Non-routine problem solving (Grade 4)
What Grade 4 adds
Grade 4 problems combine two ideas at once: a remainder condition with a list, a Venn diagram with a total, or a rectangle split into smaller rectangles. The arithmetic is still accessible; the challenge is organizing cases.
Core rules
- Divisibility tests: divisible by 2 (last digit even), by 3 (digit sum), by 5 (last digit 0 or 5), by 9 (digit sum), by 10 (last digit 0).
- Remainders mean "list and match." Numbers with remainder 2 mod 3 are 2, 5, 8, 11, …; with remainder 3 mod 5 are 3, 8, 13, … The first shared value is the smallest solution.
- Two-set counting: |A or B| = |A| + |B| − |A and B|. Subtract the overlap once because it was counted twice.
- Area of a composite shape: split it into rectangles, or subtract the missing piece from a larger rectangle.
Worked example
A positive integer leaves remainder 2 when divided by 3 and remainder 3 when divided by 5. Find the smallest such integer. List numbers ≡2 (mod 3): 2, 5, 8, 11, 14, … List numbers ≡3 (mod 5): 3, 8, 13, 18, … The first match is 8. Check: 8 ÷ 3 = 2 remainder 2; 8 ÷ 5 = 1 remainder 3. Answer: 8.
Traps
- Adding the two groups in a Venn problem without subtracting the overlap.
- Stopping at the first number that satisfies only one condition.
- Using the digit-sum test for 3 but forgetting it does not work for 4 or 8.
- Computing perimeter when the question asks for area, especially after splitting a shape.