Hello again, remember to make sure you check your answers against the solutions provided each week. For last week's Gr. 7 question most students seemed to obtain the correct answer of....well, you'll just have to scroll down below. As for the Gr. 8 question I did see a few different answers so make sure you check the correct solution below and try to see where/if you did or did not go wrong.
Further, I want to point out the great efforts by Vivian and Alan to explain their work very thoroughly. And special props to Alan for usually trying to complete both questions each week.
POTW #8 Grade 8 Question
POTW #8 Grade 7 Question
POTW #8 Grade 8 Solution
POTW #8 Grade 7 Solution
Great job on the Grade 8 question Alan. Do other students see why his solution of 20 minutes is correct and finding it to be 12 minutes was actually an error somewhere in your reasoning?
ReplyDeletePOTW 8:
ReplyDelete( and welp I was off by 8 minutes for the last POTW, but it's okay, because Dante should leave 8 mins early so that traffic and such does not make him late ^_^ )
Here's how I solved it:
So this question asks for the maximum amount of people who don't need to call
Now after doing a quick tree diagram to look for patterns in the increasing number I found that the number of employees totally contacted increases by an exponent of three every time, and this exponent triples every time. So I used this to find out how high and can get the total number of employees contacted without going over:
1 + 3^1 = 4
4 + 3^3 = 13
13 + 3^9 = 40
40 + 3^27 121
By doing this I know that a maximum of 40 people can be contacted if everyone contacted 3 people.
To find the maximum amount of people who didn't need to call anyone I had to figure out how many people should each employee call, and to maximize the number of employees who don't need to call anyone, I made the number of employees that an employee should call 3. I then figured out how many employees still need to be contacted:
100 - 40 = 60
Because 40 employees have already been contacted 60 more need to be contacted.
Now I just divided 60 by 3 to find the number of employees that will contact 3 more employees:
60 / 3 = 20
So only 20 more employees have to contact someone. Which leaves the 20 employees and the 60 more employees who don't need to contact anyone,
60 + 20 = 80
80 employees do not need to contact anyone ( I believe at least)
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A/N: I did this POTW so early, and I don't know for sure if I did this correctly XD
and about the last POTW... well I haven't reviewed my method to Alan's or the solution so I'm not sure yet, but I will reply to this about what I find out later ^_^
Don't fret about right or wrong answers in the weekly POTW. It's more about working through the problem in the way you choose, then checking the solutions the following week.
DeleteI made a mistake at the last part my answer, So ignore the part after "So only 20 more employees have to contact someone. "
Deleteand heres what should be after it:
Since 27 people are left to notify others you would subtract 20 from 27 to see the amount of people who don't contact anyone out of the 27, this leaves you with 7 and since 60 more employees were contacted and don't need to contact anyone, you add the numbers together
60 + 7 = 67
67 people do not need to contact anyone.
For what I did wrong on the last POTW
ReplyDeleteI believe I just did the wrong method for the second half, none of my calculations were wrong, just my method.^3^
And here you go Alan, reward for solving the last POTW correctly while every grade 8 couldn't XD ( insert gift box here >>>) 🎁
...welp XD
Yay, thanks Vivian!
DeleteGrade 8 POTW:
ReplyDeleteHow I did it was actually very simple, but I might have done it wrong(I don't really know)
The total number of employees is 100, including the principle. So, if the principle starts off, he/she gives 3 people the information. Then, these three people must give the information as well, making 3*3=9 more people. These 9 people must keep going, so there will be 9*3=27 more people notified. But, not all of these 27 people need to notify other people because 27*3=81, which if I add up all the values will not give me 100, but 121 people! So, some people will not call anyone.
Ignoring 81, I add up all the values before, which is 1+3+9+27.
This gives me a value of 40, which are the people who were contacted. To give the rest of the people a notification, I subtract 100 by 40, which gives me 60. 60 people have not been notified yet.
Since each person notifies 3 people, I divide 60 by 3. giving me 20 people. So, the people who have notified other people by calling is 1+3+9+20 people.
This is a total of 33 people. So, I subtract 100 by 33 which gives me 67 people who don't need to contact anyone else.
-Alan
Grade 7 POTW:
ReplyDeleteI may have miscalculated on this question since I am not very good at these types, but I guess I'll have to try, though I might edit this answer later on.
I think that the answer is 13 boxes.
Because I can't have the same amount of pigeons in each cage, I would start with 1 and go consecutively.
Adding up to 13, I would get 91. But, since this value is still less than 100, the last one must me 9 more. The last box would be 22 pigeons.
There might be more than this because I have a large difference between the last cage and second last cage, but I think this is it. Hopefully my answer is right, because I don't want to go through all that time to change my answers again!
-Alan
I think the maximum amount of cages is 13.
ReplyDeleteSince I knew the minimum amount of birds in a cage is 1, to get the most cages, they should be consecutive starting from 1. So adding up the consecutive numbers from 1 to 13 is 91. However, the next consecutive number does not add up to 100, it adds up to 105. So that doesn't work. And also, you can't have a cage with 9 pigeons to make 100 pigeons because there was already one cage with that amount of pigeons. So the next number of pigeons in a cage is 22. Adding the consecutive numbers from 1-12 is 78, the 22 will add up to 100.
So in the end, there are 13 cages. (I couldn't really explain it well).
The minimum amount of birds a cage can hold is 1 so I put only 1 bird in the first cage. The next smallest amount is 2 so I put 2 birds in the second cage. I did this until I reached 13 birds in 13 cages. That got a total of 91 birds. I can't put 14 birds in the next cage because there are only 100 birds and 91+14=105 which is too many birds. I can't any number of birds higher than 14 because they would also exceed 100. This means that I have to add an extra 9 birds (amount of birds left after I use 13 cages) to any cage that has 4 nor more birds in it. If I add it to the last one,(the 13th one) I would have 22 birds in it instead of 13. I used 13 cages.
ReplyDeleteThe maximum amount of cages required to house the pigeons is 13.
The maximum number of pigeons the cages can hold is 13. The rest of the work is hard copy.
ReplyDeleteFor this I think that it'll be 13 cages. I did this because by first just having each cage go up by 1. Then when you get to 13 birds in 1 cage, you have 91 birds. But then if you where to put 14 in the next cage you would've got 105. Then since we're 5 over if we were to get rid of the cage, it would go 1,2,3,4,6,7,8,9,10,11,12,13,14. That all added together is 100 and in 13 cages.
ReplyDeleteIn the question, it states that you must at least have 1 pigeon in each cage, but then the amount of pigeons in each one would have to be completely different. So, since we have 100 pigeons, I start at the smallest number of pigeons, which is 1. Then, since I cant have the same number, I go to 2 and then 3 and on. But then when we get to 13 cages, you get 91 birds. This means that you cant go to 14 pigeons in a cage, so you have to put in more pigeons. So then 100-91 would be 9 more pigeons to put inside the cage as then it would be 22 pigeons in the 13th cage.
ReplyDeleteTherefore, you need 13 cages.
The maximum number of cages is 13.
ReplyDelete1 cage-1 pigeon
2 cages- 2 pigeons
3 cages-3 pigeons
...
11 cages-11 pigeons
12 cages-12 pigeons
13 cages- 22 pigeons
If there was 14 cages then the total of pigeons would be 105 which is over 100 so the extra 9 pigeons could be added to the thirteenth cage making it twenty two which is still a unique number.
The maximum amount of cages is 13.
ReplyDeleteWork on hard copy.
1+2+3+4+5=15 6+7+8+9+10=40 55+11+12+13=91 13 cages but...
ReplyDelete13+9 [Leftover]= 22. 13 Cages contain: 1,2,3,4,5,6,7,8,9,10,11,12,22.
Therefore, the maximal amount of cages would be 13.
This is how I solved the grade 7 POTW. I have to find out the maximum amount of cages for 100 pigeons, but with three restrictions:
ReplyDeleteOnly one pigeon per cage, no cages inside other cages, and no cages can have the same amount of pigeons.
so, I added one pigeon per cage everytime:
C1 has 1 pigeons
C2 has 2 pigeons
C3 has 3 pigeons
C4 has 4 pigeons
C5 has 5 pigeons
C6 has 6 pigeons
C7 has 7 pigeons
C8 has 8 pigeons
and so on, until 13 cages.
1+2+3+4+5+6+7+8+9+10+11+12+13=91. That means that there are nine more pigeons to be caged, but I already used 9, and 14 is too much, so I added 9 to 13 to make 22 for the last cage. So, the maximum amount of cages to cage 100 pigeons with those three restrictions, is 13.
13, because 1+2+3+4+5+6+7+8+9+10+11+12+13=91. There is still 9 left over so 5+9=14, so you would need 13 cages
ReplyDelete67 people do not need to make a call, as doing the math 33 people would. I completed this on paper.
ReplyDeleteGrade 7 POTW:
ReplyDeleteBecause each cage must contain at least one pigeon, we know automaticlly there is 100 cages. And because there can't be 2 pigeons with the same amount, we add up consecutive numbers, starting from 1-13. At 13, there would be 91 pigeons in 13 cages, which means that on 14, it would go over above 100 to 105. So, I brought it down to 12 cages, leaving me with 78 pigeons. I then added one more cage that held 22, giving me a total of 100 pigeons.
Grade 7 POTW
ReplyDeleteSo first we know that the maximum amount of cages needed would mean the least amount of birds in each cage as that would guarantee a large total of cages. So we'd start with 1 in 1 cage. The next would be 1+2+3+4 ect so that added up there would be a limited amount in each cage (I'm pretty sure that does not make any sense whatsoever) Anyways I continued to do that until i reached 100 or closes to 100 number.
1+2+3+4+5+6+7+8+9+11+12+13=91
That brings us to 13 birds in the 13 cage. Obviously you can't have 14 in the next as there is only 9 birds. So you add the 9 into any cage that brings to total to a number that doesn't repeat. AKA any number over 4 (4+9=13 but anything over that would bring it too 14-15-16-17 ect) So i'll just add it too 13 so that it brings the 13th cage's total to 22. So the max would be 13 cages.
13 Cages Max.
ReplyDeleteAnswer on Papaer
First let's see how many people will be called, and keep going until we go over or get the right answer.
ReplyDelete1+3+9+27+81=121. That's over, so let's see how many people were notified if we take away the 81.
1+3+9+27=40
Now we're under. We need to find out how many people still need to be contacted.
100-40=60
60 people need to be contacted. I used guess and check to see how many people would have to call 3 other people and notify everyone (no over, no under)
25*3=75
20*3=60
So, instead of 27 people who need to make calls in the last group of people, it would be 20. Those 7 people would then not need to make any calls. The equation would now be:
1+3+9+20=33. This means that 33 people called 3 other people, and together everyone was notified.
100-33=67.
67 people did not have to make calls to notify everyone.
I did my POTW #8 in my math notebook and got an answer of 67.
ReplyDeleteI did the POTW on hard copy and my answer for the grade 8 one is that 60 people do not need to make a call
ReplyDeleteTo solve this question I made a large fan like chart then I counted how many times the call would be passed on until I hit 100 people. Then I subtracted that amount which was 33 from 100 to get 67. Thus 67 people don't have to make a call.
ReplyDeleteFirst, i read through all the criteria to understand what to boundaries are. The main two points in the success criteria were that there has to be at least 1 pigeon in each cage and, that no two cages can have to same amount of pigeons. This means that if we want to find the maximum amount of cages, we would use consecutive numbers. First, we'd start of with 1, then keep going up until the total of all the numbers added is 100 or the highest number before going over 100. I drew this out to make it easier.
ReplyDelete1+2+3+4+5+6+7+8+9+10+11+12+13=93 1=1 pigeon
This all fits the criteria because each cages has at least 1 pigeon, no two cages have the same amount of pigeons, and no cages are inside other cages. This means that the max we could go up to is 13 cages. If we added one more cage, the minimum amount of pigeons would be 14. If we did that, our total would go over 100. Ths means that the maximum amount of pigeon cages would be 13.
In order to determine the maximum number of cages required, whilst complying with all the restrictions, we must have a consecutive number of pigeons in each cage, starting with the number 1.
ReplyDeleteI decided to add these consecutive numbers up, to see if/when they equalled 100.
1+2+3+4+5+6+7+8+9+10+11+12+13+14=105
Since the number of pigeons is higher than 105, a cage must be taken away, and the total would equal 91. However, this would not work, since according to the rules, the number of pigeons in the cage must be over 14. You then must take away another cage, which would mean that the maximum amount of pigeon cages would be 13
Here is how I went about solving the Grade 7 POTW:
ReplyDeleteThe two main components of the "rules" were that there had to be at least one pigeon to a cage, and no two cages could have the same number of pigeons. The simplest way to solve this question, would be to put one pigeon in the first cage, followed by two in the next, three in the one after that, and so on and so forth. You do this until you reach the twelfth cage. At that moment, the sum is equal to 78 pigeons. After putting a final 13 pigeons in the thirteenth cage, you are left with a total of 91. That means that there are 9 left. However, we already have a cage with 9 pigeons in it. So we are left with two solutions. We could add 9 to the final cage (13) and get 22, or just add one pigeon to each of the last nine cages and match our total of 100. Therefore, the maximum amount of cages needed is 13 because if you have 14, then 91 plus 14 equals 105 and that is five over the given amount of pigeons.
63 people don't need to be contacted
ReplyDeletehttps://scratch.mit.edu/projects/128910829/#player