Since most of my students assume fractions with the greater denominator is the greater fraction. I decide to place different size fraction pieces on the board. Sometimes that visual connection to learning helps students adapt to learning new material.
Then, I invite students to the carpet. I ask students to examine the fractions on the board and to pay close attention to their size. (3/8, 4/8, 5/8, 1/8)
I want to see if students will notice that the fractions with the greater denominator are the smaller pieces.
As students begin to respond to the given task, I take anecdotal notes. I use these notes to adjust the complexity of this lesson and to assess students’ prior knowledge.
We will be using the following Mathematical Practices in this lesson:
MP.2. Reason abstractly and quantitatively.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.7. Look for and make use of structure.
Some students were still unable to make the necessary connections. To correct this, I review the definition of denominator. I explain that the more equal parts there are in the whole, the smaller each part will be.
Then, I illustrate and repeat this process until a level of understanding is reached. Students need to see how the structure/size of a fraction changes depending on the fraction. I may illustrate, or give them some fractions to examine for about 3-5 minutes.
My focus here is helping students conceptually understand that the fraction with the greater denominator is not necessarily the greater fraction.
Modeling how and why the more equal parts there are in the whole, the smaller each part will be helps correct students' misconceptions.
To check students' understanding so far, I draw a large rope on the board. I explain that this rope itself is a whole; however the lines indicate how this rope will be divided into parts.
Then, I place four large lines equally apart, and point to each one. I ask students to tell me how many equal parts the rope is divided into. (4 equal parts) I ask them to think of a way you can illustrate how to divide a whole into four equal parts. I give each student a small dry erase board and tell them to create their mathematical model of a whole divided into four equal parts.
I also ask them to write the fraction that represents their math model. I tell them they have about five minutes or so to complete the given task. As students are working I circle the carpet area to check for understanding.
Material: note taking paper.pdf
I ask students to move into their assigned small groups. I tell them that we will be working together on comparing parts of a whole. I guide students to see that the shaded parts of each fraction represent the amount that was divided out of the whole part. I do this by drawing a large circle on the board and dividing it into 5 smaller parts. I point to each divided section and explain that each part of the whole circle is a piece of that whole. As I point to each part I erase just that part, and continue the process until all parts of the whole circle are gone.
Then, I tell students that they will draw two fractions on the crate paper provided for each group. I tell them to draw one fraction twice as long as the other. Then, I ask them to divide the shorter fraction into four parts and the larger fraction model into eight parts. Then I tell them to shade one part of each strip. As they are working I ask, what fraction they are making? ¼ and 1/8. I also ask how are these fractions alike? how are they different? Students should understand that they are different because the amount is different. (The amount 1/4 is more than the amount 1/8). However, it is CRITICAL that students understand that they both show the unit fraction, meaning 1/4 is 1 out of four and 1/8 is 1 out of 8. They are both showing the simplest form of fourths and eighths.
I ask students to explain how the smaller fraction of a bigger whole can be the same size as or larger than a bigger fraction of a smaller whole. Students should realize that they cannot compare fractions of different-sized wholes. Allowing students to work with fractions hands-on helps them think deeper about how to compare fractions.
Materials: Comparing Fractions.docx
Now that students have had the opportunity to work hands-on with fraction models to compare them, I ask them to move back into their assigned seats. I give each student a sheet of construction paper and ask them to draw three equal-sized rectangles.
Then, I ask them to divide one into halves, one into fourths, and one into eighths.
Then, I ask students what do you notice about the number of equal parts, as the size of each part decrease. ( students should be able to point out that as the number of equal parts increases, the size of each part decreases) I do this small task to assess students’ ability to apply what they have learned so far.
Students are then given their final independent task sheet. Students are asked to compare three different fractions to see which part is greater. Then I ask them to explain why it is important that two of the given fractions are the same size. As students are working I circle the room to check for understanding and to correct any misconceptions as needed. Most students seem to understand how parts of whole works. Students can identify that the fraction size is the same, however, the parts are different.