5 mistakes learners make with standard form, each one named and explained.
One digit before the point, and the power counts how many places the point has moved, positive for big and negative for small.
(3 x 10^5) times (2 x 10^-2) comes out with a 5 at the front rather than a 6.
There are two operations in one expression and the hand does the easier one.
Question. 3 times 2, with powers of ten attached
A common answer. 5
The answer. 6
Why. The front numbers multiply and the powers add, which is one operation each, not the same one twice.
Multiply the fronts, add the indices. Two different jobs in the same sum.
0.0046 is written as 4.6 x 10^-4, because there are four zeros to look at.
Counting zeros works for whole numbers like 4000, so it gets carried across to decimals where it is one out.
Question. 0.0046 in standard form
A common answer. 4.6 x 10^-4
The answer. 4.6 x 10^-3
Why. The point moves three places to sit after the 4, and it is the places moved that the power counts.
Count the jumps of the point, not the zeros on the page.
0.0046 is written as 4.6 x 10 to the power 3, or 2.4 x 10^5 is read as a number below one.
The power looks like a size label, and a small number invites a small looking power.
Question. 0.0046 in standard form
A common answer. 4.6 x 10^3, which is 4600
The answer. 4.6 x 10^-3
Why. The point has to travel three places to the right to make 4.6, so the power records that as -3.
Above one, positive power. Below one, negative power. Check that before anything else.
(3 x 10^5) times (2 x 10^-2) is answered as 6 x 10^7.
Multiplying two things usually makes the power bigger, and -2 quietly gets treated as 2.
Question. 10^5 times 10^-2
A common answer. 10^7
The answer. 10^3
Why. Multiplying adds the indices, and 5 add -2 is 3.
Add the indices, signs and all. Then sense check: multiplying by 10^-2 makes it smaller.
The answer is given as 46 x 10^-4 or 48 x 10^2, which are correct amounts in the wrong form.
The arithmetic is right and the form is a separate rule that nobody applies until it is pointed out.
Question. 48 x 10^2 in standard form
A common answer. 48 x 10^2
The answer. 4.8 x 10^3
Why. Standard form allows exactly one digit before the point, so the 48 becomes 4.8 and the power goes up by one.
One digit before the point. If you move it, the power moves with it.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.