Welcome to our lesson on multiplying fractions. To multiply fractions, we follow three simple steps. First, multiply the numerators, which are the top numbers. Second, multiply the denominators, which are the bottom numbers. Third, simplify the result if possible. Let's look at an example: two-thirds times one-fourth. First, we multiply the numerators: 2 times 1 equals 2. Then, we multiply the denominators: 3 times 4 equals 12. So we get two-twelfths. Finally, we simplify by dividing both the numerator and denominator by their greatest common factor, which is 2. Two divided by 2 is 1, and 12 divided by 2 is 6. So our final answer is one-sixth.
Now let's learn how to divide fractions using the 'Keep, Change, Flip' method. When dividing fractions, we follow four steps. First, keep the first fraction as it is. Second, change the division sign to multiplication. Third, flip the second fraction upside down - this is called finding the reciprocal. Fourth, multiply the fractions and simplify. Let's look at an example: three-fourths divided by one-half. First, we keep three-fourths. Then, we change the division sign to multiplication. Next, we flip one-half to get two-over-one. Now we multiply: three-fourths times two-over-one. Multiplying the numerators: 3 times 2 equals 6. Multiplying the denominators: 4 times 1 equals 4. So we get six-fourths. Finally, we simplify by dividing both the numerator and denominator by their greatest common factor, which is 2. Six divided by 2 is 3, and 4 divided by 2 is 2. So our final answer is three-halves, which can also be written as one and one-half.
Let's visualize fraction multiplication using an area model. When we multiply fractions like two-thirds times one-fourth, we can represent this with a square divided into equal parts. First, we divide the square into thirds vertically, and shade two-thirds of it blue. Then, we divide the square into fourths horizontally, and shade one-fourth of it yellow. The area where these two regions overlap, shown in green, represents the product of the fractions. This overlapping region is two-twelfths of the whole square, which simplifies to one-sixth. So, two-thirds times one-fourth equals one-sixth. This visual model helps us understand that when we multiply fractions, we're finding a fraction of a fraction, resulting in a smaller part of the whole.
Now let's visualize fraction division using a model. When we divide fractions, like three-fourths divided by one-half, we're asking how many halves fit into three-fourths. Here, the blue rectangle represents three-fourths of a whole. The yellow rectangle below shows one-half. To find out how many halves fit into three-fourths, we place copies of the half inside our three-fourths region. We can fit one complete half, shown in green, and then another complete half. That gives us one whole. But we still have a small piece of the three-fourths left over, which is one-fourth of the whole. Since one-fourth is half of one-half, this remaining piece represents one-half of our divisor. So, three-fourths divided by one-half equals one and one-half. This confirms our calculation using the 'Keep, Change, Flip' method: three-fourths times two-over-one equals six-fourths, which simplifies to three-halves or one and one-half.
Let's summarize what we've learned about multiplying and dividing fractions. To multiply fractions, we multiply the numerators, multiply the denominators, and then simplify if possible. For example, two-thirds times one-fourth equals one-sixth. To divide fractions, we use the 'Keep, Change, Flip' method: keep the first fraction, change division to multiplication, and flip the second fraction. For example, three-fourths divided by one-half equals one and one-half. It's important to understand that when we multiply fractions, the result is often smaller than either fraction because we're finding a part of a part. However, when dividing fractions, the result can be larger than the first fraction because we're finding how many times one fraction fits into another. Visual models, like the ones we've explored, help us understand these operations conceptually, not just as procedures to follow. These skills are fundamental for more advanced math concepts and for solving real-world problems involving fractions.