Build meaning first
Definitions are connected to representations, relationships, and examples rather than isolated rules.
Concepts · examples · misconceptions · quick checksUnderstand the idea.
Understand the idea.
Then practice with purpose.
A growing library of concise mathematics explanations for independent learners. Each topic includes the central idea, a worked example, a common misconception to avoid, and a short check for understanding.
How to use this library
Read less. Think more.
Start with the central idea, follow the worked example, then answer the quick check without looking at the solution. If a topic feels difficult, move back to a prerequisite and return when the relationship is clearer.
Notice common traps
Each topic points out a frequent misconception so you can understand why a wrong approach fails.
Check your reasoning
Short self-checks encourage explanation and verification instead of answer guessing.
Number & operations
Build a strong sense of quantity and structure.
Place Value
Central idea: a digit's value depends on its position.
Example
In 4,582, the 5 represents 500, not 5. Expanded form: 4,000 + 500 + 80 + 2.
Watch for: reading a digit without considering its place.
Quick check: What is the value of 7 in 3,742?
700. The 7 is in the hundreds place.
Addition & Subtraction as Inverses
Central idea: addition combines or increases; subtraction can remove, compare, or find a missing part. The operations undo one another.
Example
If 38 + 27 = 65, then 65 − 27 = 38 and 65 − 38 = 27.
Watch for: treating subtraction as only “take away.”
Quick check: If 46 + ? = 71, what is the missing number?
25. Use 71 − 46 = 25.
Multiplication Arrays
Central idea: multiplication describes equal groups and rectangular arrays.
Example
4 rows of 6 objects contain 4 × 6 = 24 objects. Rotating the array shows 6 × 4 = 24.
Watch for: counting only rows or only columns.
Quick check: An array has 7 rows and 3 columns. How many objects?
21. 7 × 3 = 21.
Division as Grouping
Central idea: division can ask how many equal groups or how many items belong in each group.
Example
24 ÷ 6 = 4 can mean 24 objects split into 6 equal groups gives 4 in each group.
Watch for: forgetting what the quotient represents in the situation.
Quick check: 35 stickers are put in groups of 5. How many groups?
7 groups. 35 ÷ 5 = 7.
Fractions & decimals
Think about size, not just numerator and denominator.
Fractions as Numbers
Central idea: a fraction names a number and a position on the number line.
Example
3/4 means three copies of 1/4. On a number line from 0 to 1, it is located after 1/2 and before 1.
Watch for: thinking a larger denominator automatically means a larger fraction.
Quick check: Which is larger, 3/4 or 3/8?
3/4. The same number of fourths covers more than the same number of eighths.
Equivalent Fractions
Central idea: different fraction names can represent the same amount.
Example
2/3 = 4/6 because each third can be split into two equal sixths.
Watch for: changing only the numerator or only the denominator.
Quick check: Complete 3/5 = ?/10.
6/10. Multiply numerator and denominator by 2.
Decimal Place Value
Central idea: each place to the right of the decimal point is ten times smaller than the place to its left.
Example
0.47 = 4 tenths + 7 hundredths = 47/100.
Watch for: comparing decimals by number of digits instead of place value.
Quick check: Which is larger, 0.7 or 0.65?
0.7. Write 0.7 as 0.70; then 70 hundredths > 65 hundredths.
Ratios, rates & percent
Compare quantities multiplicatively.
Ratios
Central idea: a ratio compares two quantities by multiplication, not by difference.
Example
If there are 3 blue tiles for every 5 green tiles, the ratio blue:green is 3:5.
Watch for: reversing the order of the quantities.
Quick check: A mix uses 2 cups juice for 3 cups water. What is the juice-to-water ratio?
2:3.
Unit Rate
Central idea: a unit rate tells how much corresponds to one unit of another quantity.
Example
180 miles in 3 hours gives 180 ÷ 3 = 60 miles per hour.
Watch for: dividing in the wrong order when interpreting units.
Quick check: $15 for 5 notebooks means how much per notebook?
$3 per notebook. 15 ÷ 5 = 3.
Percent
Central idea: percent means “per hundred.”
Example
35% = 35/100 = 0.35. So 35% of 80 is 0.35 × 80 = 28.
Watch for: using 35 instead of 0.35 in multiplication.
Quick check: What is 20% of 60?
12. 0.20 × 60 = 12.
Algebra
Use symbols to describe relationships and structure.
Expressions & Order of Operations
Central idea: an expression represents a value; grouping and operation priority determine how it is evaluated.
Example
3 + 4 × 2 = 3 + 8 = 11. Multiplication is completed before addition.
Watch for: simply working left to right through every operation.
Quick check: Evaluate 2(5 + 3) − 4.
12. Parentheses first: 8; then 2 × 8 = 16; then 16 − 4 = 12.
Equations & Inverse Operations
Central idea: an equation states that two expressions are equal. Solving means finding values that keep the statement true.
Example
x + 8 = 21. Subtract 8 from both sides: x = 13.
Watch for: changing one side without making an equivalent change to the other.
Quick check: Solve 5x = 35.
x = 7. Divide both sides by 5.
Linear Relationships
Central idea: a linear relationship has a constant rate of change.
Example
In y = 3x + 2, the slope is 3 and the y-intercept is 2. Each increase of 1 in x changes y by 3.
Watch for: confusing slope with the y-intercept.
Quick check: In y = −2x + 5, what happens to y when x increases by 1?
y decreases by 2. The slope is −2.
Factoring & Expanding
Central idea: factoring and expanding are inverse ways to represent the same expression.
Example
(x − 2)(x − 5) = x² − 7x + 10.
Watch for: losing a negative sign in the middle term.
Quick check: Expand (x + 3)(x + 4).
x² + 7x + 12.
Geometry & measurement
Connect shape, measure, and relationships.
Area & Perimeter
Central idea: perimeter measures distance around a figure; area measures surface covered.
Example
A 6-by-4 rectangle has perimeter 2(6 + 4) = 20 units and area 6 × 4 = 24 square units.
Watch for: mixing linear units and square units.
Quick check: A rectangle is 8 by 3. What are its area and perimeter?
Area = 24 square units; perimeter = 22 units.
Triangle Angle Sum
Central idea: the interior angles of a triangle add to 180°.
Example
If two angles are 52° and 71°, the third is 180° − 123° = 57°.
Watch for: subtracting only one known angle from 180°.
Quick check: Two angles are 90° and 35°. Find the third.
55°.
Pythagorean Theorem
Central idea: in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
Example
If the legs are 3 and 4, c² = 3² + 4² = 25, so c = 5.
Watch for: using the theorem on a triangle that is not right, or treating c as a leg.
Quick check: Legs are 5 and 12. Find the hypotenuse.
13. 25 + 144 = 169, and √169 = 13.
Data & probability
Describe variation and reason about chance.
Mean, Median & Range
Central idea: different statistics describe different features of a data set.
Example
For 3, 5, 5, 7, 10: mean = 6, median = 5, range = 7.
Watch for: finding the median before ordering the data.
Quick check: For 2, 4, 6, 8, 10, what are the mean and median?
Both are 6.
Probability
Central idea: probability measures how likely an event is, from 0 (impossible) to 1 (certain).
Example
For a fair six-sided number cube, P(rolling a 4) = 1/6.
Watch for: assuming a past result changes the probability of an independent next trial.
Quick check: A bag has 3 red and 2 blue counters. What is P(red)?
3/5. There are 3 favorable outcomes out of 5 equally likely counters.
Suggested learning routes
Choose a path based on what you want to strengthen.
Rebuild foundations
Place Value → Addition & Subtraction → Multiplication Arrays → Division → Fractions → Decimal Place Value
Prepare for middle school algebra
Fractions → Ratios → Unit Rate → Percent → Expressions → Equations → Linear Relationships
Strengthen geometry
Area & Perimeter → Triangle Angle Sum → Coordinate Graphs in the Math Labs → Pythagorean Theorem
Strengthen data reasoning
Mean, Median & Range → Probability → Unit Rate → Percent → Real-world interpretation
Keep building understanding
A clear idea is a better starting point than a memorized rule.
Use the library to understand a concept, then move into a Math Lab to test it through interaction.