60 = 0.2 = 20%.

H3: Percent Increase & Decrease

Percent increase: Find the increase, divide by original, multiply by 100%. 40 to 50: increase 10, 10/40 = 0.25 = 25% increase.

Percent decrease: Find decrease, divide by original. 50 to 40: decrease 10, 10/50 = 0.2 = 20% decrease.

H3: Real-World Applications

Discounts: $80 shirt on sale 25% off. Discount = 0.25 × 80 = $20. Sale price = 80 − 20 = $60.

Sales tax: $60 item, 8% tax = 0.08 × 60 = $4.80. Total = 60 + 4.80 = $64.80.

Tips: $45 meal, 15% tip = 0.15 × 45 = $6.75. Total = $51.75.

H3: Slope and Rate of Change

Slope measures steepness and direction of a line. Slope = rise/run = change in y / change in x = (y₂ − y₁)/(x₂ − x₁).

Positive slope: Line goes up as x increases. Negative slope: Line goes down as x increases. Zero slope: Horizontal line. Undefined slope: Vertical line.

H3: Slope-Intercept Form

y = mx + b, where m = slope, b = y-intercept (where line crosses y-axis). To graph: start at (0,b), use slope to find next point.

Point-slope form: y − y₁ = m(x − x₁). Useful when given a point and slope.

Standard form: Ax + By = C. Find intercepts: set x=0 for y-intercept, y=0 for x-intercept.

H3: Parallel and Perpendicular Lines

Parallel lines: Same slope, different y-intercepts. y = 2x + 3 and y = 2x − 5 are parallel.

Perpendicular lines: Slopes are negative reciprocals (multiply to −1). m₁ × m₂ = −1. If m₁ = 2, then m₂ = −1/2.

H3: Linear Modeling

Use linear equations to model real situations: cost = fixed + rate × quantity. y = mx + b where b is fixed cost, m is cost per unit.

Example: A plumber charges $50 service fee plus $40 per hour. Equation: C = 40h + 50. For 3 hours: C = 40(3) + 50 = $170.

H3: Solving by Graphing

Graph both equations on the same coordinate plane. The intersection point is the solution (x,y) that satisfies both equations.

Three possibilities: One solution (lines intersect once), no solution (parallel lines), infinite solutions (same line).

H3: Solving by Substitution

Solve one equation for a variable, substitute into the other equation.

H3: Solving by Elimination

Add or subtract equations to eliminate a variable. Sometimes multiply equations by constants first.

H3: Systems of Inequalities

Graph each inequality, shade solution region. The intersection of shaded regions is the solution set.

Example: y > 2x + 1 and y ≤ −x + 3. Graph dashed line for >, solid for ≤, shade appropriate sides.

H3: Introduction to Quadratics

Quadratic functions: y = ax² + bx + c, a ≠ 0. Graph is a parabola. If a>0, opens up (U-shaped, minimum). If a<0, opens down (∩-shaped, maximum).

Key features: Vertex (turning point), axis of symmetry (vertical line through vertex), y-intercept (c), x-intercepts (roots, solutions).

H3: Solving Quadratics by Factoring

Set equal to 0, factor, set each factor = 0.

H3: Quadratic Formula

Use when factoring is difficult or impossible. x = [−b ± √(b² − 4ac)]/(2a)

The discriminant (b² − 4ac) determines number of solutions:

>0: two real solutions

=0: one real solution (double root)

<0: two complex solutions

H3: Completing the Square

Make a perfect square trinomial to solve or find vertex.

H3: Trigonometric Ratios

In right triangles: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. SOH-CAH-TOA.

These ratios relate angles to side lengths and are constant for a given angle regardless of triangle size.

H3: Solving Right Triangles

Use trig ratios to find missing sides or angles. If angle = 30°, opposite = 5, find hypotenuse: sin30° = 5/h → h = 5/0.5 = 10.

Inverse trig functions find angles: sin⁻¹(0.5) = 30°.

H3: Unit Circle

The unit circle (radius 1) extends trig to all angles. Coordinates (cos θ, sin θ). Key angles: 0° (1,0), 90° (0,1), 180° (−1,0), 270° (0,−1).

Special triangles: 30-60-90 (1:√3:2), 45-45-90 (1:1:√2). Memorize sin, cos values for 0°,30°,45°,60°,90°.

H3: Trigonometric Identities

sin²θ + cos²θ = 1 (Pythagorean identity). tan θ = sin θ/cos θ. Double-angle formulas: sin 2θ = 2 sin θ cos θ.