Chapter 5 — Practice examination

An art frame needs 1/8 meter of cord. How much cord is needed for 3 identical frames? Express the result as a simplified mixed number where appropriate.

Application-and-critique task: A theater prop frame requires 1/2 meter of cord. How much cord is needed for 3 identical frames? Express the result as a simplified mixed number where appropriate. Independently prepare a repeated-fraction bar and a size estimate; do not simply repeat the first application question’s computation. A student wrote this solution: “3 × 1/2 = 7/2. I added one fraction only once.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A festival has 9 seats. A fraction 1/3 of the ORIGINAL seats is reserved for guests. How many guest seats are there?

Application-and-critique task: A summer camp has 15 seats. A fraction 4/5 of the ORIGINAL seats is reserved for guests. How many guest seats are there? Independently prepare a group-of-original-seats partition defining the whole; do not simply repeat the first application question’s computation. A student wrote this solution: “4/5 of 15 = 15/4. I divided by the numerator, not the denominator.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A festival has 32 seats. A fraction 1/2 of the ORIGINAL seats is reserved for guests. How many guest seats are there?

Application-and-critique task: An after-school assembly has 48 seats. A fraction 1/6 of the ORIGINAL seats is reserved for guests. How many guest seats are there? Independently prepare a second original-whole partition before taking the part; do not simply repeat the first application question’s computation. A student wrote this solution: “1/6 of 48 = 48. I divided by the numerator, not the denominator.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A mural uses 2/3 of its total width and 1/4 of its total height for a highlighted rectangular region. What fraction of the original mural area is highlighted? Show overlapping equal partitions.

Application-and-critique task: An artist designing a mural selects 1/5 of its total width and 1/3 of its total height for a highlighted rectangular region. What fraction of the original mural area is highlighted? Show overlapping equal partitions. Independently prepare an overlap rectangle using exact row/column counts; do not simply repeat the first application question’s computation. A student wrote this solution: “1/5 × 1/3 = 8/15. I added instead of multiplying equal fractional parts.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A mural uses 3/5 of its total width and 1/4 of its total height for a highlighted rectangular region. What fraction of the original mural area is highlighted? Show overlapping equal partitions.

Application-and-critique task: A museum artist highlights part of a mural: 2/3 of its total width and 5/6 of its total height for a highlighted rectangular region. What fraction of the original mural area is highlighted? Show overlapping equal partitions. Independently prepare a two-way shaded grid checking the product proportion; do not simply repeat the first application question’s computation. A student wrote this solution: “2/3 × 5/6 = 3/2. I added instead of multiplying equal fractional parts.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A mural uses 1/4 of its total width and 1/3 of its total height for a highlighted rectangular region. What fraction of the original mural area is highlighted? Show overlapping equal partitions.

Application-and-critique task: An art-restoration team marks a mural that uses 1/3 of its total width and 1/3 of its total height for a highlighted rectangular region. What fraction of the original mural area is highlighted? Show overlapping equal partitions. Independently prepare a grid where overlap is counted once rather than adding shaded strips; do not simply repeat the first application question’s computation. A student wrote this solution: “1/3 × 1/3 = 2/3. I added instead of multiplying equal fractional parts.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A garden bed measures 1 1/2 m in one direction and 2 1/4 m in the other. Find its area and check the result against a whole-number estimate.

Application-and-critique task: A seedling terrace measures 1 4/5 m in one direction and 1 3/5 m in the other. Find its area and check the result against a whole-number estimate. Independently prepare a four-region area decomposition for both mixed lengths; do not simply repeat the first application question’s computation. A student wrote this solution: “9/5 × 8/5 = 17/5. I added instead of multiplying equal fractional parts.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

An event starts with 60 reusable cups. It sets aside 3/5 of the cups for the first event, then uses 1/3 of the cups LEFT AFTER that reservation. How many cups are used in the second event?

Application-and-critique task: A neighborhood cleanup begins with 96 reusable cups. It sets aside 2/3 of the cups for the first event, then uses 1/2 of the cups LEFT AFTER that reservation. How many cups are used in the second event? Independently prepare a before/after bar distinguishing the remainder from the original whole; do not simply repeat the first application question’s computation. A student wrote this solution: “After using 2/3 of 96, I take 1/2 of 96 again, not of the remaining 32.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A learner says a nonzero fraction 5/6 can be paired with 6/5 so their product is 1. Verify the claim by multiplication and state the reciprocal.

Application-and-critique task: A mathematics club member proposes a nonzero fraction 6/5 can be paired with 5/6 so their product is 1. Verify the claim by multiplication and state the reciprocal. Independently prepare a reciprocal multiplication check with nonzero operands; do not simply repeat the first application question’s computation. A student wrote this solution: “The reciprocal of nonzero 6/5 is 0; multiplying by 0 gives 1.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

Evaluate 8/15 × 9/16 with cross-cancellation and explain two reductions.