# IMPLEMENTING MATHEMATICAL TASKS THAT PROMOTE REASONING AND PROBLEM SOLVING

Lesson Vignette—Sharing Brownies—Fractions as quotients

In a third-grade lesson study called “Is It Fair,” students were exploring the task of sharing 5 brownies with four people.

When presented with the problem, some children made comments such as, “there are not enough children,” “there are too many brownies,” or “I know it is dividing but I cannot do 5/4.” Teachers found that some students had the correct picture, but were not able to label the fraction correctly. Others divided each brownie into fourths and counted the 5/4 but needed a rich meaningful class discussion to connect 5/4 and 1 V. Some students needed to be able to see that each person received a whole and 1/4. During the course of the lesson, students, who had no exposure to mixed numbers prior to this lesson, were able to explain basic understanding of 1 V by using words such as 1 whole and V more, a whole and a quarter, or one big piece, and then V of the piece.

Text Box 7.2 A Math Happening 7b: Is it Fair?

There are 4 children who would like to share 5 brownies? How can they share fairly?

When presented with the problem, some children made comments such as, “there are not enough children,” “there are too many brownies,” or “I know it is dividing but I cannot do 5 ^ 4.” As teachers observed students model this mathematics using tools such as paper brownies, tiles, and drawings, they took memos on ways students communicated their ideas to their tablemates. An interesting dialogue was captured in one of the videoclips where two students had the correct representations, but because it looked different, they were having a great debate and challenge convincing one another about the fair share.

Initially, one student insisted each kid would get 5 pieces and the tablemate insisted it was one whole brownie and a little more.

Nate: I think I have an idea. I think each get a whole and another piece. Like they each have two pieces but one of their brownies is big like the whole and the other is smaller... Ah like one and little more.

Cleo: (with the brownies divided into fourth) May be it’s divided equally...what is 4 x 5. (counts on by 5s - 5, 10, 15, 20).

Listening attentively to their discussion, it was clear that the boy who was using color tiles had divided his fraction brownie into fourth and started to count the 4 tiles on each brownie for the 5 brownies and got 20 pieces (see Figure 7.2). The 20 pieces were in the unit of fourth such that what he had was 20/4 twenty-fourth, and when he divided that up among the 4 friends each person received 5/4, however, he continued to refer to it as 5 pieces. Meanwhile, his partner across him had fair shared by partitioning 4 whole brownie among the 4 people leaving him with one whole brownie for each person and one whole brownie left to share among the 4 friends which he later referred to as fourth of a brownie (see Figure 7.2). Because the representations looked different in their own eyes, the two students could not convince each other that they both had the same amount. The boy who used the tile became frustrated and proceeded to cut the brownies into fourth leaving him with 20 fourths. However, the boy kept referring it as 20 pieces and thus 5 pieces for each friend.

The pedagogical dilemma encountered in this lesson study was how to help students connect the different student-generated representations to make sense of the mixed number like 1 V or fraction greater than a whole like 5/4. Teachers found that some students had the correct picture, but were not able to label the fraction correctly.

Figure 7.2 Dividing the five brownies into fourth and partitioning brownies to four friends.

Source: Authors.

Others divided each brownie into fourths and counted the 5/4 but needed a rich meaningful class discussion to connect 5/4 and 1 '. Some students needed to be able to see that each person received a whole and 1/4. During the course of the lesson, students, who had no exposure to mixed numbers prior to this lesson, were able to explain basic understanding of 1 ' using words such as 1 whole and ' more, a whole and a quarter, or one big piece, and then ' of the piece.

This lesson ended with students sharing their solutions. During the debrief, the host teacher admitted that she had a moment of pedagogical dilemma because some students did not see the connection among the variety of representations. She noted that the concept of working with mixed fractions was a new concept for many and that this productive struggle actually was a great place to start the lesson the following day. Sometimes, the teachable moment can wait another day if the teacher is able to revisit the work from the prior day and take the opportunity to have the class debate and convince one another that the classmate that said, “5 pieces” and the classmate that said “one and a little bit more” could have clarified their mathematical language to say “5 pieces of the fourth or 5/4” and “one and a quarter.” The connection needed was to show how 5/4 could be also 1 '. Using manipulatives, numbers, and words, students can argue in the mathematics classroom to make sense of the new concepts.