# Decomposing Submarine Sandwiches

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## Objective

SWBAT decompose a whole into a sum of fractions.

#### Big Idea

Students will decompose paper submarine sandwiches into fraction equations.

## 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. 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 and marked 0, 1, and 2 on the line. I asked students to do the same on their own number lines.

Next, I gave students each of the following numbers and asked students to identify where each number would be located on the line. After students had time to place each number, I asked students to turn and talk about their thinking. I also asked for a student volunteer to explain to the class where the number would be placed and why. I started by giving students decimal numbers in the form of money (\$1.00) as students tend to be more comfortable with money than with fractions. Here's what the end result will look like: Class Number Line.

• \$1.00
• \$2.00
• \$0.50
• \$1.50
• \$2.50
• ½
• 2/2
• 0/2
• 3/2
• 2  ½

Conversations

Here, Student Finding \$1.50, a student explains that \$1.50 is the midpoint between 1 and 2 wholes.

This student explained that \$0.25 is half of \$0.50 so it would be located between \$0.00 and \$0.25: Locating \$0.25

Later on, this student explained how she used her understanding of money to find the location of 1/2 on the number line: 1:2 = \$0.50.

Here are a couple examples of student work:

## Teacher Demonstration

30 minutes

Vocabulary

Before students returned to their desks, I introduced the following vocabulary posters: fractionnumerator & denominatortypes of fractions, proper & improper fractionsdecomposing, and composing. After introducing each poster, I asked students to either give me examples of the vocabulary word or to turn and talk about the difference between two words (such as decomposing and composing). I wanted students to be able to refer to these posters throughout today's lesson when constructing mathematical arguments.

To get ready for today's lesson, I asked students to get five sheets of lined paper and staple them together. Next, I passed out a copy of the Fraction Questions to each student and showed students how to paste one question at the top of each page (turned horizontally).

Then, I passed out 10 strips of paper to each student and modeled how to quickly construct submarine sandwiches! I had originally planned on copying the following page for students Submarine Sandwiches, but decided to ask students to create sandwiches instead.

Goal & Problem

To begin today's lesson, I explained the goal: I can decompose a whole into a sum of fractions. I asked students: Who has eaten a submarine sandwich before? Almost every hand was raised and students were eager to describe their experiences! After a few students shared, I introduced the following Problem on the board: Emily bought one whole submarine sandwich. She wants to share her sandwich with one or more friends. How can she share her sandwich if it has been cut into _____?

Decomposing the Sandwich into Halves

I asked students to flip to the front page of their lined paper packet. I held up a paper submarine sandwich and asked: What if Emily cut her sandwich into halves? (I modeled how to fold and cut accordingly.) How might Emily share this sandwich with others? I then showed students how to glue the sandwich on the top paper, how to organize their papers, and how to decompose 1 whole into a sum of halves: Teacher Model, Halves

One student said, "She could eat 1/2 and share the other 1/2 of the sandwich." I drew a bar diagram (also called tape diagram) to represent the student's thinking. Then I modeled how to write an equation: What I hear you saying is that Emily could decompose the whole into a 1/2 + 1/2. What type of fraction is 1/2 again? (a unit fraction) Why is it called a unit fraction? (Because the numerator is a one!) How else could the halves be eaten? With time, one student said, "Emily could eat both halves!" I then modeled how to express this student's idea: 2/2 + 0/2 = 2/2 =1.

Decomposing the Sandwich into Thirds

We then moved on to modeling thirds: Teacher Model, Thirds. Following the same process as above, I asked: What if Emily decomposed the whole sandwich into thirds? One student offered, "One person could eat 2/3 and another person could eat 1/3." I then represented this student's reasoning on my paper: 2/3 + 1/3 = 3/3 = 1.

To encourage students to begin thinking about the multiplication of fractions by whole numbers (4.NF.B.4.A), I asked: How else could we write this same equation? Instead of writing 2/3, could we write 2 times... Students continued my sentence, "2 x 1/3!" I then modeled how to write the equation using multiplication: (2 x 1/3) + 1/3 = 3/3.

Next, a student said, "The sandwich could also be split evenly between three people. Each person would get a third." I asked: How would I use an equation to represent your thinking? The student said, "1/3 + 1/3 + 1/3 = 3/3 =1." Another student said, "You could also write 3 x 1/3 = 3/3!"

Decomposing the Sandwich into Fourths

Next, we cut a submarine sandwich into fourths. Again, using the bar diagram and an equation, we modeled two ways the sandwich could be decomposed. One student offered, "2/4 + 2/4 = 4/4 =1." Another helped with the multiplication equation: (2 x 1/4) + (2 x 1/4) = 1. Then, another student said, "We could just write 2 x 2/4 =1." I loved watching my students naturally multiply fractions by wholes without teaching them a formal lesson on this topic!

Then, a student said, "We can also split the sandwich into 3/4 + 1/4." After modeling this on our papers, another student said, "That's the same as (3 x 1/4) + 1/4 =1!"

## 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: For continued practice today, I'd like for you to continue decomposing your submarine sandwiches into a sum of fractions, just as we have practiced together. For your next problems, you'll be cutting the sandwich into fifths... then sixths... then sevenths... all the way up to tenths...

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. How are you going to decompose the whole?
3. What's one way that Emily could share her sandwich?
4. How else could you write this equation?
5. What does 7/7 equal? Why?
6. What do you always add when working with fractions... the numerator or the denominator?
7. Does that feel right?
8. What makes more sense to you?

Folding into Fifths

Some students struggled with folding their sandwiches into fifths. I was so proud of this student for being a leader in our classroom and helping students with folding secrets: Folding into 5ths!

Conferences

Representing Fifths A: During this conference, an amazing teachable moment happened! This student was adding: 1/5 + 1/5 = 1/10. Instead of teaching her the rule for adding fractions, I asked: Does that feel right? I was able to then say later on: Now does that feel right... that 2 "fifths" is equal to 2/5?

Representing Fifths B: This student explained the correlation between the addition and multiplication equations beautifully. To help him connect with the problem further, I asked: Which person would you rather be?

Representing Sixths A: Here, a student split her sandwich into six equal pieces. She did a great job explaining why 6/6 = 1 whole.

Representing Sixths B: When conferencing with this student, I happened upon another teachable moment. He got confused with representing the number of sub sandwich parts in fraction form. Again, I asked guiding questions: Can you explain the thirds? Does that sound right? I then provided explicit instruction on adding fractions following this conversation.

Completed Work

Here are examples of student work. This activity clearly lays the groundwork for fraction computation in the future. You'll also see that some students felt more comfortable with including a multiplication equation than others.