Cesira Giacobassi Michelle Ruotolo Kristen Swanson Laura Casper-Teague Planning for Teaching Through Problem-Solving: a Guide GLCE(s): N.ME.03.16: Understand that fractions may represent a portion of a whole unit that has been partitioned into parts of equal area or length; use the terms “numerator” and “denominator”. N.ME.03.17: Recognize, name and use equivalent fractions with denominators 2, 4, and 8 using strips as area models. N.ME.03.18: Place fractions with denominators of 2, 4, and 8 on the number line; relate the number line to a rule; compare and order up to three fractions with denominators, 2, 4, and 8. N.ME.03.19: Understand that any fraction can be written as a sum of unit fractions, e.g., ¾ = ¼ + ¼ + ¼. N.MR.03.20: Recognize that addition and subtraction of fractions with equal denominators can be modeled by joining and taking away segments on the number line. 1. Selecting/creating and analyzing a task(s). A Fraction Task Using Apples. The lesson was adapted from A Collection of Math Lessons from Grades 3-6 by Marilyn Burns. The students will be instructed to work collaboratively in groups of four to complete the lesson. There will be three worksheets that will be passed out one at a time. Different colored apple cut-outs will accompany each worksheet. The apple cut-outs will be used as manipulatives by the students to solve the fraction problems on the worksheets. The first worksheet will be the easiest of the problems and the next two worksheets will increase in difficulty. The first worksheet will ask the students to share X number of apple pieces equally among the four of them and state how much of the apple(s) each student received. The second worksheet will ask the students to solve a more challenging problem than the first worksheet. The third worksheet will ask the students to solve a problem using division of apples. The students will be given X number of apples and are asked to divide the apples into Y number of pieces. The final part of the problem will ask the students to determine how many apples were shared. 2. Anticipating problematic aspects of task. It is anticipated that some students may have difficulty working with fractions and writing fractions in the correct format. There may be problems using scissors to cut the apple cut-outs into equal parts to best represent the fraction. Students may have difficulty understanding the connection between the dimensional (3-D) apples and the flat (2-D) cut-out apples. In addition, each student in the group of four will have individual job roles such as the person who cuts the paper apples, the person who manipulates the apple cut-outs, the person who glues the apple cut-outs and the speaker. Some of these assigned jobs may be challenging for some students. Time management is of concern for the lesson. Students are expected to understand the physical representation of fractions, to have the ability to understand part-to-whole relationships, and to pay attention to detail such as cutting precisely. Finally, we expect the students to improve and further develop their team work skills. 3. Learning goals for the lesson. Our learning goals for this lesson include helping the students represent fractions with models as well as in written form and to begin to gain conceptual knowledge of fractions. Additionally, the students will learn to verbally express their solutions to problems and to explain the methods they used to solve the problems. The students will practice manipulating fraction problems by using the mathematical operations of adding, subtracting, multiplying, and dividing of fractional parts. The students will practice following directions, paying attention to details, and working effectively as a team. 4. Arranging materials and students. Three worksheets will be used that ask the students to solve problems involving fractions. Each worksheet will have a different color apple cut-out (red, yellow, or green) that will be used to represent apples and will be used as manipulatives to solve the fraction problems on the individual worksheets. Some of the supplies the students will need include glue, scissors, pens, the worksheets, the apple cut-outs, the circular pie pieces, and real apples. An introductory activity, which includes real apples, cutting boards, and knives, will be provided by the interns. We will arrange the students in groups of four. We have chosen this arrangement due to the fact that the desks are arranged in pairs in the classroom. It is easy for students to turn their desks and make groups of four. Our worksheets are designed with four sections (one section for each student) that will require the students to glue their apple-cut out answers onto their section of the worksheet. 5. Anticipating what children are bringing with them to the lesson. The students will have experienced fractions in their everyday lives, such as drinking half of a glass of milk, understanding that four quarters equal one dollar. We also presume that most of the students will have eaten apple slices before and have gone to apple orchards. Lastly, students have worked with scissors and glue in art class as well as in their every day classrooms. Due to our limited time in the classroom, we have not seen fraction lessons and are unsure about the students' level of knowledge or what formal instruction the students have had regarding fractions. 6. Beginning the lesson We will begin the lesson by splitting the class into groups of four. Fraction manipulatives of three different colored apple cut-outs will be passed out to each group. One intern will ask the students, “Have you ever been to an apple orchard?” We will then have a two minute discussion about their experiences at apple orchards. “What kinds of apples are there?” “Do you have a favorite apple?” From this discussion we will launch the topic of our lesson and introduce the book Apple Fractions by Jerry Pallotta. We will tell the student that throughout the reading we will be working with our apple pieces to make fractions which were talked about in the story. 7. Monitoring students in the middle part of the lesson. We will read the story Apple Fractions. As the story is read aloud to the students by one of the interns, another intern will be demonstrating the fractions that are being read. The fractions will be demonstrated on the overhead projector with circular pie pieces supplied by the CT. Students will also be modeling the story fractions with their fraction manipulatives. The other two interns will walk around the room monitoring the work of the different groups and addressing misconceptions or problems that the students are having. Once the story has been read, the interns will verbally address problems with the whole class that were encountered during the problem solving task. This will be done as a review and also to clarify any misunderstandings that might cause problems in the next part of the lesson. After misconceptions have been addressed, we will pass out the first worksheet. The students will work in their groups to solve the problem on the first worksheet. Groups that complete the worksheet will be given a second worksheet. The second worksheet will be a more difficult problem than the first worksheet. The last worksheet is a challenge problem that involves division of apples. 8. Bringing the class back together to discuss the mathematics. The students will be directed to focus their attention to the front of the room. The interns will conduct a discussion about the various worksheet answers that the students developed. The group leader will act as a speaker for the group who will share with the whole class the group's solutions to the worksheets. As the speakers from each group present their answers, the interns will model their answers by cutting real apples into the same number of pieces suggested by the group. The worksheet solutions will be reviewed and the interns will pose impromptu questions to the students as the speakers present their fraction solutions. This part of the discussion will use “teachable moments” to extend learning and to help the students to connect their knowledge in a deep conceptual understanding of the fraction tasks. The lesson will be closed by reviewing the concepts that were introduced during the lesson. Each worksheet solution will be reviewed using the problem solutions that the individual groups created. The students will be prompted to share their solutions as the interns review the answers to help them connect further with the fraction task and to offer practice with repetition and recall of the problem solving strategies and solutions.