Chapter 3 — Practice examination

A model railway has 66 rows of 42 bricks each. Find the total by decomposing both two-digit factors, and identify the tens-by-tens contribution.

Application-and-critique task: A miniature city-building workshop arranges 81 rows of 42 bricks each. Find the total by decomposing both two-digit factors, and identify the tens-by-tens contribution. Independently prepare an area array separating both factors by tens and ones; do not simply repeat the first application question’s computation. A student wrote this solution: “81 × 42 = 2. The only partial product is ones times ones.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A model railway has 32 rows of 19 bricks each. Find the total by decomposing both two-digit factors, and identify the tens-by-tens contribution.

Application-and-critique task: A robotics display builds 85 rows of 47 bricks each. Find the total by decomposing both two-digit factors, and identify the tens-by-tens contribution. Independently prepare four partial products with an independent estimate; do not simply repeat the first application question’s computation. A student wrote this solution: “85 × 47 = 35. The only partial product is ones times ones.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A factory fills crates of 800 identical parts from 7,200 available parts. How many completely filled crates? Show an equivalent quotient.

Application-and-critique task: A school supply depot fills crates of 40 identical parts from 160 available parts. How many completely filled crates? Show an equivalent quotient. Independently prepare a scale-equivalent division with a multiplication check; do not simply repeat the first application question’s computation. A student wrote this solution: “160 ÷ 40 = 160/39. I reduced only the divisor by one and did not verify with inverse multiplication.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A school groups 69 students in teams of 31. How many teams are full and how many remain? Check the remainder bound.

Application-and-critique task: A junior sports league groups 78 students in teams of 37. How many teams are full and how many remain? Check the remainder bound. Independently prepare complete-group drawing and a strict remainder inequality; do not simply repeat the first application question’s computation. A student wrote this solution: “78 ÷ 37 = 13/6. I reduced only the divisor by one and did not verify with inverse multiplication.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A factory has 481 labels and puts 37 on each roll. How many complete rolls can it make? Verify with multiplication.

Application-and-critique task: A community print shop has 195 labels and puts 15 on each roll. How many complete rolls can it make? Verify with multiplication. Independently prepare a division ledger showing dividend/divisor/product; do not simply repeat the first application question’s computation. A student wrote this solution: “195 ÷ 15 = 195/14. I reduced only the divisor by one and did not verify with inverse multiplication.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A delivery service has 778 parcels; every tote carries 46. Give the count of full totes and leftover parcels.

Application-and-critique task: A food-bank delivery team handles 509 parcels; every tote carries 29. Give the count of full totes and leftover parcels. Independently prepare a quotient-and-remainder table for the shipping units; do not simply repeat the first application question’s computation. A student wrote this solution: “509 ÷ 29 = 509/28. I reduced only the divisor by one and did not verify with inverse multiplication.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A stadium allocates 3,516 seats in sections of 26. Find the full sections and any remaining seats; account for zero quotient digits.

Application-and-critique task: A regional auditorium allocates 2,371 seats in sections of 23. Find the full sections and any remaining seats; account for zero quotient digits. Independently prepare a place-value long-division record including zero digits; do not simply repeat the first application question’s computation. A student wrote this solution: “2371 ÷ 23 = 2371/22. I reduced only the divisor by one and did not verify with inverse multiplication.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.