Is 4 9 Greater Than 1 3? The 139 Latest Answer

Question:

How many 1/3 cups do you need to make 1/2 cup?

Divide fractions:

When we have a number that is divided by a fraction, we need to multiply by the inverse of the fraction. This means that when we divide by 2/7, we are actually multiplying by 7/2 to get our result. The product of the multiplication can then often be reduced by factoring both numbers.

Answer and explanation: 1

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I recently read the book Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense by Julie McNamara and Meghan Shaughnessy.

I posted the following image on Twitter while reading during my daughter’s swim class.

My colleague Hedge replied that he was challenged by a middle school teacher on this very subject.

I let them know that a few years ago, when I was a digital curriculum developer, I was also challenged for this idea. The argument I heard at the time was that using context to check the correctness of fractional comparisons would be against the fact that fractions are numbers. Therefore, 1/2 is always greater than 1/3, regardless of context. I wondered about that at the time, but I still felt it was important to show context.

Jump ahead and I’ve been thinking about this idea all day. I think I finally understand why we need to be careful about what we say about the role of context when comparing fractions. I may be completely off the mark, but I’ll still share my thoughts and let you decide in the comments whether you want to challenge my thoughts or share an alternative point of view.

Let’s start with whole numbers. If I told you to compare 3 and 6, you would probably say “3 is less than 6” or “6 is greater than 3”. This is how the numbers 3 and 6 are related.

Now what if I showed you these two pictures of 3 and 6: (As illustrated by my daughter’s toys.)

Technically the 3 dolls are bigger and therefore make up more stuff, but does that really mean that 3 is now bigger than 6? In the end, my daughter has fewer dolls (3) than figures (6). The context does not fundamentally change the relationship between the numbers 3 and 6.

In this case I don’t even know how to justify that she has more when I refer to the dolls. Sure, they’re bigger, but she might prefer to have more things to play with, and chooses the 6 figures even though they’re smaller overall.

Let’s continue by looking at this from a fractional perspective. Now I take 1/3 of the dolls and 1/2 of the figures.

Consistent with the idea that context should determine when one number is larger than another, I should be confident that 1/3 of the puppets is larger than 1/2 of the figures because 1 puppet is so much larger than the 3 characters. Oh wait, or should I be thinking that 1/2 of the figures is larger than 1/3 of the dolls because I ended up with 3 figures, which is a larger number of things than 1 doll? It’s not that clear cut, even if I try to let the context dictate how to interpret the fractions.

It boils down to breaks representing a relationship. When I think about the relationships each fraction represents, 1/2 is always greater than 1/3 no matter how I try to rotate it. Looking back at my examples, taking 1/2 of the figurine group means I’m taking a larger proportion of that group (this whole) than if I take 1/3 of the puppet group (a different whole, but still a whole) . The size of things in my group (total) doesn’t matter because the relationship represented by 1/2 is larger than the relationship represented by 1/3.

Does this mean that we should ignore contexts altogether? no There’s still a lot of debate about who ate more pizza when one person eats half of a small pizza and another person eats a third of a large pizza. Context is still interesting to discuss and helps students use math to interpret the world around them. However, if our goal is to compare fractions, then 1/2 is greater than 1/3 every time.

That’s the argument I came up with today, trying to understand the criticism I’ve heard. Now that you’ve read it, what do you think?

Lesson 2: Compare and reduce fractions

/en/factions/introduction-to-factions/content/

compare fractions

In Introduction to Fractions, we learned that fractions are a way of representing a part of something. Fractions are useful because they tell us exactly how much we have of something. Some fractions are larger than others. For example, which is larger: 6/8ths of a pizza or 7/8ths of a pizza?

In this image we can see that 7/8 is larger. The figure makes it easy to compare these fractions. But how could we have done that without the pictures?

Click through the slideshow to learn how to compare fractions.

We have already seen that fractions have two parts.

One part is the top number or numerator.

The other is the bottom number or denominator.

The denominator tells us how many parts a whole has.

The counter tells us how many of these parts we have.

When fractions have the same denominator, it means they split into the same number of parts.

This means we can compare these fractions just by looking at the numerator.

Here 5 is more than 4…

Here 5 is more than 4… so we can say that 5/6 is more than 4/6.

Let’s look at another example. Which of these is bigger: 2/8 or 6/8?

If you thought 6/8 was bigger, you were right!

Both fractions have the same denominator.

So we compared the counters. 6 is greater than 2, so 6/8 is more than 2/8.

As you’ve seen, when two or more fractions have the same denominator, you can compare them by looking at their numerators. As you can see below, 3/4 is larger than 1/4. The larger the numerator, the larger the fraction.

Comparing fractions with different denominators

On the previous page we compared fractions that have the same lower numbers or denominators. But you know that fractions can have any number as a denominator. What if you need to compare fractions with different lower numbers?

For example, which of these is larger: 2/3 or 1/5? It’s hard to tell just by looking at her. After all, 2 is greater than 1, but the denominators are not equal.

If you look at the picture, the difference becomes clear: 2/3 is larger than 1/5. With an illustration it was easy to compare these fractions, but how could we have done that without the picture?

Click through the slideshow to learn how to compare fractions with different denominators.

Let’s compare these fractions: 5/8 and 4/6.

Before we compare them, we need to change both fractions so that they have the same denominator, or bottom number.

First we find the smallest number that can be divided by both denominators. This is what we call the lowest common denominator.

Our first step is to find numbers that are divisible by 8.

Using a multiplication table makes this easy. All numbers in the 8-series can be divided by 8 without a remainder.

Now let’s look at our second denominator: 6.

We can use the multiplication table again. All numbers in the 6-series can be divided by 6 without a remainder.

Let’s compare the two lines. It looks like there are some numbers that are divisible by both 6 and 8.

24 is the smallest number that occurs in both series, so the lowest common denominator.

Now let’s change our fractions so that they both have the same denominator: 24.

To do this, we need to change the numerators in the same way we changed the denominators.

Let’s look at 5/8 again. To change the denominator to 24…

Let’s look at 5/8 again. To change the denominator to 24…we had to multiply 8 by 3.

Since we multiplied the denominator by 3, we also multiply the numerator, or top number, by 3.

5 times 3 equals 15. So we changed 5/8 to 15/24.

We can because every number over itself is equal to 1.

So if we multiply 5/8 by 3/3…

So when we multiply 5/8 by 3/3, we’re really multiplying 5/8 by 1.

Since every number times 1 is equal to itself…

Since any number times 1 equals itself, we can say that 5/8 equals 15/24.

Now let’s do the same with our other fraction: 4/6. We also changed its denominator to 24.

Our old denominator was 6. To get 24, we multiplied 6 by 4.

So we also multiply the numerator by 4.

4 times 4 is 16. So 4/6 equals 16/24.

Now that the denominators are the same, we can compare the two fractions by looking at their numerators.

16/24 is greater than 15/24…

16/24 is greater than 15/24… so 4/6 is greater than 5/8.

reduce fractions

Which of these is bigger: 4/8 or 1/2?

If you did the math or just looked at the picture you might have been able to tell that they are the same. In other words, 4/8 and 1/2 mean the same thing, even though they’re spelled differently.

If 4/8 means the same as 1/2, why not just call it that? One half is easier to say than four eighths, and it’s also easier for most people to understand. After all, when dining out with a friend, split the bill in half, not eighths.

If you write 4/8 as 1/2, reduce it. When we shorten a fraction, we write it in a simpler form. Reduced fractions are always equal to the original fraction.

We have already reduced 4/8 to 1/2. If you look at the examples below, you can see that other numbers can be reduced to 1/2 as well. These fractions are all the same.

5/10 = 1/2

22.11 = 1/2

36/72 = 1/2

These fractions have also all been reduced to a simpler form.

4/12 = 1/3

14/21 = 2/3

35/50 = 7/10

Click through the slideshow to learn how to reduce fractions by dividing.

Let’s try to reduce this fraction: 16/20.

Since the numerator and denominator are even numbers, you can divide them by 2 to shorten the fraction.

First we divide the numerator by 2. 16 divided by 2 equals 8.

Next we divide the denominator by 2. 20 divided by 2 equals 10.

We reduced 16/20 to 8/10. We could also say that 16/20 equals 8/10.

If the numerator and denominator are still divisible by 2, we can further reduce the fraction.

8 divided by 2 is 4.

10 divided by 2 is 5.

Since there is no number by which 4 and 5 can be divided, we cannot reduce 4/5 any further.

This means that 4/5 is the simplest form of 16/20.

Let’s try reducing another fraction: 6/9.

While the numerator is even, the denominator is an odd number, so we can’t reduce by dividing by 2.

Instead, we need to find a number by which 6 and 9 can be divided. A multiplication table makes this number easy to find.

Let’s find 6 and 9 in the same row. As you can see, 6 and 9 can both be divided by 1 and 3.

Dividing by 1 doesn’t change these fractions, so we use the largest number that 6 and 9 can be divided by.

That’s 3. This is called the greatest common divisor, or gcd. (It can also be called the greatest common factor or GCF.)

3 is the gcd of 6 and 9 because it’s the largest number they can be divided by.

So we divide the numerator by 3. 6 divided by 3 is 2.

Then we divide the denominator by 3. 9 divided by 3 is 3.

Now we’ve reduced 6/9 to 2/3, which is the simplest form. We could also say that 6/9 equals 2/3.

Irreducible fractions

Not all fractions can be reduced. Some are already as simple as they can be. For example, you can’t cancel 1/2 because there is no number other than 1 that can divide both 1 and 2. (For this reason, you cannot reduce a fraction that has a numerator of 1.)

Also, some fractions with larger numbers cannot be reduced. For example, 17/36 cannot be reduced because there is no number by which both 17 and 36 can be divided. If you can’t find common multiples for the numbers in a fraction, it’s probably irreducible.

Try this!

Reduce each fraction to its simplest form.

Mixed numbers and improper fractions

In the previous lesson, you learned about mixed numbers. A mixed number has both a fraction and a whole number. An example is 1 2/3. You would read 1 2/3 like this: one and two thirds.

Another spelling would be 5/3 or five thirds. These two numbers look different but are actually the same. 5/3 is an improper fraction. It just means that the numerator is greater than the denominator.

There are times when you might prefer to use an improper fraction instead of a mixed number. It’s easy to convert a mixed number to an improper fraction. Let’s learn how:

Let’s convert 1 1/4 to an improper fraction.

First we need to figure out how many parts make up the whole number: 1 in this example.

To do this, we multiply the whole number , 1, by the denominator 4.

1 times 4 equals 4.

Now let’s add this number, 4, to the numerator 1.

4 plus 1 equals 5.

The denominator stays the same.

Our improper fraction is 5/4, or five quarters. So we could say that 1 1/4 equals 5/4.

This means that in 1 1/4 there are five 1/4s.

Let’s convert another mixed number: 2 2/5.

First we multiply the whole number by the denominator. 2 times 5 equals 10.

Next we add 10 to the numerator. 10 plus 2 equals 12.

The denominator remains the same as always.

So 2 2/5 equals 12/5.

Try this!

Try converting these mixed numbers to improper fractions.

Convert improper fractions to mixed numbers

Improper fractions are useful for math problems that use fractions, as you’ll learn later. However, they are also more difficult to read and understand than mixed numbers. For example, it is much easier to visualize 2 4/7 in your head than 18/7.

Click through the slideshow to learn how to convert an improper fraction to a mixed number.

Let’s turn 10/4 into a mixed number.

You can think of every fraction as a division problem. Just treat the dash between the numbers like a division sign (/).

So we divide the numerator 10 by the denominator 4.

10 divided by 4 equals 2…

10 divided by 4 gives 2… with remainder 2.

The answer 2 becomes our integer because 10 can be divided by 4 twice.

And the remainder, 2, becomes the numerator of the fraction because we have 2 parts left.

The denominator stays the same.

So 10/4 equals 2 2/4.

Let’s try another example: 33/3.

We divide the numerator 33 by the denominator 3.

33 divided by 3…

33 divided by 3…is 11 with no remainder.

The answer, 11, becomes our integer.

There’s no remainder, so we can see that our improper fraction was actually an integer. 33/3 equals 11.

Try this!

Try converting these improper fractions into mixed numbers.

/en/fractions/add-and-subtract-fractions/contents/

When comparing two fractions, the denominator of the first fraction must be equal to the denominator of the second fraction. In this case, the two denominators are different, resulting in fractions and unequals. The first step is to find the lowest common denominator (LCD) for both fractions and .

A fraction represents a part of a whole. When working with fractions, you may need to determine which fractions are larger or larger than other fractions.

No, a third is NOT more than half. Half is more than a third.

Since the two fractions 1/3 and 1/2 have the same numerator (remember the numerator is the number above them), they are easy to compare. If two fractions have the same numerator, the fraction with the smaller denominator is the larger fraction. Therefore 1/2 is greater than 1/3.

Let’s use pizzas to visualize the problem. If you cut a pizza in half, then the 1/2 fraction would represent exactly half the pizza. If you cut the pizza into thirds, you get three slices of pizza. As you can see in the picture, half of the pizza is a larger slice than a third of the pizza.

What is fraction ordering?

When ordering fractions, we arrange a set of fractions starting with the smallest, followed by the next smallest, and so on. This is called ascending order.

To do this, we rewrite the fractions so that they have the same denominator, which we can then compare.

We can order any type of fraction including proper fractions, improper fractions and mixed numbers.

E.g.

Write these fractions in ascending order:

In ascending order:

If we want to compare two fractions, we must have the same denominator. So we need the LCM (least common multiple) of #4# and #3#, which is #12#. So we need to multiply 4 by #3# and 3 by #4#

Remember that whatever you do below must be done above.

For #3/4# we multiply 3 down (denominator), so we need to multiply #3# up (numerator).

For #2/3# we multiply #4# down, so we need to multiply #4# up.

#(3xx3)/(4xx3)=9/12# and #(2xx4)/(3xx4)=8/12#

Now just look at the counts of the two answers. #9# is bigger than #8#, so #9/12# is bigger! #9/12# was originally #3/4# .

So #3/4# is greater than #2/3# .

I hope you understood this and my source is my knowledge!

I recently read the book Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense by Julie McNamara and Meghan Shaughnessy.

I posted the following image on Twitter while reading during my daughter’s swim class.

My colleague Hedge replied that he was challenged by a middle school teacher on this very subject.

I let them know that a few years ago, when I was a digital curriculum developer, I was also challenged for this idea. The argument I heard at the time was that using context to check the correctness of fractional comparisons would be against the fact that fractions are numbers. Therefore, 1/2 is always greater than 1/3, regardless of context. I wondered about that at the time, but I still felt it was important to show context.

Jump ahead and I’ve been thinking about this idea all day. I think I finally understand why we need to be careful about what we say about the role of context when comparing fractions. I may be completely off the mark, but I’ll still share my thoughts and let you decide in the comments whether you want to challenge my thoughts or share an alternative point of view.

Let’s start with whole numbers. If I told you to compare 3 and 6, you would probably say “3 is less than 6” or “6 is greater than 3”. This is how the numbers 3 and 6 are related.

Now what if I showed you these two pictures of 3 and 6: (As illustrated by my daughter’s toys.)

Technically the 3 dolls are bigger and therefore make up more stuff, but does that really mean that 3 is now bigger than 6? In the end, my daughter has fewer dolls (3) than figures (6). The context does not fundamentally change the relationship between the numbers 3 and 6.

In this case I don’t even know how to justify that she has more when I refer to the dolls. Sure, they’re bigger, but she might prefer to have more things to play with, and chooses the 6 figures even though they’re smaller overall.

Let’s continue by looking at this from a fractional perspective. Now I take 1/3 of the dolls and 1/2 of the figures.

Consistent with the idea that context should determine when one number is larger than another, I should be confident that 1/3 of the puppets is larger than 1/2 of the figures because 1 puppet is so much larger than the 3 characters. Oh wait, or should I be thinking that 1/2 of the figures is larger than 1/3 of the dolls because I ended up with 3 figures, which is a larger number of things than 1 doll? It’s not that clear cut, even if I try to let the context dictate how to interpret the fractions.

It boils down to breaks representing a relationship. When I think about the relationships each fraction represents, 1/2 is always greater than 1/3 no matter how I try to rotate it. Looking back at my examples, taking 1/2 of the figurine group means I’m taking a larger proportion of that group (this whole) than if I take 1/3 of the puppet group (a different whole, but still a whole) . The size of things in my group (total) doesn’t matter because the relationship represented by 1/2 is larger than the relationship represented by 1/3.

Does this mean that we should ignore contexts altogether? no There’s still a lot of debate about who ate more pizza when one person eats half of a small pizza and another person eats a third of a large pizza. Context is still interesting to discuss and helps students use math to interpret the world around them. However, if our goal is to compare fractions, then 1/2 is greater than 1/3 every time.

That’s the argument I came up with today, trying to understand the criticism I’ve heard. Now that you’ve read it, what do you think?

Lesson 2: Compare and reduce fractions

/en/factions/introduction-to-factions/content/

compare fractions

In Introduction to Fractions, we learned that fractions are a way of representing a part of something. Fractions are useful because they tell us exactly how much we have of something. Some fractions are larger than others. For example, which is larger: 6/8ths of a pizza or 7/8ths of a pizza?

In this image we can see that 7/8 is larger. The figure makes it easy to compare these fractions. But how could we have done that without the pictures?

Click through the slideshow to learn how to compare fractions.

We have already seen that fractions have two parts.

One part is the top number or numerator.

The other is the bottom number or denominator.

The denominator tells us how many parts a whole has.

The counter tells us how many of these parts we have.

When fractions have the same denominator, it means they split into the same number of parts.

This means we can compare these fractions just by looking at the numerator.

Here 5 is more than 4…

Here 5 is more than 4… so we can say that 5/6 is more than 4/6.

Let’s look at another example. Which of these is bigger: 2/8 or 6/8?

If you thought 6/8 was bigger, you were right!

Both fractions have the same denominator.

So we compared the counters. 6 is greater than 2, so 6/8 is more than 2/8.

As you’ve seen, when two or more fractions have the same denominator, you can compare them by looking at their numerators. As you can see below, 3/4 is larger than 1/4. The larger the numerator, the larger the fraction.

Comparing fractions with different denominators

On the previous page we compared fractions that have the same lower numbers or denominators. But you know that fractions can have any number as a denominator. What if you need to compare fractions with different lower numbers?

For example, which of these is larger: 2/3 or 1/5? It’s hard to tell just by looking at her. After all, 2 is greater than 1, but the denominators are not equal.

If you look at the picture, the difference becomes clear: 2/3 is larger than 1/5. With an illustration it was easy to compare these fractions, but how could we have done that without the picture?

Click through the slideshow to learn how to compare fractions with different denominators.

Let’s compare these fractions: 5/8 and 4/6.

Before we compare them, we need to change both fractions so that they have the same denominator, or bottom number.

First we find the smallest number that can be divided by both denominators. This is what we call the lowest common denominator.

Our first step is to find numbers that are divisible by 8.

Using a multiplication table makes this easy. All numbers in the 8-series can be divided by 8 without a remainder.

Now let’s look at our second denominator: 6.

We can use the multiplication table again. All numbers in the 6-series can be divided by 6 without a remainder.

Let’s compare the two lines. It looks like there are some numbers that are divisible by both 6 and 8.

24 is the smallest number that occurs in both series, so the lowest common denominator.

Now let’s change our fractions so that they both have the same denominator: 24.

To do this, we need to change the numerators in the same way we changed the denominators.

Let’s look at 5/8 again. To change the denominator to 24…

Let’s look at 5/8 again. To change the denominator to 24…we had to multiply 8 by 3.

Since we multiplied the denominator by 3, we also multiply the numerator, or top number, by 3.

5 times 3 equals 15. So we changed 5/8 to 15/24.

We can because every number over itself is equal to 1.

So if we multiply 5/8 by 3/3…

So when we multiply 5/8 by 3/3, we’re really multiplying 5/8 by 1.

Since every number times 1 is equal to itself…

Since any number times 1 equals itself, we can say that 5/8 equals 15/24.

Now let’s do the same with our other fraction: 4/6. We also changed its denominator to 24.

Our old denominator was 6. To get 24, we multiplied 6 by 4.

So we also multiply the numerator by 4.

4 times 4 is 16. So 4/6 equals 16/24.

Now that the denominators are the same, we can compare the two fractions by looking at their numerators.

16/24 is greater than 15/24…

16/24 is greater than 15/24… so 4/6 is greater than 5/8.

reduce fractions

Which of these is bigger: 4/8 or 1/2?

If you did the math or just looked at the picture you might have been able to tell that they are the same. In other words, 4/8 and 1/2 mean the same thing, even though they’re spelled differently.

If 4/8 means the same as 1/2, why not just call it that? One half is easier to say than four eighths, and it’s also easier for most people to understand. After all, when dining out with a friend, split the bill in half, not eighths.

If you write 4/8 as 1/2, reduce it. When we shorten a fraction, we write it in a simpler form. Reduced fractions are always equal to the original fraction.

We have already reduced 4/8 to 1/2. If you look at the examples below, you can see that other numbers can be reduced to 1/2 as well. These fractions are all the same.

5/10 = 1/2

22.11 = 1/2

36/72 = 1/2

These fractions have also all been reduced to a simpler form.

4/12 = 1/3

14/21 = 2/3

35/50 = 7/10

Click through the slideshow to learn how to reduce fractions by dividing.

Let’s try to reduce this fraction: 16/20.

Since the numerator and denominator are even numbers, you can divide them by 2 to shorten the fraction.

First we divide the numerator by 2. 16 divided by 2 equals 8.

Next we divide the denominator by 2. 20 divided by 2 equals 10.

We reduced 16/20 to 8/10. We could also say that 16/20 equals 8/10.

If the numerator and denominator are still divisible by 2, we can further reduce the fraction.

8 divided by 2 is 4.

10 divided by 2 is 5.

Since there is no number by which 4 and 5 can be divided, we cannot reduce 4/5 any further.

This means that 4/5 is the simplest form of 16/20.

Let’s try reducing another fraction: 6/9.

While the numerator is even, the denominator is an odd number, so we can’t reduce by dividing by 2.

Instead, we need to find a number by which 6 and 9 can be divided. A multiplication table makes this number easy to find.

Let’s find 6 and 9 in the same row. As you can see, 6 and 9 can both be divided by 1 and 3.

Dividing by 1 doesn’t change these fractions, so we use the largest number that 6 and 9 can be divided by.

That’s 3. This is called the greatest common divisor, or gcd. (It can also be called the greatest common factor or GCF.)

3 is the gcd of 6 and 9 because it’s the largest number they can be divided by.

So we divide the numerator by 3. 6 divided by 3 is 2.

Then we divide the denominator by 3. 9 divided by 3 is 3.

Now we’ve reduced 6/9 to 2/3, which is the simplest form. We could also say that 6/9 equals 2/3.

Irreducible fractions

Not all fractions can be reduced. Some are already as simple as they can be. For example, you can’t cancel 1/2 because there is no number other than 1 that can divide both 1 and 2. (For this reason, you cannot reduce a fraction that has a numerator of 1.)

Also, some fractions with larger numbers cannot be reduced. For example, 17/36 cannot be reduced because there is no number by which both 17 and 36 can be divided. If you can’t find common multiples for the numbers in a fraction, it’s probably irreducible.

Try this!

Reduce each fraction to its simplest form.

Mixed numbers and improper fractions

In the previous lesson, you learned about mixed numbers. A mixed number has both a fraction and a whole number. An example is 1 2/3. You would read 1 2/3 like this: one and two thirds.

Another spelling would be 5/3 or five thirds. These two numbers look different but are actually the same. 5/3 is an improper fraction. It just means that the numerator is greater than the denominator.

There are times when you might prefer to use an improper fraction instead of a mixed number. It’s easy to convert a mixed number to an improper fraction. Let’s learn how:

Let’s convert 1 1/4 to an improper fraction.

First we need to figure out how many parts make up the whole number: 1 in this example.

To do this, we multiply the whole number , 1, by the denominator 4.

1 times 4 equals 4.

Now let’s add this number, 4, to the numerator 1.

4 plus 1 equals 5.

The denominator stays the same.

Our improper fraction is 5/4, or five quarters. So we could say that 1 1/4 equals 5/4.

This means that in 1 1/4 there are five 1/4s.

Let’s convert another mixed number: 2 2/5.

First we multiply the whole number by the denominator. 2 times 5 equals 10.

Next we add 10 to the numerator. 10 plus 2 equals 12.

The denominator remains the same as always.

So 2 2/5 equals 12/5.

Try this!

Try converting these mixed numbers to improper fractions.

Convert improper fractions to mixed numbers

Improper fractions are useful for math problems that use fractions, as you’ll learn later. However, they are also more difficult to read and understand than mixed numbers. For example, it is much easier to visualize 2 4/7 in your head than 18/7.

Click through the slideshow to learn how to convert an improper fraction to a mixed number.

Let’s turn 10/4 into a mixed number.

You can think of every fraction as a division problem. Just treat the dash between the numbers like a division sign (/).

So we divide the numerator 10 by the denominator 4.

10 divided by 4 equals 2…

10 divided by 4 gives 2… with remainder 2.

The answer 2 becomes our integer because 10 can be divided by 4 twice.

And the remainder, 2, becomes the numerator of the fraction because we have 2 parts left.

The denominator stays the same.

So 10/4 equals 2 2/4.

Let’s try another example: 33/3.

We divide the numerator 33 by the denominator 3.

33 divided by 3…

33 divided by 3…is 11 with no remainder.

The answer, 11, becomes our integer.

There’s no remainder, so we can see that our improper fraction was actually an integer. 33/3 equals 11.

Try this!

Try converting these improper fractions into mixed numbers.

/en/fractions/add-and-subtract-fractions/contents/

In this post, we will talk about comparing fractions and present some models that might help us in this work, especially for children developing their conceptual understanding of fractions. Before we begin, I should make this disclaimer first: I really love fractions. How really. Love. speak. around. She. I know I’m probably an odd minority, but I’m always fascinated by how children use fractions, build their first understanding of fractions, and eventually learn how to become fluent with fractions. When I tell my prospective teachers about my love for fractions, there’s a clear difference in how we feel:

My students are like:

Student Reaction #3. Student Reaction #2. Student Reaction #1.

It is in order; I still love them (my students and factions).

But back to my original question, posed by my math-savvy cousin this week. Her son, who is in the third grade, brought home a math problem that said, “Compare 2/5 and 2/6. Which one is bigger?” When my cousin raised the issue, her first (and natural) reaction was to say, “Wait! Now, are we asking third graders to find common denominators? Is this a third grade math standard? What’s another possibility to answer that?” My cousin’s question I’ve heard so many parents ask, and it’s perfectly normal; math classes have changed over the years, and it’s always good to pause and say, “Wait , what?” As we unpack this specific problem issue, let’s establish some ground rules:

You have the right to use your existing knowledge of division, fair sharing and any other knowledge related to fractions. If you look at 2/5 and 2/6 and think, “Well, I need to find common ground,” that’s one way to solve this problem. They probably have a different strategy for comparing these fractions that I haven’t talked about explicitly. Cute! (Due to time and space constraints, I only wanted to unpack three models for this blog). We should recognize that our children come into school already with an INCREDIBLE amount of math knowledge and can use that knowledge if we have teachers who are willing to elevate and build on that prior experience. Our kids aren’t born to know the difference between a denominator and a numerator, but they’re certainly willing to talk about dividing things up fairly so everyone gets an equal piece.

When we look at 2/5 and 2/6 to see which is larger or smaller, we have to realize that children might have some assumptions about these fractions given their knowledge of whole numbers:

For example, a child might think that 2/5 and 2/6 are the same (because 2=2). Or that 2/5 is smaller because 5<6. Each of these explanations makes sense when you think about how kids are just learning to see numbers like 2/5 as a complete and singular value, not just 2 and 5 with a line separating them. Don't discard that thinking - it really makes sense for kids! Let's move on to discuss some ways to compare these fractions and draw a conclusion about their relative value, starting with the most complex method some of us have learned in school. Strategy 1: Find common denominators, get equivalent fractions, then compare numerators: As adults, we may find common denominators, find fractions with equivalent values, and then compare the numerators. If we do that, then we can 30.12. with 30.10. to compare. That means the difference between 2/5 and 2/6 is only 2/30... which leaves a really small margin for error, especially when you're trying to draw an accurate surface model. This is a valid strategy, but we know from research that children need images, concrete sets, and/or actions to represent their thinking. #WhereMyCGIFolksAt? Strategy 2: Part-to-whole models, displaying 2/5 and 2/6 This strategy allows a child to draw two rectangles of the same shape and size. To name the fractions 1/5 and 1/6, the wholes should be the same size. Remember that 1/6 of a large pizza is another 1/6 of a small pizza. In order to compare fractions, we have to deal with whether or not the wholes you are cutting these pieces out of are the same size. So the kids just have to cut one whole into 5 equal parts and another into 6 equal parts and then shade two and compare? Sounds easy right? But before you take my word for it, give it a try. It's no joke to draw by hand (WITHOUT A RULER) a rectangle in fifths and sixths while drawing precisely! Super challenging because these factions are very close in value. (I cheated and used the PPT table function). This signals to me that the curriculum (whether intentional or not) may not have provided for children to use a surface model - perhaps too much room for error? One that would have been more obvious might have been 2/10 and 2/100, but who the hell wants to cut a rectangle into hundredths. #BYE #RedDressLadyEmojiWalkoutNow Strategy 3: Use existing knowledge about 1/5 and 1/6 with fractions as a division Children have experience with dividing things up. It's true. They have shared food or toys before and tried to make things "fair" for everyone. While they learn to handle it accurately (sometimes their "half" helping can be quite "generous" compared to mine), this is knowledge that we shouldn't ignore or disregard. When you're dividing objects, you probably won't hear a kid say, "Now I'll do fractions as division, y'all," but that doesn't mean they're not actually doing that math in front of you. You might ask a child, "Which kids would get more of a cookie: a cookie shared by 5 kids, or a cookie of the same size shared by 6 kids?" What would you like to share so you and your friends get the most cookies?” The book The Doorbell Rang is an AMAZING example of this idea. As children model fractions for themselves as division, they can begin to explore what that denominator really means when you have the same numerator. They can also make sense of why a larger denominator (like the 6 in our example) might not imply larger values ​​when you want to share the same object. Moving from whole numbers to fractions is challenging and shouldn't be taken for granted. And I can't stress it enough: We don't have to wait until 4th and 5th grade to teach our students fractions or the concept of fairness. They have a lot of experiences with sharing things that they can use! Once kids know that 1/5 > 1/6, they can use that to see how 2/5 > 2/6 (instead of sharing one cookie, kids now share two cookies). Problems like this lay the foundation of why we can show that 2/5 is greater than 2/6 even though 5 < 6. It depends on the size of the piece (or the number of dividers) (at least in this context and the strategies presented). The important thing about Strategy 3 is that it keeps kids from focusing on what the actual difference is between 2/5 and 2/6 (which is 2/30) and gets them to make sense of what the numbers actually represent is related to their experiences with sharing and fair sharing. Are these the only ways to show a comparison of two fractions? Certainly not! I would highly recommend these books if you want to read about the research (which is accessible to multiple audiences) here and here on how children learn fractions. But you know I'm not just going to leave it like that, right? #BAHAHAHAHAHAAAA What if I presented you with this picture? I divided one rectangle into 5 equal parts and another one into 6 equal parts. I shaded two from each rectangle to represent 2/5 and 2/6. Did I just prove that 2/5 and 2/6 are equal? How might this image mislead children? (The shoe assignment and this image above are my favorite images that have such amazing conversations about fractions.) I'll leave you to think about that for a second. bye references Empson, S.B., & Levi, L. (2011). Extension of children's math: fractions and decimals. Portsmouth, New Hampshire: Heinemann. Hutchins, P (2014). The doorbell rang. New York, New York: HarperCollins. McNamara, J. & Shaughnessy, M.M. (2015). Beyond Pizza and Cake: 10 Essential Strategies to Support Hernia Sense, Grades 3-5. Sausalito, CA: Mathematical Solutions. (This quote refers to the latest edition). Thanks to my cousin for letting me share our conversation with you all!