Depth · one concept from first principles · step 2 of 4
A is in the row headed 4 and the column headed 6, so A = 4 + 6 = 10.
Each addition-table cell is just its row head plus its column head, and along a row the answers grow by the header step.
3.OA.D.9Look For A PatternCell A in the addition table equals its row's left number 4 added to its column's top number 6.
An addition table writes in each cell the whole you get by putting its two head numbers together, and A's heads are the left number 4 and the top number 6.
Joining a group of 4 with a group of 6 makes one whole built from both parts, and adding is exactly how that combined whole is counted.
The row's steady step backs this up: across the row the top numbers rise by 2 (2, 4, 6), so each entry sits 2 above the one before, matching 4 added to a top number that is 2 larger.
Each extra 2 in the top number lifts the sum by exactly 2, because 4 + 6 regroups as (4 + 4) + 2 — the previous entry with one more step of 2 added on.
Each multiplication cell is (left number) x (top number): row 1 gives 1, 3, 5; row 3 gives 3, 9, 15; row 5 gives 5, 15, 25.
Laying out the row and column heads as a grid makes every product easy to read off.
3.OA.D.9Draw A DiagramThe two blanks are B = 3 x 3 = 9 and C = 5 x 5 = 25, the interior cells the row and column rules land on.
Once you know each cell is row head times column head, every blank is determined.
3.OA.D.9Look For A PatternEvery table cell is just its row head combined with its column head, so once you spot the steady step you can fill any blank!
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →