Complete addition and multiplication tables

3.OA.D.9 · take · grade 3

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

The grids below are parts of an addition table and a multiplication table. Find the rule for how the numbers change as you move right (across a row) and down (down a column), then find the numbers that belong in A, B, and C.

Addition table (top grid): the top header row (the numbers being added on) reads $2,\ 4,\ 6$ from the left, and the leftmost column (the numbers being added to) reads $2,\ 4,\ 6,\ 8$ from the top. Inside the grid the entries $4,\ 6,\ 8$ / $6,\ 8,\ A$ / $8,\ 10,\ 12$ are filled in, and one cell is left blank as A.

Multiplication table (bottom grid): the top header row (the numbers being multiplied by) reads $1,\ 3,\ 5$ , and the leftmost column reads $1,\ 3,\ 5$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Show solution

Understand

Two grids are shown: part of an addition table and part of a multiplication table. Using the rule that values grow by a fixed step as you move right and as you move down, find the missing numbers in cells A, B, and C.

Givens

Addition table header row (the numbers added on): 2, 4, 6.
Addition table left column (the numbers added to): 2, 4, 6, 8.
Filled addition entries: row for 2 is 4, 6, 8; row for 4 is 6, 8, A; row for 6 is 8, 10, 12.
Multiplication table header row: 1, 3, 5; left column: 1, 3, 5, with two cells blank as B and C.

Unknowns

The number in cell A (addition table).
The numbers in cells B and C (multiplication table).

Constraints

In the addition table each entry is (left number) + (top number).
In the multiplication table each entry is (left number) x (top number).
Across a row and down a column the values change by a constant step.

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

Each table cell follows a simple rule (sum or product of its row and column heads), and reading along a row or column shows a constant step, so I find the pattern and fill each blank by its row and column heads.

Execute

#5 Look for a Pattern 3.OA.D.9

A sits in the row whose left number is 4 and the column whose top number is 6, because that row reads 6 (=4+2), 8 (=4+4), then A. So A = 4 + 6 = 10. (Across the row the values go up by 2 each step: 6, 8, 10.)

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.

#1 Draw a Diagram 3.OA.D.9

Each multiplication cell is (left number) x (top number). With heads 1, 3, 5 across and 1, 3, 5 down, the full grid is row 1: 1, 3, 5; row 3: 3, 9, 15; row 5: 5, 15, 25.

1{\times}1{=}1,\ 3{\times}3{=}9,\ 5{\times}5{=}25

Laying out the row and column heads as a grid makes every product easy to read off.

#5 Look for a Pattern 3.OA.D.9

The two blank cells are the interior growing cells of this part of the table: B is 3 x 3 = 9 and C is 5 x 5 = 25. These are the cells the row/column step rule lands on (3, 9, 15 across; 5, 15, 25 across).

3 \times 3 = 9,\quad 5 \times 5 = 25

Once you know each cell is row head times column head, every blank is determined.

Answer: A = 10, B = 9, C = 25

Review

In the addition table the row 6, 8, 10 and column 8, A, 10... all step by 2, so A = 10 is consistent. In the multiplication table 9 and 25 are the products 3x3 and 5x5, both matching the row-times-column rule.

Instead of the sum/product rule, you could extend each row and column by its constant step (addition table +2, multiplication rows +1x, +3x, +5x) and read where the blanks fall.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing the constant row/column steps in the addition and multiplication tables to fill the blanks.

💡 Every table cell is just its row head combined with its column head, so once you spot the steady step you can fill any blank!