Exploring Unit Fractions 1/8, 1/9, 1/10, 1/100

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.

Tasks

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. At this point, I didn't need to ask students to provide corresponding decimals. They automatically explained, "50/100 is located here because 50/100 is equal to $0.50!"

• 100/100
• 200/100
• 50/100
• 3/2
• 1/4
• 3/4
• 7/4

Powerful Conversations

Throughout today's number talk, there were many ah-ha moments! A discussion about 1/2 led us to counting by halves. Then, students began to notice and explain patterns: Discussion About Halves.

Next, a student modeled how to place 7/4 on the number line: Locating 7:4. I loved watching him make sense of the number line.

Finally, our investigation of the number line led us to Finding Equivalent Fractions. At first, only a few students noticed the pattern. Then more and more students excitedly raised their hands to share. Here's the end result: Equivalent Fractions for 1:4. 

Finished Product 

Here's what the class number line looked like at the end of today's number talk: Class Number Line. Also, here are a couple student work examples: Student Number Line and Student Number Line 2. 

Vocabulary Posters

Although I didn't specifically review each of the following vocabulary posters, students referred to them throughout this math period. Once in a while, I'd ask questions like: What is a fraction again? Other times, students would refer to the posters and say, "That's decomposing!" 

• fraction
• numerator & denominator
• types of fractions
• proper & improper fractions
• proper & improper fractions
• composing
• decomposing

Lesson Preparation: Google Presentations

For today's lesson, I created four Google Presentations for representing Examples of 1/8, Examples of 1/9, Examples of 1/10, and Examples of 1/100. I wanted to continue providing students with an avenue to share their thinking and an opportunity to collaborate and comment on each other's work.

After creating the above presentations, I shared each of them with students. Here's further information on How to Create a Google Presentation for Student Practice. Sometimes I have students copy the presentations in order to make them their own. However, for today's lesson, I wanted all students to collaborate and work together on the same presentation (all at one time).  

Prior to today's lesson, I assigned each student a slide number in order to provide each child with a workspace. For example, Student A was assigned the first blank slide, #4. Student B was assigned slide #5. Students used the same slide with each presentation. In order to communicate assigned numbers quickly, I created and shared the following Google document with students: Student Numbers.

Ultimately, the goal was for students to follow this process:

1. Represent unit fractions (such as 1/8) by choosing appropriate tools.

2. Take a picture of their representation using their computers.

3. Insert the picture on their assigned slide.

4. Use arrows and text boxes to show how many 1/8 units are equal to one whole. 

5. Collaborate with other students to make their work better. 

Goal & Introduction

Once students had opened all the shared documents, I asked them to join me on the front carpet. 

To begin, I reviewed our math Goal: You have now investigated how to represent several unit fractions, such as 1/2 and 1/5. Today, we are going to continue working toward the same goal: I can show and explain how many unit fractions are in a whole. Only this time, we are going to move on to representing unit fractions with larger denominators, 1/8, 1/9, 1/10, and if you have time... 1/100! 

2. Explain: your representation by labeling the unit fraction and using an equation to show how many of this unit fraction equals a whole.

3. Make Comments: on other student work that are mathematical, thoughtful, respectful, and helpful. 

4. Respond to Comments: and clarify your thinking so that others can understand your work better. 

Math Tool & Math Talk

In order to review how to strategically choose appropriate tools (Math Practice 5) and critique the reasoning of others (Math Practice 3), I used the Appropriate Tools Anchor Chart as well as the Math Talk Anchor Chart. 

Just like yesterday, I selected two students to watch for and eventually celebrate students choosing appropriate tools and making thoughtful comments by giving the Outstanding Tool Selection Award or the Exceptional Collaborator Award. Here are printable versions of these awards: Tool Awards and Exceptional Collaborator Award. 

In this video, Reviewing Tools & Comments, you'll hear a class discussion about choosing tools and collaborating effectively. I explained that I was looking for students to look for comments made on their work (by me or by other students). This was an important teaching point today because I wanted students to not only comment on others' work, but to also respond to comments by changing their thinking, changing their representation, or by simply replying back. 

Modeling with Student Examples

At this point, I wanted to recognize students' work: Modeling Using Student Work. I specifically focused on what I wanted to see more of:

• I wanted students to try using a number of items or parts greater than the denominator. For example, instead of using 8 pennies to represent 1/8, I wanted students to experiment with using multiples of 8, such as using 24 or 16 sticky notes.
• I liked seeing students specifically naming the whole, such as "5 wiggle cushions equals one whole."
• I also wanted to see students using an addition and a multiplication equation to show the number of unit fractions in one whole (1/8 x 8 = 8/8 and 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 8/8 = 1). Some students chose to use simpler addition equations, such as 5/8 + 3/8 = 8/8 = 1. 
• I pointed out the importance of labeling, such as 1/5 of 20 books is 4 books.  

Challenge Items

Finally, I wanted to inspire students to use new and more challenging tools so I held up several "challenge items," such as a clock, money, a box of 1000 staples, a gallon jug, a graduated cylinder, a 21 oz bottle of lotion, a protractor, a can of pop, and art supplies.

Math Partners

Students continued working with the same math partners as yesterday. Even though students were working on their own slides, they'd often check with their partner before taking a picture of their representation. Also, some students needed their partner's support to take a picture. One student would hold up the computer or fraction model while the other student took a picture.  

Different Starting Points

While many students were ready to begin representing fractions using the 1/8, 1/9, 1/10, and 1/100 presentations, others were still finishing up their representations of fractions from yesterday (1/5, 1/6, and 1/7). Students were able to advance when they were ready! 

Monitoring Student Understanding

Once students began working, I conferenced with every student. 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. How did you choose this tool?
2. Is this tool working out for you?
3. Have you run into any problems with this tool? 
4. Can you show me 1/8? 
5. How many units of 1/10 will it take to get to a whole? 
6. How do you know? 
7. Where is one whole? 
8. Can you explain your thinking? 
9. Which comment was the most helpful? Why? 

Conferences

While conferencing with this student, Representing 1:10 of 20 Pencils, I tried to help this student use prior knowledge (an understanding of 1/10) to complete the equation. 

This student, Representing 1:100 Box of Staples, did an amazing job using division to explain why 5 is 1/100 of 500. 

Completed Work

Here are a few examples of student representations of 1/8:

• 1:8 Gallon
• 1:8 Geoboard
• 1:8 Watercolor Paint
• 1:8 Ziplock Bags

Here are a few examples of student representations of 1/9:

• 1:9 Box of Clips
• 1:9 Pennies
• 1:9 Unifix Cubes A
• 1:9 Unifix Cubes B

Here are a few examples of student representations of 1/10:

• 1:10 Box of Paper Clips
• 1:10 Clock
• 1:10 Clock
• 1:10 Pattern Blocks

Here are a few examples of student representations of 1/100:

• 1:100 Base Ten Blocks A
• 1:100 Base Ten Blocks B
• 1:100 Box of Staples
• 1:100 Money