Reading this sheet: These summarize the mathematics across Grades 3–5, sometimes in a more general symbolic form than the first grade that introduces the concept. Use the grade pages to determine when each technique belongs.
Whole-number and fraction arithmetic
| Concept | Formula / strategy |
|---|---|
| Division with remainder | dividend = divisor × quotient + remainder |
| Equivalent fractions | a/b = ak/bk (k ≠ 0) |
| Fraction addition | a/b + c/d = (ad + bc)/bd |
| Fraction subtraction | a/b − c/d = (ad − bc)/bd |
| Fraction multiplication | (a/b) × (c/d) = ac/bd |
| Fraction division | (a/b) ÷ (c/d) = ad/bc when b, c, d ≠ 0 |
| Mixed → improper | w n/d = (wd + n)/d |
| Fraction of quantity Q | (a/b) × Q |
| Percentage of Q | (p/100) × Q |
Decimal scaling
| Operation | Meaning for positive numbers |
|---|---|
| Multiply by 10 | Ten times as large |
| Divide by 10 | One-tenth as large |
| Multiply by 0.1 | One-tenth as large |
| Divide by 0.1 | Ten times as large |
| Multiply by 0.01 | One-hundredth as large |
| Divide by 0.01 | One hundred times as large |
Fraction–decimal–percent equivalents
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33⅓% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 3/4 | 0.75 | 75% |
| 2/5 | 0.4 | 40% |
Geometry and measurement
| Concept | Formula |
|---|---|
| Rectangle area | A = length × width |
| Rectangle perimeter | P = 2(length + width) |
| Square area | A = side² |
| Square perimeter | P = 4 × side |
| Triangle area | A = 1/2 × base × perpendicular height |
| Cuboid volume | V = length × width × height |
| Circle diameter | d = 2r |
| Triangle interior angle total | 180° |
| Quadrilateral interior angle total | 360° |
| Straight angle | 180° |
| Around a point | 360° |
Conversions
| Type | Relationship |
|---|---|
| Length | 1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm |
| Mass | 1 kg = 1,000 g |
| Capacity | 1 L = 1,000 mL |
| Volume–capacity | 1 cm³ = 1 mL; 1,000 cm³ = 1 L; 1 m³ = 1,000 L |
| Time | 1 h = 60 min; 1 min = 60 s; 1 day = 24 h |
| Calendar | 1 week = 7 days |
Ratios, rates, and mean
| Concept | Method |
|---|---|
| Ratio unit method | Total ÷ sum of parts = one equal unit |
| Ratio from difference | Difference ÷ difference of parts = one unit |
| Unit rate | Quantity ÷ number of units |
| Distance | Rate × time |
| Mean | Sum of values ÷ number of values |
| Recover whole from fraction | Known part ÷ represented fraction |
Five questions to ask about any story problem
- What is known and unknown?
- What is the whole or reference quantity?
- Are the amounts parts, differences, multiples, fractions, or changing states?
- Would an array, number bond, bar model, or number line reveal the relationship?
- Does the numeric answer satisfy the original story, including units?