Problem
Count outcomes with a table
Two dice numbered 1 to 6 are rolled and we take the sum. Order counts, so (1,5) and (5,1) are different, and the sums run from 2 to 12. Fill the 6x6 table, count the ways to make 6, and name the most frequent sum.
Reasoning · Grade 2-2 Problem Solving with Tables
Your answer
Watch out Eleven sums from 2 to 12 look equally likely, but each sum fills a different number of cells — one for 2, six for 7.
How to solve
Strategy Make a Systematic List — Filling the 6 by 6 table is a complete, organized list of all 36 outcomes, so no sum is missed. Once the table is full, counting how many cells hold a 6 answers part (2), and spotting which sum appears in the most cells answers part (3).
1STEP 1
Fill in the sum table
Adding the two dice per cell fills from 2 down to 12.
cell = (row die) + (column die)
Adding the two numbers in each cell is just basic addition within 20, done 36 times in an orderly grid.
2.OA.B.2Make A Systematic List2STEP 2
Count the ways to make 6
The pairs making 6 run (1,5) to (5,1): 5 ways.
1+5, 2+4, 3+3, 4+2, 5+1 → 5 ways
Each cell that shows a 6 is one way to roll a sum of 6, so we just count those cells.
2.OA.A.1Make A Systematic ListEvery cell of the table showing 6 is one way to roll a sum of 6, so counting those cells counts the ways.
Why?
One cell stands for one roll of the two dice and one roll fills one cell, so cells and outcomes match up exactly.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The first die's face and the second die's face do not restrict each other, so all six by six pairings really occur.
🧱Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.
3STEP 3
Find the most common sum
Counting cells, 7 takes six — the whole diagonal — the most of any sum.
#(7) = 6 ＞ #(6) = #(8) = 5 ＞ …
The counts climb to a peak in the middle and come back down; the middle sum, 7, has the most ways.
2.OA.A.1Look For A PatternAnswer
5 ways to make 6, and 7 is the most frequent sum
There are 6 x 6 = 36 cells in total. Adding up the counts 1+2+3+4+5+6+5+4+3+2+1 gives 36, which matches, so no outcome was missed or double-counted. The counts are symmetric and peak at 7, which makes sense because 7 is the middle sum and can be made the most ways.
Takeaway
Fill the grid neatly and the answers fall out by counting - the middle sum 7 wins because it has the most ways, all with Grade 2 addition!
- Fill in the sum table
- Count the ways to make 6
- Find the most common sum