Incredible frequency of careless mistakes

@Rei, I've gone through pretty much the same problems as you in school, and also made the same sort of mistakes! And I'll assume yours isn't a medical problem. You mentioned a big list of mistakes as being your problem. You probably need to ask yourself first: is there something fundamentally wrong in your understanding, or is it that you just make mistakes when pressed for time, etc.? If it's the former, get the concept right first, saving the calculations for later. If it's the latter, it's a much more common problem that we all get into. I can't give you a more specific answer in that case without knowing more about when you make these careless mistakes.

If that seems too abstract, here's some concrete help about your $x=480\times 72$ problem. Don't do it multiple times in the same way - instead check whether the answer makes sense in different ways! (This is an extremely important step that I've seen many people neglect, because they get a correct answer. That's really not understanding maths.) First off , $x = (480 \times 72) > (y=480 \times 70)$, where $y$ should be very easy to do :$ y= 48 \times 7 \times 100 = 33600$. This alone should tell you that , if you got the correct answer in your worksheet, it must be 34560!

Next step: (I assume you know basic algebra) We know that $x = 480 \times 70 + 480 \times 2 = 480 \times 70 + 960$. If that seems too difficult to compute, just take $x \approx 480 \times 70 + 1000 = 34600$. Now, we finally get your answer of $34560$, by subtracting 40!

Ok, suppose you're still not sure. That's fine, we'll do another quick check. It's easier $z= 500 \times 70$ which is just $5\times 7$ with three zeros to follow, so $z=35000$. And $x$ should be a little less; if you're interested you can find the difference. Otherwise, you can check your computations by finding which answer you got was closest to $35000$, and discarding the rest.

What have we gained in the process?

• We admitted that certain computations are hard to do by hand.
• Hence, we looked for concepts to simplify those computations, and looked for approximate answers.
• We checked our answers several times, in multiple ways.

Here're some thumb rules to keep in mind:

• Human beings make errors in a variety of situations, so you're no exception!
• Since you mentioned errors in multi-digit addition, subtraction , etc. you may want to try those first,ignoring everything beyond.It's not important that you compute very fast, but it's very important that you get the algorithms very clearly.
• When you get an answer, just stop and ask, "how can I check this"? Check your answer any way you want - use a calculator, computer, etc.

Here're some simple exercises to try out; you may want to try finding an approximate answer first, without doing any pen-and-paper calculation:

• $99 \times 99$
• $3.14 \times 2.99$
• $-1 - (-1)$

As an example, I'll give some hints for the first one. Let $a=99^2$. I find it hard to square 99, so I'll just square 100 and say $a$ is slightly less than 10000. Next, I'll maybe use an identity like $a=(100-1)^2$ and zoom in on how much $a$ is less than 10000. I'll perform a crude check by stating that $a>90 \times 90 = 8100$. So $a$ lies between $8100$ and $10000$.

Hope I've taken the keen edge off your despair!

Does that help?

Try working out problems with someone else who can watch over your shoulder and point out mistakes in real time as you make them. Then you can correct the mistake right away, before it propagates into rest of your work. This will allow you to notice and appreciate incremental improvements where you need fewer corrections, as opposed to having one less reason your final answer was wrong.

It may also help to reverse the process, to look for mistakes others make as they work, to train your ability to notice mistakes, which you can then apply to your own work.

If you know you are having certain problems, try focus on the specific problem. Also, don't work on the problem in isolation -- work on it in context too.

Also, work out your own way of doing things. From looking at your notebook paper, my first instinct is that part of your difficulty is "skipping steps" -- trying to do several steps in your head when you still have difficulty with the individual steps is a surefire recipe for disaster.

Look at your work there -- all of your errors occur in the same spot, where it looks like you are doing two things:

• You are multiplying $4 \times 7$ in your head
• You are adding $5$ to the result (because it carries over from $8 \times 7$)

You may have better luck working out the problem in a way that doesn't require you to do two things at once. For example, you might compute $480 \times 7$ separately in a separate diagram: $$\begin{matrix} & & 4 & 8 & 0 \\ \times & & & & 7 \\ \hline & & & & 0 \\ & & ? & ? & \\ + & ? & ? \\ \hline & ? & ? & ? & ? & \end{matrix}$$

Or you might try "lattice multiplication" which lets you do the individual single-digit products independently of the addition steps.