Adding and Subtracting Fractions

The lesson below is still images taken from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. If the PowerPoint is not one of my freebies, you may also head over to my YouTube channel to see the slideshow fully animated and can pause as needed to be sure you grasp each concept before moving forward (each lesson will always be free there!).

In this lesson we go over how to add and subtract fractions. The slides dive into what a denominator is, what it means for a fraction, and how only “like things” can be combined. Students get the chance to explore improper fractions and mixed numbers and have the opportunity to switch between the two.

So without further ado, read through the slides below to get a feel for how to add and subtract fractions with like and unlike denominators as well as how to convert between improper fractions and mixed numbers!

Add 1Add 2Add 3Add 4Add 5Add 6Add 7Add 8Add 9Add 10Add 11Add 12

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Multiplying and Dividing Fractions

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • Fractions
  • Adding and subtracting
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Finding a least common multiple of two numbers to find a least common denominator
  • Adding and subtracting fractions with like denominators and unlike denominators
  • Like Denominators
  • In order to add or subtract fractions, we must add or subtract the same things.
  • For example, if I add $5 and $8, the sum is $13.
  • If I add €5 and €8, the sum is €13.
  • But what if I add $5 and €8? I can’t just combine the two currencies into some odd new thing.
  • Instead, I’d have 13 paper notes that represented currency.
  • So for fractions:
  • Like Denominators
  • Here our denominators are the same, so we can simply add or subtract our numerators like so:
  • Notice that our denominator stays the same.
  • Essentially we are adding 2 somethings and 1 something, which gives us 3 somethings. The somethings just happen to be fifths.
  • Likewise, subtraction works in the same way!
  • And just like that, you can add and subtract fractions with like denominators!
  • LCM
  • To add and subtract fractions without a common denominator, you will need to find the least common multiple of the denominators.
  • The least common multiple is the smallest multiple shared by two or more numbers.
  • For example:
  • Our first multiples are just our numbers times 1. No matches yet.
  • Second? Nope
  • Ah, on the third we see a match, so 12 is our LCM!
  • Common Denominator
  • So for fun, let’s say we want to add these two fractions:
  • We cannot add these two fractions in their current forms as they do not have the same name.
  • As we saw in the last slide, however, we can find the LCM to find a common denominator.
  • But how did we change?
  • We know now that multiplying by 1 does not change a fraction, so we can multiply by a form of 1.
  • Common Denominator
  • On the left, we need to transform our denominator 4 into 12.
  • Thus we need to multiply by 3.
  • As long as we multiply both the top and bottom by the same number (3/3 = 1) it will not change the value!
  • On the right, we need to transform 6 into 12, which means:
  • 3 x 3 on the left is:
  • 1 x 2 on the right is:
  • Now we are back to what we know, just adding across the top!
  • Improper Fractions to Mixed Numbers
  • What happens when you see something like this?
  • We found our sum, but 5 is bigger than 4, making this an improper fraction.
  • While this form is sometimes useful, it is considered proper to convert this fraction into a mixed number.
  • A mixed number is a number with both a whole part and a real part.
  • To do this, we ask how many 4’s go into 5:
  • Since 5 divided by 4 is 1 with a remainder of 1, the remainder becomes the numerator.
  • Improper Fractions to Mixed Numbers
  • Let’s take a deeper look visually at what is going on here.
  • Our improper fractions is:
  • Thus we need to color in fourths:
  • But we have 5 fourths, which means we need another whole circle to shade the rest of our pieces.
  • But wait, didn’t we shade a whole circle.
  • And then one fourth after that.
  • Thus you can see how both improper fractions and mixed numbers represent the same quantity!
  • Mixed numbers to improper fractions
  • Alright, so what if I wanted to go back from a mixed number to an improper fraction in case I need to add or subtract some mixed numbers?
  • Well, let’s start with our same mixed number:
  • Our pieces are fourths, and we have 1 whole with four pieces in it, so first we multiply our denominator by the whole number.
  • Then add the numerator, so 4×1=4 and 4+1=5.
  • Then of course our denominator will remain the same!
  • Tada! Back to our improper fraction!
  • Working with Mixed Numbers
  • The good news is that when you want to add or subtract mixed numbers, you only have to deal with these numbers in parts. For example:
  • As our fractions already have a common denominator, we may add them as we have before:
  • Then we are left to simply add our whole numbers 3 and 1:
  • And there is your answer!
  • As with all the other instances, subtraction is exactly the same, except of course for the fact that you would subtract!
  • Special cases
  • The only difference that may arise in adding or subtracting mixed numbers are:
  • When adding, you may have an improper fraction as the fractional part of your answer.
  • When subtracting, you may not be able to subtract the second fraction from the first.
  • In the first instance, you would need to “carry” the extra whole and add 1 to your whole part of the answer.
  • In the second case, you would need to “borrow” a whole from the first number.
  • Both of these are covered more deeply in previous classes, so we will not go over examples here.
  • conclusion
  • Adding or subtracting fractions requires a common denominator.
  • To find a common denominator, you must find the LCM of both denominators.
  • To add or subtract fractions, you simply add or subtract the numerators and leave the denominator the same.
  • When finding a common denominator, you must multiply both the top and bottom by the same number to preserve the value.
  • Improper fractions are fractions with a numerator larger than the denominator.
  • Mixed numbers have both a whole part and fraction part.
  • Improper fractions and mixed numbers can represent the same quantity.
  • You use division to convert from an improper fraction to a mixed number and multiply then add for the reverse.
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