Session Focus

This session focuses on addition and subtraction with fractions and mixed numbers. You will explore contexts and representations students use to make sense of adding and subtracting fractions, as well as different strategies students use to solve addition and subtraction problems that involve fractions.

Getting Started: Reasoning Through Addition and Subtraction with Fractions

The topic “adding and subtracting fractions” may bring to mind experiences of manipulating numerators and denominators and following procedures with little understanding. Research has found that even in middle school, students who have not developed meaning for fractions may interpret them as two separate whole numbers. For example, in an assessment question from the National Assessment of Educational Progress (NAEP), students were asked to pick the closest estimate for the following problem:

Take a moment to consider what reasoning (correct or incorrect) a student might use to choose each of these responses.

Middle school students most often [55% of 13-year-olds and 36% of 14-year-olds] chose 19 or 21. These students most likely added either only the numerators or only the denominators. They did not recognize that the fractions being added are each close to 1, and thus did not recognize that 2 was the closest estimate.

By using a variety of representations for fractions, students in Investigations’ classrooms learn to visualize the meaning of these numbers, their equivalents, and their relationships to landmarks numbers such as 12 and 1. Representing fractions, comparing and ordering fractions, and finding fraction equivalents lays the conceptual foundation for adding and subtracting fractions.

Closest Estimate

In the Ten-Minute Math activity, Closest Estimate, students choose the closest estimate to the actual answer to a computation problem and explain their reasoning.  Students are also expected to discuss whether the estimate is more or less than the actual answer.

Without solving the problem, find the closest estimate.

Is the actual answer more or less than your estimate? How do you know?

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

• 4 x 6 Grids

Activity 2

• Fraction Clock Student Activity Sheet

Activity 3

• Large Clock Face
• 4 x 6 Grids
• Video Note Taking Template

In this activity, you will learn about different contexts and representations students use to decompose fractions.

Making Sense of Addition and Subtraction with Fractions

In order to add and subtract fractions with understanding, students need to develop a solid understanding of fractional quantities and their relationships and connect the operations of addition and subtraction with whole numbers to their work with fractions.

In the early grades, young students often solve open-ended addition problems that involve finding addends for a given sum, such as:

I have seven vegetables. Some are peas and some are carrots. How many of each could I have?

Problems like these help students learn that a quantity can be broken apart and expressed in different ways using addition (e.g. 1 + 6, 5 + 2, 4 + 3, 3 +4) and explore relationships among the different combinations (e.g., 3 peas + 4 carrots and 4 peas + 3 carrots).

Similarly, using representations to find combinations of fractions that equal whole numbers and fractional quantities supports students in exploring how numbers can be expressed as the sum of fractional parts, and visualize how fractions relate to each other and to a whole.

Different Ways to Make 1

In 3rd grade, students explore different ways to make 1 using fractions. Using the context of cookies, they work to find different ways to make a yellow hexagon (a cookie) using pattern blocks and record their solutions using equations.

Students may observe:

As students consider how these equations model the combinations they are making with the pattern blocks, they begin to see that the same operations they have been using with whole numbers can also be carried out with fractions, and that the same notation can be used to model these operations with fractions.

Different Ways to Make 56

In 4th grade, students extend this work as they find different ways to decompose fractional amounts on 4 x 6 grids.

Using the 4x6 grids, find as many ways to decompose 56 as you can. Show each of your solutions and label each with an addition equation as shown in the example below:

Below are some student observations about decomposing 56:

How does knowing that a fraction can be “broken apart” into smaller fractions in the same way that a whole number can be broken apart into smaller numbers help students as they begin adding and subtracting fractions?

Students make sense of addition and subtraction with fractions by using a variety of representations. In this activity, you will explore two visual models that students use when adding and subtracting fractions. Each model is accompanied by a game that students play to make sense of computation with fractions.

Clock Fractions

Students have been using two models for fractions- area (rectangles) and linear (number lines). In 5th grade, in addition to these two models, students use a clock face as a rotational model for their work with fractions. For example, this clock shows a rotation from 12 to 3.

This rotation could be represented using the following fractions: 312 of the way around the clock, 14 of an hour or 1560 (15 minutes out of 60 minutes in one hour).

Although the rotation of a clock’s hand does sweep out a part of the area of the clock face, it is important for students to think about rotation of the clock hand, rather than the area the hand sweeps out. This idea of rotation lends itself to representing sums that are greater than 1. When the hand goes all the way round the clock, it shows 1212 or 1, and can then continue around, showing more than one whole revolution.

Record the fraction names and fraction equivalencies (halves, thirds, fourths, sixths, and twelfths) for each of the clocks on Clock Fractions.

Roll Around the Clock

After exploring equivalencies, students use the clock face to play Roll Around the Clock. This game provides students with practice finding fractional parts of the rotation around a circle and using a rotation model to add fractions.

Watch a 5th grade teacher introduce this game to her class. As you watch, consider the students’ reasoning as they discuss how to select which fraction cube to roll and determine how far they are from 1.

Play Roll Around the Clock. As you play, think about how you are using the clock face to add the fractions you roll, and how you are determining what you would need to roll to get to 1.

How does using the clock face support students in developing an understanding of addition with fractions?

Fraction Track

In the previous session, you used equivalencies to label fraction tracks. You will now learn about how students use the tracks to play Fraction Track. This game supports students in comparing fractions on a number line, adding and subtracting fractions, and finding combinations of fractions with sums between 0 and 1.

The goal of this game is to win chips by landing them exactly on 1 on the fraction tracks. To start the game, players place one chip on each fraction track at any point less than ¾. They take turns drawing fraction cards that are 1 or less and moving a chip (or multiple chips) the total amount on the card.

In the example below, the card drawn is 810. There are several ways a player can move 810 on the tracks. They can move on one track or on several, however, they cannot “loop back” on the same number line once they have reached one. For example, a player could use part of 8/10 to move on the tenths track and then use the remaining amount of 810 to move on one or more of the other tracks.

Determine the best possible moves this player can make to land as many chips on one as possible using the card drawn. Consider the following questions:

• If a player wanted to get to 1 on the tenth track how much would they need to move? (710 + ? = 1) How much of the eight tenths would they have left? What other moves could they make on other tracks that would get another chip to 1?
• Could the player use the full 810 to move a chip to 1?

Once you have determined the best moves this player could make, write an addition equation that represents those moves.

Play Fraction Track. As you play, consider the strategies you are using to determine the best possible moves. What understandings about fractions, fraction equivalences and the operations are you using to determine which chips to move?

As with many games in Investigations, learning to play Fraction Tracks strategically takes time. Read the 5^thgrade Dialogue Box, Playing Fraction Track to learn more about how students may approach this game as they are first learning to play it.

How can the two different models that you have explored in this activity help students visualize and solve addition and subtraction problems that involve fractions?

In this activity, you will learn about different strategies students use to add and subtract fractions and mixed numbers, solve subtraction problems that involve fractions and watch as students share a variety of solutions to these problems.

The strategies that students develop for adding and subtracting fractions in Investigations are based on their own understanding of fractions and of the operations of addition and subtraction. Even when not using a representation to solve a problem, students draw upon the visual images of these numbers that they have developed by using multiple representations of fractions, as well as their knowledge of equivalents.

Read the 5^th grade Teacher Note, Adding and Subtracting Fractions to learn about some of the different strategies students use to add and subtract fractions.

Solve the following problems. Show your work. You may use the clock face, 4x6 rectangles or a number line if you wish. Try not to use the procedure of multiplying the numerator and the denominator by the same amount to find an equivalent fraction in order to solve these problems.

Watch 5th grade students share their solutions to these problems. After watching each student, reflect on the strategies and representations they use. You may use the video note taking template to jot down your thinking about each clip if you wish.

Reflect on this student’s solution strategy. How does this student use her knowledge of equivalent fractions, the operations of addition and subtraction and the clock face representation to make sense of the problem and solve it?

Reflect on this student’s solution strategy. How does this student use her knowledge of equivalent fractions and the operations of addition and subtraction to make sense of the problem and solve it?

Reflect on this student’s solution strategy. How does this student use her knowledge of equivalent fractions, the relationship between addition and subtraction, and the number line model to make sense of the problem and solve it? What is this student still working to make sense of?

Reflect on this student’s solution strategy. How does this student use her knowledge of equivalent fractions, the operations of addition and subtraction, and an area model to make sense of the problem and solve it?

Reflect on the strategies the students featured in these video clips use. Share your thinking on the Video Forum. You may wish to write about something that stood out to you about a particular strategy one of these students used, compare the strategies shared in these video clips to the way(s) you solved these problems, or you may wish to share some general observations or questions about addition and subtraction with fractions and mixed numbers that relate to these videos.

Based in their understanding of adding and subtracting fractions using representations and, 5th grade students then move towards adding and subtracting fractions without representations. To do this, students often use the strategy of replacing the fractions with equivalent fractions with common denominators and then adding or subtracting the numerators.

Once you have completed the work in this session, go to the Session 4 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 Video Forum. Make sure you have responded to at least 2 of your colleagues on this forum.

• Students use their understanding of fractional quantities and their relationships, and the operations of addition and subtraction to solve addition and subtraction problems with fractions and mixed numbers.


• Representing fractions, comparing and ordering fractions, and finding fraction equivalents lay the conceptual foundation for adding and subtracting fractions.


• Representations and story problem contexts help students make sense of adding and subtracting fractions and mixed numbers.