Teaching Tips –Fraction Operations
Finding the LCM Efficiently
An efficient method of finding the LCM of smallish numbers is to mentally check successive multiples of the larger number until you find one that is also a multiple of the smaller. For example, to find the LCM of 12 and 9, think:
12 x 1 = 12; is 12 a multiple of 9? X
12 x 2 = 24; is 24 a multiple of 9? X
12 x 3 = 36; is 36 a multiple of 9? Ö
\ LCM(12, 9) = 36
Fraction Strips
I like fraction strips for teaching fraction concepts and fraction operations. They are hands-on, versatile and lead nicely to the number line representation of fractions.
The photo shows how you might use the fraction strips to show that , and are equivalent.
A nice investigation when introducing fractions is to ask students to find all fraction combinations that add to 1 whole. In the first investigation, restrict them to two colors. In subsequent investigations, you allow three or more colors.
You can use fraction strips to show that
Fraction strips can be used to develop the addition algorithm, and to check answers found using the algorithm.
By overlaying strips, you can also use fraction strips to show that .
Add and Subtract Vertically!
We add and subtract natural numbers vertically, so why not fractions as well? The concepts are the same. When the fractional part of our sum is greater than 1 then we “carry” into the whole numbers. When we subtract mixed numbers, sometimes we have to “borrow” one whole from the whole numbers to subtract the fractional part.
Students already know about borrowing and carrying, so why not build on it?
Multiplying Fractions
I like the area model that the textbook uses. The only changes I would suggest are:
· for consistency, always use a square to represent one whole.
· Add more information to the diagram. Here is my diagram for :
The diagram shows that the square is 1 unit x 1 unit and hence has an area of 1 unit2. It also shows that the shaded area is found by Area = Length x Width = 3/4 x 2/3. Each of the 6 rectangle has an area of 1/12 units2 , so the shaded area is also 6/12 units2. It follows that 3/4 x 2/3 = 6/12.
Students can draw their own diagrams for other fraction multiplication questions. The pattern quickly becomes clear – to multiply fractions, you multiply numerators and you multiply denominators.
At first, multiply first, and then simplify. After the students have mastered this, show them the “shortcut” of dividing out common factors first. Use an extreme example, e.g. , to illustrate it. I often have students do some problems both ways, and choose which they prefer (though I encourage simplifying first).
Don’t forget to teach the reciprocal and then let the students discover that a number times its reciprocal equals 1.
Dividing Fractions
We start be reminding students of what division means. The question “How many 2” lengths can be cut from a 6” length?” is answered by 6 ¸ 2 = 3.
Students can use Fraction Strips to solve similar fraction division problems. For example, the question “How many foot lengths can be cut from a foot length?” can be solved as shown alongside. Since there are four strips of length in a strip of length , we can write .
The diagram also shows that .
Students should also be given some fraction division questions such as where they can use fraction strips to estimate the answer, or derive the exact answer through some clever reasoning.
Spend some time having students estimate the answers to fraction questions, first using Fraction Strips, then using a number line, and finally “in their heads”.
Now you are ready to show the students the “shortcut”. Have each student or pair of students solve a pair or related questions, e.g. (solved with Fraction Strips) and (using the multiplication algorithm). Give each pair of students a different question to solve. List the answers on the board, with multiplication questions in one column and division questions in the other. The pattern is soon evident – the answers are the same. So we can turn any division question into a multiplication question by inverting the second fraction and multiplying.
Even after the students learn the “shortcut”, give the students plenty of questions that can be solved mentally, e.g. can be solved by thinking that if a foot long stick is divided into 3 pieces, each piece is foot long, so . Like all arithmetic questions, encourage students to try to solve the question mentally first.