Area Model Comparison

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Objective

SWBAT compare fractions using area models.

Big Idea

Students will explore how to use a variety of strategies to compare fractions, but will focus primarily on the process of decomposing area models to compare fractions with uncommon denominators.

Opening

20 minutes

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!

Getting Started

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 7/4 to 7/10

For the first task, I asked students to compare 7/14 to 7/10. 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! 

After students had time to compare these fractions on their number lines, a couple students volunteered to share their thinking in front of the class. I encouraged students to think about money conversions to help them make further sense of fractions.

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 35/100 to 3/5

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! 

Again, students did a great job showing their thinking on their own mats: Student Number Line 1:10 & 2:20 and Student Number Line & Grid 1:10 = 2:20.

Teacher Demonstration

60 minutes

Comparison Strategy Posters

Knowing that I didn't have the instructional time needed to delve deeply into each method of comparing fractions, for yesterday's lesson and today'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 (landmark fractions) and moved toward less familiar strategies (such as comparing the missing whole). 

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 area models. Just like yesterday, we're going to discuss a couple other ways to compare fractions before delving deeply into our goal. Who wants to only know about one strategy to compare fractions? And who wants to know several strategies? Every hand shot up!

Landmark Fractions

We began by discussing how to use landmark fractions to compare fractions, Landmark Fractions Poster. I explained: Landmark fractions are also called benchmark fractions. A landmark fraction is a familiar landing place on the number line, such as 1/2 or 3/4. When we look at the fraction 24/100, I know that this is close to 25/100 which is close to 1/4. When we think about 3/8, we could round up to 4/8, which  is the same as 1/2. I drew a model of this on the board to aid conceptual understanding: Teacher Model 3:8 is About 4:8. I know that 1/4 is less than 1/2 so 24/100 must also be less than 3/8. Students completed the same work on their boards as I modeled the strategy: Student Work 24:100 < 3:8.

I then provided students with another opportunity to apply this strategy: Now let's try comparing 4/10 and 98/100 using landmark fractions! Most students identified that 4/10 is close to 5/10, which is the same as 1/2 and 98/100 is close to 100/100, which is equal to one whole. Using this reasoning, they explained to nearby partners, 4/10<98/100: 4:10 < 98:100.

Compare the Missing Piece

I slowed the pace of this activity while introducing the most complicated strategy, Compare the Missing Piece: Compare the Missing Piece Poster. I explained: If you're comparing 9/10 and 3/4, both fractions are one way from a whole. For example, how many more tenths would you need to get to one whole if you have 9/10? Students responded, "1/10!" I wrote 9/10 +1/10 = 10/10 on the poster. Then I asked: How many more fourths do we need to get to a whole? Students responded, "1/4!" I then wrote 3/4 + 1/4 = 4/4 on the poster. Again, students modeled this own their own white boards as well: Student Work 9:10 > 3:4.

To make the comparison even more clear, I also made a money connection (9/10 + 1/10 = 0.90 + 0.10) and (3/4 + 1/4 = 0.75 + 0.25). I then asked the most challenging question of all: Which fraction is closer to one whole? Many students struggled with this question. I could see some retracing our steps and negotiating the value of each fraction. With time, students agreed, "9/10 is greater because it is only 1/10 away from a whole and 3/4 is 1/4 away."

I then asked students to use this same method to compare 4/5 and 99/100. After solving this problem on their boards, 4:10 < 98:100, students discussed their thinking with partners. Thereafter, a student explained her thinking in front of the class: Student Modeling 4:5 < 99:100. You'll see that this method is quite challenging! 

Area Model

Next, we moved on to the goal of today's lesson, comparing fractions using an area model, Area Model Poster. We discussed how to compare 1/3 and 4/6 by representing 1/3 and 4/6 using the same size whole. I drew a dotted line to show how 4/6 could be decomposed into 2/3 for easy comparing. 

I then asked students to compare a final set of fractions using the area model: 1/2 and 7/8. Most students successfully justified 1/2<7/8 using an area model on their whiteboards: 1:2 < 7:8

Powerpoint Presentation

To provide students with further practice using the area model to compare fractions, I created a Powerpoint Presentation called Comparing Fractions Using the Area Model. Throughout the presentation, I built a progression of learning, starting with less complex tasks and moving toward more complex tasks.

I knew that this presentation would engage students in Math Practice 7: Look for and make use of structure. Students would look for patterns between slides and they would make use of this structure to solve more complex tasks. 

To begin the presentation, I intentionally provided students with two warm up tasks. Both were designed to address two common fraction misconceptions. I wanted students to identify correct and incorrect visual fraction models, with attention to the number and size of the parts. I also wanted students to recognize that comparisons are valid only when the two fractions refer to the same whole.  

Which Model Represents 1/4?

After reviewing the Goal on the first slide, we went on to the second slide of the presentation,  Analyzing 1:4. I placed three correct models for 1/4 and two incorrect models for 1/4 on this slide. I asked: Who sees a correct area model of 1/4? Who sees an incorrect area model of 1/4? One at a time, I'd like for you to come to the board and circle correct representations and cross off incorrect representations. Students were excited to participate and the discussion that followed was amazing! 

Here, Circling Fourths A the first student selected a correct representation of 1/4. 

Then, another student explained a non-example of 1/4: Circling Fourths B.

This student also did a great job identify another correct example: Circling Fourths C.

Next, this student identified the last non-example: Circling Fourths D.

This is when the converstation became interesting! Some students struggled when some of the lines were missing: Circling Fourths E and Circling Fourths F. You'll see that my students really had a hard time with this concept! Finally, I stepped out of the coaching role and provided explicit instruction on this concept by showing students how to add one more line to further partition the whole into fourths: Dividing into Fourths and later showing students how the larger piece really just represented 2/4: Teacher Model 2:4 + 1:4 + 1:4. Many students responded, "Oh! Now I get it!"

Which is  bigger... 1/4 or 1/2?

On the next slide, Analyzing Wholes, students collaborated to decide if 1/4 of a large rectangle is less than 1/2 of a smaller rectangle. This is another important student misconception to address as students often overlook the importance of using the same size whole when comparing fractions. I loved listening to students debate back and forth on the larger fractional piece: Examining Wholes.

Teacher Modeling

Next, we went on to Slide 4 and Slide 5. I modeled how to compose (tape pieces together) and decompose (cut parts into pieces) to compare fractions using common denominators. We also discussed how to multiply the numerator and denominator by the same number to get an equivalent fraction. This will provide a nice segue into tomorrow's lesson: Teacher Demonstration 1:2 = 2:4

While I demonstrated how to compare the fractions in the first two tasks, students completed the same work on their white boards: Student White Board 1:2 = 2:4 and Student White Board 1:4 > 1:8.

Guided Practice

For the next two slides, students compared the fractions ahead of me. After giving students the opportunity to turn and talk, we discussed how to compose or decompose the area model to compare the fractions. While comparing 3/4 to 5/8, Teacher Modeling 3:4 > 5:8, I demonstrated how $10 could be decomposed into $5 + $5. I then connected this idea to fractions: Just like decomposing $10 into two parts, you can also decompose fractions into parts. For example, you can decompose 3/4 (shaded rectangle) into 3/8 + 3/8 (dotted line across the rectangle)

We moved on to comparing 3/6 to 6/12. Again, students compared these fractions on their own to begin with: Student Work, 3:6 = 6:12. Then, we discussed the task as a class: Teacher Modeling 3:6 = 6:12. Again, we discussed how 3/6 could be decomposed into 1/6 + 1/6 + 1/6 or (drawing a dotted line) 3/6 could be decomposed into 3/12 + 3/12. 

Student Modeling

For the last three slides, I stepped out of the teaching role and allowed students to take on this role instead. While students modeled their thinking, others continued to complete each comparison on their own white boards. I encouraged students to use the area model alongside of equivalent fractions to compare each set of fractions. 

Following this activity, I wanted to provide students with practice comparing fractions using the area model.

Student Practice

20 minutes

Choosing Partners

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! 

Practice Page

To provide students with further practice, I printed a Comparing Fractions Area Model Practice Page from New York Engage Module 5. I explained: For continued practice today, I'd like for you to continue comparing fractions using the area method! I then modeled the first two problems to help students make the connection between the Powerpoint activity and area model practice with a paper and a pencil. 

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). 

  1. What did you do first?
  2. What do you need to always remember?
  3. Does that feel right? 
  4. What makes more sense to you?
  5. Does that always work?
  6. What's your next step?
  7. Why did you find common denominators?

Conferences

Here, Finding Common Denominators, a student determined that 42 was the least common denominator. I was proud of him for carefully determining that 42 was the least common multiple for 6 and 7. Next, Comparing Fractions, he continued by creating area models to represent 3/7 and 2/6. I have to say that in the past, I didn't teach students to conceptualize the process of finding equivalent fractions using area models. I was so proud of my kids for successfully modeling their thinking using visual models. 

While some students needed direct instruction on finding common multiples to complete this practice page, most students were able to remember this procedure from a prior unit on multiples. 

Completed Work

Most students were very successful with this activity, especially once they got the hang of it. Here's an example of completed work: Example of Student Work.