Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using a number line model, Student Number Line, and a hundreds grid, Hundred Grids. For each task today, students shared their strategies with peers (sometimes within their group, sometimes with someone across the room). It was great to see students inspiring others to try new methods and it was equally as great to see students examining each other work for possible mistakes!
Prior to the lesson, I placed magnetic money and fractions on the board to help students conceptualize our number talk today.
I invited students to join me on the front carpet with their number lines. I then drew a number line on the board, Number Line on the Board, and marked 0, 1, and 2 on the line. I asked students to do the same on their own number lines.
Task 1: Compare 2/4 to 4/2
For the first task, I asked students to compare 2/4 to 4/2. I asked: Which fraction is greater? Which is smaller? How do you know? Please show your thinking on your number line. Prove it to me!
I conferenced with a student using a hundreds grid: Student Hundreds Grids 2:4 & 4:2. He explained how two sets of 1/4 = 2/4. He also explained how 2/4 = $0.50 and 4/2 = $2.00.
After students had time to show their thinking their own mats, a couple students came up to the front to show where each number is located on the number line: Students Modeling 2:4 < 4:2.
Here are a couple examples of student work during this time: Student Number Line 2:4 & 4:2 and Student Hundreds Grids 2:4 & 4:2. On the hundreds grid, I noticed that the student labeled 1/2 as 2/2 and another 1/2 as 3/2... I normally try to catch this and ask students to place 1/2 in each part instead.
Task 2: Compare 1/10 to 2/20
For the final task, students compared 1/10 to 2/20. Here, a student explains how he used money to determine 1/10 = 2/20: Student Conference 1:10 & 2:20.
Students then came up to the board to model their thinking for the class: Confusion 10:20 & 20:20. A great conversation resulted!
The Criss Cross Method
In today's lesson, students will learn how to use the Criss Cross Method to compare two fractions. For example, if you were comparing 1/2 to 3/4, you could follow these steps to complete the criss cross method. Refer to the picture, Criss Cross Method.
1. Multiply the numerator of the first fraction (1/2) by the denominator of the second fraction (3/4): 1 x 4 = 4.
2. Multiply the numerator of the second fraction (3/4) by the denominator of the first fraction (1/2): 3 x 2 = 6.
3. Multiply the denominators of both fractions (1/2 & 3/4) together: 2 x 4 = 8. Place this product under the first two products, to create two equivalent fractions: 4/8 and 6/8.
4. Compare: 4/8 < 6/8 so 1/2 < 3/4.
Comparison Strategy Posters
Knowing that I didn't have the instructional time needed to delve deeply into each method of comparing fractions, for today's lesson and tomorrow's lesson, I decided to expose students to a range of comparison strategies using posters. Even though we only had time to develop a full understanding of a couple comparison methods over the next couple of days (using the criss-cross method, using the area model, and finding equivalent fractions), I wanted students to know that fraction comparison is not limited to only a couple strategies.
Also, I wanted to support Math Practice 5: Use appropriate tools strategically. In the future, students will look back on these posters to determine which comparison strategy is most useful, depending on the comparison scenario.
For today's lesson, I began with the strategies students were most comfortable with (such as using hands-on tools) and moved toward less familiar strategies (such as the criss-cross method).
Goal & Introduction
For today's lesson, I invited students to the front carpet with their whiteboards. I began by introducing today's goal: I can compare fractions using the criss-cross method. Before we learn how to use the criss-cross method to compare fractions, we are going to first discuss other ways to compare fractions. Why is it important to learn more than one strategy? One student said, "Because if one strategy doesn't work, we can try a different strategy."
First, we discussed how we can use hands on tools to compare two fractions. Some students decided to follow along, recreating the poster, Using Hands-on Tools Poster, on their own white boards, while others chose to just observe.
Prior to today's lesson, I glued four beans (2 brown, 2 speckled) on one side of the poster and four beans on the other side of the poster (1 brown, 3 speckled. Covering up the right side of the poster, I asked students: What fraction of the beans are speckled? Students responded, "Two fourths... because two out of four beans are speckled." "That's the same as one half." So I wrote 1/2 above the beans on the left side. Next, I covered up the left side and asked: What fraction of these beans are speckled? Students responded, "Three fourths... because there are four beans and three are speckled." I then wrote 3/4 on the poster above the beans on the right side.
I asked students to turn and talk: Which fraction is bigger? 1/2 or 3/4? Following about a minute of conversation, students shared: 3/4 is bigger because 1/2 of four beans is 2/4. We then discussed the importance of referring to the same whole when comparing fractions. At this point, I added labels and explained: If we are using beans to compare 1/2 and 3/4, then we have to use the same number of beans to show 1/2 as we use to show 1/4. In other words, if we use four beans to show 1/2 on this side, then we have to use four beans to show 3/4 on this other side.
Next, we moved on to comparing fractions that have the same denominators. I wrote 1/4 and 2/4 on the poster, Same Denominator Poster, and asked students to do the same on their boards: Turn and talk with a nearby student. Which fraction is greater and how do you know? Many students immediately went to work drawing visual representations for both fractions: 1:4<2:4. Students pointed out, "The 2 is bigger than the one so 2/4 is bigger than 1/4." I asked: Is the fraction with the greater numerator always bigger? What if I had 1/4 and 2/8? Students then explained and I added to the poster, "Just compare the numerators if the denominator is the same." Some students explained their thinking further on their white boards: 1:4 < 2:4 Explanation.
I asked students: Can you show me two more fractions that can be compared using this strategy?One student showed 1/2 < 4/2: 1:2 < 4:2 Student Model. Another student showed how 5/15 < 10/15: 5:15<10:15.
I then asked: Well, what if we have different denominators, but the numerator is the same? I wrote 1/10 and 1/2 on this poster, Same Numerators Poster, and students did the same on their own white boards. Students immediately began talking with a partner. It was interesting to watch how many students drew models to explain their thinking: 1:10 < 1:2 Circle Model. The model students gravitated to the most was the area model.
Again, students explained, and I wrote on the poster, "1/10 is less than 1/2 because the smaller the denominator, the bigger the fraction." One student added, "Yeah, it's just like on our conjecture chart, "The smaller the denominator the bigger the fraction, like 1/2 and 1/4." Then, another student added, "But not all the time... the numerators have to be the same." This student, 1:10 < 1:2, did a great job explaining this concept, "If the numerator is the same, the fraction with the smaller denominator will be greater." (I just wished her rectangles were the same size.)
We then moved on to the money model, Money Model Poster. Since we have been practicing this model regularly during our daily number talks, students were already quite comfortable using this method. We first discussed 1/4 and 1/10. I asked: Turn and talk. Which fraction is greater? How can you use money to justify your thinking? During this time, students drew beautiful representations on their boards to help explain their thinking. Then, students anxiously said, "1/4 of a dollar is the same as $0.25 and 1/10 of a dollar is $0.10." "We know that 25 cents is greater than 10 cents, so 1/4 is the greater fraction." This student, 1:4 > 1:10, showed how 1/4 = 25/100 and 1/10 = 10/100.
Finally, we discussed the criss-cross method, Criss Cross Poster. I knew the criss-cross method was a fun trick for students to use as a second strategy (in order to check their visual models later on in this unit).
I began by modeling the criss-cross method with two simpler examples: 1:2 < 2:3 & 3:5 > 1:2. During the first task, students compared 1/2 to 2/3. I then modeled how to simply multiply the numerator from one fraction with the denominator from the other fraction: 1 x 3 and 2 x 2. Then, we multiplied the denominators together to get 3/6 and 4/6. I then asked: Which is greater... 3/6 or 4/6? Students excited said, "That's so cool!" We then moved on to comparing 3/4 and 1/2. Students caught on quickly and completed the task on their own white boards, many one step ahead of me. With time, students explained, "6/10 is more than 5/10 so 3/5 is greater than 1/2!"
At this point, we practiced the criss-cross method using more challenging tasks:
In no time, most students were ready to practice this strategy on their own!
Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. Before students began working, I asked them to discuss how they would like to support each other today. I gave them many examples: Do you want to take turns talking out loud? Do you want to solve quietly and then check with each other? Or do you want to turn and talk anytime you get stuck? Students always love being able to develop a "game plan" with their partners!
To provide students with further practice, I printed two fraction comparison pages from Math-Aids. Knowing that some students would finish faster than others, I printed two pages ofcomparison tasks, front to back.
I explained: For continued practice today, I'd like for you to continue comparing fractions using the criss-cross method! I then modeled the first five problems for the class to make sure students understood the steps.
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students by providing them with the opportunity to explain their thinking and by asking guiding questions. I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3).