First I did it the old school way: 8 + 7 = 5, carry the 10. 10 + 20 + 40 = 70. 70 + 5.
Then I thought, well that's a nice round number, 75, three quarters (three 25s). I looked back at the original addends. Take 3 from 28 to make one quarter; add it to 47 to make two quarters. Same result, three quarters, 75.
It's similar to the thought process when I'm paying cash and trying to get rid of my pennies, nickels, and dimes and get quarters for change. (Our washers and dryers take quarters, so I'm always hoarding quarters.) If the bill is 87 cents, I try to pay with a dollar and (87 - 75 = ) 12 cents in change (dime and two pennies or nickel and seven pennies). Announce to the cashier that I'm giving "a dollar, twelve" and hope that he or she enters $1.12 into the register and lets the register calculate change due (one quarter). Otherwise the cashier may freeze in utter confusion. (Why is he giving me 12 cents when he is already giving me a dollar, and the bill is only 87 cents?? Help!)
Occasionally, I make a mistake in my head and confuse the cashier and myself. Example: the bill is 97 cents. "Here is a dollar, twelve." Uhhh...
My brain recognizes the numbers on the calculator. Using a combination of both the numbers involved and the operators also on the touchpad, my brain instructs my finger to punch the ones necessary for the equation. Once completed it recognizes and utilizes the answer displayed on the screen. Hope that helps.
I've seen this a couple of times, and my approach was a little different each time.
The first time, I did it like this:
20 + 40 = 60
8 + 7 = 15
60 + 15 =75
The second time, I broke it down differently:
(rounded up to the nearest ten, then subtracted the difference)
30 + 50 = 80
2 + 3 = 5
80 - 5 = 75
When I was young, I probably would have done it the old way:
(add up the ones place first, then the tens place)
7 + 8 = 15 (hold the 5 ones, carry the 1 to the tens)
2 + 4 + 1 = 7 tens
7 concatenated 5 = 75
(or, more accurately, 7 * 10 + 5 = 75)
I know a lot of people in their mid-30s and older complain about new math concepts and techniques, but as an adult I started doing math in my head the so-called new way before I knew what students today are learning. Someone I used to work with complained that she can't assist her child with his math homework anymore because the methods for arithmetic have changed, and I understand the concern, but I also recall how most students when I was in school (even in college) hated math, didn't understand the concepts, and avoided anything involving numbers later on in their coursework (and in life). I was quite good at math and was rather lucky that the old methods didn't deter me from pursuing higher math, science, and programming. Teaching kids to be able to think about numbers in the dynamic ways they're learning now, and the way many of us have learned over time to break numbers down to do math in our heads, is good for them. The aren't confused by places, the aren't carrying numbers from one column to another, they aren't trying to figure out abstract concepts; instead, they're playing with the numbers in a more natural way, thinking of them in ways that are naturally mutable, easily broken down and recombined. The underlying principles remain, but how they're applied is much more intuitive.
I made 28 30 and subtracted two from 47 to get 45 then I added 45 and 30 to get 75.
It sounds like there are many paths to the same location. I was the most popular shopping companion at work because I could calculate rapidly in my head the cost of items that were Marked as certain percentage down. I explained repeatedly how I did it but interestingly not one of my friends ever even tried. for example 60% off meant I multiply the rounded cost by four making a 49.99 item cost $20 or if you needed the actual cost 19.96.
I checked groceries in high school and college and I am so old tax had to be figured in my head. Fortunately they had not discovered percentage point taxation in those days. In other words I practiced lots of mathematical calculations without a pencil in my youth and practice makes perfect.
Unfortunately I have to either write it down on paper or imagine writing it down .Increasingly as I age the second option is used less frequently. That does show the power of the written word, but I managed to avoid that in the Bible.
Children really need to learn that showing your working really does get you marks. I used to advise that pupils should always fill every answer space on an exam paper because sometimes even they do not know that what they are writing is NOT rubbish.