In today's lesson, the students use visual models to solve a real-world problem, then write to explain their answer. This relates to 4.NF.A1 AND 4.NF.A2 because the students must determine equivalency and compare fractions in the real-world problems.
I let the students know that today we are talking about writing with fractions. We are going to explain our answers. I remind them that we have worked in the past on writing and restating the prompt. I have a word problem displayed on the Smartboard. This is used as a Teaching Tool - Explain Your Answer.docx to instruct the students.
Rhonda loves eating pumpkin pie. She said that she will get 3/7 of the pie. Her sister, Becky, loves pumpkin pie too. She said that she will get 2/7 of the pie. Will there be enough pie for both sisters? Explain your answer.
We are going to look at the scenario, solve for the answer, then write to explain. In math, when you have to write to explain, you must solve the problem first. Do not just start writing if there is a math problem within the context that needs to be solved. Based upon what you did to solve it, then you can write to explain. We're going to look at this real-world scenario, then we are going to solve it. Then we will write to explain our answer.
I read the scenario aloud to the students. I ask the students, "What is the first thing we should do in order to solve the problem?" I give the students a few seconds to think about their answer. I call on one student. Her response is put them in order from least to greastest. I explain to her that I am trying to figure out if there is enough pie for both sisters. Do I need to put them in order from least to greatest to do that? She responds, "No." I call on another student, "See how much pie they both will eat." How can I do that? I call on another student. (I try to call on as many students as possible so that I can hear their thinking.) The student responds, "We can look at the denominator." I continue questioning the students because I want them to provide the answers. As a teacher, we sometimes want to tell the students information instead of allowing them to struggle to learn it. If they struggle, then they are more likely to understand the material. "What is that denominator telling you?" Student response: It tells you how many pieces of pie you have. I need to know if I have enough pie. Rhonda says that she is going to get 3/7 of the pie, and her sister says that she wants 2/7 of the pie. Do they have enough pie? Tell me what strategy you can use based on what we have been doing. They all know that they should draw out the fractions because this is what we've being doing in class. As the students draw their fractions, I draw my model on the board. I let them know that they are not comparing the two fractions because we are trying to see if there is enough pie in the one whole. (I do not want them confused because they see two fractions in the problem.) We divide the pie into 7 pieces because they have already told me that the denominator tells us how many to cut them into.
I discuss iterations at this point because I feel that it is a great opportunity to remind them of what each of the pieces represent. "How many sevenths do each one of these pieces represent?" A few students call out incorrect responses. Finally, one student says 1. Below each fraction piece on my model, I write 1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7 = 7/7, which equals 1.
We shade in Rhonda's 3/7. Next, we shade Becky's 2/7. Now that we have the model, we can answer the question. Below your model, put yes or no, then explain your answer. I remind the students that they should restate the prompt when answering the question. (A lot of students have difficulty with writing. One problem that I have found is that the students do not write in complete sentences when answering questions in math. One of my goals this year is to have the students write in complete thoughts.) When you explain your answer, what are you going to refer back to? Student response: Our model. Even though they did not ask, you could also tell how much pie is left after Rhonda and Becky get their pie. This will show that you have really examined the information and know what you are talking about.
The students work independently to write to explain the answer to this scenario. Upon completing their explanations, I take a volunteer to share their answer.
I drew out the whole, and I cut it into 7 pieces. I shaded 3/7 for Rhonda. I shaded 2/7 for Becky. They had a total of 5/7, and there were 2/7 left.
For this activity, I put the students in groups of 4. In each group, I have students on different academic levels. By grouping them this way, it allows the students to learn from each other. I give each student an Explain Your Answer (Fractions).docx. I give the students about 5 minutes to read and solve the problems independently first. This will allow the students to contribute to the conversation when the students work as a group. After solving the problems independently, the students must work together to solve the real-world problem by drawing a model(MP4), then write to explain their answer. This gives the students practice with Math Practice 6 because mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
I remind the students to first read the problem, plan a strategy to solve, then solve the problem. After the problem is solved, explain the answer. Be sure to answer the problem with a complete thought.
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 explanation to the problem, 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. What are you asked to explain?
2. What does you model show?
3. Are these two fractions equivalent? How do you know?
I have learned that "talk" is very important in a classroom. In the past, it was perceived that you had a really good class that was on task if the students were in their seats working quietly. Now, if that is happening often, then the students may not be having a rich learning experience. Students can learn from other students. I have learned to be more of a facilitator in my classroom, standing back and listening to what the students have to say. Only interjecting with questions that guide the students to the answer.
As I listened in on a group as they discussed the 3/5 of a large cake and 3/5 of a small cake, I heard a really good discussion. One student, who is usually sleepy, was very vocal during this lesson. I think that because she did not have to do math calculations, it was less threatening to her. I find that my lower level students gain more confidence when they are given an activity that allows them to show that they can solve a problem just as the "perceived" high level students.
Some of the comments from this group:
1. It doesn't matter who has the most because they are both equal.
2. But it says that Teresa has a large cake and Maria has a small cake.
3. But they are the same.
4. Teresa has more cake because in the model hers is bigger than Maria's cake.
After they came up with the solution, I asked, "When will 3/5 and 3/5 always be the same?" They knew that it had to be the same size cake.
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 one group to share their explanations. This is a sampe of student work for question 3 on the activity sheet which says, "Jeremy said that 3/8 is equal to 1/2. Is she correct? Explain how you know." (Student Work - Fractions.jpg). 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 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 with the whole class.
Writing is difficult for some students. By grouping the students the way I did, it allowed for the students to work together to come up with the solutions. This is important because students learn from students. Hearing the justification of answers from their classmates may help a student more than hearing information from the teacher. We have been drawing models for a while. I can tell that they understand how to compare fractions with models. It is evident in their models that they know to compare the same "wholes." Also, they understand the meaning of the denominator because they could divide the "whole" into even pieces.
A misconception I noticed from some of the students in each group was that they were not comparing by using the numerator. For example, with the fractions with a numerator of 1, some students did not take into consideration that because each fraction has a 1 for a numerator, they could compare by thinking about the size of the denominator. We have discussed that the larger the denominator, the smaller the pieces.