Marisa created a game called Paper Pool. Her pool tables were rectangles drawn on grid paper. The pockets at each corner were labeled A (lower left), B (lower right), C (upper right), and D (upper left). Marisa described each table by its size, giving the horizontal length first and the vertical height second. The figure below shows a 6 × 4 table.
How to Play Paper Pool
• The lower-left corner is always corner A, and the labeling continues counterclockwise
with B, C, and D.
• The ball always starts in corner A.
• The ball is hit with an imaginary cue (a stick for hitting a pool ball) so that it travels at a
45° diagonal across the grid.
• If the ball hits a side of the table, it bounces off at a 45° angle and continues its travel.
• The ball continues to travel until it hits a pocket.
Link to webpage:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=28
Link to handouts:
http://juliesgotmail.googlepages.com/home
Questions:
1. In what corner will the ball stop?
2. How many hits will have occurred by the time the ball stops?
3. Write rules (equations) that you could use to determine what will happen to the ball as it travels on a table of any size. Your rules (equations) should tell you, without drawing the path, the corner at which the ball will stop and the number of hits that occurred. (These might be separate equations.)
Support your answers with examples in the form of patterns, spreadsheets, tables, graphs and or illustrations.
If you have any questions please see me.
Ms. Leckman
Here is a short video with some of your responses. My video will go fast so be prepared to use the pause button to spend a minute or two on a solution. I will be posting these on our class webpages to see if we can get a larger view. :-)
K.A. Video
Enjoy. I will try to post more video this year. :-) Yes, I truly am a nerd, I try to learn new "stuff" (sorry Mrs. Campbell) all the time. :-)
Ms. L.
31 comments:
DGT
I solved this equation by using a trial and error method, and these are some things that I came up with to tell where the ball will end up, and how many hits it will take in order to get there.
First off, the easiest thing to tell is that the ball will always end up on a grid line, never on an open space.
X=odd number, Y=even number: ball will end up in “D” pocket
X=odd number, Y=odd number: ball will end in “C” pocket
X=even number, Y=odd number: ball will end up in “B” pocket
X and Y are same=ball will end up in “C” pocket
Non even-even “X” and “Y” axis will cause the sum of “x” and “y” to be the total number of hits, if the numbers are not the same. If the numbers are the same, the ball will only make two hits, the beginning and the ending.
If both the “X” and “Y” are equal, the ball will make one line, and will enter the pocket diagonal from it without bouncing.
AM
1. D
2. 5
3. If at least one number is odd the with times height will equal the number of hits if there both even it will be a number in between, so if its 18 and 20 there will be 19 hits so if one number is odd the formula is:
H+W= hits
If even
High number – low number= the range of numbers between the two it can be
For example:
16 hits and 18 hits equal 17 hits
RW
1.d
2.5
3. If at least one number is odd the width times height will equal the number of hits if their both even it will be a number in between, so if its 18 by 20 there will be 19 hits so if one number is odd the formula is:
H+W= hits
If even
Higher number- lower number= the range of numbers between the two it can be
For Example
16 and 18 hits is 17
K.A.
In my video, for about 10 seconds the actual equations are shown. After that it shows examples proving the equation works then showing the equation again to make sure the watcher understands how it works one by one. After I show all the examples I show the equations again for the watcher to double check or to see if there are any errors.
I got my equations by testing multiple lengths and widths and observing what’s happening. I wrote down any equation that would work for the first height and width then did another set of height and width with the same multiples and cancelled out any equations that didn’t work for that set. Pretty much trial and error at first, then elimination. After a while I ended up with three equations that covered all of the possible heights and widths.
A.L.
This week’s math blog is about paper pool. The game is played by having the ball roll at a 45 degree angle and every time it would hit a side of the table it would bounce again in another 45 degrees. Doing this, it will land in pocket B, C, or D.
In the 6*4 grid, the ball will always land in the D pocket. On this grid, the ball will bounce five times when it hits the pocket. The same is true for the grid sizes 12*8 and 3*2. This is because they are variations of the same
There are some equations or rules that can help determine which pocket the ball will land in.
If both dimensions are the same, the ball will always land in the C pocket…
When H=L, the Pocket will be C
If the length is even, and the Height is odd, it will land in the B pocket…
L=Even*H=odd, the pocket will be B
If the Length is odd, and Height is even, the pocket the ball lands in will always be D…
L=odd*H=even, the pocket will be D
If the Length and Height are both odd, but not equal, the ball will always land in the C pocket…
L=odd*H=odd (but not same to L), the landing pocket is C
If the Length and Height are both even, but not equal, the ball will always land in a pattern of either C or D, depending on the difference between the two numbers. The Higher you raise the Height and lower the Length, there is a pattern made. One result will be C and The next will be D and back to C and so on.
C.V.D.V.
When I was looking at my graphs I found that often times the length and the with added together equaled the number of hits. This fact can be displayed in the equation l+w=h
However this equation only works on tables where the side lengths were not factors of each other. I noticed that when I divided the side length that was the factor of the other into it that I would come up wit the number of hits minus one. This fact can be described in the equation l/w+1=h or w/l+1=h.
However I found that these two equations still didn’t cover all the tables for instance 10x4. This is when I noticed that if the side lengths shared a common factor then the number of hits would need to be determined with a different equation. To figure out the number of hits on this type of table you need to divide the side lengths by their common factor.
When I was determining the pocket that the ball would go into I immediately ruled out A. for the odd widths with odd lengths the pocket is c for odd lengths with even widths the pocket is d.
J.B
1. In corner D
2. 5
3. 1: EN=1+h/2-when the dimensions are both even, the number of hits will equal the mid point number in between.
2: N^2=C -If both dimensions are exactly the same, the number of hits will equal 2 and the ball will end up in pocket C
3:OD+Od=H - When you have 2 odd dimensions, you just need to add them together to get the number of hits.
E. Mu.
The ball will stop in corner D for a 6x4 grid. The ball will have 5 total hits by the time the ball stops. Since the ball travels in a 45 degree angle, it will travel in a path that will always cut the square, which it travels in, in half, diagonally.
Length (L) Height (He) Hits (Hi)
L + He = Hi Unless L and He is equivalent to a smaller L and He
Examples
4 x 5 rectangle 4+5=9 9 hits
2 x 4 rectangle has 3 hits because 1+2=3 hits, both 2 and 4 are divisible by 2, making it
1 x 2 rectangle
1 x 1 rectangle has 2 hits because 1+1=2 hits
3 x 3 rectangle has 2 hits because 3 and 3 are divisible by 3, making it 1 x 1 instead of 3 x 3,
7 x 35 rectangle has 6 hits because both 7 and 35 are divisible by 7. The rectangle now is 1 x 5 1+5=6 hits
If both the length and height are equal, the ball will land in C
L=H ball will land in C
If the length is even, but the height is odd, the ball will land in B.
L=even and H=odd ball will land in B
If length is odd and the height is even, the ball will land in D,
L=odd and H=even ball will land in D
If the length and height are even, it will depend on what the length and height simplified will be, based on above.
L=even and H=even it will depend on what those are simplified to
If the length and height are odd, it will depend on what the length and height simplified will be, based on above.
L=odd and H=odd it will depend on what those are simplified to
CT
This POW deals with pool and dimensions. Marissa created a paper pool table. She wanted to know what kind of rules she could use to find what pocket the ball would go into and the number of hit it would take to get there.
Marissa's ball will stop in corner D. It will take 5 hits for the ball to finally go into the corner.
One equation to determine what will happen for the ball is when the dimensions are both even, the number of hits will equal the mid point number in between them. The equation is EN=l+h/2. EN represents even number and l, h represent the length and height.
The second rule is if dimensions are exactly the same, the number of hits will equal 2 and the ball will end in pocket C. The equation for the rule is N^2=C and N^2=2 hits.
The final rule is when you have two odd dimensions; you just need to add them together to get the number of hits. The ball will go into corner pocket C. The equation for this rule is OD (l) +OD (w) =H. OD stands for odd dimension. H represents the number of hits.
D.M.
The POW for this week asked us to look at a grid and find out where it would stop and where it would land on a 6x4 grid. After looking at the grid, the ball would move at a 45 degree angle. Moving it, it would hit five different times. (You count the starting point.) After that, follow the grid. If you were to look at it like a coordinate plane, the first point or stop would be the origin (0, 0). Then it would go (4, 4), (6, 2), (4, 0), and (0, 4). The end of the grid would be (0, 4) or D. We also discovered if you add the length and the height together, you will get the total number of hits. This can be expressed in the equation, L+H=Hi
1. In what corner will the ball stop?
The ball will stop at D
2. How many hits will have occurred by the time the ball stops?
There will be five hits.
3. Write rules (equations) that you could use to determine what will happen to the ball as it travels on a table of any size. Your rules (equations) should tell you, without drawing the path, the corner at which the ball will stop and the number of hits that occurred. (These might be separate equations.)
L+H=Hi
Length=L Height= H Hits= Hi
Ex. 4+6=10 but, if you can simplify 4x6 then it would be 2X3 which would equal 5. So the distance would not equal 10, but it would equal 5.
If the LxW is equal to 1, 1 then it will be in C. If it equals 1, 2 then it will be in D. For backwards, such as 8, 4 then it will be B.
If it is 1, and an odd number, it would end in C. If it is 1, and an even number, it will be D.
1X 5 equal 6 and it would end in C. 1X 4 and it would end in D.
If both the length and with were equal, it would land in C.
Ex. 7X 7 would hit 1 and would end in C.
If the length is even but the width is odd the ball would land in B.
2X 5 would equal 7 and would land in B.
If length is odd but width is even, it will land in D.
5X 6= 11 this would be in D.
If the length and height are odd, it will depend on what the length and height simplified will be.
Same with even
M.G.
This weeks P.O.W. gave us a paper pool problem in where you can adjust the table’s length and height. The ball always starts in pocket A and there is pockets B, C, and D. You can also see how many hits the ball did before it stopped. The question was in what pocket the ball will stop if the table is 6x4, how many hits the ball did before it stops. Marissa’s ball will stop in pocket D. The number of hits before the ball stops is going to be 5.
When the dimension are both even for length and width the number of hits will equal the midpoint.The equation is: en=l+h/2.
If both dimensions are exactly the same, the number of hits will be equal 2 and the ball will end in the pocket C.This equation is:
n^2=pocket c
n^2=2 hits
When you have two odd dimensions, you just need to add them together to get the number of hits.The ball will go into pocket C. This equation is od(l)+od(h)=H. H stands for number of hits.
Key
en- even number
od- odd dimension
l-length
h-height
RW
1.d
2.5
3. if the dimensions are both odd numbers the ball lands in the c pocket. If length is even and width is odd it falls into b pocket. If length is odd and width is even the balls lands into the d pocket. If both are even the balls lands in c.
If at least one number is odd the width times height will equal the number of hits if their both even it will be a number in between, so if its 18 by 20 there will be 19 hits so if one number is odd the formula is:
H+W= hits
If even
Higher number- lower number= the range of numbers between the two it can be
For Example
16 and 18 hits is 17
A.H.
For all odd lengths, there results
A series of pockets going C, D, C, D...
And, with the exclusion of a length
Of one, continue on to have the
Number of hits equal length +
Width, except for multiples
And factors of the number
This can be expressed as -
l = length
w = width
h = hits
h = l + w
For an even height,
H also = l + w, except that
The length must be odd and not a
Factor or multiple of
The height
For a length that is a factor,
Divide by the greatest common
Factor, and then add together
To get the total number of hits
For numbers that are equivalent,
(4 x 4)
It will always be 2 because it goes
Straight to the opposite hole.
For finding the hole it will fall in, odds will always be a C, D, C pattern, and evens will always follow a pattern to be found by the first few length changes
A.H> :):):):):):):):):):):):)
DG
The POW for this week explains the process of Paper Pool. Marisa is experimenting with the height and length of the table. The always starts in corner A. Marisa is creating rules that will determine which pocket the ball will land in and the total amount of hits before the ball will end in the pocket. If her dimensions were set to 4x6 her ball will have 5 hits before it lands in pocket D.
When the dimensions of the table are both even, the number of hits will be equal to the midpoint number that is in between the two dimensions. The equation that represents this rule is EN= l+h/2 where EN represents even, l is equal to length, and h is equal to height.
If both dimensions are equal to each other, the number of hits will always be 2 and the ball will end in pocket C. The equation that represents this is:
N^2=c and N^2+2 hits
The third and final rule for Paper Pool is when you have two dimensions and add them together it will equal the number of hits OD(1)+OD(w)=h. In the equation OD represents odd dimensions.
K.T.
1) It will stop in corner D with 5 hits.
2) On a 6x4 table, the ball would hit 5 times before stopping.
Rules:
1) If you had made a square with even lines, the ball would end up in the pocket C.
2) If you combine any two odd numbers together, like 9*5 the ball will end up in the C pocket.
3) If the length is an odd number and the width is an even number, the ball would end up in pocket B.
4) If the length is an even number and the width is an odd number, then the ball would end up in pocket D.
1. D
2. 5
3. If they are either odd or the same they will go into the c pocket, if they are both even they will go into the c pocket, if there an even and an odd it will go into the d pocket if the odd number is on the left and if the even is on the right it will go into the b pocket.
If at least one number is odd the with times height will equal the number of hits if there both even it will be a number in between, so if its 18 and 20 there will be 19 hits so if one number is odd the formula is:
H+W= hits
If even
High number – low number= the range of numbers between the two it can be
For example:
16 hits and 18 hits equal 17 hits
D.C.
1) If the square is a 6x4 the ball would stop at corner
2) If the square is a 6x4 there would be five hits.
3) -If it is a square it will land in the opposite corner an have two hits.
- if it is not a square it will either land in any corner but A.
AR
1.The ball will stop at corner D on a 6x4 pool table.
2.The ball will hit 5 times before it hits to corner D.
3.There are five main rules, or equations, that we came up with determining the dimensions. (L= length, W=width, H=hits)
They are:
1.If L=W then H=2
2.L+W=H (if one or both of the dimensions are odd.)
3.½ w=L then H=3
4.½ L+ ½ W=H (if both the dimensions are even.)
P.O.W.
Pool C H
1. The ball will stop in pocket D if the grid is 6X4.
2. The ball will bounce of the walls three times before landing in pocket D.
3. If the dimensions for the length and width are the same, the ball will go straight from A to C. If the width is twice the length the ball will bounce once and then go into D. If the length is twice the width the ball will bounce once and then go into C. If the length is a third the length of the width, then the ball will bounce twice and go into C, this will also happen if the length is three times the length of the width. A formula that will work in almost any dimensions is that if you add both dimensions, then you get the number of hits.
Rule/Equation= L+W/GCF of the length and width.
EX: 5+11/1=16
21+5/1=26
20+10/3=3
If both length and width are even they can land anywhere except for A.
If the length is even then the ball will land in pocket B.
If the height is even it will land in pocket D.
If both the same number it will land in pocket C.
If there is even and odd it will land in B.
POW MR
Formulas
If length and width are both odd but not equal, if you add them it = the number of hits (for example 9x7=16 hits because 9+7=16)
If both length and width are the same number they will land in C
Also if both are odd they will land in C.
If the length is even and width is odd it will land in B
And if length is odd and width is even it will land in D
It will never land on A because the pool ball can’t follow its own path.
Also if length and width 2 even numbers as long as they are not equal, or if the length and width are not factors. Then when added and then subtract the number between length and width.
For example (2+4-3=3 hits)
If there is and odd number in place of length or width then L+W=Hits unless length or width is a multiple or factor.
L+W/greatest common factor of the length and width
Examples
3+4=12/1=12
20+20=40/20=2
POW February 15, 2008 J.K.
1. My first conclusion is that if either length or width is 1, than the ball will land in either D or C.
Out of the possibilities of D or C being the pocket that the ball goes in is if the sum of the length and the width is even than the ball will land in C. If it is odd the ball will land in D.
If the length and the width of the pool table is the same then the ball will land in pocket C.
If the length is even and the width is odd, than the ball will land in either pocket A or B.
The ball will never land in A. No matter what you do you cannot prove this wrong.
2. If the length is twice as large as the width then the number of hits will be 3. This is also true for if the width is twice as large as the length.
If the width is 1 and the length is an even number, then the ball will land in pocket B. If the width is 1 and the length is odd, then the ball will land in pocket C.
3. Equation is number of hits= length plus width divided by GCF = hits. Or h=l+w/GCF
Also this: Pocket B=length even
Pocket C=Length and Height either both even or both odd
Pocket D=height is even
EK
• Whenever the pool table is a number by the same number, it will always land in C with 2 hits. For example: 6x6. This is because of the 45 degree angle
• The ball will never land in A because that is where it starts.
• If the table is odd number by odd number, it will end in C.
• If the horizontal length is half the vertical length, the ball will land in the D pocket.
• If the x is odd and the y is even it will land in D.
• Whenever the x is one and the y is any odd number it will end in C.
• If the vertical length is half the horizontal length, it will land in the B pocket.
JL1
• The ball will never go in the A hole because of the 45 degree angle that it has to start out with.
• When ever the pool tables x and y are the same than the ball will end up in the C hole.
• When ever the x is one and y is any odd number the ball will end up in the C hole
• When the x is odd and y is even it will end in the D hole.
• If the horizontal length is half of the vertical length the ball would end up in the B hole
• When it is a odd and a odd the ball will end up in the C hole
• If the vertical length is half the horizontal then the ball will land in B.
SV
• The ball will never end in the “A” pocket because of the 45 degree angle it starts with.
• When the x and y are the same number it will end in the “C” pocket.
• When the x is odd and y number is even it will end in the “D” pocket.
• If the x and y are odd and odd it ends in “C”.
• If the horizontal is half of the vertical length it will end in the “D”.
• Whenever the x is 1 and the y is any odd number it will end in “C”.
• If the vertical length is half the horizontal length it will end in “B”.
Length (horizontal) Height (vertical) Number of Hits Pocket
4 2 3 B
4 4 2 C
5 3 8 C
6 4 5 D
6 5 11 B
6 7 13 B
7 7 2 C
4 9 13 B
8 5 13 B
1 2 3 D
1 3 4 C
1 4 5 D
2 4
JK POW February 15, 2008 part two
D.E.
POW for Feb 15
1. On the pool table Marisa drew, it first stops at D.
2. It hits five stops before going into D
3. There are many different equations for many different lengths and widths.
• For odd numbers it’s the length added to the width which equals the hits.
• When there are 2 numbers that are the same, they always have the outcome of 2 hits, because then the pool ball goes straight and always into the C hole.
• An odd and even number always go stop at D.
• Any pool table with the widths of odd and even numbers besides 3, added together will become the number of hits before it stops.
• With even numbers any length and width that have even numbers can have their number of hits found by finding the median between the two.
• Any pool table with the length or width of an even with another number twice as much, it is always 3 hits and ends up in the D hole.
• It doesn’t matter what length or width a pool table it is, but it will never go stop at A because of the 45 degree angle it travels in.
• A formula is to find how many hits there all for all length and width is length plus width divided by GCF.
• If it’s odd and even it lands in hole B.
• If it’s odd and odd it lands in hole C.
• If it’s even and odd it will land in hole D.
MS
Questions:
1. In what corner will the ball stop?
The ball will start in corner A and end in corner D.
2. How many hits will have occurred by the time the ball stops?
The ball will have hit the sides of the table four times until it lands at the point.
3. Write rules (equations) that you could use to determine what will happen to the ball as it travels on a table of any size. Your rules (equations) should tell you, without drawing the path, the corner at which the ball will stop and the number of hits that occurred. (These might be separate equations)
(Odd number)(Odd number) =C
(Even number)(Odd number)=B
(even number) (even number) =D or C
E.H.
After The ball fires out from pocket a, it ricochets off of the line two squares left from pocket c, then it diagonally strikes 2 squares down from pocket c, then striking 2 squares left from pocket b, before diagonally flying into pocked d.
Only counting the hits that have occurred away from the pockets, the ball will hit the grid outline 4 times.
There could be multiple equations
For this type of situation, let’s let the height of the board equation be expressed by h and the length of the board be expressed by l, and finally the number of strikes by s. If either the h or the s is an odd number
The sum of both will equal the s. if both the h and the l are the same the s will be zero, and if both of the numbers are even, your s will be in-between your l and h
h + l=s odd and odd, odd and even
h + l= 0 both the same
9h + l0 / 2 = s even and even
P.O.W NH
Pool:
1. The ball will go into D if it’s 6x4
2. The ball bounces three times and hits pocket D id 6 x 4
3. If the dimensions are the same number than the ball will go straight from A to C. If the length is half the width than the ball will go into D. If it’s the opposite way around than the ball will go into C. If the length is equal to a third of the width than the ball will bounce twice and go into pocket C. If the number of hits is an odd number than the ball will go into pockets B and D. If the hit number is even than you’ll get the ball in C. If the Length or width is an odd number than the length and the width added together will equal the hits unless it’s a factor or multiple of the number.
4. L +W/GCF of the length and width.
EX. 5 + 11 / 1 = 16
21 + 5 / 1 = 26
20+10/3= 3
5. Length is even than it will land in pocket = B
If the height is even it = D
If there both the same number = C
If there odd it lands in C
If there both even it = B, C, D
If there even and odd it goes in B
D.E.
POW for Feb 15
On the pool table Marisa drew, it first stops at D.
It hits five stops before going into D
There are many different equations for many different lengths and widths.
For odd numbers it’s the length added to the width which equals the hits.
When there are 2 numbers that are the same, they always have the outcome of 2 hits, because then the pool ball goes straight and always into the C hole.
An odd and even number always go stop at D.
Any pool table with the widths of odd and even numbers besides 3, added together will become the number of hits before it stops.
With even numbers any length and width that have even numbers can have their number of hits found by finding the median between the two.
Any pool table with the length or width of an even with another number twice as much, it is always 3 hits and ends up in the D hole.
It doesn’t matter what length or width a pool table it is, but it will never go stop at A because of the 45 degree angle it travels in.
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