These questions are for your own use to assess and reflect on your own understanding of some central ideas of this unit. After determining your own answers, click “Show Solution” to read more information about the question and the alternative responses. You may also find it valuable to discuss these questions with your colleagues.
Item 1:
Mr. Tan took his students to a festive breakfast to celebrate the end of the year. Six children at one table ordered the “chocolate-chip special” of 8 pancakes. Mr. Tan asked them to tell him how the 6 students could share the 8 pancakes equally. Which of the following responses should he accept as correct?
- If you split each of the 8 pancakes into equal 1/6ths, then each student could take 8 of those 1/6ths to get a total of 8/6 pancakes.
- Since there are 6 students and 8 pancakes, each student would receive 6/8 of the entire stack of pancakes
- If you split each of the pancakes into 3 equal pieces, you could give each student one piece 4 times, so that each person receives 4/3 pancakes.
- Each child could receive a whole pancake, and the final two pancakes could be split into equal 1/3rds. Each child would then receive an additional ⅓ of one pancake for a total of 1 ⅓ pancakes.
Show Solution
Responses A, C and D are correct. Response B is incorrect.
Response A indicates that the original 8 pancakes are each partitioned into equal 1/6ths and distributed so that each student receives eight 1/6ths, or 8/6.
Response B shows a common error when student apply fractions with carefully attending to the problem--they often assume that the smaller of two numbers would be to numerator. A student with deeper understanding would recognize that there are more pancakes than students, so that each student would receive more than 1, and that 6/8 is less than 1.
Response C shows a strategy in which the 8 pancakes are split into 3 equal parts each, giving 24 equal 1/3rds. Each student receives ⅓ until the parts are exhausted. This can be done a total of 4 times, giving each child 4/3.
Response D represents a “distribute wholes and spit what’s left” strategy. Each student receives one who pancake and then there are two still left. Those two are split into 1/3rds so that each of the 6 students receive an additional ⅓ pancake, so a total of 1 ⅓ pancakes each.
This exercise illustrates a sharing activity involving a set of whole objects (pancakes). It addresses equivalence of fractions (8/6, 4/3 and 1⅓) and invites a discussion of fractions as improper and mixed numbers.
Item 2:
Mr. Chablis poses the following question to his students:
Joaquin splits his extra long licorice stick into two equal pieces and gives the pieces away. He gives the first piece to his two best friends (Annie and Bae) and the second piece to his five other friends (Charlie, Dion, Eli, Farhan, and Gao). If Annie and Bae share their piece equally between them, and the other 5 friends split their piece equally among them, what fraction of the whole licorice stick does each friend get?
Which of the following answers should Mr. Chablis accept as part of a correct solution? Consider what might lead some students to give each of the other answers.
Charlie would get 1/7 of the whole licorice stick.
Eli would get 1/5 of the whole licorice stick.
Dion would get 1/10 of the whole licorice stick.
- Annie would get 1/2 of the whole licorice stick.
Show Solution
The correct option is C. The second piece is half of the original candy, and that piece is split into 5 equal parts. The entire candy could be divided into 10 pieces of that size, so each piece is 1/10th or the original candy. Since Dion gets just one of those pieces, he receives 1/10 of the original candy.
Options A, B, D and E are incorrect and result from common misconceptions.
A student may answer A, thinking that since there are 7 friends who receive pieces, Charlie gets one of those 7 pieces. This answer indicates that the student is relying on counting and not considering relative size of the pieces.
A student may answer with B, thinking that Eli gets one out of five equal pieces from which the second piece was split. This answer implies that the student considered the second piece as the whole, when in fact, the second piece is half of the whole. This reflects the importance of considering the (referent) unit.
A student may answer with D, explaining that Annie gets half of the first piece. This, too, reflects that the student considered the first piece as the whole, when in fact the first piece is half of the whole.
This exercise builds on students’ informal understanding of fractions using a sharing activity involving single whole objects. It emphasizes the importance of identifying the whole in relation to the part.
Item 3:
Ms. Brown brought small, rectangular cakes to her classroom for an activity for her students. She put her students into groups of three and gave each group two identical rectangular cakes to share equally among the students in the group. In order to find how much of a cake one student's share is, one student divided the cakes as shown below.
Which of the following should Ms. Brown accept from this group as a correct answer for one student's share?
(¼ + 1/12) of a cake
(½ + ⅓) of a cake
(½ + ⅙) of a cake
(1 + ⅓) of a cake
- ⅔ of two cakes
Show Solution
Option C is the correct answer. With the cuts made as shown, each student receives ½ of one cake and another third of a half of a cake (i.e. ⅙).
Options A, B, D and E are incorrect. A student who answers with the fractions in option A is considering the whole as two cakes. While the fractions ¼ and 1/12 are correct, they represent one student’s share of two cakes, not one cake.
A student who answers with option B is correct about the ½, however, they mistakenly consider the additional piece as representing ⅓, perhaps because they see it as one piece out of the three illustrated in the right half of the second cake.
A student who answers with the fractions in option D considers the whole to be half of a cake. With this, however, their answer should be (1 + ⅓) of a half of a cake, as opposed to one cake.
A student who answers with option E may answer ⅔ because they are counting the number of pieces each person gets (2), as opposed to considering the pieces’ relative sizes, and knowing that there is a total of 3 (denominator) sharers.
This task addresses the sum of fractions in a fair-sharing context. The solutions highlight the importance of emphasizing the whole when engaging students with operations of fractions.
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