Now that our family health crisis has abated (my daughter is doing well), I guess it's time to jump back in with both feet. A math program leader in a district with which I am consulting, asked for my opinion on an important issue of curriculum and instruction.
How much time should middle school teachers spend on the traditional vertical algorithm for adding and subtracting mixed numerals vs. converting to improper fractions immediately?
I assumed that both methods are still commonly taught with about equal time given to each, but I wasn't all that sure about how that was across the country. This is where I need the help of my informed readers.
First, my thoughts. From a practical perspective of those who utilize fractions in their occupation, I would guess that the mixed numeral form is most commonly employed. Whether it's the carpenter tak! ing measurements to see how many board feet of wood must be ordered (or for precise measurement to the nearest sixteenth of an inch) or someone following a recipe in the kitchen, I can't imagine that converting to improper fractions would be their first choice. On the other hand when I personally need to add fractions in a math problem, I usually use improper. I took an informal survey of one of the groups I'm working with and the majority stated they were taught both methods and some preferred working with mixed numerals and others said it's more complicated that way.
Are the number of steps roughly the same?
Mixed Numerals Algorithm for Subtraction:
(1) Convert the proper fractions to common denominator form.
(2) If needed, regroup, i.e., "borrow" 1 from the whole nu! mber part of the larger mixed number, convert the 1 into commo! n denomi nator form and combine this with the other fraction (of course students are shown short-cuts for this which they blithely and mechanically follow without much thought).
(3) Subtract the whole numbers and the proper fractions.
(4) If the resulting fraction is improper convert it and add the whole number part to the previous result.
Improper Fraction Algorithm:
(1) Convert each mixed numeral to an improper fraction by the traditional algorithm (again blithely and mechanically without much thought).
(2) Determine a common denominator (or the lcd) and convert each fraction.
(3) Subtract the fractions.
(4) Convert the answer to mixed numeral form by the traditional division algorithm.
Now I may have combined steps or there are oversights but essentially they appear to be roughly the same number of steps. However, the difficulty or complexity level of th! e steps
may not be equivalent.
I also feel that the mixed numeral form requires somewhat more conceptual understanding even if the child does it routinely. It may also prepare the youngster for working with algebraic expressions like A + B/C, but that's debatable. Further there seems to me to be a strong connection between the Mixed Numerals Algorithm and adding and subtracting denominate numbers. For example:
Subtract
15 hr 37 min
9 hr 46 min
I doubt that we would encourage students to convert both to minutes first, subtract, then convert back to hrs and min. I could be wrong there!
I feel there are arguments on both sides here. My instinct is that both need to be taught but it's not clear to me how much time should be spent on each method. Certainly some youngsters could handle both with facility while some would struggle mightily with at least one of these methods.
Further, I suspect there are some youngster! s who convert mixed numerals to improper fractions procedurall! y withou t full conceptual understanding that a mixed numeral is an addition problem!
Your experiences and thoughts...
How to add and subtract mixed fractions
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