Goldbach's Conjecture POW: Was Goldbach correct in
the idea that every 2 digit even number can be written as the sum of two prime numbers.
LIst the even numbers beginning with 10
10 = 5 + 5
12 = 5 + 7
14 = 7 + 7
16 = 11 + 5
18 = 13 +5 etc....What do you notice? Are there any patterns. Go all the way to 98. Would the Sieve of Erastosthenes help this problem?
8 4 2 1 POW....Sure prooves how important following the Order of Operations is!
Steps to the Problem of the Week (points may vary according to the particular problem that week...............)
1. State the question in a way that makes sense.
Yes, this means to rewrite the question 10points
2. Make a plan to solve
Plans can be drawing a diagram, making a chart, guess and check, finding a pattern 10 points
3. Show your work completely
This means everything you do....even the scratch work. 30-40 points
4. Give your solution
This is your answer (s) 10- 20 points
5. Comment on mathematical patterns and discoveries
Very important part where you make observations, and reveal
your thinking. You may also extend the problem here. 30-40 points
6. Rating of the problem
On a scale of 1-10 how does this problem rate for interest and learning? 1 point
Next weeks POW asks: What are the chances that your last two digits of your phone number would be = to the sum of 12? Hint: Think of all the possibilites first. The chance that something occurs is written like a fraction. My last two digits are 33 so, 3 + 3 = 6. Could a graph help?
. SUM FUN To Be Assigned 9/10
Can you find a quick and easy way to add the counting numbers from 1-100.
Plan: Try a quick and easy way to add 1-10 and let that guide you.... Don't forget to comment on the patterns you discover. (I wonder if any addition strategies we've discussed could help?) HINT: WHAT IS 9+1 and 8+2 and 7+ 3 and 6+4 and 5 more? Study that pattern......apply it to a strategy to add 1-100!
Wow We had some great solutions:
There was the pattern of 1-10= 55
11- 20 =155
21-30 = 255 so then you
can see that the next group is = 355 followed by
555 etc. for a
final total when all done as 5050!
One stategy was to add 49 pairs of 100 with
1 + 99 2 + 98 3+ 97 4 + 96 5 + 95 etc all the way to the last pair of 49 + 51 which 49 pairs of 100 = 4900 plus a 100 on the end and a 50 in the middle for 5050
Another student found out the coolest idea...take 1 + 100 and get 101, then times that by 50 (the middle number) WOWEEE
An economic genius named Gauss figured the formula for any group of numbers
from 1- n , simply do n/2 (n+1) = Sum. This means take the last number, divide it by 2 then multiply that times the last number plus one. He did that in second grade! THIS YEAR BRUCE FOUND THIS OUT TOO! WAY TO GO!
To THE PROBLEM OF THE WEEK PAGE:
Throughout the website you will discover some great problems. If you see a colored rectangular box, that could mean that I have hidden an answer. Just keep checking to see if the answer has been revealed.
Beginning the first POW is always the hardest, but then you will get the hang of it!
It is a very good idea to follow all the steps which are clearly stated below. Click on the rubric link to see how the points are calculated for your POW.
Wow! Did this ever have a great shortcut. First if you list the numbers of the possibilities in an organized table you would see:
0 + 0 = 0
0 + 1 =1
0 + 2 = 2
0 + 3 = 3
0 + 4 = 4
0 + 5 = 5
0 + 6 = 6
0 + 7 = 7
0 + 8 = 8
0 + 9 = 9
You can see that for the last two digits, the sum makes a definite pattern of 0-9
That would be a series of 10. So you know the ones as in 1 + 0 =1 AND so on til 1 + 9= 10 would also have a series of 10. So for all of the combos that would be 10 x 10 for a 100 total possibilites.
The new Pow for Nov 15 : Suppose a mean spirited person decided to start a rumor that school would be cancelled for the week of Thanksgiving. The person started the rumor on November 8th by telling one person, then in the way that rumors work the second person told someone as did the first person.so now the rumor mill has started. If each person always told one other person, then how many people would miss the week of THANKSGIVING?: Try a chart, use a calculator too! WHAT PATTERNS CAN YOU FIND....Look for an exponent connection!
Day 1 November 8 First person told one 1=1 = 2 total
Day 2 November 9 Those two each told one =2 + 2 =4 Day 3 November 10 Four people all told another =4 + 4=8
Day 4 November 11 8 people each tell another = 8 + 8 = 16 Day 5 .November 12 16 people each tell another =16+16= 32
Day 6 ETC....finish up until November...21. ON NOV 22 How many are missing from school? (or schools?)
Day 7 .64 1.27
Day 8 1.28 2.55
Day 9 2.56 5.11
Day 10 5.12 10.23
Do you notice any patterns? Look at each column and look across too? You should get a total of :
The magic Square asks you to fill in the numbers 1-9 in a 3x3 grid so that every row, column, and diagonal makes the answer of 15. Be sure to think of great combinations, nice numbers and this will be a snap. Remember Comment!
What is the significance of the median for this problem?
The results found evens in the corners, odd numbers in the t zone, patterns in the adjacent blocks and a major connection to five in the middle .
The Handshake problem asks for you to find a pattern of number of people in the room to how many handshakes occur if everyone shakes hands with every person only once. Do no more than 10 and start with 1 person . Make a chart to help you. You may also use geometric diagrams. Example:
A B means for 2 people there is one handshake (one line between them.
A B C would look more like a triangle with A shaking with B , Awith C and B and C have to shake too . A
Predict what would it be for 100, is there a
The BASEBALL POW....
Using the diagram and the logic clues find out who is on first and all the other positions. For a little bonus where does the little joke "Who's on First" originate? Be sure to explain your comment how you solved the puzzle....
Follow the order of operations (Parenthesis first) (Exponent next, Multiply or Divide, then Add or subtract)
Make the numbers 8 4 2 1 become an equation which equals 0-10 and then 11-20 Finished in Jan of 03
Pascals TRIANGLE POW
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Continue this pattern to the tenth row and then find all kinds of patterns...Try adding each row. Look diagonally ....look in sections....there are so so many!
#2 Can you devise a way of using the five digits 1 3 5 0 0 in such a way that when you write a five column addition problem you end up with the smallest possible sum. You must also never have the decimal point in the same spot, and you may never repeat any number. AND you have to use each of the digits in each of the 5 numbers.
THEN KEEPING THE SAME RULES YOU MUST DO IT AGAIN, THIS TIME WITH THE LARGEST NUMBER POSSIBLE SUM as the answer.
THE FORMATION WOULD BE SOMETHING LIKE . _ _ _ _ _
_ . _ _ _ _ etc but something is wrong.....( I didn't line up my decimal point.)
Keep all POW's in your binder all year!
Make sure your folder is the proper color.
Using 4 numbers with at least three digits each, can you make the sum of your birthday as month, day and year, but do not use a 0 as your first digit.
August 15, 1992 would then be 81592
NEWSPAPER FRONT PAGE
This is to match our December project....you are to create the front page of a newspaper using a catch headline and articles that talk about the math you have learned so far this year. You can incorporate graphs too if you would like. For this POW your question, plan and comment are on a separate piece of paper in the pow folder and the Newspaper part goes in the December project after your graph!
Strategy of 13....Fill in the 8 circles with numbers 1-8 one time each to make 13 in all connective ways.
What patterns do you discover?
Take the digits 0001234 and organize these as four separate numbers to add first to get the smallest sum possible and secondly, to get the largest sum possible. Remember this is four separate numbers. The decimal points must line up properly. You must use all the digits in each number. For example 3,124,000 and .0034210 use all the digits and are different from each other so they would be acceptable. Don't forget Q P W S C ! (The 5 steps to doing a POW are listed above)