3/9 Is Bigger Than 2/12 Rhetorical Analysis

Teacher: Would anyone like to share their solution with the class? (TR7)

(Brandon raises his hand)

Brandon: 6/9 is bigger than 10/12 because 3/9 is bigger than 2/12. (SR4)

Teacher: What do you mean that 3/9 is bigger than 2/12? (TR3C)

Brandon: 1/9 is a bigger piece than 1/12. So 3/9 is going to be bigger than 2/12. (SR6)

Teacher: Comment? Suzanne. (TR7)

Suzanne: I disagree. (SR4)

Teacher: Why do you disagree with Brandon? (TR3J)

Suzanne: Because you have to compare 10/12 to 6/9, not only 2/12 to 3/9. (SR4)

Teacher: Could you come up to the board and show us a picture of what you are saying.

(Suzanne comes up the board and draws the picture below)

Suzanne:

Suzanne: So you can see that 10/12 is bigger than 6/9.

Brandon: But

3/9 is still bigger than 2/12. So why would 10/12 be bigger than 6/9? (SR3)

Suzanne: We are comparing the pieces to a whole. 2/12 has smaller pieces, so it is going to take less pieces to reach a whole of 12/12. (SR6)

Teacher: Suzanne can you explain why 2/12 would take less pieces to reach the whole, and thus make 10/12 the bigger fraction? (TR3J)

Suzanne: 10/12 is only 2 pieces under the benchmark of 1, so to reach the whole for 10/12 it needs two more pieces. But, 6/9 is 3 pieces under the benchmark of 1, so to reach the whole for 6/9 it needs 3 more pieces. (SR6)

Teacher: Did you have something to add Emily? (TR7)

Emily: I agree with what Suzanne is saying. But, I was thinking about it in a different way. (SR4)

Teacher: What do you mean by that? (TR3C)

Emily: So I was thinking about how many eggs it would take to fill up the egg cartons. My first carton has 10 eggs and only needs two more eggs to get to a whole. My second carton has 6 eggs and needs three more eggs to get to a whole. The eggs in the carton of 10 has smaller eggs in order to fill in the spots, and the the eggs in the carton of 9 has bigger eggs in order to fill in the spots. So the carton of 9 has very big eggs, so these big eggs create a larger space away from the whole. But, the carton of 12 has little eggs, so these little eggs create a smaller space away from the whole. (SR6)

Teacher: Could you come up to the board and show us what you mean? (TR3C)

(Emily goes up to the board)

Emily:

Jackie: Why does it matter that the eggs are different sizes? (SR3)

Teacher: Emily thinks that size is important to consider, but Jackie is not so sure. Mark, do you have anything to add? (TR7)

Mark: Because we want to find the distance under 1. So the size (of the pieces) under 1 is important. (SR4)

Teacher: Can you explain what you are saying about how much distance under 1 is important? (TR3C)

Mark: For Emily’s example, the missing amount of eggs would represent the distance.

Emily is saying that 6/9 needs 3 more big eggs to fill the carton. Whereas, 10/12 needs 2 little eggs to fill the carton. So, like Brandon was saying 3/9 is bigger pieces than 2/12. But since, 2/12 has smaller pieces then it requires less to fill the whole. So, 10/12 would be the bigger fraction because it’s closer to its whole since there are less, smaller pieces. (SR6)

Teacher: Can anyone figure out how you would use a number line to help support your thinking? (TR1)

Jimmy: Can I draw it on two number lines, or does it have to be on one number line? (SR3)

Teacher: What does everyone think? (TR7)

Rebecca: I think it is better to draw two number lines for each fraction and compare to 1 on each number line. (SR4)

Teacher: Why do you think it is better to have two number lines, Rebecca? (TR3J)

Rebecca: Because then you can put each fraction on its own number line and still compare to one. So it doesn’t matter how you split it up one the number line. It just might make it easier to look at each of the fractions side by side. (SR6)

(Rebecca goes up to the board)

Rebecca:

Rebecca: So now you can see that 10/12 is bigger than 6/9 when comparing both fractions to 1. Also, Brandon can see that 3/9 is bigger than 2/12, but that only makes that fraction farther away from 1. So 10/12 would still be bigger.

(SR6)

Teacher: Does everyone understand why 3/9 makes the fraction 6/9 farther away from 1 when comparing each fraction to a whole? (TR1)

Lilly: I like using the number line because then I can see that 3/9 is still bigger than 2/12. Also, drawing the number line I needed to go backwards 3 in order to get to 6/9. But, for 10/12 I only needed to go backwards 2 in order to get to 10/12. So it makes sense that 6/9 is farther away from 1 when comparing to the benchmark of 1. Since 6/9 is farther away from one then it would make it the smaller fraction. (SR4)

Brandon: Yeah, I like this idea of using the number line too. It makes it easier to see that 10/12 is the bigger fractions.

Teacher: So how does this idea of 3/9 being farther away from the whole relate back to our previous thoughts about the size of the pieces? (TR1)

Brandon: So like I was saying earlier 3/9 is bigger than 2/12. But by using the number line, I saw that 3/9 makes 6/9 further away from its whole. So in order to complete the whole of 9/9 I would need 3 big pieces. Whereas, 10/12 is 2/12 away from its whole, and since I said 2/12 is smaller then 10/12 requires smaller pieces to get to its whole. So 10/12 would be bigger than 6/9 because 10/12 needs smaller pieces to get to its whole. This whole time we have been trying to compare each fraction to its whole, 1. (SR6)

Teacher: It is very important to remember that we have been comparing these two fractions to the benchmark of 1.

So we must keep in mind what we are comparing the fractions to. Brandon, you did a great job realizing that 3/9 is bigger than 2/12, but we need to remember to relate those fractions back to the original problem.

Part 4: “Book ends” for your lesson

4a. Launch:

Materials: Small whiteboards and markers.

Give each student a whiteboard and a marker. Write 2 fractions on a larger classroom whiteboard. Ask the students to write which faction they think is larger on their smaller whiteboard. The students only have 3-5 seconds to produce an answer. The teacher will call on students to explain their thinking without telling the students which answer is correct. This way, the teacher is able to gauge where the students thinking is at before the lesson begins. Remind the students that there are many strategies when comparing two fractions in order to motivate their thinking.

4b. Closure:

The lesson will be ended with a journal entry. Each student has their own math journal where they can write their thoughts down at the end of the lesson.

Question; Does everyone understand that the size of the pieces is important to consider when the fractions that are being compared are under the

benchmark?
Response: Today I learned that comparing fractions involves a lot of thinking. I have to remember to always think about the original fractions that I am comparing. When I’m comparing two fractions that are under the benchmark I need to think about the size of those pieces for each fraction. Also, I need to think about how many of those pieces I have under the benchmark, and depending on the size, that could make my fraction either closer or farther away from the benchmark. I hope we keep practicing fractions more in class then I think I’ll be able to compare fractions to benchmarks without a problem.
Response: Today I had to work on dumb fractions again. I am too scared to say anything in class, but luckily Brandon had the same answer as I did, so I didn’t have to speak up. I totally thought that 6/9 was the bigger fraction all along too. I still don’t know what everyone was talking about when they said they wanted to compare the size of the pieces of the fractions. Also, I’m not sure that I understand why the size of the pieces is so important. I know that 3/9 is bigger than 2/12, so 6/9 should be bigger than 10/12 too. Maybe that’s what they meant when they said the size of the pieces, because 3/9 does have larger pieces. Since 3/9 has larger pieces than that must mean 6/9 should be bigger. I am still not sure why everyone thought 10/12 is the bigger fraction.