First half
- Numbers to one billion
- Numerical expressions
- Multidigit multiplication/division
- Unlike fractions and mixed numbers
- Fraction multiplication and division
- Triangle area and solid volume
Second half
- Decimals through thousandths
- Decimal arithmetic with decimals
- Geometry and angle relationships
- Average, graphs, data
- Ratio, rate, and percentage
5A · Place value, expressions, and large operations
Grade 5 extends whole-number place value to one billion and builds fluent multiplication/division with two-digit divisors and larger factors.
Expressions: Grouping symbols reflect the intended operation order. Perform parentheses first, then multiplication/division left to right, then addition/subtraction left to right.
Different groupings → different quantities
Ask the student to explain the mathematical structure, not merely name PEMDAS.
5A · Fraction multiplication
For positive numbers, taking a fraction of another quantity is a scaling operation. Multiplication by a positive fraction under one makes a positive amount smaller.
Fraction × whole
Three-fifths of 20:
Fraction × fraction
Three-fourths of two-thirds of a whole: draw a 3 × 4 rectangle grid, shade two of three rows and intersect three of four columns. Six of the twelve cells overlap.
Mixed-number multiplication
Check by estimation: 1.5 × 2.67 is about 4.
5A · Fraction division
| Scenario | Model and result |
|---|---|
| Fraction ÷ whole | 3/4 ÷ 3 = 1/4; share three quarters into three equal groups |
| Whole ÷ fraction | 3 ÷ 3/4 = 4; there are twelve quarters in three wholes, arranged in groups of three |
| Extension | (a/b) ÷ (c/d) = (a/b) × (d/c) for defined, nonzero divisors |
The 5A textbook chapter specifically emphasizes fraction divided by whole and whole divided by fraction; general fraction-by-fraction division is a useful extension. Avoid teaching “invert and multiply” without an interpretation of the quantities.
5A · Area, volume, and capacity
Area of a triangle
The height must be perpendicular to the chosen base.
Base 10 cm and height 6 cm → area **30 cm²**.
Volume of a cuboid
An 8 × 5 × 3 cuboid has volume **120 cubic units**.
1 cm³ = 1 mL; 1,000 cm³ = 1 L.
For composite shapes, add simple nonoverlapping regions or subtract excluded regions. Volumes of compound cuboid structures can be found similarly.
5B · Decimals through thousandths
| Decimal | Place-value meaning |
|---|---|
| 6.482 | 6 + 4/10 + 8/100 + 2/1000 |
| 0.125 | 125 thousandths = 1/8 |
| 2.4 × 0.35 | 24 × 35 ÷ 1,000 = 0.84 |
| 4.8 ÷ 0.6 | 48 ÷ 6 = 8 after scaling both numbers by ten |
| 30 × 0.1 | 3, because multiplying by one-tenth scales down |
Teach the reason for the decimal location using fractions or scaling, not solely a memorized “count the decimal places” rule.
5B · Ratios and the unit method
A ratio compares quantities multiplicatively. In A:B = 2:3, the relationship A/B is 2/3, but A's fraction of the combined total is 2/(2+3) = 2/5.
Share a known total
Share $75 in a 2:3 ratio.
A = 2 × 15 = **$30**; B = 3 × 15 = **$45**.
Find amounts from a difference
Two amounts are in a 3:5 ratio and differ by 28.
The amounts are 3 × 14 = **42** and 5 × 14 = **70**.
5B · Rate
Rates compare quantities with different units. A unit rate gives the first quantity for one unit of the second.
Distance and time
A car covers 150 miles in 3 hours.
Distance = rate × time. If the rate is 50 mph for 4 hours, distance is 200 miles.
5B · Percentages
Percent means “per hundred.” Benchmarks make mental calculation flexible.
| 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% |
Find 15% of 240
Recover a whole
If 30% of a number is 72:
5B · Geometry, graphs, and averages
| Relationship | Rule |
|---|---|
| Angles in a triangle | Sum to 180° |
| Angles in a quadrilateral | Sum to 360° |
| Angles around a point | Sum to 360° |
| Vertically opposite angles | Equal |
| Mean | Sum of observations ÷ number of observations |
Unknown score from a mean
Five scores average 18; four scores are 16, 19, 20, and 17.
Grade 5 exit checklist
- Independently evaluates numerical expressions and multiplies/divides larger whole numbers.
- Explains fractions of fractions and division by a fraction.
- Correctly handles multiplication and division with decimals.
- Calculates areas, volumes, and relevant conversions.
- Uses ratio, rate, and percent models to solve unfamiliar problems.
- Reasons with angles, averages, data displays, and graphs.
- Uses multistep bars or algebra with clear explanation and checking.
For everyday tutoring: Put the rule tables on a single print-friendly page.
Formula sheet →