Lesson on Comparing Fractions: Is 2/3 More Than 1/2?

Is 2/3 More Than 1/2?

Okay, let’s start with the “easy” answer. The answer to "is 2/3 more than 1/2" is yes. The real number $\frac{2}{3}$ is greater than $\frac{1}{2}$. Here are some ways of seeing it:

Extending the Math

But hold up! Why would we publish an entire blog post about a math problem that can be answered with one word? Remember that mathematics is not about “getting the right answer” any more than literacy is about “conjugating verbs.”

Let’s unpack the models shown above. It is true that on a standard number line, $\frac{2}{3}$ is to the right of $\frac{1}{2}$. But what makes that justifiable proof? It is only a convention, after all. There is no law that number lines must go horizontally from left to right. It is also not clear what the fractions mean. Consider this: two-thirds of the population of Connecticut is a lot smaller than one-half of the population of the Texas.

In this blog, I will walk through two different aspects of this question:

1. I will show how fractions are contextual mathematical objects, so there can be instances where $\frac{1}{2}$ is greater than $\frac{2}{3}$.
2. I will explore different ways of comparing fractions so you can teach a lesson to 4th and 5th grade students.

Fractions and Context

If we start with the beginning of the number, it’s obvious that 2 is less than 3, right? Even a statement as obvious as that runs into problems if you end up comparing items that cannot be compared (are 2 cats “less than” 3 dogs?) or are missing important qualifiers (2 meters is not less than 3 inches).

If you asked your students to show why 2 is less than 3, how would students generate their justification? Consider asking students how to show that 2 is less than 3 and comparing students’ representations. Did anyone use a number line? If they drew multiples of an object, how did they make it clear that the objects could be compared?

After discussing the concept of comparing numbers, you may then want to review your students’ understanding of fractions. Prompt your students to illustrate the fraction $\frac{2}{3}$ and spend time comparing representations. How are students’ models similar or different? Did anyone change the numerical representation, for example by writing “0.666…?” Did anyone use a number line?

Classroom Activity: Ask students is 2/3 always, sometimes, or never greater than 1/2? Probe students’ thinking, especially those who say “sometimes.” What are your students already thinking about?

Real-World Fractions Lesson

Every math teacher must occasionally contend with the question, “why does this matter?” There are plenty of places where a deep understanding of comparing fractions can help you in the real world.

• Fractions are found when precise measurements are needed, for example when measuring medicine or baking pastries.
• Combining the correct fractions of ingredients is critical to making all sorts of materials such as slime, cement, and paint.
• Chemists must be extraordinarily precise with their fractions in order to create chemical reactions like testing for a virus.

Not only is fraction arithmetic important in the real world, but so is what a fraction means in the first place. The fraction $\frac{2}{3}$ is greater than $\frac{1}{2}$ when they are fractions of the same thing, but it is important to also understand what the fraction is measuring or how is it is being presented. If a news report claims that $\frac{2}{3}$ of Americans support something, which Americans? How was the poll created? Or consider the claim that one theater is at $\frac{2}{3}$ capacity whereas another is at $\frac{1}{2}$ capacity. Are the theaters the same size?

Let’s get to a lesson for Grades 4–5 that you can implement in your classroom to teach this concept. Begin by having students hunt for fractions in the real world. You can lead them towards particular fractions if you want to connect it to something topical, for example the U.S. Senate needing a supermajority for a vote.

Step 1: Assign students a place to look for a fraction. This can be done in just a few minutes online with direction like “everyone, search for a news article containing the word ‘supermajority.’” This can also be done as a longer assignment, for example, “your homework is to find a fraction in a book or webpage that’s not about math.”

Step 2: Have students independently write down their responses to the following questions. Note that these questions may have multiple correct responses.

• How is the fraction being used in the book or webpage? (for example, “describe a supermajority”)
• Is it being compared to another fraction? If so, what fraction(s)? (for example, “yes, a typical majority”)
• What is the numerator? What is the denominator?
• What is the part? (for example, “senators voting yes”)
• What is the whole? (for example, “all U.S. senators”)

Step 3: Compare students’ responses. Begin by identifying similarities and differences among responses and having students explain their responses. Facilitate a discussion specifically around what assumptions the source makes about the fraction. Some sample questions are provided below.

• Does the fraction represent a large or small number? The magnitude of a fraction is relative. Two-thirds of all senators is a lot when only half the senators are expected to vote yes, but two-thirds is not much when every single senator is expected to vote yes.
• Does the fraction represent an increase, decrease, or neither? The direction of a fraction is also relative—an idea you can connect to negative numbers if students are ready. Comparing a fractional gain to a fractional loss can be different from comparing a whole number gain to a whole number loss. For example, if a sum of money increased by a factor of two-thirds then decreased by a factor of two-thirds, it would not be back where it started. (It would be five-ninths of where it started.)

Differentiating Instruction

The above lesson is flexible. The initial task can be made simpler by directing students to a source where they are likely to see specific fractions that you curate (for example, a news story including “supermajority”). The initial task could also be made more advanced, perhaps by having students look through scientific journals, which are likely to have many fractions that are difficult to understand in context.

For students struggling with why fractions even matter, begin by focusing on the “what.” Each fraction can be modeled as a whole subdivided into parts. For instance, you could use the common model of dividing a circle into equal-sized slices. This approach can also show the importance of context in comparing fractions. While $\frac{2}{3}$ is greater than $\frac{1}{2}$ when the circles are the same size, it may no longer be true if the circles have different radii.

If students draw the circle models on graph paper, you can have them count the number of units in each circle to investigate which is greater. Consider facilitating a discussion around the visual models:

• What are real-world examples of drawing circles on graph paper? (Possible answer: when gardening in a circular plot of land)
• When would one model of fractions be more useful than another? (Possible answer: bar models are better when comparing distance along a road; circle models are better when displaying survey results)

Extending Instruction

For students who are familiar with percent, you can ask students to connect the fractions to percent. It is another way to think about parts and wholes, and it is also a helpful way to think about why as a number on a number line, $\frac{2}{3}$ is greater than $\frac{1}{2}$, as 66.666...% is greater than 50%.

In addition to 0.5 and 50%, the number $\frac{1}{2}$ can be thought of as the ratio of 1:2. From that perspective, it’s not that 2:3 is “greater” than 1:2; rather, it’s that the slope of the relationship $y=\frac{2}{3}x$ is greater than the slope of the relationship $y=\frac{1}{2}x$. Slope is a way to illustrate the connection between the slope of a line and a number placed along a real number line.

In this way, you can introduce ideas of algebra and connect fractions to variables. Have students multiply both $\frac{2}{3}$ and $\frac{1}{2}$ by the same number. What do they observe? Do all students reach the same conclusion?

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Go beyond whether two-thirds is more than half in math class. HMH Into Math is a core curriculum for Grades K–8 that will inspire students to see the value of fractions in their daily lives through real-life activities and lessons.