Exploring Unit Fractions 1/2, 1/3, 1/4
Lesson 2 of 16
Objective: SWBAT show and explain how many unit fractions are in a whole.
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!
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:
- $0.10 (At this point, I showed students how to divide the number line between 0 and 1 whole into 10 equal parts.)
- If one whole is 10/10, then where is 20/10?
- How many tenths would equal $0.70?
- How many tenths would equal $1.20?
- How many tenths would equal $1.50?
- How many tenths would equal $1.80?
Throughout this number talk, I encouraged students to observe and share patterns. Here's a student who noticed the connection between 0.10 and 1/10: Noticing Patterns.
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 the number on top called again? Other times, students would refer to the posters and say, "Hey! That's an improper fraction!"
- numerator & denominator
- types of fractions
- proper & improper fractions
Lesson Preparation: Google Presentations
For today's lesson, I created three Google Presentations for representing Examples of 1/2, Examples of 1/3, and Examples of 1/4. I wanted to provide 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).
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 1/2) 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 1/2 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 today's lesson, I invited students to get their laptops out on their desks and to open the shared presentations and student numbers document above. I explained: Today we will all be working on the same presentations! Each of you will be assigned a slide number. To find out your number, open the Student Numbers document.
I introduced the Goal: I can show and explain how many unit fractions are in a whole. I explained: Today, we are going to begin a project that will take several days to complete.
Math Tool Ideas
Showing the Math Tool Ideas slide of the first presentation (Examples of 1/2), I continued: Think about all the tools in our classroom you could use to represent one half. For example, you could use the gallon measuring set, or a clock, or colored tiles! I've laid some tools out on the counter (Tools on Counter), but you can also look for other tools as you see fit! Once you have represented 1/2 with a tool, I'd like for you to take a picture of your thinking and insert it onto your assigned slide.
Inserting a Picture
I then modeled how to take a picture of a fraction representation using a laptop computer and how to insert a Labeled Photo into the class presentation. We also reviewed how to insert an arrow and text box.
Choosing Math Tools
To engage students in Math Practice 5 (Use appropriate tools strategically), I wanted to explicitly teach students how to successfully choose appropriate tools to represent fractions. So I shared the following anchor chart that I created before the lesson: Appropriate Tools Anchor Chart. We discussed the importance of asking key questions such as, "Which tool will help you visualize and represent a math situation BEST?" or "What are the limitations of certain tools... and the strengths?"
At this time, I introduced the Outstanding Tool Selection Award and randomly chose a student to watch for a student to celebrate for choosing appropriate tools. Here's a copy of the Tool Awards. I was hoping that this would encourage students to choose tools strategically.
Building Representations of 1/2
Even though I had several more steps that I wanted to teach students, I tried to provide one step at a time in order to ensure undestanding. So, at this point, I asked students to return to their desks to represent, capture, and insert photos of representations of 1/2 using appropriate tools.
I asked students to join together on the carpet again so that we could discuss the next step. First, I projected the class presentation so that we could look at the many ways that students chose to represent 1/2. Then, I asked: How many halves are in a whole? Students responded, "2!" Who can tell me how to represent this with an equation? Following a brief discussion and teacher guidance, we came up with two ways: 1/2 + 1/2 = 2/2 = 1 and 1/2 x 2 = 1. I then modeled how to use arrows and text boxes to label how many halves are in a whole using one of the above equations: Photo & Equation.
Students quickly returned to their desks to label 1/2, label the whole, and to explain how many halves are in a whole using an equation. Here are several examples of students adding labels to explain their thinking:
Next, I asked students to join me once more on the carpet. To provide students with an opportunity to engage in Math Practice 3 (Construct viable arguments and critique the reasoning of others), I wanted to teach students How to Insert a Comment on other students' slides. To ensure comments were helpful and based on math, I took the time to teach students to make mathematical, thoughtful, respectful, and helpful comments. I shared the following anchor chart to help teach how to collaborate effectively: Math Talk Anchor Chart.
Just as before, I introduced the Exceptional Collaborator Award and randomly chose a student to watch for a student making mathematical, thoughtful, respectful, and helpful comments. Here's a copy of the Exceptional Collaborator Award.
Again, students were excited to complete the assigned task! They couldn't wait to comment on others' work and to get comments as well! Here are a few examples of the collaboration that took place:
Half Dollar Comments: On this slide, you'll see where one student commented, "Can you explain how this is a half?" Then the student who created the slide replied back, "It is half because it is a half dollar."
Unifix Cubes Comments: I enjoyed reading these comments as well. One student asked, "How did you think of that?" Then, the student replied, "I got this because there are 10 cubes and 5 of each color and 5 is half of 10." What great conversations!
Ruler Comments: This was one of my favorites! Some students were confused and commented, "I kind disagree because a ruler is 12 inches long and you marked 5 inches." The student then clarified by adding more labels and replying, "I stopped at ten because I forgot that 12 could be split in half."
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! 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 the fraction model or computer up while the other student took a picture. Here's an example of how an extra hand can be helpful: Taking Pictures.
To provide students with more practice representing unit fractions, I explained: Today, you get to continue on by representing other unit fractions, 1/3 and 1/4 using the next two shared presentations (Examples of 1:3 and Examples of 1:4). Once you are done representing and explaining the number of unit fractions in a whole, please take the time to comment on at least two other students' work before moving on to the next presentation.
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).
- How did you choose this tool?
- Is this tool working out for you?
- Have you run into any problems with this tool?
- Can you show me 1/3?
- How many units of 1/4 will it take to get to a whole?
- How do you know?
- Where is one whole?
- Can you explain your thinking?
- Which comment was the most helpful? Why?
Several amazing conferences took place today and there were many opportunities for learning. For example, I found a few students representing 1/3 by showing 1 yellow cube and 3 red cubes. This was a great opportunity to teach the meaning of the numerator and the denominator. I would begin by asking: Can you show me 1/2? This is because students understand 1/2 and it is important to link what they know with more complex situations.
During one conference, Playing Cards, a student had laid out 8 cards in 4 even piles. To encourage him to explain his thinking, I said: Show me 1/4.... How do you know that's 1/4? The student then took away 4 cards, leaving him with one card in each pile, almost as if this was a more sensible representation. I then asked: If you put those back down, how many groups do you have? He said, "Four groups." Pointing to one group of 2 out of 8, I asked: So is this equal to 1/4? He smiled and said, "Yeah." I wanted to push his thinking a bit more so I asked: What if you had 100 kings, 100 jokers, 100 queens, and 100 jacks? Would you still have one fourth? He replied, "Yes, because it is still one group of four."
Also in the above video, you'll see some students taking pictures of fraction representations, some listening to my conference with this student, and others are finding new tools to represent their thinking. One student asked if he could use our wiggle cushions to represent 1/5. Without my knowing, he sat on all five wiggle cushions in the back of the room, trying to be silly. It was wonderful to see that all my other students were so engaged that it didn't even phase them and it was also great to see this student get right back on task without needing teacher guidance!
Here are a few examples of student representations of 1/3:
Here are a few examples of student representations of 1/4: