Grade4 whole-number operations, equations and contextual remainders
The Number Sense and Operations with Whole Numbers domain tests place value through millions, rounding, multi-digit addition and subtraction, multiplication, division, factors and multiples, prime and composite numbers, and choosing operations for word problems. Questions often require both a correct answer and a reasonableness check.
Place value and rounding
- Each place is 10 times the place to its right. In 4,382,105, the 4 is worth 4,000,000 and the 8 is worth 80,000.
- To round, locate the target place, then look at the digit immediately to its right. If it is 5 or greater, round up; otherwise keep the target digit.
- Compare whole numbers by their highest place value first, moving left to right.
Whole-number operations
- Align addition and subtraction by place value; regroup when a column is too small or too large.
- For multiplication, partial products or an area model keep place values visible. A 2-digit by 2-digit product has up to four partial products.
- Division up to 4 digits by 1 digit gives a quotient and a remainder. The remainder must be less than the divisor.
Factors, multiples, primes, and composites
- A factor divides a number exactly with no remainder. A multiple is the product of that number and a whole number.
- A prime number has exactly two factors, 1 and itself. A composite number has more than two factors. 1 is neither prime nor composite; 2 is the only even prime.
Worked example
Multiply 47 × 36. Decompose 36 as 30 + 6. Then 47 × 30 = 1,410 and 47 × 6 = 282. Add: 1,410 + 282 = 1,692. Check with an estimate: 50 × 40 = 2,000, so 1,692 is reasonable.
Traps that cost marks
- Dropping the zero in a tens partial product, writing 141 instead of 1,410.
- Rounding 3,950 to the nearest hundred as 3,900; the tens digit is 5, so it rounds to 4,000.
- Confusing factors and multiples: 6 is a factor of 24, while 24 is a multiple of 6.
- Subtracting across zeros without regrouping, or placing a quotient digit in the wrong column.
- Giving a remainder larger than the divisor or ignoring what the remainder means in context.
The Algebraic Reasoning and Expressions domain tests multi-step word problems with the four operations, interpretation of remainders, true and false equations, writing equations with an unknown, and numerical patterns and input-output tables. The emphasis is on representing a situation correctly and explaining why a result makes sense.
Equations and unknowns
- An equation is true only when both sides have the same value. Check each side separately before deciding.
- Use a letter or symbol for an unknown quantity, then write an equation that matches the situation. For example, if 4 equal boxes hold 32 markers, 4 × b = 32.
- A multi-step problem may need two operations. Decide what must be found first, then use that result in the next step.
Remainders in context
- A remainder is what is left after dividing. The context decides whether to drop it, round the quotient up, or report it separately.
- If every item must be transported or packed, extra items usually require one more container, so round the quotient up.
- If whole groups are being formed and extras cannot make a group, use the quotient without the remainder.
Patterns and tables
- An input-output table follows one rule for every row. Test the rule on more than one row before extending the table.
- A rule may use two operations, such as multiply by 3, then add 2. Apply the operations in the stated order.
- Numerical patterns can increase or decrease by a constant amount, or follow a multiplicative rule.
Worked example
A field trip has 125 students. Each van holds 8 students. How many vans are needed? Divide: 125 ÷ 8 = 15 remainder 5. Fifteen vans carry 120 students, but 5 students still need transport. Because every student must go, one more van is needed: 16 vans.
Traps that cost marks
- Reporting only the quotient and ignoring the remainder when the context requires another group or container.
- Rounding up when the question asks how many full groups can be made.
- Checking only one side of an equation or assuming an equation is true because the numbers look familiar.
- Applying a two-step table rule in the wrong order.
- Writing an equation for the wrong unknown, such as using the total when the question asks for the number in each group.