This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational.
My wife and I once took dancing lessons to learn how to do the jitterbug and other steps. There was a basic dance step which we learned during the first session. In subsequent sessions we combined the basic moves with things like bending our knees, twirling around, and so on—adding more moves on top of the main moves. In the end, the foundational dance steps had served as the gateway to more involved moves which, to casual observers, appeared to be complicated, but to us was a mixture of mastered steps and moves.
Students are doing something similar math-wise when, after learning and (we hope) mastering the basics of working with negative integers, they apply this to working with negative fractions. Initially, they started with gains and losses, and determining whether, say, a loss of $10 and a gain of $4 represented a good or bad day. In other words, they learned to compute -10+4. In so doing they saw intuitively that there was a loss of $6, represented as -6.
The original steps students learned for adding and subtracting with negative integers now take on variations. Namely, students now learn to work with negative rational numbers (i.e., fractions and decimals). The next steps are solving problems like -3/7 + 2/9, -15 2/3 + 8 3/5, and -2.34 + 2.099. To prevent overloading of information, mixed numbers and decimals should be taught on a separate day than common fractions.
Prior Knowledge. In addition to knowing how to work with negative integers, students must have knowledge of and fluency in determining order of fractions and decimals. Students are given problems in which they must order fractions and decimals from least to greatest or vice versa, as well as plotting the rational numbers on a number line. A typical problem might be to order from least to greatest the following: -2 5/6, -3 3/5, 1.7, -2.25, 8/9, 2/3.
Warm-Ups for Lesson on Common Fractions. Warm-ups should review ordering fractions, addition and subtraction with negative integers, and addition and subtraction of fractions. Typical problems may include:
- 3/7 ____ 5/9 (Fill in blank with < or >)
- –15 +23 = ___
- 1.2 = ___/100
- 3/5 – 7/20
- 2/3 + 5/8 – 3/12
Scaffolding the Procedure for Common Fractions. The scaffolding strategy is to take what students know how to do, namely working with negative integers and extend that procedure as shown by the examples below:
Negative integers: a) 7 – 12, b) -5 – 7, c) -10 + 15
Fractions with same denominator: a) 7/20 – 12/20 = (7-12)/20 = -5/20 = -1/4
Putting the two integers together over the denominator now makes the problem into one they’ve seen before, namely 7-12. Students should now do other examples independently.
b) -5/8 – 7/8 = (-5-7)/8 = -12/8 = -1 ½
c) -10/17 + 15/17 = (-10+15)/17 = 5/17
More than two fractions: -2/13 +5/13 – 6/13 -3/13 =
(-2+5-6-3)/13 = -6/13
Fractions with unlike denominators: a) -1/3 – 7/15 b) 3/7 -2/3 c) -7/8 – 5/12
Continue to place the integers together over the common denominator. As before, work through one example together; students should then work the others independently.
a) (-5-7)/15 = -12/15 = -4/5
b) (9-14)/21 = -5/21
c) (-21-10)24= -31/24
Warm up for Lesson on Mixed Numbers and Decimals. Warm-ups should review ordering mixed fractions and decimals, addition and subtraction with negative integers, and addition and subtraction of fractions. Typical problems may include:
- 3 5/8 ___ 3 2/3 Indicate whether > and < goes in the blank.
- 0.23 __ 0.099 Indicate whether > or < goes in the blank.
- 2 5/7 + 3 3/14
- -5/8 + 3/20
Scaffolding the Procedure for Mixed Numbers. The goal here is to extend students’ understanding of adding and subtracting mixed numbers that are positive to ones that include negative values.
Review of Adding and Subtracting by Wholes and Fractions: Students are familiar with problems such as 15 2/5+5 1/5. These can be solved by first adding the whole numbers and then the fractions: 20 +3/5 = 20 3/5.
In some cases values must be carried such as 15 2/3+5 2/3. To obtain the total, the 4/3 is changed to the mixed number and then added to 20 for the final answer of
For subtraction, the same procedure applies: whole numbers subtracted and then the fractional portion. When we have problems where the second fraction is larger than the first, regrouping is necessary:
17 2/9-5 1/3 becomes 17 2/9-5 3/9. This requires “borrowing”, in which 1 is borrowed from the 17. I have found the easiest way to explain this is by rewriting the problem as: 16+1+2/9-5 3/9
Then the “1” is written in terms of the common denominator. Since 1 can be written as 9/9, the problem can be rewritten as 16+9/9+2/9-5 3/9 which becomes 16 11/9-5 3/9. This now allows us to subtract whole numbers, and fractions separately: 16-5=11; 11/9 – 3/9=8/9. The answer is 11 8/9.
Using “Improper Fraction” Form: The above method would be a complex way to solve problems that have negative numbers for reasons that will be shown. A less confusing, though sometimes labor intensive method is to express the mixed numbers in improper fraction form. In the problem above 17 2/9 becomes 155/9; 5 3/9 becomes 48/9 (155-48)/9=107/9. Converting to a mixed number results in 11 8/9.
This method is generally easier to work with when there are negative numbers involved. For example, consider the problem 8 2/9 – 10 2/3: the problem becomes 8 2/9 – 10 4/9
8 2/9 – 10 2/3 = 8 2/9 – 10 4/9 = (74-94)/9
This is now solved as done with common fractions: -20/9, which can then be expressed as a mixed number: – 2 2/9
The problem can be solved using the “wholes and fraction” method, but students tend to find it confusing. The above problem 8-10 + (-2/9) = -2 – 2/9 = -2 2/9.
Suppose, however, that the problem were 8 7/9 – 10 2/3. Now it becomes 8 7/9 – 10 4/9, which in turn is expressed as (8-10) + (3/9). The final form is -2 + 1/3, which is solved as -6/3 + 1/3 = -5/3 = -1 2/3. Students need to keep track of a number of things—when is the fraction part added, and when is it subtracted. When it is added, -2 + 1/3 becomes – (2-1/3).
At this stage we want to keep things straightforward. More advanced students may be given a few problems to be solved with the wholes and fraction method for extra credit. In general, however, problems with mixed numbers use numbers small enough that the improper fraction method is the most efficient way to go.
For large numbers such as 130 5/18 – 231 7/22, a more efficient way to solve it would be to express the fraction portion as a decimal, so the problem becomes 130.278 – 231.318. Which brings us to the topic of working with decimal fractions.
Adding and Subtracting Decimals. Problems like 0.015 – 0.05 follow the same principles as integers. That is, we have to figure out whether there is a gain or a loss. This entails determining whether 0.05 is less than or greater than 0.015. A common mistake is that students seeing the 15 in 0.015 and 5 in 0.05, assume that because 15 is greater than 5 that 0.015 is greater than 0.05. One way to avoid this mistake is to fill in the decimals with zeroes so they are now comparing 0.015 with 0.050. It’s evident that 15/1000 is less than 50/1000. There will be therefore be a net loss: – (0.050-0.015) = -0.35.
Another technique to use is to ask “If something costs $0.015 per gallon and another costs $0.05 per gallon, how much does 1,000 gallons of each cost?” $15 is clearly less than $50.
Students should be given practice with identifying which decimals are larger, along with the addition and subtraction problems.