Session Summary

This session focuses on fraction concepts introduced in the early grades, and the different contexts and representations students use as they develop meaning for fractions.

Early Fraction Concepts

Even before fractions are formally introduced, students begin to informally explore fraction concepts as they engage in activities focused on two-dimensional geometry. As students compose and decompose two-dimensional shapes in different ways, they begin to notice relationships between the different shapes.

Play the first-grade game, Fill the Hexagons. As you play, think about the relationship between the different shapes.

What fractional ideas might students notice as they play this game?

Read the 1st grade Teacher Note Understanding Fractions to learn more about the important fraction concepts students explore in first and second grade.

Below is a list of materials you will need for each activity in this session. You will also find these materials listed at point of use in the session.

Activity 1:

A blank sheet of paper and a pencil

Activity 2:

Optional: A printed copy of the video note taking template and a pencil

Activity 3:

Optional: Pattern blocks, or paper pattern block cut outs

In this activity, you will explore various contexts and activities students use in 1st and 2nd grade to develop an understanding of fractions as equal parts of a whole.

In 1^st and 2^nd grade, students see, describe, and create images that provide opportunities to explore the meaning of fractions as equal parts of a whole. Much of this work builds on and complements students’ previous experience with geometry. As they partition geometric shapes into equal parts, students learn that one half describes one of two equal parts, one third describes one of three equal parts, and that one fourth describes one of four equal parts. They also explore how halves, thirds and fourths of the same shape can look different.

Exploring Halves of 2-Dimensional Shapes

In 1^st grade, students work with images of two-dimensional shapes that have been partitioned into halves and fourths as part of the classroom routine Quick Images: Shapes. During this routine, students briefly see images of a two-dimensional shapes that has been partitioned into halves or fourths and draw what they see. Then, with the image displayed, students describe the image and explain how they remembered it. This routine provides students with an opportunity to develop language to describe equal parts of a whole.

Students begin by looking at circles partitioned into halves during the routine. During subsequent sessions, they look at squares partitioned into halves, circles partitioned into quarters and squares partitioned into fourths.

Complete the following grade 1 Quick Images: Shapes routine by watching the video. Note: Have a pencil and paper available so that you can draw the images displayed during the routine.

Read the 2^nd grade Dialogue Box, Halves of Squares, in which students discuss the shapes displayed in the Quick Images routine that you just completed.

How might viewing and discussing halves of squares extend first grade students’ understanding of the concept of one half from their previous work with circles partitioned into halves?

Halves of Geoblocks: Recognizing that Different Looking Halves of the Same Shape Are Equal

In Grade 2, students continue to partition familiar shapes into equal parts. They also consider whether equal parts of the same whole must look the same. As part of their work with three-dimensional shapes, students encounter the idea that halves of the same shape can look different.

In the Halves of Geoblocks activity, second grade students describe what one half of rectangular prism K, pictured below, would look like. They then work to prove that various other geoblocks make up half of rectangular prism K.

As students work on this activity, they come across different shapes that are half of rectangular prism K, including rectangular prism J, cube D, and triangular prism N. They develop mathematical arguments to prove that these different geoblocks are each half of block K and contain the same amount of wood as one another.

Some second-grade students may have clear arguments about whether some or all of the blocks contain the same amount of wood as one another, while other students may be unsure about whether any of the blocks contain the same amount of wood. Read the 2^nd grade Math Practice Essay to learn more about how students explore these questions during this activity, and how this activity supports student engagement in Math Practice 3: Construct viable arguments and critique the reasoning of others.

In this activity, you will examine a context that second grade students use to identify and describe non-unit fractions—fractions with a numerator greater than 1 and view some video of students working with the context.  

Over the course of grades 1 and 2, students begin to think about how to describe more than one fractional part of a whole using fraction vocabulary and associated notation. Though it is not expected that students will use this fraction notation accurately and consistently until Grade 3, learning about and seeing fraction notation can support students in grades 1 and 2 in developing an understanding of fractions as numbers.

Fraction Flags: A Context for Exploring Non-Unit Fractions

In second grade, students are introduced to Fraction Flags—an activity that develops over several sessions in which they color in the outlines of flags partitioned into halves, thirds and fourths and use words and/or fraction notation to label their work. This activity supports students in identifying halves, thirds, and fourths of regions and in exploring non-unit fractions.

Feature - Math Words and Ideas

Math Words and Ideas is a digital resource which can be used flexibly and that provides a closer look at the ideas and problems students encounter, examples of student work and questions that students and families can talk about together.

Math Words and Ideas

Watch Math Words and Ideas, More Than ¼ to learn more about the Fraction Flags activity. Complete the “Try It” activity at the end of the presentation.

Making sense of non-unit fractions and using fraction notation to describe more than one fractional part of a whole is complex. Watch as second-grade students make sense of and describe non-unit fractions during the Fraction Flags activity.

As you watch each video, consider the following questions:

• What do you think the students understand about fractions and fraction notation?
• What do you think students are working to make sense of?
• What questions might you ask to learn more about their understandings?

You may wish to print and take notes on this template as you consider these questions.

Respond to the video questions above.

Respond to the video questions above.

In the Fraction Flag Video Forum share your responses to the following questions about one of the students in the videos:

• What do you think the student understands about fractions and fraction notation?
• What do you think the student is working to make sense of?
• What questions might you ask to learn more about the student’s understandings?

In this activity, you will consider how students explore the relationship between the size of a fractional amount and the size of a whole using pattern blocks.

Pattern Block Fill-Ins

Throughout the early grades, students engage in activities where they fill in different shapes using pattern blocks. While the focus of this work is on describing, comparing, and naming 2-dimensional shapes, counting shapes, and on finding combinations of shapes that fill a region, students often notice relationships between the pattern blocks, and between the blocks and the shape they are working to fill that help lay the foundation for work with fractions.

For example, when filling in the following shape during the Ways to Fill activity, second grade students might notice that…

What pattern-block relationships do these students notice? How might these relationships support them in making meaning for fractions?

Half Yellow Designs

In third grade, students build on these experiences as they create pattern block designs in which the size of a fractional amount is determined by the size of the whole.

In the activity Half Yellow Designs, students are challenged to create designs using pattern blocks in which exactly half the design is yellow as in the example below:

Create 3 half yellow designs. As you work, think about why making a design that is half yellow is more complex than determining half of an already drawn shape. You may use paper pattern blocks, or digital pattern blocks if you don’t have access to wooden or plastic blocks. You might want to take a picture or screenshot of one of your designs to share in the discussion forum.

Once you have completed the work in this session, go to the Session 1 Discussion Forum. Respond to one of the discussion threads in the forum, or start your own thread about a question, idea, or topic of interest that is related to the content of the session.

After you have posted your comments, take time to read others’ posts and respond to at least 2 of your colleagues.

Return to the Fraction Flag Video Forum. Make sure you have responded to at least 2 of your colleagues on this forum.

As students work with fractions in the early grades, they develop an understanding of the following mathematical ideas:

• Fractional parts are equal. When finding the fraction of a region, like a circle or rectangle, equal parts means equal in area.


• Different fractions can represent the same quantity.


• A region can be divided into non-congruent parts that have equal area.


• The size of a fractional amount is determined by the size of the whole.

These foundational ideas are complex. Students develop an understanding of them by working with multiple contexts and representations over time.