In today's lesson, the students learn to compare fractions by using visual models (4.NF.A2).
I let the students know that today, we are going to build upon what we have learned about fractions so far. We will continue to model with fractions. We just talked yesterday about comparing fractions. We're going to need that information today. Someone tell me what you learned about comparing two fractions with different numerators and denominators. I give the students a few minutes to think. Student responses: 1) You can draw models of both fractions, 2) you need to make sure it is divided right, and they are equal pieces, and 3) the two wholes need to be the same size.
I share with the students that in today's lesson, we are ordering fractions. When you order fractions, you can order them from greatest to least or least to greatest. What does least mean? Smallest. What does greatest mean? Largest. Pointing to the Ordering Fractions.pptx power point on the board, I let the students know that our directions say to put the fractions in order from least to greatest. (I point out to the students that they should always pay careful attention to the directions. Often, from past experiences, I know that some students tend to not pay attention to "least to greatest or greatest to least" in the directions.)
The students see in the power point that I have three fractions: 1/2, 4/5, and 2/3. We have already learned about the denominator. Someone raise your hand and tell me what does the denominator tell you to do when you are dividing your fraction? Student response: It tells you how many pieces are in the whole. In this fraction, we have 1/2, so how many pieces will we have? 2. How many am I going to shade out of the 1/2? 1. I reiterate to them that the numerator tells us how many to shade. I have 1 out of 2. In 4/5, how many pieces do I need to divide the whole into? 5. How many do I need to shade? 4. So, I have 4 out of 5. In 2/3, how many pieces do I divide the whole into? 3. How many am I going to shade: 2. Notice that your wholes should be the same length, and you should try your best to divide them into equal sized pieces. Once you have the model, the fractions can be put in order. I continue to question the students by asking, "What is the first fraction that is the smallest?" 1/2. "What is the second smallest fraction?" 2/3. "Therefore, the largest fraction is 4/5." I tell them that based upon the models, we have put these fractions in order from least to greatest.
On the previous day, one student said that she learned that the larger the fraction, the smaller the pieces. I want to make sure the students know to take the numerator into consideration when comparing fractions. I explain to them that if you have a common numerator of 1, you can put the fractions in order without drawing them out. Why can you put them in order without drawing them out if the numerator is 1? (The students do not respond as I want them to, therefore, I continue probing with questions.) If you cut a cake into 3 pieces, 4 pieces, or 7 pieces, and you want a big piece of cake, which piece would you want? The majority said 1/3. (When students can relate to the problem, they tend to get a better understanding of the problem.) From that point, they could tell me that 1/4 was the next largest piece, and 1/7 was the smallest piece of cake. I conclude by sharing, "What we are saying is that the larger the denominator, the smaller the pieces. Even though we know that the bigger the denominator, the smaller the pieces. You must take the numerator into consideration if all the numerators are not the same."
For this activity, I put the students in groups of 4. I give each group an activity sheet and a bag with 4 fractions (Fraction Pieces.docx). The students must work together to put fractions in order from least to greastest by using the models (MP4).
Each student pulls one fraction from the bag. Before the students work in collaborative groups, I want them to take a few minutes to draw a model of their fraction. I point out to the students that they must be precise in drawing their fractions because it can effect the ordering of the fractions. (I made a decision to let the students draw the fractions instead of using fraction strips in this lesson. The rationale for doing this is that the students will not be allowed to use fraction strips on our formative assessment. As well as, it gives the students a better understanding of creating models.) After a few minutes of independent work time, I allow the students to work together. I find that collaborative learning is vital to the success of students. Students learn from each other by justifying their answers and critiquing the reasoning of others.
Students must look at all 4 models for the shaded part of the fractions, and agree upon the order of the fractions. Because each student is drawing a separate fraction, I give them a rectangle to use because all of the wholes must be the same size (Fraction Wholes.docx). By the end of the lesson, each student should know how to order fractions based upon a visual model.
The students are guided to the conceptual understanding through questioning by their classmates, as well as by me. The students communicate with each other and must agree upon the answer to the problem. Because the students must agree upon the answer, this will take discussion, critiquing, and justifying of answers by both students. As the groups discuss the order of the fractions, they must be precise in their communication within their groups using the appropriate math terminology for this skill. As I walk around, I am listening for the students to use "talk" that will lead to the answer. I am holding the students accountable for their own learning.
As they work, I monitor and assess their progression of understanding through questioning.
1. How can you use a model to represent the fraction?
2. Is this fraction greater than or less than this fraction?
3. How do you know that your answer is correct?
4. How does the size of the pieces change with each denominator?
As I walk around the classroom, I hear the students communicate with each other about the assignment. I hear the classroom chatter and constant discussion among the students. Before Common Core, I thought that a quiet class working out of the book was the ideal class. Now, I am amazed at some of the conversation going on in the classroom between the students.
Any groups that finish the assignment early, can go to the computer to practice the skill at the following site until we are ready for the whole group sharing: http://www.mathsisfun.com/numbers/ordering-game.php
As I walked around listening in and questioning students, the discussion between one group of students caught my attention. Out of the group of 4 students, there was one student who seemed to have become the group leader. This student just recently became a student at my school. This group was to put the following fractions in order from least to greatest: 4/7, 2/3, 3/5, and 2/8. As she ordered the fractions, I heard her say, "This one has 4 pieces shaded, so it is last. But, I'm confused about these two because they both have 2 pieces shaded." I waited for the other 3 members of the group to jump in and correct her. They did not. So I began questioning, "How do you put fractions in order by the shaded part?" The student that emerged as the leader began answering the question incorrectly. The students were trying to put the fractions in order according to how many pieces were shaded. This surprised me because of all of the class discussions and modeling that has taken place in class. I corrected that group then, but this misconception was also addressed whole class because I wanted to reach all students who may have the same misconception.
To close the lesson, I bring the class back as a whole. I feel that it is important to close the class out by discussing the activity. As the students were working in their groups, I walked around to listen in and to identify good work samples to share with the class. I allow two groups to share their models of the fractions in order from least to greatest (Student Work.jpg Student Work 2). This gives those students who still do not understand another opportunity to learn it. I like to use my document camera to show the students' work during this time. Some students do not understand what is being said, but understand clearly when the work is put up for them to see.
I feel that by closing each of my lessons by having students share their work is very important to the success of the lesson. Students need to see good work samples as well as work that may have incorrect information. More than one student may have had the same misconception. During the closing of the lesson, all misconceptions that were spotted during the group activity will be addressed whole class.
Overall, I feel as if this was a good lesson. The students seem to have enjoyed the activity and working collaboratively. Most of the groups took care in arranging the models in order from least to greatest. Some of the fractions were very close in size, but the students used the shaded part to help order the fractions. This made for an excellent discussion among the group members to determine how to order the fractions. The students knew that if one person's drawing was incorrect (not cut into equal-sized pieces), it could cause the ordering of the fractions to be wrong. That person had to get a new fraction piece and draw the fraction over again.
For homework, the students must order the following fractions by drawing models: 3/4, 7/8, and 2/3.