Fractions in math are hard because of how we think about them. Is there another way to think about fractions? Yes! We can by talking about fractions differently.

A short demonstration on how to use the fraction cards to show dividing

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Title: Mixed Numbers as Improper Fractions - Fast And Easy Math Learning Videos

To change a mixed number into an improper fraction, first, multiply the whole units by the number of parts in each whole unit.
Then add the left over part
Seven and one-third is equal to twenty-two thirds
Here's another one. First, multiply the whole units by the number of parts in each whole unit.
Then add the left over part
Three and two-fifths is equal to seventeen-fifths.
Last one. First, multiply the whole units by the number of parts in each whole unit.
Then add the left over part
One and three-sixths is equal to nine-sixths.

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https://youtu.be/XtiTk1GmHCI

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Title: Dividing Fractions

You know that the multiplication of both whole numbers and fractions can be represented using an array that looks like this. This example shows four-fifths multiplied by two-thirds.
The product of multiplying the fractions is eight-fifteenths.
You also know that division is just another way of writing multiplication when one of the factors is unknown. That’s easy to see using this array. Let’s assume that you know the total size of the array, which is eight-fifteenths, and you know one of the factors in the multiplication, four-fifths.
But you don’t know the other factor, which we’ll now call X.
X is the number you need to multiply by four-fifths to get eight-fifteenths.
We can represent this situation with two different number sentences.
The first number sentence is written as multiplication. We know that four-fifths is multiplied by some number on this side called X.
And the result is eight-fifteenths.
We can also describe exactly the same situation with a division number sentence like this.
We know that we have eight-fifteenths, which is divided by four-fifths.
But we don’t know the length of this side of the array, which is the value for X.
So, these equations tell us exactly the same thing because they represent the same situation shown by the array.
The X in both equations represents the same number – it’s the missing value for the length of one side of the array.
While this array is helpful in seeing what a division problem represents, it’s not very helpful in finding out the number we need on this side, because we don’t know how many parts to divide each whole unit into. So how do we find this number? The best way is to use what we know about algebraic reasoning, using these two equations to describe the array.
We’ll start by finding the value of X in the multiplication equation.
This will help us later in understanding how to solve the division equation.
You know that you can multiply both sides of an equation by the same number and it doesn’t change the equation. We can use this to our advantage if we choose the right number to multiply by. We want a number that will give us one X instead of four-fifths X on the left side of the equation.
You know that multiplying any number by its reciprocal gives you 1. Let’s multiply fourth-fifths by its reciprocal like this.
Five-fourths times four-fifths is one, so the left side of the equation now becomes 1 times X, which we write as just X. You also know that if you multiply one side of an equation by a number you have to multiply the other side by the same number.
So, we multiply the right side of the equation by five-fourths.
When we multiply, we get forty-sixtieths.
We can divide both terms by 20, to see that X equals two thirds.
When we compare this to the multiplication problem we started with, we see that two-thirds is the correct answer, because two-thirds times fourth-fifths is indeed eight-fifteenths.
But how does this help us understand how to divide fractions? It’s actually very simple. Let’s look at these two statements from the equations.
Remember that X represents the same number in both equations. It’s the length of this line in the aray.
The equation on the right says you can find X by dividing eight-fifteenths by four-fifths. The eqution on the left says you can find X by multiplying eight-fifteenths by the reciprocal of four-fifths.
Using what you already know about algebra, you can prove that this same approach always works, but we won’t go through that proof here. It means that dividing any number by a fraction and multiplying that number by its reciprocal gives you the same result.
We’ll just look at a couple of examples to verify that this works.
We’ll start with this multiplication statement. Three-fourths times five-eighths equals fifteen-thirty-seconds.

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https://youtu.be/ncad2bziX5s

Decompose a fraction (x/y) into a sum of unit parts (1/y) using visual models.

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Title: Fractions as Division - Fast And Easy Math Learning Videos

You already know that you can write the remainder in a division problem as a fraction. In this lesson you’ll learn that all division problems can be written as fractions. First let’s review writing remainders as fractions:
Thirteen divided by 3.
Remember: to solve this problem we make an array.
The array we made is 3 by 4, and there’s 1 left over.
Now we break each of our remaining blocks into 3 parts.
And we continue making the array with the parts.
So, now the array we’ve made is 3, by 4 and one-third.
So, 13 divided by 3 is equal to 4 and one-third.
Instead of just splitting the remainder blocks into thirds, we could have split all of the blocks into thirds before forming the array, like this.
Now we’ll make an array. Remember, since we split the blocks into thirds, each whole block will end up being 1 column in the array.
So the array we formed is 3 by thirteen-thirds.
So 13 divided by 3 is equal to thirteen-thirds.
We can do the same thing for all division problems. Any division problem can be rewritten as a fraction with the numerator as the dividend and the denominator as the divisor.
Let’s look at another example.
Three divided by 4.
Since 3 is less than 4 we cannot make an array without splitting each of the blocks into fourths, like this.
Now we can make an array.
So the array we formed is 4 by three-fourths
So 3 divided by 4 is equal to three-fourths.
Let’s look at another example: 56 divided by 9. We can rewrite this as a fraction with 56 as the numerator, and 9 as the denominator.
Let’s look at one last example
Four-sixths. Just as we can rewrite division problems as fractions, we can also rewrite fractions as division problems. So four-sixths is equal to 4 divided by 6.
In the remainder of this lesson you’ll rewrite division problems as fractions and fractions as division problems.

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https://youtu.be/l4I7Hj-Taxc

This video covers how to find the "Highest Common Factor" of a group of numbers. We cover 2 methods. The first is the easier method, which involves listing out the factors of each number and picking the highest factor shared by all the numbers. The second involves using prime factors.

This video is suitable for maths courses around the world.

UK:
KS3 - Only need to know the first method (up to 1:55)
GCSE Foundation - Only need to know the first method (up to 1:55)
GCSE Higher - All suitable

Maths Playlist:
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Dr. Powell multiplies fractions using both fraction tiles manipulatives as well as via a drawing demonstration.

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Materials found in the classroom were used to create this animation. Students used a hamburger to illustrate the concept of fractions to explain this to other students. The planning of this project was integral to their understanding of fractions. Allowing students to be "hands-on" with the materials to create this movie, created a deeper understanding of the process of animation and justifies their knowledge of fractions.

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Title: Multiplying Fractions and Whole Numbers - Fast And Easy Math Learning Videos

When we multiply whole numbers, we can show the multiplication using an array like the one shown here. This array shows the multiplication, 2 times 3.
The array is two blocks tall and three blocks wide.
We can show multiplication of a fraction times a whole number using the same kind of array.
Let’s change the first factor in this multiplication to a fraction.
Here we have the fraction 3 halves times the whole number 3. To show the fraction 3 halves, we start by dividing each whole unit on this side into halves, like this.
We show 3 halves here.
We show the whole number 3 here.
Now we draw the array.
We have three halves here, another three halves here, and three more halves here.
We can show that using repeated addition like this.
When we do the addition, we get 9 halves. That means that 3 halves times 3 is 9 halves.
Another example. Two times one third.
Here we have 2 blocks tall times one third of a block wide.
This can also be written as repeated addition. This time, we have the fraction one third, two times.
We write that like this.
One third plus one third.
We know this is equal to 2 thirds.
So, 2 times one third is two thirds.
Another example.
Six fifths times 4. We write this as repeated addition like this.
When we add, we get 24 fifths.
So six fifths times 4 is 24 fifths.
You probably noticed a pattern here. When we do the repeated addition in each case, the result of adding the numerators is the same result as multiplying the numerator of the fraction by the whole number.
Here, we added the numerator three, three times, and got 9.
That’s the same as multiplying 3 times 3 which is also nine.
Here, we added the numerator one, two times, and got 2.
That’s the same as multiplying 1 times 2, which is also 2.
Here, we added the numerator 6, four times and got 24.
That’s the same as multiplying 6 times 4, which is also 24.
So, to multiply fractions, we don’t have to do the repeated addition, we can just multiply the numerator times the whole number.
That gives us the numerator of the result.
The denominator stays the same.
Here’s an example. Four times three fifths. We multiply the numerator, which is three, times the whole number four, which gives us a numerator of 12.
And we keep the denominator.
So 4 times 3 fifths is 12 fifths.
Last example. Two fifths times 3. We multiply the numerator, 2, times the whole number 3, which gives us a numerator of 6, and the denominator stays the same.
The result is 6 fifths.
In the remainder of this lesson you’ll multiply whole numbers and fractions like these.

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https://youtu.be/BV-T6b7S9dc

Students learn to find equivalent fractions with multiplication by relating the equations to length, area, and number line models. For more videos and instructional resources, visit TenMarks.com. TenMarks is a standards-based program to complement any math curriculum with scaffolded lessons, guided practice, inquiry-based tasks, assessments and interventions.

Use this shortcut to add fractions easily.
Steps of the Smiley Method
1. Multiply the denominators
2. Starting with the fraction on the left, cross multiply the numerator times the denominator.
3. Add the two solutions from step 2 for your numerator.
4. Reduce if possible

Transcript
Here is a shortcut method for adding fractions. Let's start with 1/2 plus 1/3 I call this method the smiley face because you can tell this looks like a smiley face. The first step is to multiply the two denominators together. 2 times 3 equals 6 and next start on the left and cross multiply so 1 times 3 is 3 and move to the other side and 1 x 2 = 2 and add those together and you get 5/6 Pretty simple. Add the fractions 2/5 + 1/3 Go with Mr. Smiley Start on the left and cross multiply, 2 x 3 = 6 Move to the other side and 1 x 5 = 5 and that equals 11/15 and you can't reduce any further. Next let's add 3/4 + 2/3 go with the smiley method , there is the smiley fraction, 4 x 3 = 12,start at the left and 3 x 3 = 9 , 2 x 4 = 8 and that equals 1712 and that is an improper fraction which is in it's simplest form. This method also works with a mixed fraction. Let's add the mixed fractions, 21/2 + 31/3. The first thing to do is to go ahead and convert it to an improper fraction. 2 x 2 = 4 + 1 = 5/2, and the other side is 3 x 3 =9 + 1 = 10/3. Next apply the smiley method. 2 x 3 = 6 and start on the left and 5 x 3 = 15 move across, and that is 15, cross multiply again and that becomes 20 which equals 35/6 which is an improper fraction and if you would like you 30 by 6 which equals 5 and 5/6 left over. So the smiley face method works on mixed fractions. Thanks for watching and please subscribe to MooMooMath. We upload a new Math video everyday.

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Title: Writing Fractions from Drawings - Fast And Easy Math Learning Videos

In this lesson you’ll learn to write the fraction that’s shown in a picture.
For example suppose you want to write the fraction shown by the shaded part of this picture. In this case the square is the whole.
And the whole is divided into 4 equal parts.
So 4 is our denominator.
3 of the parts are shaded.
So 3 is our numerator.
The shaded part of the picture shows three-fourths.
Let’s look at another example.
In this case the whole is the triangle.
The triangle has been divided into three parts.
However, the parts are not equal in size, so we can’t write a fraction from this picture. To write a fraction, the figure would have to be divided into equal-size parts.
Let’s look at another example.
In this case the whole is a circle.
Notice that we have two wholes shown.
The wholes are each divided into five equal parts.
So the denominator is 5.
7 of the parts are shaded
So the numerator is 7.
The shaded part of the picture shows seven-fifths.
Let’s look at one last example.
In this case the whole is a rectangle.
Notice that there are 4 wholes shown.
Each whole has been divided into three parts
So the denominator is three.
10 of the parts have been shaded in
So the numerator is 10.
The shaded part of the picture shows ten-thirds. In the remainder of this lesson you’ll write the fraction shown by the shaded part of a picture. Now do the rest on your own.

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https://youtu.be/AksVei-5guU

In this video, the ratios involve decimals and fractions. You will learn how to simplify such creatures.

This webinar was sponsored by the National Center on Intensive Intervention (NCII) and National Center on Systemic Improvement (NCSI) and presented by Drs. Russell Gersten, Sarah Powell, and Robin Finelli Schumacher. The webinar discusses 1) the importance of fractions instruction and typical challenges faced by students, 2) share recommendations for fractions instruction, and 3) provide considerations for supporting students within secondary or Tier 2 and intensive intervention.

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Title: Multiplying Fractions - Fast And Easy Math Learning Videos

When we multiply two fractions, we can show the multiplication using the same array we’ve used before, like this.
Since we’re dealing with fractions of numbers, we need whole units that are bigger than these that we’ve used before. We can zoom in a little like this.
Now we can see each whole unit in more detail.
We start by showing the first fraction in the problem, two-fifths, on this scale.
So, we divide each unit into five equal parts to show fifths.
We show the second fraction on this scale.
So, we divide each unit into four equal parts to show fourths.
Next, two show the first fraction, we count two-fifths here, and show the second fraction by counting three-fourths here.
The product of these fractions is shown by this array.
We’re multiplying fractions, so the first thing we need to know is how many parts are in each whole unit in the answer. A whole unit is shown here
We have five parts per unit times four parts per unit.
When we multiply these two numbers, we see that we have twenty parts in a whole unit.
This means the denominator of the answer is 20.
Next, we need to know how many of these parts we have in the array we created by multiplying the fractions. Two parts are counted on this side, and three parts are counted on this side.
Two times three is six, so when we connect the lines we see that the array has six of these parts in all.
This means the numerator of the answer is 6.
Again, there are twenty parts in each whole unit, so the denominator is 20.
We have six parts in the array, so the numerator is six.
So the product of the two fractions is six-twentieths.
Now notice that the numerator, six, is equal to the product of the two numerators in the fractions we multiplied.
Two times three equals six.
Also notice that the denominator, twenty, is equal to the product of the two denominators in the fractions we multiplied.
Five times four equals twenty.
Here’s another example.
We can show the problem in an array.
We will show two-thirds on this side.
So, we divide the whole units into thirds, and count up two-thirds.
We will show the second fraction, 5/2 on this side.
So, we divide the whole units into halves, and count up five-halves.
The array for our problem looks like this.
So now we have whole units that have three parts times two parts.
So, we have six parts in each whole unit.
This means the denominator in the answer is 6.
To find the numerator of the answer, we count two parts times five parts.
So, there are a total of 10 parts counted in the array.
This means the numerator is 10.
Again, each unit is divided into six parts, so the denominator is six.
We count ten of these parts.
So, the numerator is ten.
Again, notice that the numerator in the answer is the product of the numerators in the fractions we multiplied.
The denominator is the product of the denominators in the fractions we multiplied.
We could solve every multiplication problem with fractions using diagrams like this. But, that would take a long time. In every case, the answer would be the same as we get if we just multiplied the numerators and denominators as we did in these examples.
We saw that when we multiply the denominators, we find how many parts are in each whole unit, which tells us the size of the parts in the array that represents the answer.
When we multiply the numerators, we find how many parts we have in the array.
Now that we know what multiplying fractions means, we can use these facts to solve all multiplication problems with fractions.
Here we have four-fifths times one-half. To multiply these two fractions, first we multiply the numerators.
Four times one equals 4.
Next, we multiply the denominators.
Five times two equals ten.
So the answer is four-tenths.

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Title: Rewriting Fractions with LCM as Common Denominator - Fast And Easy Math Learning Videos

You’ve already learned how to find the least common multiple of two numbers. In this lesson, you’ll learn to use the least common multiple of two numbers to find a common denominator for two fractions.
Here’s an example, one sixth and four ninths. One way to find a common denominator for these two fractions is to use the least common multiple of the denominators. First, let’s find the least common multiple of 6 and 9.
We start with six and list some multiples, as shown here.
Then we take 9 and list some multiples.
The least common multiple is the smallest number included in both of these lists of multiples. In this case, it’s 18.
Now let’s use that as the common denominator to rewrite both fractions.
One-sixth becomes three eighteenths.
And four ninths becomes eight eighteenths.
Another example, three fourths and three tenths. We start by finding the least common multiple of the denominators, 4 and 10.
We start by listing some multiples of four, as shown here.
Then we list some multiples of 10.
The least common multiple is 20.
We now can rewrite the fractions with a common denominator of 20.
Three fourths becomes fifteen twentieths.
And three tenths becomes six twentieths.
Last example, three fourths and one sixth. To find the common denominator, we find the least common multiple.
We list some multiples of 4 as shown here, then some multiples of 6.
The least common multiple is 12.
So, the least common denominator is 12.
We now rewrite each fraction with a denominator of 12.
Three fourths becomes 9 twelfths and one sixth becomes 2 twelfths.
In the remainder of this lesson, you’ll rewrite fractions with a common denominator after using the least common multiple of the denominators to find a common denominator.

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