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Long and Short Term Goals

- To develop a responsible and positive attitude we chose Respect for Self, Others and Learning for the long term goal.

- Our students are weak in number sense, numeration and problem solving. We chose Compare unit fractions in the context of solving a real world problem for the short term goal.

Questions Asked During the Planning of the Lesson

- What do students already know about fractions?

- How do we think students will answer the question: Which is bigger 1/3 or 1/4?

- What will be the context of the problem?

- How can we design the lesson so students will discover the concepts we are teaching?

- What materials do we need?

The Lesson Problem

- Mr. Bateman, the principal, is willing to share a candy bar with me. He said I could either have 1/3 or 1/4 of the candy bar. Seeing that I really like chocolate I am wondering, which is larger, 1/3 or 1/4?

The More Difficult Question

- The Hershey Bar is divided into 12 smaller pieces.
- Should we ask a second question – How many pieces would you get?
- When should we ask this question?
- How should we ask this question – individually or in the small groups?

Lesson Booklet

A work booklet was designed. The question, “How many pieces would I get?” was inside the booklet on the second page. After children worked independently for about 5 minutes, the teacher would ask them to share their ideas with their group. She would suggest that thinking about the number of pieces you would get in 1/3 or 1/4 may help you decide which is the bigger part.

We picked ¼ because the bottom number is 4 and the other bottom number is smaller so ¼ is bigger.

1

4

1/4 is more than 1/3.

Students’ IdeasSome students still didn’t understand.

Me and my partner think the 1/3 is bigger because the less pieces the bigger the pieces are.

Students’ IdeasSome children used multiple representations to show this concept and they could explain it.

Sharing Ideas pieces the bigger the pieces are.

- At the beginning of the lesson 18 children thought 1/4 was larger and 6 children thought 1/3 was larger.

- After working in small groups, 4 more children thought 1/3 was larger.

- Then children came to the front of the room to share their ideas.

- After listening to other children during the large group discussion, 12 children seemed to be convinced that 1/3 was larger than 1/4.

Feedback Session pieces the bigger the pieces are.

- Joan’s Question: How do you facilitate children’s understanding when one child in the group says 1/4 is larger and the other says 1/3 is larger and they both think they are correct?

Feedback Session pieces the bigger the pieces are.

- Problem: Students did not have enough room on their worksheet. Some got confused when they opened the booklet during the independent work time and saw the second question; for others, seeing the twelve parts of the candy bar opened up their thinking.

- Solution: Revise the worksheet to one large sheet, keep the image of the candy bar with the twelve sections, but eliminate the second question from the paper.

Feedback Session pieces the bigger the pieces are.

- Problem: Not all children used manipulatives during the independent working time.

- Solution: Remind children that they can use any materials in the basket to help them solve the problem.

Feedback Session pieces the bigger the pieces are.

- Problem: Children had a hard time explaining their correct ideas.

- Solution: Teacher could help the children choose words when necessary so the rest of the class can better understand what the child is saying.

Feedback Session pieces the bigger the pieces are.

- Problem: With pairs, little progress is made if both children are confused.

- Solution: Make the groups larger so there is a better chance that one of the children will have productive ideas.

Second Implementation pieces the bigger the pieces are.

At first, all students said 1/4 was larger than 1/3.

Second Implementation pieces the bigger the pieces are.

Their ideas began to change during small group work.

1/4 is bigger or larger because there is 4 pieces. pieces the bigger the pieces are.

1/3 is smaller because there is 3 pieces.

1

3

1

4

1

4

1

3

is larger

is smaller

Second ImplementationAt the beginning of the class discussion, some students were still convinced that 1/4 was larger.

Second Implementation pieces the bigger the pieces are.

In this group, all the students agreed that 1/3 was larger, but one child said 1/3 was 9 pieces and another child said it was 4.

1 pieces the bigger the pieces are.

3

1

4

1 whole

We think it is 1/3 is because you want as much as you can get. Just because the bottom number is biggest doesn’t mean you get more.

.

The Turning PointAfter this group argued that just because the bottom number is biggest it doesn’t mean you get more, students began to raise their hand to say they changed their mind.

The Turning Point pieces the bigger the pieces are.

At the end of class all except one student agreed that 1/3 was bigger than 1/4. This student said, “If you subtract 1 from 1/3 it would be two big pieces. If you subtract 1 from 4, it would be 3.” He was still thinking about whole numbers.

Feedback pieces the bigger the pieces are.

- The larger student work groups seemed to stimulate a better exchange of information and discussion and prevented the “stalled” groups we had observed in the first setting.

- The elimination of the second page and additional question did away with the confusion that it had caused in the AM session.

- We found that the paper manipulatives helped some children and confused others.

Reflections pieces the bigger the pieces are.

- I learned from watching other teachers teach and watching students learn.
- I saw the importance of giving enough time for students to think-- whereas teachers often rush to give answers.
- I saw the value in encouraging students to share among themselves and explain their thinking-- whereas teachers often do most of the talking.
- These observations made an impression on me that will affect how I approach both the teaching and learning in my classroom.

Reflections pieces the bigger the pieces are.

- I really enjoyed watching all of us contributing so many ideas and then being able to pull it all together and focus on our final lesson study plan.
- It provided an opportunity to encourage and support one another as we tried new instructional methods aligned with the teaching and learning standards from the Michigan Curriculum Framework and NCTM.
- It is never easy to find the amount of extra time needed to accomplish such a task as we took on but we did.

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