I. Standard Algorithms
A. Multiplying Fractions Algorithm To multiply fractions:
1. Multiply the numerators to get the new numerator.
2. Multiply the denominator to get the new denominator.
3. Simplify the result if necessary.
Example:
B. Multiplying Mixed Numbers
To multiply mixed fraction:
1. convert to improper Fractions
2. Follow the general algorithm for multiplying fractions (above).
3. Convert the results back to mixed fractions.
II. Conceptual Understanding
1. Whole number times fraction (Fraction times whole number)
Number Line
To grasp the concept, we start with multiplying a whole number times a fraction.
Example 1:
4 x 2/3
We could think of this problem in terms of repeated addition where we are asking, 'what is the total of 4 groups of 2/3'? To demonstrate this, the following number line shows each number 1 through 4 divided into three equal parts as reflected by the denominator in 2/3. Starting at zero, we move to the right 2 one third units (represented by the numerator) 4 times. This leaves us at 8/3 or 2 2/3.
solving using the rules of multiplying fractions:
4 x 2/3 = 4/1 x 2/3 = 8/3 = 2 2/3
Alternatively, instead of thinking of the problem as 4 groups of 2/3, we could also have thought of it as asking 'what is 2/3 of 4'.
Example 2:
2/3 x 4
This number line shows 4 divided into three equal parts (4 ÷ 3 = 4/3 = 1 1/3). Starting at zero, we move to the right 2 of those 4/3 units ending at 8/3 = 2 2/3.
2. Fraction times fraction
Number Line
Example 3:
3/4 x 1/2
We can think of this problem as asking 'what is 3/4 of a 1/2' or, since we are using the number line, we can read it as '3/4 of the distance to 1/2.' We first draw a number line which shows 1/2 of 1.
Next, we are going to figure what is 3/4 of this 1/2. To do this, we divide the 1/2 into four equal sections as indicated in the denominator of the first fraction. We can now visualize the distance, but we don't know the actual length of this distance in terms of the whole from 0 to 1. To figure this, divide the other 1/2 of the line also into four parts. Now, the entire whole from 0 to 1 has been divided into 8 parts. Finally, starting at 0 we move three of these units to the right ending on 3/8.
II. Conceptual Understanding
1. Whole number times fraction (Fraction times whole number)
Number Line
To grasp the concept, we start with multiplying a whole number times a fraction.
Example 1:
4 x 2/3
We could think of this problem in terms of repeated addition where we are asking, 'what is the total of 4 groups of 2/3'? To demonstrate this, the following number line shows each number 1 through 4 divided into three equal parts as reflected by the denominator in 2/3. Starting at zero, we move to the right 2 one third units (represented by the numerator) 4 times. This leaves us at 8/3 or 2 2/3.
solving using the rules of multiplying fractions:
4 x 2/3 = 4/1 x 2/3 = 8/3 = 2 2/3
Alternatively, instead of thinking of the problem as 4 groups of 2/3, we could also have thought of it as asking 'what is 2/3 of 4'.
Example 2:
2/3 x 4
This number line shows 4 divided into three equal parts (4 ÷ 3 = 4/3 = 1 1/3). Starting at zero, we move to the right 2 of those 4/3 units ending at 8/3 = 2 2/3.
*Note: Dividing 4 into 3 parts equals 1.333.. which equals 1 1/3. It would have been much clearer to mark the first jump as 1 1/3 then the second as 2 2/3. Instead I marked each jump as 4/3 as a movement of 4/3. Just realize that 4/3 + 4/3 + 4/3 = 12/3 = 4.
Number Line
Example 3:
3/4 x 1/2
We can think of this problem as asking 'what is 3/4 of a 1/2' or, since we are using the number line, we can read it as '3/4 of the distance to 1/2.' We first draw a number line which shows 1/2 of 1.
As examples 2 and 3 above illustrate, the idea of multiplication as repeated addition doesn't work very well as a concept when multiplying fractions. It's at this point that some educators introduce the idea of multiplication as scaling.
To scale a number is to make it larger or smaller by a certain factor. The concept is taught in geometry where an object is enlarged or reduced by a certain magnitude but still retain it's shape. Conceptually, repeated addition only allows you to think of multiplication as an operation that makes things bigger. The benefit of thinking of multiplication as scaling is that you can conceptually also see multiplication as a process of making a number smaller.
A good definition I came across is from the book "Putting Essential Understanding of Multiplication and Division into Practice"
As the above definition indicates, one number of the factors in a multiplication problem is viewed as a scaling factor and the other number is seen as the number being scaled up or down. Some tutorials I've seen refer to the first number as the scaling factor and some refer to the second number as the scaling factor. Ultimately it doesn't matter given the commutative property off multiplication but I will stick to what is generally taught when multiplying whole numbers. That being the first factor is the multiplier (scaling factor) and the second number is the number being multiplied (the multiplicand)
Practice:
IXL: Multiply fractions by whole numbers II
IXL: Multiply fractions by whole numbers: word problems
IXL: Multiply two fractions
IXL: Multiply two fractions: word problems
IXL: Multiply two mixed numbers
Does the order of multiplicand and multiplier matter in a multiplication equation?
Example 2 (above)
2/3 x 4
The 2/3 is the scaling factor. Since it is less than 1 we know the answer will the less than 4. The multiplicand, 4, is being scaled down.
Example 3 (above)
3/4 x 1/2
Again, the scaling factor, 3/4, is less than 1 so we know the answer will be less than 1/2.
Practice:
IXL: Multiply fractions by whole numbers II
IXL: Multiply fractions by whole numbers: word problems
IXL: Multiply two fractions
IXL: Multiply two fractions: word problems
IXL: Multiply two mixed numbers
Scaling
Multiplication as Scaling
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