4 mistakes learners make with surds, each one named and explained.
Simplifying a surd means pulling out the largest square factor, and a root only leaves the root sign when you square it.
6 over root 3 is left as it is, or given as 2 over root 3.
It is already a correct value, and the instruction to rationalise sounds like tidying rather than a step.
Question. 6 over root 3
A common answer. 2 over root 3
The answer. 2 root 3
Why. Multiplying top and bottom by root 3 turns the bottom into 3, and 6 root 3 over 3 is 2 root 3.
Rationalising changes the form, never the value. Multiply top and bottom by the root.
(2 + root 5)(3 - root 5) is answered as 11 - root 5.
Root 5 times root 5 is 5, and the minus sign in front of it gets left behind in the rush.
Question. -root 5 times +root 5
A common answer. +5
The answer. -5
Why. A negative times a positive is negative, and the roots cancel to give 5, so the term is -5.
Square the root and keep the sign. The sign belongs to the term, not to the root.
The square root of 50 is simplified to 2 root 5 or 10 root 5.
Any pair of factors of 50 looks like a fair start, and 5 times 10 is the pair everybody reaches for.
Question. the square root of 50
A common answer. 2 root 5, from 50 = 2 x 25 with the parts swapped
The answer. 5 root 2
Why. 50 = 25 x 2, and only the 25 is a square number, so only the 25 comes out, as 5.
Look for the largest square factor. Nothing else will come out of the root.
The square root of 50 is simplified to 25 root 2.
The 25 was correctly spotted as the square factor, and then it walked out of the root unchanged.
Question. the square root of 50
A common answer. 25 root 2
The answer. 5 root 2
Why. The 25 leaves the root as its own square root, which is 5.
Sense check the size. A root of 50 cannot be 35.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.