Well, life got a little crazy and #MathLovinMonday got postponed a week, I’m sure none of you know how that goes…
Anyway, to start off our math lessons I want to focus on a topic that, in my experience, students of all skill level dislike to some degree or another.
As soon as I say that word in my classroom it is followed by a chorus of “No!”, “Can’t we just use our calculators?”, and the inevitable “I hate fractions.”
Fractions aren’t scary, they only seem that way because many don’t understand how to use them correctly. Generally fractions are introduced in grade school, I don’t know exactly what age but I do know it’s before Jr. High.
If a student doesn’t understand what they are taught in grade school and is never retaught the skills associated with using fractions they are always going to be in the “I Hate Fractions” camp.
This week we are going to focus on how to determine if two fractions are equivalent.
Alright, before we can get into the math we need to start with the vocabulary.
By definition, a fraction is a numerical quantity that is not a whole number. Fractions are written in the form:
To be a fraction both the numerator and the denominator need to be whole numbers, no decimals allowed.
Equivalent fractions are fractions that look different but are actually equal to each other.
For example, and are equivalent fractions. We would write this as
Alright, that’s all fine and good, but is a pretty common fraction to use for such a lesson, right?
So, what if it’s not such a common fraction?
Take the following problem:
How do we determine if the two fractions are the same? We simplify.
To simplify a fraction, we first find the greatest common divisor (GCD) for the numerator and denominator of each fraction. The greatest common divisor is the largest number that evenly divides, meaning leaves no remainder, both the numerator and the denominator.
So, let’s simplify the fractions from the problem above. First, we will start with the fraction
We’ll start with 5. To make the factor tree you simply write your starting number, 5, with two branches off for the first two factors. I always start with 1 and the number I’m dividing, just in case we have a prime number.
Next we will make the factor tree for 15. We start the same way, with the factors of 1 and 15, but, in this case, 15 has other factors so we add another level to our tree.
As you can see above, the second level of the factor tree for 15 and the first level of the factor tree from 5 both contain a 5 as one of the factors. Since there are no other common factors this means we have found the GCD of our numerator and denominator.
So, to simplify [pmath size=12]5/15 we divide both the numerator and denominator by 5, giving us .
We are halfway to solving our original problem, do you remember what it was?
So next we need to simplify . That means we need to make our factor trees for 15 and 45. We can reuse the one for 15 from above, so we just have to make one for 45.
Remember to start with 1 and 45 because we know that a set of factors with 1 is going to be a part of every factor tree.
Now we can look at each level of the factor trees. In level one of the tree for 15 and in level two of the tree for 45 we see a common factor of 15. There are other common factors between 15 and 45, we see a 3 and a 5 in both trees, however, 15 is the largest of these factors and so it is our GCD.
So now we will divide the numerator and the denominator by our GCD, (15÷15)/(45÷15) to get .
Therefore, since and both simplify to we know they are equivalent.
Please post any questions or comments below, and remember if you have a topic you’d like to see on #MathLovinMonday let me know!