A folded mirror: 4th grade math

A folded mirror

Take every number from 1 through hundred. Say 5. For that number write down every combination of addition facts. It is simple, Kavitha said. "Just do this", she illustrated:


0 + 5
1 + 4
2 + 3
"You try for 6", she said

0,6
1,5
2,4
3,3

wow that is a lot of work. Will you ever be done with it? How long will it take. I needed to know how many pages will this fill if this is done for all numbers from 1 to 100.

Roughly half

I know roughly that the number of addition factors are about half the given number. So the total number of facts will be the addition of all numbers from 1 through 100 divided by 2.


1 + 2 + .... + 100

How much will that be? Although I know there is a formula for that, why not have some fun and derive it


1   + 2 + .... + 100
100 + 99 + .... + 1
________________________

101 + 101 + ..... + 101 (100 times)

_________________________ 

So the total is 100 x 101

Say that is roughly 10,000.

Half of that will be 5,000. I need to do this because I have added the series itself. The addition facts will be further half of that. that would be 2500.

How many pages

So she needs to write 2500 facts. Assuming 25 lines on a page and each line accomodating 10 facts, she can accomodate 250 facts for one page. Total is 2500 facts. So if you divide 2500 with 250 facts for each page you end up with 10 pages.

Not too bad. The teacher does not have too-ill but only slightly-ill of intentions.

How long will it take?

it could take 30 minutes to complete a page. In a day you may push a kid to do two pages. So it will take a week to do this for a number like 100.

Say, there is a kid that deserves a twice as much enlightenment and the temptation is to assign to do this exercise for 200. Well 200 is double that of 100.

But if you figure out the math the kid instead of working for 2 weeks actually will have to work for 4 weeks to complete this. If I were to create a table of this relationship between numbers and weeks you will see, perhaps expected, but nevertheless unsettling reality.


100, 1 week
200, 4 weeks (1 month)
300, 9 weeks (2 months and 1 week)
400, 16 weeks (4 months)
500, 25 weeks (6 months and 1 week)
600, 36 weeks (9 months)
700, 49 weeks (almost a year)
800, 64 weeks (more than a year)

what is between 100 and 800, they seem close enough but with respect to this exercise one kid would have wrapped it up in 1 week (however grudginlgy) but the kid with 800 will have to work for an entire year

What if they share

Say you have allocated two students to do this for a number like 200. One might be inclined to give kid1 to do the 1st 100 and give the second kid from 100 to 200. By looking at the above time frames the 1st kid would spend 1 week for her 100 and the second kid will end up spending 3 weeks on her 100 (which is from 101 to 200).

Folded mirror

Imagine a string where you write these numbers from top to bottom: say for 5


0
1
2
3
4
5

If I fold it right smack in the middle


 |
2|3
1|4
0|5
 |

See if the line in the middle were to be a mirror those are your addition facts. where each addition adds up to the same number which is 5.

So for number 6, say, one can turn the addition facts exercise into a folded-number-writing exercise by writing


0 1 2 3
6 5 4 3

and refrain from writing the sum because the sum is always 6.

But again doing so might turn the work pointless.

Looks like a case of elite curiosity killing the cat.