Decomposing Pizza Fractions

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Objective

SWBAT prove two fractions are equivalent using decomposing.

Big Idea

Students will fold rectangular paper "pizzas" to prove that two fractions can look differently but have the same value.

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 and Hundred Grids (found at this link). 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 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 5/10 to 5/100

For the first task, I asked students to compare 5/10 to 5/100. 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!  

As students completed their work using the number lines and hundreds grids, I encouraged them to turn and talk about their findings. This is an important time for me to conference with students to monitor student understanding. Here, Student Conference 1:10 vs 1:100, I discussed the value of 5/10 and 5/100 with one student. It's amazing how much learning can take place during these conferences! 

While filming the above conference, there was a heated debate between a group of students that can be heard in the background. I couldn't wait to hear what they were disagreeing about! Here's the conversation that took place: Disagreement 5:100?.

Also, here's an example of a Student Number Line 5:10 & 5:100 at this time. 

Task 2: Compare 6/4 to 4/2

Next, we moved on to comparing 6/4 to 4/2. Here, Placing 6:4 on the Number Line, a student explains the how he located six fourths on the number line using his knowledge of one fourth. I loved hearing him say that 6/4 is equal to 3/2.

Then, another student showed how to locate 4/2 on the number line: Placing 4:2 on the Number Line. Students could hardly wait to show their thinking on the white board! 

Here's an example of a Student Number Line 6:4 & 4:2 and Hundreds Grid 4:2 & 6:4.

Teacher Demonstration

50 minutes

Goal & Problem

To begin, I introduced the goal: I can explain why two fractions are equivalent. I then explained a real world problem to help students see the connection between fractions and every day life (Math Practice 4): Kayla and Taylor ate a whole pizza! They each ate the same amount, but Kayla ate more pieces that Taylor. How could this be? 

Students excitedly responded, "I know how!" This was a great opportunity for students to turn and talk to share their thinking. After a few minutes, we discussed the problem further as a group: 

Student #1: Different Sizes

Student #2: Cutting into more Pieces

Student #3: Circle Model

This conversation was especially important as students were defending the fact that fractions can have the same value, even though they may look different (equivalent fractions).

Getting Ready

I passed out three yellow rectangles and a sheet of lined paper to each student. Then, I modeled how to fold each rectangle in half and how to color half of each rectangle: Coloring 1:2. I explained: Let's say that the shaded part of our rectangles could represent the amount of pizza that Taylor ate and the unshaded part represents the amount of pizza Kayla ate. 

Decomposing 1/2

Next, I showed students how to fold each yellow rectangle so that the additional folds were in the opposite direction as the original fold: Decomposing 1:2.

1. We folded one paper in half again, creating 4 rectangles.

2. We folded the next paper into thirds, creating 6 rectangles.

3. We folded the last paper into fourths, creating 8 rectangles. 

I wanted students to truly experience how 1/2 is equal to 2/4, 3/6, and 4/8.

Proving Equivalency

After pasting the rectangles down on the lined paper, I asked: Would everyone agree that each of these rectangles represent 1/2? We then wrote 1/2 next to each rectangle. I continued: If Taylor ate 1/2 of a pizza, and Kayla ate the same amount, but actually ate more pieces, looking at this first pizza, what fraction could Kayla have eaten? Students said, "Two fourths!" I then said: So what you're saying is that 1/2 is equal to 1/4 + 1/4? I modeled how to write: 1/2 = 1/4 + 1/4. 

We continued in this manner, writing 1/2 = 1/6 + 1/6 + 1/6 next to the second rectangular pizza and 1/2 = 1/8 + 1/8 + 1/8 + 1/8 next the the third rectangular pizza. 

Next, we further named the equivalent fractions by simplifying each of the above equations: 

1/2 = 2/4

1/2 = 3/6

1/2 = 4/8

Here's what a student's paper looked like at this point: Naming Equivalent Fractions.

Constructing a Conjecture 

Before going any further, I wanted to refer back to our conjectures poster from yesterday's lesson: Conjectures Poster Before. I explained: Yesterday, we began to write another conjecture at the bottom of our poster... To find equivalent fractions, you... Can anyone explain what they found during your investigation yesterday? Students hands shot up in the air as if they had been just waiting for the moment that we would return to the poster! 

I called on a student (whose name starts with M, so I placed and "M" next to the conjecture to encourage ownership of learning). The student explained and I wrote while a few students chimed in to encourage precise wording: To find equivalent fractions, you multiply the numerator and the denominator by any number. After discussing further, students decided we should in "or divide" above the word multiply. They also thought that should clarify that the number must be the "same" so we added that in as well: Conjectures Poster After.

The process of making conjectures is one of the best ways to engage students in Math Practice 3: Construct viable arguments and critique the reasoning of others.

Applying Conjecture

We then went back to the equivalent fractions that we had just created using the rectangular pizzas. I said aloud: I wonder if this conjecture is always true! Let's look at our pizza fractions and see! Look at the first rectangle and equation, 1/2 = 2/4. Can anyone explain how this equation supports M's conjecture? A student excitedly shared, "You can multiply the denominator by 2 to get 4 and then you can multiply the numerator by 2 to get 2." We then checked the other equations using the same method and added more notes to show our thinking: Student Example of Labeled Pizzas

Making Observations

As often as possible, I try to incorporate opportunities for students to look closely for patterns (Math Practice 7). This is important because identifying patterns helps students make sense of complicated math concepts. I asked students if any students would like to share any observed patterns. Here's the conversation that resulted:

After the last student shared her thinking, I took the time to restate her thinking to make sure students heard and understood her observation: Restating Student Observation

Dividing

This led us to a conversation about "taking rows out by dividing." I wrote off the side... 2/4 = 1/2 and asked: What if we take the rows back out by dividing the 4 rectangles into 2 rectangles? What would I divide the 4 by to get to 2? (2) And what would I divide the 2 by to get to 1? (2) We then modeled and discussed how to divide to prove equivalency for 3/6 = 1/2 and 4/8 = 1/2. Here's how the board looked after this conversation: Dividing to Find Equivalent Fractions

More Observations

Soon after, more hands went up without any prompting. Students exclaimed, "Oh! I see a pattern!" Here are the last couple of observations: 

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

I explained: For practice today, I'd like for you to continue drawing horizontal lines to decompose fractions! I passed out a double-sided Equivalent Fractions Practice page from the Georgia Department of Education 4th Grade Equivalent Fractions Unit. I modeled how to complete the first problem and then students were ready to go! As a side note, I didn't teach students to multiply in fraction form: 2/3 x 3/3 = 6/9.... I wanted to keep it simple: 2/3 = 6/9 because 3 x 3 = 9 and 2 x 3 = 6. 

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. How many rows did you end up with when you drew ____ horizontal lines?
  3. What's the shortcut? What could you multiply the denominator and numerator by? 
  4. What does this show you?
  5. Does your equation match the are model representation?
  6. Does this always work? 

Conferences

Here, Student Conference 2:3 = 12:18, a student explained how he cut the pizza "five times" to get 18 pieces. I then asked him to show the "shortcut" if he didn't have the model. I wanted students to connect the area model representation to the abstract equation. 

This student, Student Conference 3:4 =6:8, showed how 3/4 is the same as 6/8 because you are just splitting in the 3/4 in half. She then confidently said that she could just multiply the numerator and denominator "times 2." 

Completed Work

Here are examples of student work: