In today's lesson, the students learn to find equivalent fractions using number lines. This aligns with 4.NF.A1 because the students will explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size.
We begin by reviewing what the students have already learned about equivalent fractions. On the Smart Board, I show models of two equivalent fractions (Equivalent Fractions Opener.pptx). "What can you tell me about equivalent fractions?" I give the students a few minutes to think about the question. One student responds, "They have to be the same whole." I ask for other responses. The students are reluctant to answer the question. This lets me know that we have much work to do with equivalent fractions. (Because of the previous days lesson, I thought that the responses would flow easily. Sometimes, as teachers, we may think that students fully understand a concept, when maybe they are having difficulty connecting all of the pieces. At this point, I am glad that the students will have an opportunity to use another tool to master this skill.) I let the students know that in today's lesson, we will learn to find equivalent fractions in a different way. We can use a number line to show equivalent fractions.
To begin, I review important information discussed in the previous day's lesson. On the Smart board, I model for the students how to divide a circle into equal size pieces. I remind the students that in order to compare fractions, we must be referring to the same whole. I make this point clear by drawing two rectangles of the same size. When you divide your region into pieces, they must be even size pieces. This same concept applies to a number line. A number line represents the whole. If I want to use number lines to see if 1/3 and 2/6 are equivalent, I can't draw one long number line, then the second number line shorter than the first. They must be the same length and with pieces of equal length.
I let the students know that I am exposing them to different tools that they can use to be successful with fractions. You can find equivalent fractions with fraction strips as we did the previous day. I let them know that they may also find equivalent fractions with number lines.
I read the slide in the Number lines and Equivalent Fractions.pptx displayed on my Smartboard aloud. "Peter drove 1/4 of the distance to his mom's house. What is another name for 1/4? Let's try eighths." In order to use a number line, first divide the number line into fourths and eighths. You will divide the number line into the number represented by the denominator. If the denominator is an even number, you can divide the number line in half for your benchmark.
I model by dividing the number line into fourths. (The number line in the power point is already divided, but I want the students to see how it is done step by step, therefore, I used the white board to model.) Modeling for students is very important because some students are visual learners. They need to see what output is expected of them. I put 1/4 on the number line. I ask, "What comes next?" The students respond chorally. A lot of the students said 1/5. I remind them that the number line was divided into fourths, therefore, we should count by fourths. I remind the students that a fraction is part of a whole, therefore, it falls between the numbers 0 and 1. I label these two numbers on the number line. Together, we labled 1/4, 2/4, 3/4, and 1. I explain to the students that we do not have to write 4/4 because any number over itself equals 1. If you have 4 out of 4, you have 1 whole. (I model this by drawing a circle, dividing it into fourths, then shading all four pieces.)
Next, on the white board, I draw another number line below the first one. As the students watch, I divide this number line into eighths. I explain that because the denonimator is an 8, which is an even number, I can divide the number line in half. I label 0 and 1 on the number line. Together, we name the other fractions that go on the number line.
I ask the students to look at the two number lines. "Which two fractions are equivalent?" They tell me 2/8 and 1/4. I let the students know that they are correct. I further explain (for those few students that did not respond chorally with the others) that the way we know that they are equivalent is because they are lined up at the same point on the number lines. (I circle the two numbers so that there will be no misunderstanding about what I mean.)
I let the students know that they will have the opportunity to use a number line in groups. I encourage them not to get discouraged because number lines can be a difficult tool to master. I remind them of the fact that the more you practice on anything, the easier it becomes.
I give the students practice on this skill by letting them 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.
For this activity, I put the students in pairs. I give each group a Number Lines and Equivalent Fractions Group Activity Sheet.docx. The students must work together to find equivalent fractions on a number line. By using a number line, the students are using their tools strategically to master a concept (MP5). The students must communicate precisely to others within their groups. They must use clear definitions and terminology as they precisely discuss this problem.
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 pairs discuss 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. How do you represent each of these fractions on the number line?
2. How do the fractions compare?
3. If they are equivalent, how do you know?
As I walked around the classroom, I heard the students communicate with each other about the assignment. From the video, you can 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.softschools.com/math/fractions/equivalent_fractions/games/
To close the lesson, I have one or two students share their answers. 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.
Number lines are very difficult for students to work with. From this lesson, only about 6 students actually could divide the number line correctly and label the fractions. I expected this to be a difficult concept that is why I asked the students not to be discouraged early in the lesson. Even though the majority did not master number lines, the experience of participating in this activity provided them exposure that increased their knowledge. It may not be mastery, but some understanding was gained based upon my observation and reviewing their activity sheets. In the video of student work (Equivalent Fractions with Number Lines Video.mp4), you can see that there is much knowledge gained. In previous lessons, we used number lines to compare and order numbers according to their place value. Because I know the importance of students gaining a conceptual understanding, I will continue to create more lessons that require students to practice with number lines.