Do your students do this? What went wrong?
Addition and Subtraction of Fractions is an important topic, and can be divided into stages:
- Like fractions,
- Unlike fractions,
- Related fractions,
- Unrelated fractions,
These are generally introduced in the 2nd, 3rd and 4th grade respectively.
In part 1 of this series of blog posts, we are going to talk about addition and subtraction of like fractions, how to introduce the topic and common pitfalls such as the above negative example.
Like fractions refers to proper fractions with common denominators, such as
1/4 + 1/4
When first introducing addition and subtraction of fractions, it is good to start with unit fractions, where the numerator is 1, with an answer that is less than or equal to 1. Here are some of our observations and recommendations.
No Rules at the Beginning
Many teachers dive straight into teaching the procedural and mechanical aspects of fraction addition and subtraction, e.g. “add the numerator, but not the denominator” etc, and usually present the abstract form in the first lesson, e.g.
1/4 + 1/4 = 2/4
In Singapore Math, Fractions is introduced with the Concrete-> Pictorial-> Abstract (C-P-A) progression.
First start with some familiar objects, such as round pizzas, chocolate bars etc. Simple questions such as “you have 1/4 of a pizza and I give you another 2/4 of the same pizza, how much of the pizza do you have?” will start kids thinking about adding fractions in a concrete manner.
Then transfer the examples to pictorial and abstract forms:
Have the kids derive their own “rules” regarding fractions, based on their own observations. Then let them verify their rules using more advanced examples, e.g. 3/5+2/5 = 5/5 =1.
Deducing their own rules from concrete and pictorial manipulation is not only fun for kids, it strengthens their foundation when it comes to abstract representation.
Concept of the same whole
In the above examples, notice that we always draw the fractions as part of the same object or the “same whole“. This is a very important concept that students need to understand. If fractions are presented pictorially as referring to different wholes, it is easy for the students to get confused. In the opening example, which is a very common mistake, students see 8 equal parts, and therefore, use 8 as the denominator!
In the US Common Core, this fact is emphasized in 4th grade before formally introducing addition and subtraction of fraction:
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. – (CCSS.Math.4.NF.3a)
Before formally teaching addition and subtraction of fractions, it is good to do the reverse as a preparation and ask the students to decompose a fraction into sums of fractions with the same denominator, in as many ways as they can think of e.g.
5/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 2/8 + 3/8 = etc
In addition to preparing the students to mentally see adding and subtracting fractions as joining and separating parts, this exercise also prepares the student for mixed numbers and improper fractions.
Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by
using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. – (CCSS.Math.4NF.3b)
The Importance of One
The bar model is good pictorial representation to illustrate adding like fractions.
Don’t forget examples of subtraction involving 1 as well:
These cases of fraction addition and subtraction involving the whole number is very important when we’re dealing with addition and subtraction of mixed numbers and improper fractions later.
Adding and subtracting mixed numbers
After introducing mixed numbers and improper fractions, students are ready to add and subtract fractions involving mixed numbers.
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. – (CCSS.Math.4NF.3c)
Same whole again
Here it is also important to emphasize the concept of the “same whole”, e.g. in the pictorial representation for 5/8 + 6/8 = 11/8, we have to present the parts on the same whole, and since the sum is greater than 1, the result should occupy more than 1 whole unit.
Writing everything as improper fractions?
Very often, we see students doing this:
That is, they write everything as improper fractions, perform rudimentary operations, converting the result back to mixed numbers, and then simplify.
Instead, students can be taught to see “regrouping” as a viable strategy, i.e.
Kids need to make sense of what they are doing, and applying appropriate strategies for different questions. It is also not uncommon to see students doing this:
instead of simply,
Kids visualize what we say and draw
Teacher’s role in interpreting the curriculum
There is a limit to what curriculum designers can show in static print. Much of the responsibility lies in the teacher, and we need to interpret and translate the curriculum accurately through teaching. We sometimes feel that curriculum and lesson plans should be complemented by video instructions to avoid some of the pitfalls we observed below.
In Math In Focus, the following example is presented in Practice 3 (page 53) of workbook 2B:
The intention of the curriculum designer is to show that the addition is with reference to parts of the same whole (hence shaded parts are different). If the teacher/parent left out the right half of the question (a single whole shape), it is not clear the two shapes refer to the same whole, thus leading to the above confusion.
Sometimes, we see teachers label their parts like the diagram on the left:
It can be confusing for kids: “Are the parts of different values?” A better way would be to label the parts like the diagram on the right.
Instead of saying 1/4 as “one over four”, we can use “one-fourth” to stress the fact that the entity is a fraction and not the result of an operation of two numbers. Kids visualize what we say, and if we say 1/4 as “one-fourth”, they would be able to visualize 1/4 plus 1/4 as adding two “one-fourth”, therefore the answer is “two-fourths”!
Addition and subtraction of like fractions is the first step in the introduction of the big topic of fraction arithmetic, and is a great chance for teachers and parents to set things right and lay down the right foundation for the future. At the same time, it is an opportunity for students to exercise their deduction and reasoning skills by coming up with the “rules” for adding and subtracting fractions through hands on observations.
This article is part of a series of blog posts on Fractions:
- >> Read the next post on Fractions: Addition and Subtraction of Fractions Part 2 – Related Fractions
- << Read the previous post on Fractions: Mixed Numbers and Improper Fractions
- Or start from the beginning: Understanding Fractions as Equal Parts
More Fraction Resources
For more fraction resources, refer to our main fractions page.