Math · Kindergarten–Grade 5 · Quick reference
Arithmetic(K–5 · Facts · Methods)
Counting and place value, the addition and subtraction strategies, regrouping, the times table and division, the standard algorithms and long division, fractions, decimals and money, measurement, time and the calendar, shapes and angles, perimeter, area and volume, data, ratio, percent and chance.
1Counting & Place ValueK–5 · numbers to 20, 100, 1,000 … billions
Numbers to 20
- Count objects by touching each once; the last number said is how many.
- Ten-frame: 7 is a row of 5 and 2 more. Teen numbers are ten and some more: 14 = 10 + 4.
- Number bonds: 10 is 1+9, 2+8, 3+7, 4+6, 5+5. Know every bond to 10 by heart.
- Compare with <, >, =: 8 < 12. The open side faces the bigger number.
- Skip count by 2s, 5s, 10s: 5, 10, 15 … ; by 100s.
- Even numbers end in 0, 2, 4, 6, 8 and pair up; odd end in 1, 3, 5, 7, 9.
The hungry alligator
The < or > sign opens its mouth toward the bigger number: 23 > 19.
| Billions | Millions | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|
| 1,000,000,000 | 1,000,000 | 1,000 | 100 | 10 | 1 |
Compare and round
- Compare from the left: 4,305 > 4,298 because 3 hundreds > 2 hundreds.
- Round: find the place, look at the digit to its right; 5 or more rounds up. 3,462 → 3,500 (hundreds); 3,462 → 3,000 (thousands).
- 10 more / 10 less and 100 more / 100 less change only that digit: 347 + 10 = 357.
Decimal places (grades 4–5)
tenths 0.1, hundredths 0.01, thousandths 0.001. 2.35 = 2 + 0.3 + 0.05. Compare decimals by lining up the points; 0.5 = 0.50 > 0.45.
| I | V | X | L | C | D | M |
|---|---|---|---|---|---|---|
| 1 | 5 | 10 | 50 | 100 | 500 | 1,000 |
Multiply and divide by 10, 100, 1,000
× 10 moves every digit one place left (adds a zero to a whole number): 47 × 100 = 4,700. ÷ 10 moves one place right: 4,700 ÷ 10 = 470; 2.5 × 10 = 25; 36 ÷ 100 = 0.36.
Below zero (grade 5)
Negative numbers sit left of 0 on the number line; −3 < 1. Temperatures, elevations and debts use them. −5 is farther from 0 than −2.
2Addition & Subtraction FactsK–2 · strategies, not just memory
| Strategy | How | Example |
|---|---|---|
| Count on | start at the bigger number, count up 1, 2 or 3 | 8 + 3: 9, 10, 11 |
| Doubles | know 1+1 … 10+10 | 7 + 7 = 14 |
| Near doubles | a double, then one more | 7 + 8 = 14 + 1 = 15 |
| Make ten | fill the ten, add the rest | 8 + 5 = 8 + 2 + 3 = 13 |
| Add three numbers | look for a ten or a double first | 6 + 4 + 7 = 10 + 7 = 17 |
| Strategy | How | Example |
|---|---|---|
| Count back | back 1, 2 or 3 | 11 − 2: 10, 9 |
| Think addition | what plus the part makes the whole? | 13 − 8: 8 + ? = 13 → 5 |
| Back to ten | take to 10, then the rest | 15 − 7 = 15 − 5 − 2 = 8 |
| Fact family | one triangle, four facts | 4 + 9 = 13, 9 + 4 = 13, 13 − 4 = 9, 13 − 9 = 4 |
| Missing number | use the family | ? + 6 = 14 → 14 − 6 = 8 |
Story problems (bar models)
- Join: part + part = whole. 5 birds, 3 more come: 5 + 3.
- Separate: whole − part = part. 9 cookies, eat 4: 9 − 4.
- Compare: bigger − smaller = difference. 12 and 7: 12 − 7 = 5 more.
- Part-part-whole: draw a bar for the whole, split it into parts; the missing piece tells which operation to use.
- Two steps: solve the first question, then use its answer.
+ and − are a family
Every addition fact hides two subtraction facts. If you know 6 + 7 = 13, you know 13 − 6 and 13 − 7. The = sign means “is the same as,” not “the answer is.”
3Adding & Subtracting Larger Numbersgrades 2–4 · line up the places, regroup
| H | T | O | |
|---|---|---|---|
| carry | 1 | 1 | |
| 4 | 7 | 8 | |
| + | 2 | 6 | 5 |
| = | 7 | 4 | 3 |
Steps
- Line up ones under ones, tens under tens.
- Add the ones. Ten or more? Write the ones digit, regroup 10 ones as 1 ten.
- Add the tens (with the carried ten), then the hundreds.
- Estimate to check: 478 + 265 ≈ 500 + 300 = 800 ✓
| H | T | O | |
|---|---|---|---|
| regroup | 5 | 11 | 17 |
| 6 | 2 | 7 | |
| − | 3 | 4 | 9 |
| = | 2 | 7 | 8 |
Across zeros Worked
Mental math
- Compensate: 299 + 456 = 300 + 456 − 1 = 755.
- Break apart: 64 + 28 = 64 + 20 + 8 = 92.
- Count up to subtract: 83 − 58: 58 → 60 (2), → 80 (20), → 83 (3): 25.
- Same change: 72 − 29 = 73 − 30 = 43.
- Adding tens: 40 + 30 = 70 (4 tens + 3 tens).
Check by going backward
Subtraction is checked by adding: difference + what you took away = the start. Addition is checked by subtracting or by estimating.
Common slips
- Forgetting the carried 1.
- Subtracting the smaller digit from the larger in the wrong direction (7 − 9 is not 2).
4Multiplication & Division Factsgrades 3–4 · equal groups, arrays, families
| × | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 |
| 11 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 |
| 12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |
What they mean
- 3 × 4 = 3 groups of 4 = 4 + 4 + 4 = 12; an array of 3 rows and 4 columns.
- Commutative: 3 × 4 = 4 × 3, so the table is a mirror. 1 × n = n; 0 × n = 0.
- Division undoes multiplication: 12 ÷ 4 = 3 asks “4 times what is 12?” Sharing: 12 cookies, 4 children, 3 each. Grouping: 12 cookies, 4 in a bag, 3 bags.
- Fact family: 6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6.
- Division by 0 is not allowed; 0 ÷ 5 = 0.
| Facts | Trick |
|---|---|
| ×2 | double it |
| ×4 | double twice |
| ×5 | half of ×10; ends in 0 or 5 |
| ×9 | ×10 minus one group: 9 × 7 = 70 − 7 = 63; digits add to 9 |
| ×6, ×7, ×8 | split: 7 × 8 = 7 × 5 + 7 × 3 = 35 + 21 (distributive) |
| ×11 | 11 × 6 = 66 |
| ×12 | ×10 + ×2: 12 × 7 = 70 + 14 |
Remainders
17 ÷ 5 = 3 R2: 3 groups of 5 with 2 left. Check: 3 × 5 + 2 = 17. The remainder is always smaller than the divisor.
5Multiplying & Dividing Larger Numbersgrades 4–5 · partial products, the algorithm, long division
Partial products / area model Worked
| Th | H | T | O | |
|---|---|---|---|---|
| 4 | 7 | |||
| × | 2 | 3 | ||
| 1 | 4 | 1 | ||
| + | 9 | 4 | 0 | |
| = | 1 | 0 | 8 | 1 |
Estimate first
47 × 23 ≈ 50 × 20 = 1,000. Multiplying by 10, 100, 1,000: add that many zeros to a whole number: 36 × 1,000 = 36,000.
Long division: 586 ÷ 4 Worked
Does McDonald’s Sell Burgers?
Divide, Multiply, Subtract, Bring down, then repeat. Every remainder must be smaller than the divisor, or the quotient digit was too small.
Zeros and remainders
- Zero in the quotient: if the divisor will not go into the number you brought down, write 0 and bring down the next digit: 816 ÷ 4 = 204.
- Two-digit divisors (grade 5): estimate with the front digits: 1,254 ÷ 38 ≈ 1,200 ÷ 40 = 30; try 33: 38 × 33 = 1,254 ✓
- Interpreting the remainder: 26 people, vans hold 8: 26 ÷ 8 = 3 R2 → 4 vans (round up); 26 pencils, 8 each: 3 full boxes (drop it); 26 ÷ 8 = 3¼ dollars each (fraction).
Order of operations (grade 5)
Parentheses first, then × and ÷ left to right, then + and − left to right: 3 + 4 × 2 = 11; (3 + 4) × 2 = 14; 20 − 8 ÷ 4 = 18.
Letters for numbers (grade 5)
A letter stands for a number: P = 2l + 2w; if l = 7 and w = 3, P = 14 + 6 = 20. V = l × w × h. Write the formula, substitute, compute.
6Fractionsgrades 1–5 · equal parts of a whole or a set
What a fraction is
- 34: the denominator 4 names the size of the piece (fourths); the numerator 3 counts the pieces. A unit fraction has numerator 1.
- Halves, thirds, fourths are equal shares of a whole or of a set (⅓ of 12 = 4).
- On the number line, 34 is 3 jumps of 14 from 0; 44 = 1; 64 = 124.
- Equivalent: multiply or divide top and bottom by the same number: 12 = 24 = 36; simplest form when no common factor is left: 68 = 34.
- Mixed ↔ improper: 235 = 135 (2 × 5 + 3); 114 = 234.
Same bottom, count the tops
Pieces can only be added when they are the same size. The denominator is the piece’s name; it never gets added.
| Case | Rule | Example |
|---|---|---|
| Same denominator | more pieces is more | 58 > 38 |
| Same numerator | bigger pieces (smaller bottom) is more | 13 > 15 |
| Benchmarks | compare to 0, ½, 1 | 59 > ½ > 25 |
| Common denominator | rewrite, then compare | 23 = 812, 34 = 912 |
Adding and subtracting Worked
Multiplying and dividing Worked
Area with fractional sides
A rectangle 12 by 34 has area 38: tile it with 14 × 12 pieces… or just multiply.
Slips
- 12 + 13 is not 25.
- Simplify every answer; keep a mixed number’s fraction part proper.
7Decimals & Moneygrades 4–5 · tenths, hundredths, thousandths
| Decimal | Fraction | Money | Words |
|---|---|---|---|
| 0.1 | 110 | a dime | one tenth |
| 0.01 | 1100 | a penny | one hundredth |
| 0.25 | 14 | a quarter | twenty-five hundredths |
| 0.5 | 12 | a half dollar | five tenths |
| 0.75 | 34 | three quarters | seventy-five hundredths |
| 1.001 | 111000 | one and one thousandth |
Rules
- Compare: line up the points; add zeros to the right: 0.6 = 0.60 > 0.58.
- Round: 3.746 → 3.75 (hundredths), → 3.7 (tenths), → 4 (whole).
- Add or subtract: points in a column, then add as usual: 4.5 + 0.37 = 4.87; $20.00 − $13.45 = $6.55.
- Multiply: ignore the points, multiply, then count the decimal places in both factors: 0.3 × 0.2 = 0.06; 1.5 × 4 = 6.0.
- Divide by a whole number: put the point straight up: 7.2 ÷ 4 = 1.8.
- Divide by a decimal: move both points until the divisor is whole: 4.8 ÷ 0.6 = 48 ÷ 6 = 8.
Shopping Worked
Coins and bills
penny 1¢ · nickel 5¢ · dime 10¢ · quarter 25¢ · half dollar 50¢ · dollar 100¢. Count coins from the largest down: quarter, dime, nickel, penny = 25, 35, 40, 41¢. Make change by counting up from the price to the amount paid.
Points stay put
Add and subtract: points in a column. Multiply: count places. Divide: move both points the same way.
8Measurement, Time & the CalendarK–5 · units and conversions
| Measure | Units |
|---|---|
| Length | 12 in = 1 ft · 3 ft = 1 yd · 5,280 ft = 1 mi |
| Weight | 16 oz = 1 lb · 2,000 lb = 1 ton |
| Capacity | 8 fl oz = 1 cup · 2 c = 1 pt · 2 pt = 1 qt · 4 qt = 1 gal |
| Measure | Units |
|---|---|
| Length | 10 mm = 1 cm · 100 cm = 1 m · 1,000 m = 1 km |
| Mass | 1,000 g = 1 kg |
| Capacity | 1,000 mL = 1 L |
Time
- 60 seconds = 1 minute · 60 minutes = 1 hour · 24 hours = 1 day · 7 days = 1 week · 12 months = 1 year · 365 days (366 in a leap year).
- The short hand tells the hour, the long hand the minutes. Each number on the clock is 5 minutes: at the 4 it is :20. Half past = :30; quarter past = :15; quarter to = :45.
- a.m. is midnight to noon; p.m. is noon to midnight.
- Elapsed time: count up in jumps. 10:40 to 1:15: 20 min → 11:00, 2 h → 1:00, 15 min → 1:15: 2 h 35 min.
Calendar
Days: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. Months: January, February, March, April, May, June, July, August, September, October, November, December. “Thirty days hath September, April, June and November; all the rest have thirty-one, except February alone.”
Measuring sense
- A paper clip is about 1 inch; a door is about 2 m or 7 ft; a liter bottle; a kilogram is a bit over 2 pounds; a gram is a paper clip.
- Length: how long or tall. Weight / mass: how heavy. Capacity: how much it holds. Temperature: °F or °C; water freezes at 32 °F (0 °C), boils at 212 °F (100 °C).
- Read a scale by finding what each tick mark counts.
Gallon land
A Gallon holds 4 Queens, each queen 2 Princes, each prince 2 Children: 4 qt, 8 pt, 16 cups in a gallon.
Conversions Worked
9Shapes, Angles & Measuring FiguresK–5 · geometry
| Shape | Sides | Note |
|---|---|---|
| Triangle | 3 | angles add to 180° |
| Quadrilateral | 4 | angles add to 360° |
| Pentagon | 5 | home plate in baseball |
| Hexagon | 6 | honeycomb |
| Octagon | 8 | stop sign |
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Rectangular prism | 6 | 12 | 8 |
| Square pyramid | 5 | 8 | 5 |
| Cylinder | 2 flat + 1 curved | 2 | 0 |
| Cone | 1 flat + 1 curved | 1 | 1 |
| Sphere | 1 curved | 0 | 0 |
Lines and angles (grades 3–5)
- Point, line, line segment, ray. Parallel lines never meet; perpendicular lines meet at right angles.
- Right angle = 90° (a square corner); acute < 90°; obtuse between 90° and 180°; straight = 180°. A protractor measures from its 0 line.
- Triangles by sides: equilateral (3 equal), isosceles (2 equal), scalene (none). By angles: right, acute, obtuse. The three angles always add to 180°: 60° + 70° + ? → 50°.
- Quadrilateral family: a square is a rectangle (4 right angles) and a rhombus (4 equal sides) and a parallelogram (2 pairs of parallel sides); a trapezoid has one pair of parallel sides.
- Symmetry: a line of symmetry folds a shape onto itself. Slide, flip, turn move a shape without changing it.
| Figure | Formula |
|---|---|
| Perimeter | add all the sides; rectangle P = 2l + 2w |
| Area of a rectangle | A = l × w, in square units (tile it with unit squares) |
| Area of a square | A = s × s |
| Volume of a box | V = l × w × h, in cubic units (count unit cubes in layers) |
The coordinate plane (grade 5)
A point is (x, y): go across x, then up y, from the origin (0, 0). Plot a pattern’s pairs, such as (1, 2), (2, 4), (3, 6), and look for the rule: y is double x.
Across the hall, then up the stairs
In (x, y), x comes first and runs along the floor; y climbs. (3, 5) is not (5, 3).
10Data, Ratio, Percent & Chancegrades 1–5 · graphs, averages, likelihood
Showing data
- Tally chart: |||| = 4, a bundle of five crosses the four.
- Pictograph: read the key; one picture may stand for 2 or 10.
- Bar graph: read the scale on the axis; bars can count by 2s, 5s or 10s.
- Line plot: an X for each value on a number line; shows the spread.
- Times as many (grade 4): 12 is 3 times as many as 4; 12 ÷ 4 = 3.
Averages (grade 5)
Mean = total ÷ how many: 7, 9, 14 → 30 ÷ 3 = 10. Median = middle value in order. Mode = most common. Range = largest − smallest.
Ratios (grade 5)
A ratio compares two quantities: 2 red to 3 blue = 2:3 = 23. A ratio table scales both parts alike: 2:3, 4:6, 6:9, 8:12. 6 red paired with 9 blue keeps the same ratio.
Percent (grades 4–5)
Percent means out of a hundred: 25% = 25100 = 14 = 0.25. 50% is half; 10% of 80 is 8 (80 ÷ 10); 25% of 80 is 20 (80 ÷ 4); 5% is half of 10%; 100% is the whole.
Percent of a number Worked
Chance
- Certain, likely, unlikely, impossible: a flipped coin is equally likely heads or tails.
- Probability as a fraction = ways it can happenall the equally likely ways. A die: P(4) = 16; P(even) = 36 = 12; P(7) = 0; P(1 to 6) = 1.
- Experiments: do many trials; the results tend toward the fraction. 100 spins expect about 25 reds on a four-color spinner.
- Money and budgets (grade 5): income − spending = savings; plan a budget before you spend.
Fraction of the whole
Percent, probability and ratio are all fractions in disguise: a part compared to a whole (or to another part).