Common Sums
Collect the data
Make a sheet of paper with 12 rows. In each row, put 8 squares. The leftmost column of squares has the numbers from 1 to 12 written in order in the squares. Put 1 token on each of the 12 numbers. Start rolling a pair of dice. After each roll, move the token for the sum of the dice one square to the right. The goal for each token is to be the first to get all the way to the right across the page.
Let your child come up with some questions to investigate. Some natural questions are:
- Which token(s) will win and why?
- Which tokens do well and which ones do poorly?
- Which token is the worst?
- How will the winners change if the row lengths are made shorter or longer?
Have your child explain their ideas about the answers to these questions, and then investigate their ideas by running experiments.
Add a competitive element to this by guessing which token will win before the round starts.
Bonus Material
Introduction
Make a sheet of paper with 12 rows. In each row, put 8 squares. The leftmost column of squares has the numbers from 1 to 12 written in the squares. Put 1 token on each of the 12 numbers. Start rolling a pair of dice. After each roll, move the token for the sum of the dice one square to the right. The goal for each token is to be the first to get all the way to the right across the page.
Let your child come up with some questions to investigate. Some natural questions are:
Which token will win and why?
Which tokens do well and which ones do poorly?
Which token is the worst?
How will the winners change if the rows are changed to have fewer squares or more squares?
Have your child explain their ideas about the answers to these questions, and then investigate their ideas by running experiments.
Add a competitive element to this by guessing which token will win before the round starts.
Analysis
As with the analysis of the game Zoo Rescue, the simplest way to analyze this is to list out all the possibilities and count how many times each sum occurs.
When you count how many times each sum occurs, the frequencies are:
1 – 0, 2 – 1, 3 – 2, 4 – 3, 5 – 4, 6 – 5, 7 – 6, 8 – 5, 9 – 4, 10 – 3, 11 – 2, 12 – 1.
By the way, these are good numbers to remember for any dice game that involves summing the two dice!
So, 1 will always lose and 7 is the most likely to win. However, the difference in frequency between 7 and 6 or 8 is not very great. If you just do a few rolls, it would be very hard to predict with any certainty which one would win. It is only when you do a great many rolls that you can guarantee that 7 will win eventually.
Helping your child
Investigations are meant for your child to play with and think about. Let your child explore these looking for interesting patterns and beautiful relationships. Resist the temptation to unveil what is going on and give the answer. If your child seems to have reached a dead end, suggest that they come back to the investigation at a future time to play with it again.
Many investigations benefit from organizing results, and this is a great thing for you to help your child with. Help them make tables, drawings, or whatever may help them see more easily what is going on. And of course, it is perfectly fine to give them gentle nudges from time to time in the right direction. Remember that your child will learn a lot by developing persistence and learning how to look more deeply at things.