Decomposing & Composing Fractions

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Objective

SWBAT prove two fractions are equivalent by decomposing & composing fractions.

Big Idea

Students will add and take away lines in area models to prove two fractions are equivalent.

Opening

30 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. 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 others 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 also created and introduced a Coin Conversion Chart for students to reference. Here are the Pictures of Coins that I used to create this chart.

I invited students to get a Student Number Line and Hundred Grids. I then drew a 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 3/10 to 1/4

To begin, I asked students to compare 3/10 to 1/4 on their number lines and hundreds grids. During this time, some students chose to work alone while others worked with a partner in their math groups. I took this time to conference with students. After students had the opportunity to turn and talk, I randomly chose three students to share their thinking on the board: Students Modeling 3:10 > 1:4

Others watched carefully, checking their own number lines and hundreds grids (Student Number Line 3:10 > 1:4 and Hundreds Grid 3:10 > 1:4) to make sure they agreed with the students demonstrating their thinking.

I tried to take every opportunity to build upon students' observations. For example, in the video, one student pointed out that 1 divided by 2 equals 1/2 and that 1/2 divided by 2 = 1/4. I then asked the rest of the class... What's 1/4 divided by 2? And what's 1/8 divided by 2? And so on... 

Task #1: Compare 1/5 to 3/10

Next, we moved on to comparing 1/5 to 3/10. Again, I rotated between groups during this time. Here's an example of a conference with two students: Conferencing with Students 1:5 & 1:10. You'll hear me say: Oops! I need to not talk so loud!" During this time, I really try to encourage student thinking instead of students relying on teacher answers. 

Here are a couple examples of student number lines: Student Number Line A 1:5 > 1:10 and Student Number Line B 1:5 > 1:10. All students represented their thinking using the hundreds grids as well: Hundreds Grid 1:5 > 1:10

Once everyone had the opportunity to show their thinking to their math groups, I randomly chose three students to share their thinking on the board: Comparing 1:5 to 1:10. You can see that some students really like to use hands-on models! 

This led to a final conversation using a Circle Model. It took me a moment to figure out what the student was trying to show! It was a great opportunity to connect the number line, hundreds grid, and decimals to a circle model! 

Teacher Demonstration

50 minutes

Lesson Introduction & Goal

To begin the lesson, I shared today's goal: I can prove two fractions are equivalent by decomposing & composing fractions. I explained: Yesterday, you learned how to decompose fractions by folding paper. Today, we are going to continue decomposing fractions, however, today, we are also going to learn how to compose fractions! 

Decomposing vs. Composing

To help compare the concepts of decomposing and composing, I created a three column chart with 6 rows on a piece of chart paper. At the top, I labeled two of the columns: Decomposing 1/2 and Composing 1/2. Along the side, I labeled each row: Meaning, Area Model, Equation, Operation, and Observations. I waited to complete the rest of the chart with the class. Here's what the completed chart will look like after our lesson:Decomposing vs Composing Poster.

I asked students to draw the following chart in their math journals: Chart in Student Journal to correlate with the larger chart on the board.

Chart Demonstration

At this point, I used the chart to model the difference between decomposing and composing. Students completed the charts in their journals as I completed and explained each row.

Row 1: Decomposing & Composing

I first explained, Whenever you see the word, "decomposing," I want you to think of a pair of scissors. Using the following clipart, Scissors & Tape Pictures, I pasted a couple pairs of scissors next to the word "decomposing." And whenever you see the word, "composing," I want you to think of a roll of tape! I pasted a roll of tape next to "composing."

Row 2: Meaning

Now let's talk about the meaning of these words. Decomposing is when you are "destructing into pieces." Students wrote the meaning down in their own math journals. Then, I demonstrated the act of destructing by cutting a piece of paper up over and over with a pair of scissors.

I continued: Let's talk about composing now. Composing is when you are "connecting the pieces," like you would with tape. (Now looking back, I wished I would have taped the pieces of paper perviously cut back together!)

To aid comprehension, I asked students to turn and talk about the difference between decomposing and composing.

Row 3: Area Model

Here's what an area model for decomposing would look like. Do you see how I have one whole divided into two parts and 1/2 is shaded? Let's take a red marker and make a dotted line to show how I could cut 1/2 into fourths! 

Look at the area model for composing. We have a whole divided into four parts and 2/4 of the whole is shaded. Now, watch me tape 1/4 and 1/4 together to get 1/2. I drew two pieces of "tape" across two fourths. Students seemed to really understand this concept. 

Row 4: Equation

For the next row, I wrote out the equation for decomposing one half (1/2 = 2/4) and the equation for composing one half (2/4 = 1/2). 

Row 5: Operation

I continued: What operation did we use to change 1/2 into 2/4. (Multiplication!) What operation did we use to change 2/4 into 1/2? (Division)

Row 6: Observations

I asked students to turn and talk: What are you noticing? How is decomposing like composing? How are they different? 

After some time, a student said, "With decomposing, the pieces get smaller and with composing, the pieces get bigger." Another student student said, "Yeah, but the amount stays the same no matter what."  We added these to our charts. 

By the time we were done, students' journals looked like this: Completed Chart in Student Journal.

Presentation

To help teach today's lesson, I created a Powerpoint Presentation: Composing & Decomposing  to provide students with guided practice.

Decomposing Practice

I invited students to join me on the carpet with their white boards. Starting with the Decomposing Section of the presentation, I explained: Now we get to practice decomposing and composing fractions! First, we'll start with decomposing. Changing to the next slide, First Decomposing Example, I demonstrated how we can decompose a fraction using horizontal or vertical lines: #1 Example 1:2 = 2:4. (Interestingly, students only ended up using horizontal lines during today's lesson!)

One at a time, students volunteered to show how to decompose the fractions on each of the following slides:

During this time, other students showed their thinking on their boards. Some students drew the model, Model & Equation on Student White Board, while others listed equivalent fractions, Equivalent Fractions for 4:8

Composing Practice

Next, we moved on to the Composing Section. By this time, students were eager to have a chance to come up to the board to show their thinking. I first modeled how to compose fractions using the first example, #1 Example 2:4 = 1:2.. Then, just as before, students came up, one at a time to take turns demonstrating: 

Since we worked on decomposing during yesterday's lesson as well, at this point I wanted to move on by providing students with independent practice composing fractions.

Student Practice

50 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

I explained: Today, you get to continue composing fractions on your own. I passed out a double-sided Composing Fractions Practice Page page from the New York Engage Fraction Module. I modeled how to complete the first problem and then students were ready to go!

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. Can you explain what you did?
  2. Why is this called composing? 
  3. What happens to the numerator? 
  4. What happens to the denominator?
  5. Does the amount shaded in change? 
  6. Does this always work? 

Conferences

Here's one example of a conference with a student, Conferencing with Student, 9:15 = 3:5. This student did a great job making his work more precise. You'll also hear a heated conversation in the background! As students finished, I asked them to check their work with others. The process of respectfully disagreeing is such an important part of constructing viable arguments. 

Completed Work

Here's a Student Work Example. Most students did really well on this assignment.