Sarah’s Passing Drill

In another edition of revisiting problems from the CAMI vaults, at this month’s meeting we went back to further explore a number pattern we first looked at in January 2017 (Carl’s Basketball Problem).

We started off discussing WHAT IS SIMILAR? WHAT IS DIFFERENT? looking at these four expressions:

WHAT IS SIMILAR?

The totals are all even (10, 20, 20, 42)
All addition
Each one is made up of factors of the -largest number
Nothing gets bigger as you move to the center – they either get smaller or stay the same
Each expression is symmetrical – they are palindromes – they read the same from left to right as right to left

WHAT IS DIFFERENT?

Some numbers are even and some are odd
2 of them have repetition of the same number
The 2 that have different numbers are multiples of the other numbers
Each expression is adding a different number of terms
the first one has 3 digits, then 4, then 5, then 6
the 6+3+2+3+6 is the only one with both odd and even numbers
the first numbers are consecutive and the number of digits in each are consecutive
the first numbers are 4, 5, 6, 7
the number of terms is 3, 4, 5, 6

Then Mark explain a drill for passing a ball.

“My wife was a big soccer player growing up and she was telling me about this passing drill they used to do. The coach would put them in groups and they’d play a game called ’Rounds’. Let’s say we were a group of 4. The first round, I pass the pass to the person to my left and everyone does the same until the ball comes back to me. When it gets back to me, that is the end of the round. For the next round, I skip one person, and everyone else does the same until the ball again comes back to me. For the third round, I skip two people, pass the ball and everyone else does the same. Once again, the round ends when the ball gets back to me. The game ends when I would skip everyone else in the circle and pass the ball to myself.”

In summary:

In round 1, we just pass it around the circle. Everyone passes to the person next to them, until it comes back to me.
In round 2, we skip one person and pass it around the circle again, until it comes back to me.
For each round, we keep skipping an additional player.
The game ends when I would pass it to myself.

Then each of us created our own visual representation to make sense of how the passing drill works (and to help us ask some clarifying questions).

Then Mark shared one more visual representation to make sure it was clear to every one how the passing drill worked. Below is a representation of a group of four people passing the ball.

In this diagram, the purple kite represents the starting player and the blue rhombi represent the other players. In round 1, there are 4 passes (shown in teal). In round 2, there are 2 passes (shown in burgandy). In round 3, there are 4 passes (shown in orange). For a total of 10 passes between 4 people.

PROBLEM-POSING

Is there a connection between the game and what we saw in the beginning of the meeting? (Like the 4+2+4=10) – Deneise
If we start out with an odd number in the initial game, will the game continue into infinity? – Deneise
Does adding more players necessarily add more rounds? – Cristina
Can we find some sort of pattern if we use a different number of players?- Alex
In certain rounds, some players don’t get to kick the ball. How much does each player get to kick the ball? – Sarah
Are there any number of players that won’t work? That would “break” the game? Go infinite? – Leo
Is there a # of players that would keep the ball from returning to the start? – Deneise
Why does my picture look almost 3-D? Are these patterns found in Nature? Is this related to the Fibonacci sequence or something like that? – Leo
How can you predict the number of total passes based on the number of players? – Eric
How can you predict the sequence of passes from the number of players? – Sarah
(For example with 4, the sequence was 4,2,4)
If you start with an odd number of players is there a difference than starting with an even number of players? Is there only 1 pattern? 2 patterns? – Patricia
Is there some relationship to prime numbers and factors? – Alex

We started off with some independent exploration time and then broke into three groups.

Mark had some data to share and offered everyone a choice.

If we wanted to, we could choose to generate our own data about the different numbers of passes for different numbers of players and in different rounds. Some folks choose this option. In addition to pencil and paper, Mark shared these two tools: