Practice · Complete addition and multiplication tables · Sensim Math

Variant 1 easy

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 $3,\ 6,\ 9$ from the left, and the leftmost column (the numbers being added to) reads $3,\ 6,\ 9$ from the top. Inside the grid the entries $6$ ,\ $9$ ,\ $A$ / $9$ ,\ $12$ ,\ $15$ / $12$ ,\ $15$ ,\ $18$ 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 $2,\ 3,\ 4$ , and the leftmost column reads $2,\ 3,\ 4$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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): 3, 6, 9.
Addition table left column (the numbers added to): 3, 6, 9.
Multiplication table header row: 2, 3, 4; left column: 2, 3, 4, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

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

A sits in the row whose left number is 3 and the column whose top number is 9, so A = 3 + 9 = 12. (Across a row the values go up by 3 each step.)

3 + 9 = 12

Each addition-table cell is just its row head plus its column head, and along a row the answers grow by the header step.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 2, 3, 4 across and down, every cell is the product of its two heads.

3 \times 3 = 9,\ 4 \times 4 = 16

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

3Fill B and C

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

B is on the diagonal where left = top = 3, so B = 3 x 3 = 9. C is where left = top = 4, so C = 4 x 4 = 16.

3 \times 3 = 9,\quad 4 \times 4 = 16

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

Answer: A = 12, B = 9, C = 16

4 · Reviewdoes it hold up?

In the addition table the entries step by 3 along each row and column, so A = 12 is consistent. In the multiplication table 9 and 16 are the products 3x3 and 4x4, both matching the row-times-column rule.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Variant 2 easy

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 $1,\ 3,\ 5$ from the left, and the leftmost column (the numbers being added to) reads $2,\ 4,\ 6$ from the top. Inside the grid the entries $3$ ,\ $5$ ,\ $7$ / $5$ ,\ $7$ ,\ $9$ / $A$ ,\ $9$ ,\ $11$ 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 $2,\ 4,\ 6,\ 8$ , and the leftmost column reads $2,\ 4,\ 6,\ 8$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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): 1, 3, 5.
Addition table left column (the numbers added to): 2, 4, 6.
Multiplication table header row: 2, 4, 6, 8; left column: 2, 4, 6, 8, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

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

A sits in the row whose left number is 6 and the column whose top number is 1, so A = 6 + 1 = 7. (Across a row the values go up by 2 each step.)

6 + 1 = 7

Each addition-table cell is just its row head plus its column head, and along a row the answers grow by the header step.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 2, 4, 6, 8 across and down, every cell is the product of its two heads.

4 \times 4 = 16,\ 8 \times 8 = 64

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

3Fill B and C

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

B is on the diagonal where left = top = 4, so B = 4 x 4 = 16. C is where left = top = 8, so C = 8 x 8 = 64.

4 \times 4 = 16,\quad 8 \times 8 = 64

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

Answer: A = 7, B = 16, C = 64

4 · Reviewdoes it hold up?

In the addition table the entries step by 2 along each row and column, so A = 7 is consistent. In the multiplication table 16 and 64 are the products 4x4 and 8x8, both matching the row-times-column rule.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Variant 3 medium

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 $10,\ 20,\ 30$ from the left, and the leftmost column (the numbers being added to) reads $10,\ 20,\ 30$ from the top. Inside the grid the entries $20$ ,\ $30$ ,\ $40$ / $30$ ,\ $A$ ,\ $50$ / $40$ ,\ $50$ ,\ $60$ 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 $4,\ 5,\ 6$ , and the leftmost column reads $4,\ 5,\ 6$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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): 10, 20, 30.
Addition table left column (the numbers added to): 10, 20, 30.
Multiplication table header row: 4, 5, 6; left column: 4, 5, 6, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

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

A sits in the row whose left number is 20 and the column whose top number is 20, so A = 20 + 20 = 40. (Across a row the values go up by 10 each step.)

20 + 20 = 40

Each addition-table cell is just its row head plus its column head, and along a row the answers grow by the header step.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 4, 5, 6 across and down, every cell is the product of its two heads.

5 \times 5 = 25,\ 6 \times 6 = 36

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

3Fill B and C

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

B is on the diagonal where left = top = 5, so B = 5 x 5 = 25. C is where left = top = 6, so C = 6 x 6 = 36.

5 \times 5 = 25,\quad 6 \times 6 = 36

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

Answer: A = 40, B = 25, C = 36

4 · Reviewdoes it hold up?

In the addition table the entries step by 10 along each row and column, so A = 40 is consistent. In the multiplication table 25 and 36 are the products 5x5 and 6x6, both matching the row-times-column rule.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Variant 4 medium

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 $4,\ 8,\ 12$ from the left, and the leftmost column (the numbers being added to) reads $4,\ 8,\ 12$ from the top. Inside the grid the entries $8$ ,\ $12$ ,\ $16$ / $12$ ,\ $16$ ,\ $A$ / $16$ ,\ $20$ ,\ $24$ 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 $5,\ 6,\ 7,\ 8$ , and the leftmost column reads $5,\ 6,\ 7,\ 8$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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): 4, 8, 12.
Addition table left column (the numbers added to): 4, 8, 12.
Multiplication table header row: 5, 6, 7, 8; left column: 5, 6, 7, 8, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

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

A sits in the row whose left number is 8 and the column whose top number is 12, so A = 8 + 12 = 20. (Across a row the values go up by 4 each step.)

8 + 12 = 20

Each addition-table cell is just its row head plus its column head, and along a row the answers grow by the header step.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 5, 6, 7, 8 across and down, every cell is the product of its two heads.

5 \times 5 = 25,\ 8 \times 8 = 64

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

3Fill B and C

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

B is on the diagonal where left = top = 5, so B = 5 x 5 = 25. C is where left = top = 8, so C = 8 x 8 = 64.

5 \times 5 = 25,\quad 8 \times 8 = 64

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

Answer: A = 20, B = 25, C = 64

4 · Reviewdoes it hold up?

In the addition table the entries step by 4 along each row and column, so A = 20 is consistent. In the multiplication table 25 and 64 are the products 5x5 and 8x8, both matching the row-times-column rule.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Variant 5 medium

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 $1,\ 2,\ 3$ from the left, and the leftmost column (the numbers being added to) reads $1,\ 2,\ 3,\ 4$ from the top. Inside the grid the entries $2$ ,\ $3$ ,\ $4$ / $3$ ,\ $4$ ,\ $5$ / $4$ ,\ $A$ ,\ $6$ / $5$ ,\ $6$ ,\ $7$ 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 $2,\ 4,\ 6$ , and the leftmost column reads $2,\ 4,\ 6$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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): 1, 2, 3.
Addition table left column (the numbers added to): 1, 2, 3, 4.
Multiplication table header row: 2, 4, 6; left column: 2, 4, 6, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

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

A sits in the row whose left number is 3 and the column whose top number is 2, so A = 3 + 2 = 5. (Across a row the values go up by 1 each step.)

3 + 2 = 5

Each addition-table cell is just its row head plus its column head, and along a row the answers grow by the header step.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 2, 4, 6 across and down, every cell is the product of its two heads.

2 \times 2 = 4,\ 6 \times 6 = 36

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

3Fill B and C

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

B is on the diagonal where left = top = 2, so B = 2 x 2 = 4. C is where left = top = 6, so C = 6 x 6 = 36.

2 \times 2 = 4,\quad 6 \times 6 = 36

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

Answer: A = 5, B = 4, C = 36

4 · Reviewdoes it hold up?

In the addition table the entries step by 1 along each row and column, so A = 5 is consistent. In the multiplication table 4 and 36 are the products 2x2 and 6x6, both matching the row-times-column rule.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Variant 6 hard

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 $2,\ 4,\ 6,\ 8$ from the left, and the leftmost column (the numbers being added to) reads $2,\ 4,\ 6$ from the top. Inside the grid the entries $4$ ,\ $6$ ,\ $8$ ,\ $A$ / $6$ ,\ $8$ ,\ $10$ ,\ $12$ / $8$ ,\ $10$ ,\ $12$ ,\ $14$ 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 $6,\ 7,\ 8$ , and the leftmost column reads $6,\ 7,\ 8$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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, 8.
Addition table left column (the numbers added to): 2, 4, 6.
Multiplication table header row: 6, 7, 8; left column: 6, 7, 8, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

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

A sits in the row whose left number is 2 and the column whose top number is 8, so A = 2 + 8 = 10. (Across a row the values go up by 2 each step.)

2 + 8 = 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.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 6, 7, 8 across and down, every cell is the product of its two heads.

6 \times 6 = 36,\ 8 \times 8 = 64

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

3Fill B and C

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

B is on the diagonal where left = top = 6, so B = 6 x 6 = 36. C is where left = top = 8, so C = 8 x 8 = 64.

6 \times 6 = 36,\quad 8 \times 8 = 64

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

Answer: A = 10, B = 36, C = 64

4 · Reviewdoes it hold up?

In the addition table the entries step by 2 along each row and column, so A = 10 is consistent. In the multiplication table 36 and 64 are the products 6x6 and 8x8, both matching the row-times-column rule.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Variant 7 medium

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 $4$ ,\ $6$ ,\ $8$ / $6$ ,\ $8$ ,\ $A$ / $8$ ,\ $10$ ,\ $12$ / $10$ ,\ $12$ ,\ $14$ 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.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.
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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

#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, so A = 4 + 6 = 10. (Across a row the values go up by 2 each step.)

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.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 1, 3, 5 across and down, every cell is the product of its two heads.

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.

3Fill B and C

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

B is on the diagonal where left = top = 3, so B = 3 x 3 = 9. C is where left = top = 5, so C = 5 x 5 = 25.

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

4 · Reviewdoes it hold up?

In the addition table the entries step by 2 along each row and column, 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.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Variant 8 hard

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 $5,\ 10,\ 15$ from the left, and the leftmost column (the numbers being added to) reads $5,\ 10,\ 15,\ 20$ from the top. Inside the grid the entries $10$ ,\ $15$ ,\ $20$ / $15$ ,\ $20$ ,\ $25$ / $20$ ,\ $25$ ,\ $30$ / $A$ ,\ $30$ ,\ $35$ 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 $3,\ 5,\ 7$ , and the leftmost column reads $3,\ 5,\ 7$ from the top. In this part of the multiplication table, two cells are left blank as B and C.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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): 5, 10, 15.
Addition table left column (the numbers added to): 5, 10, 15, 20.
Multiplication table header row: 3, 5, 7; left column: 3, 5, 7, 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.

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Fill A in the addition table

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

A sits in the row whose left number is 20 and the column whose top number is 5, so A = 20 + 5 = 25. (Across a row the values go up by 5 each step.)

20 + 5 = 25

Each addition-table cell is just its row head plus its column head, and along a row the answers grow by the header step.

2Build the multiplication table rule

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

Each multiplication cell is (left number) x (top number). With heads 3, 5, 7 across and down, every cell is the product of its two heads.

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.

3Fill B and C

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

B is on the diagonal where left = top = 3, so B = 3 x 3 = 9. C is where left = top = 5, so C = 5 x 5 = 25.

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 = 25, B = 9, C = 25

4 · Reviewdoes it hold up?

In the addition table the entries step by 5 along each row and column, so A = 25 is consistent. In the multiplication table 9 and 25 are the products 3x3 and 5x5, both matching the row-times-column rule.

Another way: Instead of the sum/product rule, extend each row and column by its constant step and read where the blanks fall.

Standardsmin 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.

💡Takeaway. 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!

Complete addition and multiplication tables · 8 practice problems

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