Lesson Plan Reflection: Fractions Level: Kindergarten Context: This lesson is designed for a kindergarten class at Matthew Whaley elementary school. The class has 21 students. It is an inclusion class, and there are students with behavioral problems and students that are developmentally delayed. The students have been introduced more than once to the concept of “one half;” this lesson is intended to be a review of that fraction an introduction to “one fourth.” The whole SOL is not covered. Objectives: Students will use explore the concepts of “part” and “whole” through the fraction “one-half” and “one-fourth.” Students will be able to represent “one-half” visually. Students will recognize that fractions must be equal, that is, the split must be a “fair share.” SOL: K.5. The student will identify the part of a set and/or region that represents a fraction for halves and fourths. Materials/Resources: Construction paper Scissors Fraction worksheets (attached) Content and Instructional Strategies: 1. Remind students what they have learned in the past about fractions, especially about one-half. Remind students that when you have half of something, it’s only part of the whole. 2. Pass out pieces of construction paper. Have students write their names on the paper. Explain that they are going to pretend that the paper is a cake. First, pretend they have the cake all to themselves—explain that they get the WHOLE cake if they don’t have to share with anyone. Next, tell the students they have to share the cake with their friend. They want it to be a fair share, so how much cake should each person get? Hold up piece of paper, divided not in half— is this a fair share? Would your friend like it if you got this much cake and they got this much? When we make fractions, we always have to be fair. The parts have to be the same size. 3. Remind the students that this is called one half. Have students cut their “cake” in half. You can make it exactly half by folding it so the corners kiss—that way the pieces are on top of each other and neither one hangs over the other, so you know they’re the same. Cut the pieces. 3. Tell students to pretend that now they have three friends to share the cake with. How many people are there all together, including them? Three plus one more is four. How can we divide our cake so that four people each get the same size piece? Is it fair if one person gets a huge slice and the other three get very small slices? Remember, “fair share.” 4. Explain that when we share between four people, the fraction is called one fourth, or one quarter. There are four quarters in a dollar, just like there are four pieces in our cake. 5. Ask students how we can make four pieces (divide our cake into quarters). Put the two halves next to each other so we can see the whole cake again. How can we make four? Divide each half in half, again folding so the corners kiss. Cut. How many pieces do we have now? Can we

share between four people? Will they all be happy because they each have the same amount of cake, or will they be sad because someone has more? 6. If students show understanding, ask: How many quarters are there in the whole cake? How many halves are there in the whole cake? If your cake is cut in quarters, but you only have to share between two people, how many quarter pieces will each person get? 7. Ask the students to think of other things that they can split to share with people. They have to start with something that’s a whole, and split it so that there are parts that are the same size. Discuss ideas with children. Draw some of the suggestions on an overhead or document camera so that students can informally connect the concept of a fraction with a visual representation. Examples:

A pizza divided in half.

A sandwich divided in half.

8. Pass out worksheet to students. Read and clarify instructions: For each picture, draw a line that divides the object in half. Instruct students: before they draw, explain that their pencils have been turned into “fraction makers.” Use their pencil to practice dividing in half. That way, they don’t have to erase. Remember that each part has to be the same size to be fair. Then, draw a fraction on your own (don’t forget, a fraction is a whole object divided into equal parts). Evaluation: Informally assess for understanding during whole-group discussion. Formally assess students’ comprehension by checking the worksheets—the lines should split the object into equal parts. Differentiation and Adaptations: Most kindergarten students cannot read, so teacher should be willing to reiterate the directions for the worksheet for students who do not remember or understand the instructions. Take care to involve all students in class discussion of fractions and in coming up with examples of halves. Students with speech difficulties or developmental delays may be reluctant to volunteer, but they should be included in the discussion nonetheless to promote a sense of self-efficacy and to informally assess understanding of the concepts. While students are completing worksheet, travel through the classroom helping students where necessary. If certain students seem to be struggling with the concept, give them further practice for homework or work with them separately using manipulatives (for example, fraction strips or pattern blocks). If gifted students seem bored, challenge them to hypothesize what other fractions might be called: “If you divide a whole into two equal parts, they are called halves. What do you think fractions that are three equal parts of a whole are called? What about five?” Gifted students can also create models of fractions to show the rest of the class. Finally, they can be challenged to find more than one way to divide something in half (for example, a sandwich can be cut horizontally or diagonally). Notes: This lesson covers only a very small portion of a concept. It is important to take new fractions slowly because it is an entirely new mathematical idea to which most five-year-olds have not been exposed. After this introductory lesson, the teacher can make an effort to

incorporate mathematical language like “fraction,” “part,” “whole,” “half,” and “fourth” into everyday classroom dialogue to reinforce the concept. Be sure to use the terms correctly—do not say “half the class” when you mean about half, because then students will forget that fractions must be equal parts.

Reflection: I was nervous to teach this lesson because it was very different from math lessons I have taught previously. I have observed a lot of math, but because I didn’t start my practicum until more than half way through the year, I have watched a lot of review and practice (building basic addition and subtraction skills, for example) and not many lessons that teach something completely new. I knew that fractions were not a new topic—in fact, my teacher had been reviewing “one half” all week—but I wanted to introduce “one fourth,” and I wasn’t entirely sure how to do it to make it as cognitively easy as possible for my students. Overall, I think I did a good job teaching my lesson. I did not follow my lesson plan exactly, and I didn’t cover quite as much as I intended, but I believe that I encouraged the children to think and problem solve for themselves, which they don’t seem to get enough of in kindergarten. I followed a suggestion my teacher made to use the terminology “one fourth” instead of “one quarter,” so they didn’t instantly think of the coin “quarters.” My original plan was to introduce them both, explaining why there are four “quarters” in one whole dollar, but by the time I was ready to bridge from “one fourth” to “one quarter” it was the end of the lesson and my students had lost focus, so I think it was best that I stopped where I did. I started the lesson with all the kids gathered together on the rug, which I’ve found makes it easier to focus their attention. As per the lesson plan, I did a brief review of fractions, bringing up a lesson I know the students had done previously (folding a paper plate “pizza” in half). I encouraged discussion about the plate, emphasizing the terminology “one half,” which most students seemed able to recall with little prompting. Then I presented the students with a “problem”—needing to share my “cake” with my friend. I used a piece of paper as an example and emphasized the concept of fair share by showing divisions of the paper that were blatantly unequal and asking, “Would my friend like it if I got a piece this size and he got a piece that was only this big?” and all the students were very able to understand that that would not be fair. Then, without telling them exactly how to do it, I passed out the papers and instructed the children to split their cake so that they got pieces that were the same size. I had them go back to their desks and told them that they would have five minutes, so that they knew how long they had to split and decorate their “cakes” (I knew that I could not pass out paper without letting the children draw on it). I was not sure what would happen giving my students this minor degree of autonomy. Most of the lessons I have observed, even in math, have emphasized following directions, and I have noticed that some of the students get in the habit of copying the teacher exactly without thinking at all. I wanted to try to avoid this problem, and overall I think it went well. Most of the children did exactly what I would have expected to divide their “cake” evenly—folded it in half and cut down the middle. However, some of the students avoided doing anything or shredded their papers, which I think showed more of a behavioral problem than a lack of understanding. Two cut shapes out of their papers—two shapes that were exactly the same size, which was a “fair share,” but it’s not quite what I was trying to emphasize, which concerned me at first because I thought I might have given instructions poorly, but I realized that I hadn’t really

done my explanation of fractions yet, so I decided to wait and see what would happen with those students. We went back to the rug and I gave the students an opportunity to show off their decorated papers. Next I asked them to help me divide my own “cake” because I wanted to emphasize that you can fold paper in half to make pieces that are exactly the same size—some of the students had done this and some had estimated a line down the middle. When I asked how to make sure that my pieces were the same, most of the students said “cut it in half,” which I thought was funny—they clearly understood that “half” meant equal pieces, even if they weren’t quite sure what a fraction actually was. Finally, one student correctly told me that folding can give me two equal pieces, so I cut my “cake” and showed the class. Next I wanted to emphasize that halves always look the same as their other half, but they don’t always match other peoples’ halves—some people cut their papers in half lengthwise and others widthwise. I pointed this out to the students and explained that it was fine because the two pieces matched. Upon reflection, I realize that I should have challenged students to come up with another way to divide a paper in half (diagonally), but I did not think of it at the time. Developing and practicing my guiding questions ahead of time would have been helpful. Finally, I introduced the terminology “fractions,” emphasizing that those of us who used the whole paper had made fractions, which is when you have an equal part of the whole. I pointed out the papers of the students who had cut shapes, and explained that their pieces were equal, which was good, but that they not used the whole page so they had not made fractions. I gave them new pieces of paper and encouraged them to make fraction halves (using the whole page), because we were doing math and not just art. Then I gave the whole class a new challenge—dividing the original cake into four pieces, so that you could share with more than one friend. I sent the children back to their seats and gave them three minutes to do this. When we returned to the rug, we showed off our cake pieces again. Most of the children had made four traditional 4.25” x 5.5” rectangles, but a few had made long strips, so I pointed these out and reemphasized that all the different types were good because they were a fair share. I asked the students to remember that we had made halves when we had two pieces of cake, and challenged them to come up with the name for our four pieces of cake. They seemed reasonably familiar with the term “one fourth” from everyday life. I emphasized that “one half” and “one fourth” were both fractions, and I asked the children if they could think of any other fractions, but no one was able to. However, the students seemed attentive, so I challenged them to come up with how many pieces of the whole the fractions “thirds” and “fifths” had, and they were able to respond correctly, which I thought was a reasonable indicator that they understood. Last, I passed out the worksheets and read the students the instructions—divide the pictures at the top in half, and draw a fraction at the bottom. Most of the students did very well on their worksheets, especially dividing the objects in half at the top. Some had more difficulty drawing pictures of their own fractions. In retrospect, it may have been better to have the worksheet make students divide into halves and into fourths because we ended the lesson talking about fourths and other fractions, instead of focusing yet again on halve. However, many of the children would have forgotten the directions and gotten confused. I think the open-ended nature of half the worksheet (drawing the fraction) was good because it really allowed me to see what the children know, and it was appropriate for children at both ends of the ability spectrum—those who did not yet fully grasp the concept could at least try, and those who were able could draw multiple fractions or different types (see sample student work, attached).

I took the worksheets home and wrote comments on them, and they were returned to the students the following day. I tried to make my comments encouraging and helpful, and I emphasized behavior during class as well as simple performance on the worksheet—one student, for example, did well on the worksheet but refused to follow directions in class, and I wanted to remind this child (and parents) that this is not acceptable. Overall, I was impressed with my students’ performance during this lesson, and the worksheets really show that they understand the basic concepts. In the instances where they do not, they allow me to see what misconceptions I need to correct (for example, one student still does not understand that fractions cannot just be two equal pieces; they have to be two equal parts of a whole object). I will try to correct this idea with this student this year, and if I teach a similar lesson in the future, I will be sure to address it from the start. Overall, teaching this lesson gave me confidence in my ability to teach math. I practiced some strategies for giving directions (like bringing all the students on the rug to instruct instead of trying to focus their attentions at their separate, distracting seats), and they were successful— my students were much better behaved and more engaged than when I have attempted to work with them in similar situations in the past. In addition, the lesson allowed me to gain a better understanding for how teachers grade assignments at such a young age—there are no “grades,” but the teacher can make comments and develop an overall picture of how each individual child is performing and developing conceptually. Finally, I was able to gauge what my students already knew and did not know, in order to tailor the lesson to the individual class. I did not stick perfectly to the lesson plan, but I emphasized necessary concepts, expanded on ideas when the students seemed able to understand, and allowed the lesson to progress at a natural pace according to the needs of my students.

This student did not begin work until the teacher prompted her numerous times. She most likely has at least a basic understanding of what a fraction is, based on her performance on the top half of the worksheet, but she preferred drawing her own picture instead of doing the assignment. It is difficult to tell whether or not the picture is divided in half.

This is one of the students who does not yet understand that fractions are parts of a whole. He understand s how to divide objects in half, but when asked to draw his own fractions he alternates between dividing a whole object into equal parts and drawing two equal size pieces without showing the whole.

The open ended nature of this assessment allowed this student, who clearly understood the meaning of “one half,” to go “above and beyond.” However, he does not show understanding that any object can be divided into half, nor does he show a comprehension of fractions other than one half (which is acceptable, because this was only an introduction to that concept).

This student seemed to have the best understanding of fractions. The objects the student drew are varied, showing an understanding that a fraction can be a part of any whole, and the student divided some into halves and others into fourths, showing understanding of this more difficult concept as well.