I open with representable images of fractions. Looking at a whole amount, represented by a circle, I shade in a quantity to show half and then explain to students that this was how many people wanted to stay home from school today! As this is the first day of school, after our winter break, the students are a bit sleepy and low on energy.
I ask, "Do you have any ideas about what this diagram could represent?" Students turn and talk, while I change the diagram to show rectangles and circles with fourths and thirds. I pull the students back, to hear some of the thinking so I can use these diagrams to represent their ideas. By taking student thinking to use as the basis for our discussion, rather than a workbook problem or one of mine, students now have ownership of this lesson. My goal is to kindle an interest in fractions.
Next, I show them a number line and write a few fractions with different denominators. I explain that we are going to order these fractions on a number line. To engage them in the task I return to their examples from the diagrams and ask, "How can we find out if 4/6 of a pizza is larger or smaller than 1/3 of a pizza?"
During this section of the lesson students will be ordering fractions with denominators of halves, fourths, and eighths. They begin with identifying where half is located because it is a unit fraction that is easy for the students to comprehend.
The students have been using fraction bars created from construction paper, and they will be using these to determine the location of fractions on the number line. These fraction strips incorporate the Common Core Math Practice of using appropriate tools strategically.
The use of these fraction bars in this lesson will be critical for the students understanding and success in this lesson. During this lesson students begin to connect using a concrete model of a fraction to a numerical sequence and value. This is an important step for students developing visualization as they begin to move away from the support of the manipulative.
I start with half first, and ask the students, "Where does 1/2 go on this number line?" Looking at the number line, the students have 0 and 1 to orient them. The first tick on the number line is labeled zero, the last mark on the number line is one.
To save precious teaching time, I create the number lines in advance with the exact number of tick marks to work with unit fractions for halves, fourths, and eighths. Because this is the first time students are writing fractions on a number line, I choose this approach to support their success.
I provide the students with all of the fractions that will be included on these number line and wrote them on the board ahead of time as well. When each fraction is placed on the number line, I cross it off the list of fractions.
In determining where the fraction 1/2 is placed on our first number line, I am looking for students to explain that it goes on the middle tick mark. One of my students explained it was half because he noticed, "There are the same number of tick marks before and after the mark in the middle."
Rather than moving on to identifying a different fraction, I want to develop flexibility with manipulation of fractions from the start. So, working together I have the students use their fraction bars to find all the fractions that are the same as 1/2 . They are able to create and identify 2/4 and 4/8 are the same as 1/2.
I ask the question, "How do you find where the other fractions are on the number line?" One student recognizes 8/8, 4/4, and 2/2 would be located at the same spot as one. I ask the students to see if each team can come up with a plan to figure out where the other fractions are located. This repeated reasoning for the Common Core Math Practices is important for students to apply and demonstrate. I would include questions to students at this point if students do not realize the pattern at this point such as, "Do you see any other fractions that match this pattern?"
During their discussions, one of the students keys in on the idea that the smallest fraction piece, 1/8, would be closest to zero. Then someone else recognizes that they could fill in all the other eighth pieces on the number line. The eighth fractions are written in on the number line.
I ask the students, "Are there any fractions of fourths that are the same or equal to any of the eighths?" "How can you they are the same?"
The use of the fraction bars (MP5 - Use appropriate tools strategically) is key to their completing this task. The students have been using the fraction bars and are familiar with comparing fractions. They are able to fill in the fourth fractions under the eighths accurately.
During this section of the lesson, I want the students to use their fraction bars to determine the correct order of the fractions on the number line, or to check their work if they feel confident in moving away from the concrete model. This is a transitional point in the unit for the students to move between using manipulatives to visualizing fractional values and sequence.
The students have a second blank number line to complete using halves, thirds, and sixths. The students work in teams, and I remind them of the strategy for finding half, and then which is the smallest fraction closest to zero. The resource for this section is color coded to assist students in choosing the correct number line. It can be printed in black in white or in color. Using color coding also support language learners with this task.
As teams work, there are a few students that are stuck. I have these students begin to draw diagrams of the same size and divide them into different size fractions. Because the students are creating their own models at this time, they were able to connect the size of a half to 3/6. It was easy for them to see that 2/6 goes between 1/6 and 3/6 on the number line. Then they were able to fill in the fractions of 4/6, 5/6, and 6/6. I am observing students for use of repeated reasoning from Math Practice 8 in this area from what was demonstrated with the eighths number line and how they apply with the number line for sixths.
Some of the students used their fraction bar strips and some of the students continue to draw their own diagrams to complete the task and check their work.
To wrap up this lesson, students record in their math journal how to locate the half fraction on the number line. I want them to focus on this fraction because it is key in their comparison skills. I chose to focus on this unit fraction because it one that students will use most frequently as they move forward to compare other unit fractions. The concept of larger than half or less than half supports students developing an understanding of fractions.