4 mistakes learners make with algebraic fractions, each one named and explained.
Only a factor of the whole top and the whole bottom can be cancelled.
With x/3 = x/4 + 1 you cross multiply and turn a linear equation into a quadratic.
Cross multiplying clears the fractions and has always worked, so it becomes the first move rather than a choice.
Question. What is the quickest first step for x/3 = x/4 + 1?
A common answer. cross multiply the two fractions
The answer. multiply every term by 12
Why. The lowest common denominator clears both fractions at once and keeps the equation linear.
Count the terms. More than two means use the common denominator.
From x/2 = 8 you divide by 2 and write x = 4.
Dividing is the undoing move you reach for, and there is a 2 in the question to divide by.
Question. Solve x/2 = 8
A common answer. x = 4
The answer. x = 16
Why. x has already been halved to give 8, so x is twice 8.
Say what was done to x, then do the opposite. Not the same thing again.
You cross out the x in (x + 4)/(x + 5) and write 4/5.
The same symbol is on the top and the bottom, and cancelling matching symbols is exactly what worked with numbers.
Question. Simplify (x + 4)/(x + 5)
A common answer. 4/5
The answer. it will not simplify
Why. Try x = 1: 5/6 is not 4/5. Only something multiplying the whole top and the whole bottom can go.
Factorise first. If it will not factorise, nothing cancels.
You turn 1/a + 1/b upside down and write a + b.
Flipping each fraction is easy and looks like the same move done twice.
Question. What is the reciprocal of 1/2 + 1/3?
A common answer. 2 + 3 = 5
The answer. 6/5
Why. 1/2 + 1/3 is 5/6, and the reciprocal of 5/6 is 6/5.
Reciprocal of a sum: add, then turn over. Never one piece at a time.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.