Introducing fractions as division is an effective way to teach the concept
Do your students struggle with a deep understanding of fractions? Give this abstract concept some firm footing by introducing fractions as division. Building upon students’ existing knowledge of division, we make sure they understand the most important property of fractions - representing a certain part of a whole.
In this article, we’ll explore an effective approach to conceptualizing fractions by using division. It’s the best starting point for those who want to be sure that students will grasp the idea of fractions in the first place. We’ll show how Happy Numbers teaches students to represent fractions as division expressions and vice versa, and to rename fractions as mixed numbers using quotient and remainder.
Understanding fractions as division
As always, introducing any concept goes better when we have concrete examples to use as a reference. Simple visual analogies can show students that fractions actually represent division and can be replaced with a division expression. For example, dividing 6 cookies equally among 2 plates gives us 3 cookies on each plate:
Students are already familiar with the concept of division, and it won’t be difficult for them to record this problem in the form of the equation: 62.
Dividing 6 cookies equally among 2 plates leaves us with a whole number result. However, sometimes there may be more plates than cookies, and this is when fractions can help us to do the math!
In the problem below, for example, we have 2 pastries to divide equally among 3 plates. Now that’s a tough situation because at first sight, it may seem illogical to divide a smaller number by a greater one. However, it’s possible, and more than that, it means that the answer will be less than a whole number.
Fractions (in this case, cutting the pastries) allow students to distribute them equally onto the three plates.
Next, students record this example by identifying how many pieces are on each plate and how many pieces there are in a whole pastry. We sum up the idea by moving to fractions.
Students see that 2 pastries divided by 3 plates can also be represented as the equation: 2 3. They also see that the division can be written in the form of a fraction by changing the division sign to the line between the numerator and the denominator.
Students recognize that there are 2 3 or 23 of the pastry on each plate:
In other words, the dividend became the numerator and the divisor became the denominator of the resultant fraction. This is what fraction as division looks like.
Fractions as division with remainder
Once students grasp the idea of fractions as division, they’re ready to advance to more complex scenarios. We introduce fractions as division when there are enough objects to divide into equal groups, but the division still isn’t exact. The result is a fraction greater than one.
For example, to divide 3 pastries equally among 2 plates, students cut each one into 3 smaller pieces and get 32.
In some cases, students divide whole numbers and end up with a remainder. No reason to panic, as they already know how to show that a quotient and a remainder can easily be turned into a mixed number.
In the example below, students divide 7 flatbreads equally among 3 trays. First, they create 3 equal groups of 2:
With the help of visual supports, students put 2 pieces on each tray (which becomes the quotient of the equation). Then, they divide the remaining piece into 3 smaller pieces (which transforms the remainder of 1 into a fractional piece).
Finally, students realize that division of 7 by 3 gives a result that can be easily represented as a mixed number.
Students now see that 7 divided by 3 is the same as 73 , and that the quotient 2 and the remainder 1 can help them rename the fraction as a mixed number.
To see the full exercise, follow this link.
Examples of word problems
Gradually, students become more confident and are ready to solve fraction word problems, using models first, and then moving to word problems with no visual support.
Happy Numbers makes sure that students gradually move from concrete to pictorial representations by introducing models they can use to practice their fluency, problem solving, and creative skills.
We move from concrete to pictorial models to make sure students understand fraction as a division, not just by dividing pieces of pastries and cookies. Teachers can use pictorial models as examples and as practice as well, letting students create their own models.
Finally, students apply their knowledge by solving actual word problems, like the one below.
Understanding fractions as division can be a challenging task for students. Happy Numbers makes it simpler by introducing concrete examples and then moving toward pictorial models. Once students advance step by step from concrete to pictorial to abstract mathematics, they will be more capable and confident with more complex tasks, like converting fractions and performing operations on them.