Have you seen the amazing visuals over at www.fractiontalks.com? They’re ideal for any teacher looking to get all students talking about fractions and mathematical reasoning, regardless of ability levels. We used one of the images to introduce fractions to some 3rd grade students. We learned a lot and the students did too! We’d like to share our learning with you. (If you want to read about our experience in 4th grade, click here.)
I hope that there are other 3rd (and 4th) grade teachers out there that might find this analysis useful if they are looking for a low-floor introduction to fractions. While this open and engaging learning opportunity is ideal for students who have no previous knowledge of fractions, it’s appropriate for all students. It’s a low-floor opportunity that allows for students at all levels to participate. It also allows for rich discussion and sense-making because there are many possible representations.
Please feel free to plagiarize and make it your own as you see fit. I’d love to hear from you if you use it so we can get better together. (Special thanks to Nat Banting for curating a bank of such amazing fraction visuals. Check out the one we used here. And a hat tip to Andrew Stadel for his sensational Estimation180 activities. We used this one here.)
Some quick background:
I conducted a lesson inquiry using these visuals with a team of 3rd grade teachers. Each lesson was a unique experiences with its own twists and turns and choices. The lesson pathway below shows the main flow of the learning experience and took about 60 minutes. Further analysis and takeaways can be found at the end of this post, but I’ve tried to embed a lot of our thinking within the flow of the lesson.
This lesson was taught to 3 classrooms of 3rd grade students. Students had little-to-no prior knowledge of fractions. We used this lesson as a way to introduce the concept of a fraction using area models and describing colors in rectangles.
Our objectives (as teachers):
- Students are explaining to each other, questioning each other, engaging discourse. (How well do the visual images and the Fraction Talk work to get kids talking about fractions?)
- Students will show what they know about partitions (halves, thirds, fourths) from 1st and 2nd grade.
Learning objectives (for students):
- Students will communicate their thinking about parts and wholes using words and numbers.
(Note: We did not focus on the objective at the start of the lesson. We used the Into portion of the lesson as an opportunity for students to internalize the objective. We did not refer to the objective until we were ready to conclude the Into and begin the Through part of the lesson.)
To begin today’s lesson, please look at the image on the board. You have a black-and-white copy of this at your desks.
What do you notice about this picture? It’s a pie. And there’s a piece missing.
So you notice that I ate a piece. Hmmmm.
How many pieces do you think are left?
Andrew’s image allows for a low-floor entry point in to the lesson. We wanted to activate and engage student thinking about parts and wholes. We handed out small paper copies of the image so that students could draw on the paper.
We gave students about 3-5 minutes of work time. We anticipated that they would try to cut up the rest of the pie to estimate how many pieces were left.
So, what do you think? How many pieces do you think are left and how do you know?
We focused on the thinking, not the answer. Most students thought there were 4 or 5 pieces left, but we never concerned ourselves with the accuracy of those answers. We wanted students to have a chance to talk about parts and wholes, to internalize the objective. We never told students the “right” answer. Instead, we concluded the introduction of the lesson by pointing to the objective and stating:
So that’s what we’re doing today. You just talked about how many parts make up the whole. For the rest of this lesson, you will communicate your thinking about parts and wholes using words and numbers.
Through (part 1):
The rest of the lesson hinges around this image:
Please look at the image on the board. You’re getting a copy handed to you.
We printed up color copies of the images and had students slide the image into clear plastic sleeves. Later in the lesson students drew their rectangles and showed their thinking on the sleeves using dry-erase markers. Teachers were planning on reusing this image as part of lessons moving forward.
What do you see? Describe the image to your partner.
Squares! Tiles! Brown, Blue, White! Patterns!
I’m going to draw a shape on this picture.
What kind of shape is this? Rectangle.
How many tiles make up this rectangle? 6. Of these tiles, how many are brown? 3.
So, you’re telling me that 3 out of 6 are Brown.
I scribed this sentence clearly on the board: “3 out of 6 are Brown.”
Well, math has a tool that allows us to talk about “3 out of 6”. We can use fractions to describe parts and wholes. You’ll learn more about fractions later, but for now, let me show you how I can say “3 out of 6” using a fraction.
I then scribed 3/6 on the board. Next to it I wrote “three sixths”.
We did not want to get bogged down in vocabulary and definitions. We wanted students to have a positive and fun way to explore fractions for the first time.
I then erased my writing and posted two structures students could use for the rest of the lesson. Students could use either words or fractions in their explanation. I used both orally in my instruction, but mostly wrote answers in fraction from.
This allowed us to check for understanding and for students to practice using the structures to communicate their answers.
Through (part 2):
The rest of this lesson centers around 4 prompts using this image. Teachers will use this image throughout the next few weeks as a way to have students work on fluency of fractions using area models.
Now, draw a different rectangle that is made up of 6 squares like this one. And use numbers to describe how much of it is blue. Write it on the white part of your clear sheet.
We monitored student learning and listened to their thinking. After about 3-5 minutes, we selected a few student work samples to share. Students evaluated the accuracy of answers. We also looked for any students that drew a 6×1 rectangle and described it. If we didn’t see one, we created an example and had students discuss.
I quickly asked students to share their fractions and scribed them down.
What do you notice about all of these fractions? What do they have in common? The bottom number is always 6. (We did not mention the word denominator. They have weeks to learn the vocabulary. For now, we wanted students to use the language they wanted to use.)
Discuss with your partner. Why do all of these fractions have 6 in bottom? Because you asked us to draw a rectangle that has a size of 6.
So, you’re telling me that the bottom number tells me how many parts make up the whole? Yes.
Boom. They tell me an informal way to think about the denominator without ever being told. We scribed this down on chart paper for use in future lessons. We’ll revise the language as their understanding of fractions becomes more formal.
For now, we wanted to keep the learning about fractions contextual and visual. To push this thinking forward, I wrote 0/6 on the board.
Let’s say a student said that “0/6ths” of a rectangle was blue. Is this possible? What’s going on here? Discuss.
This allows students to see that the numerator represents the number of blues in a rectangle. We did not use the word numerator. Initially, students said it wasn’t possible, but we let that simmer for a moment. Then students started to look at the diagram to see if there were rectangles with zero blues. (There are a few.)
What does the 0 in the top of this fraction mean about the rectangle? There are no blues.
Now, draw a rectangle that contains exactly 2 blue squares. It can be any size you’d like, but it must have exactly 2 blues. Use numbers or words to describe how much of it is blue. Write it on the white part of your clear sheet.
We monitored student learning and listened to their thinking. After about 3-5 minutes, we selected a few student work samples to share. Students evaluated the accuracy of answers. I quickly asked students to share their fractions and scribed them down.
What do you notice about all of these fractions? What do they have in common? The top number is always 2. (We did not mention the word numerator.)
Discuss with your partner. Why do all of these fractions have 2 on top? Because you asked us to draw a rectangle that had exactly 2 blues.
So, you’re telling me that the top number tells me how many blue parts I have? Yes.
Let’s say a student said that “2/2” of a rectangle was blue. Is this possible? What’s going on here? Discuss.
In one class, a few students found this rectangle and we used it as a part of the discussion.
What is true about a rectangle that is ‘two halves’ blue? It’s all blue.
So you’re telling me that 2 out of 2 represents the whole thing?
Now, draw a rectangle of any size. Use numbers to describe how much of it is blue. Write it on the white part of your clear sheet.
Here is some of what we saw.
Now draw a rectangle that is half blue. Explain how you know that half of it is blue.
And here’s one of the reasons why I love this image in this lesson: The flow allows students to go from no understanding of fractions to exploring equivalent fractions in about 45 minutes. It’s such a natural and low-floor entry into a variety of fraction concepts and skills.
We let students work on this for some time. Students were encouraged to find more than one and describe it. We also encouraged them to ask each other if answers were “right” or “wrong.”
Here’s an example of some of what we saw:
We used this last prompt to begin our closure for the middle part of this lesson.
Who would like to share with me their answers for rectangles that are half blue?
I scribed all answers on the board regardless if they’re right or not. For example:
Hmmmm. I think some of these fractions are imposters! They don’t represent rectangles that are a half blue. Talk with your neighbor. Which fraction might be an imposter? Which fraction do you know is true?
Then we facilitated a discussion letting students express reasoning about whether a fraction was an imposter or not. Some students contextualized the numerators and denominators: “Six out of six means the whole thing is blue!” “Two-ninths is way less than half blue.” Some students used visual examples in the tile diagram.
We wanted to leave some tension in the air at the end of the exploration, a chance for students to continue to ponder.
There was still some unresolved debate about whether some of the fractions were imposters or not. In the example above, the verdict on 3/6ths was still out. We didn’t rush to an answer. Instead, we invited students to keep thinking about it and talk about it with each other.
Reflections, Takeaways, Analysis:
At the end of the day, we reflected on our objectives and student learning. Some things we noticed and wondered and a few professional goals moving forward.
- We heard a lot of peer-to-peer discussion because it was an easy low-floor safe conceptual way to introduce fractions.
- We could have added sentence frames in text below their image to help support their thinking.
- The first two prompts allow students to make quick contextual meaning of numerators and denominators.
- The openness in the questions/prompts creates a variety of correct answers that students can talk about.
- Students need clear structures and roles to maximize the value of discussion.
- Think time: It’s important!
- Having visuals that they can draw and write on is really important!
- Showing the visuals increased student interest and engagement compared to working out of the textbook.
As always, feedback and comments are welcome. What inspired you? What opportunities did we miss?
Help us get better together.