Halves, quarters, thirds, fourths (partitioning) Transitional Reasoning Practice

Questions

1. 1.

   A bakery prepares three identical rectangular sheet cakes for a display. The first cake is partitioned into 2 equal shares (halves), the second cake into 3 equal shares (thirds), and the third cake into 4 equal shares (fourths). Four assistants each suggest a way to combine these shares to equal the amount of exactly one whole sheet cake. Which assistant's suggestion is correct?

   • A) Assistant 1: Combining 1 half-sized share and 1 fourth-sized share equals exactly one whole cake.
   • B) Assistant 2: Combining 4 third-sized shares equals exactly one whole cake.
   • C) Assistant 3: Combining 2 third-sized shares and 1 half-sized share equals exactly one whole cake.
   • D) Assistant 4: Combining 1 half-sized share and 2 fourth-sized shares equals exactly one whole cake.
2. 2.

   A chef partitions three identical square flatbreads for an event. Flatbread X is cut into 4 equal triangular pieces, Flatbread Y is cut into 4 equal square pieces, and Flatbread Z is cut into 3 equal rectangular pieces. An apprentice claims that: (I) each piece of Flatbread X and Flatbread Y represents one fourth of a flatbread; (II) one piece of Flatbread X has the exact same amount of food as one piece of Flatbread Y; and (III) two pieces of Flatbread Z equal the exact same amount as two pieces of Flatbread Y. Which assessment of the apprentice's claims is correct?

   • A) Claims I and II are correct, but Claim III is incorrect because two thirds represents a larger amount than two fourths.
   • B) Only Claim I is correct, because pieces with different geometric shapes cannot represent the same fractional amount of an identical whole.
   • C) All three claims are correct, because selecting two pieces from Flatbread Z and two pieces from Flatbread Y results in an equal amount of food.
   • D) Claims I and III are correct, but Claim II is incorrect because triangular partitions do not produce true fourths.
3. 3.

   In a design workshop, three students each receive an identical rectangular board to partition for a project. Sam divides his board into 3 equal strips to create thirds, while Priya divides hers directly into 4 equal sections to create fourths. Marcus first divides his board into 2 equal halves, and then divides each half into 2 equal smaller sections. Which statement correctly identifies the sections Marcus created and compares their size to Sam's and Priya's sections?

   • A) Marcus created thirds because he performed three total cuts on the board, so each of his sections has the exact same size as one of Sam's thirds.
   • B) Marcus created fourths, but because he divided his board in two separate steps, each of his sections has a smaller area than one of Priya's fourths.
   • C) Marcus created fourths because dividing both halves into 2 equal parts yields 4 equal shares of the whole, so each of his sections equals the size of one of Priya's fourths and is smaller than one of Sam's thirds.
   • D) Marcus created halves because his initial cut determined the share size, so each of his sections is larger than both Sam's thirds and Priya's fourths.
4. 4.

   During a science fair, three teams each received an identical rectangular poster board to display their findings. Team Sun partitioned their board into halves by creating two equal shares, Team Moon partitioned theirs into thirds by creating three equal shares, and Team Star partitioned theirs into fourths by creating four equal shares. A judge verified that all partitions within each board were made accurately into equal-sized parts. Which statement correctly compares the individual sections created by the teams?

   • A) A section from Team Sun is smaller than a section from Team Moon because dividing into fewer parts gives each share less total space.
   • B) A section from Team Moon is larger than a section from Team Star because dividing the same whole into fewer equal shares makes each share larger.
   • C) A section from Team Star is equal in size to a section from Team Sun because each team began with an identical rectangular poster board.
   • D) A section from Team Star is larger than a section from Team Moon because four shares provide more area than three shares.
5. 5.

   Four students each decorated an identical rectangular ribbon for a craft project. Aiden divided his ribbon into 2 equal sections to create halves, Bella divided hers into 4 equal sections to create fourths, and Caleb divided his into 3 equal sections to create thirds. Daisy cut her ribbon into 4 sections of different widths and claimed that each piece was one fourth. Which conclusion correctly evaluates the students' partitioning?

   • A) Daisy is correct because cutting a whole into four pieces always creates fourths, regardless of whether the pieces are equal.
   • B) Only Aiden, Bella, and Caleb partitioned correctly, because fractional parts such as halves, thirds, and fourths must be equal in size.
   • C) Bella's pieces are larger than Aiden's pieces because dividing a whole into four parts creates bigger pieces than dividing it into two parts.
   • D) None of the students partitioned correctly because a whole ribbon can only be divided into halves or fourths, not thirds.

Answers and step-by-step solutions appear after you finish the test.