## Thursday, May 21, 2009

### M3N5 - Simple cases of fraction addition/subtraction

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.e. Understand the concept of addition and subtraction of decimal fractions and common fractions with like denominators.
Addition and subtraction of fractions are discussed in three different grades (M3N5e; M4N6b; M5N4g). Both this current standard and M4N6b involve fractions with like denominators. For M4N6b, there is a note stating that denominators should not exceed 12. So, what is the difference between M3N5e and M4N6b? If one of the reasons for developing the GPS was to minimize repetitions, why is this topic repeated in Grade 4?

One of the differences is that in Grade 3, the sum or the minuend must be less than or equal to 1 as students will not be studying improper fractions and mixed numbers until Grade 4. Thus, 2/5 + 1/5 is appropriate in Grade 3 but not 4/5 + 2/5. However, the most important reason for discussing simple addition and subtraction in Grade 3 is to help students understand fractions as numbers, just like whole numbers.

Fractions are often introduced as parts of a whole. Although this way of looking at fractions is relatively easy for students to grasp, research also shows that this is a very limiting view of fractions. In other words, if students can consider fractions only as parts of a whole, they will have difficulty dealing with fraction arithmetic. Part of a whole is a relationship, and we cannot perform arithmetic operations on relationships. We can only add, subtract, multiply, and divide numbers. Thus, students must understand fractions as numbers in order to make sense of fraction arithmetic. So, how do we help students to see fractions as numbers? Well, one way is to help students experience situations where fractions are added or subtracted. From those experiences, students can realize that fractions are numbers because they can be added or subtracted. It sounds like a circular argument, and it probably is. However, I would like to think this relationship more of reflexive, i.e., neither one is a prerequisite for the other, and an understanding of one can actually promote and deepen the understanding of the other.

What is important, though, is that experiences students will encounter are something that they can determine as addition/subtraction situations. For example, we can ask students what is the total length of a tape if a 2/5-meter segment and 1/5-meter segment are put together end to end. They can see that this situation is an addition situation - you would use addition if the lengths of the segments were 2 meters and 1 meter, respectively.

Another key idea is the unitary view of fractions. In other words, students should understand 2/5-meters as made up of 2 1/5-meter segments. Then, 2/5 + 1/5 is really 2 1/5-units and 1 1/5-unit put together, or 2+1 1/5-units. By recognizing that fractions may be added or subtracted, and having a way to reason through to find the answers, students can develop the understanding of fractions as numbers. With this knowledge as the starting point, students in Grade 4 can explore fraction addition and subtraction more formally.