Sensim Math · Depth 한국어

← 2-2 · Complete addition and multiplication tables · Generalize a Growing Pattern into a Rule

Complete addition and multiplication tables · 8 practice problems

3.OA.D.9

Generated variants — 8

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: A = 20, B = 25, C = 64

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, 124,\ 8,\ 12 from the left, and the leftmost column (the numbers being added to) reads 4, 8, 124,\ 8,\ 12 from the top. Inside the grid the entries 88,\ 1212,\ 1616 / 1212,\ 1616,\ AA / 1616,\ 2020,\ 2424 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, 85,\ 6,\ 7,\ 8, and the leftmost column reads 5, 6, 7, 85,\ 6,\ 7,\ 8 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 4 8 12 4 8 12 8 12 16 12 16 A 16 20 24 Multiplication table × 5 6 7 8 5 6 7 8 B 30 35 40 30 36 42 48 35 42 49 56 40 48 56 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): 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.

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 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=208 + 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.
#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.
5×5=25, 8×8=645 \times 5 = 25,\ 8 \times 8 = 64
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
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.
5×5=25,8×8=645 \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

Review

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.

Instead of the sum/product rule, extend each row and column by its constant step 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!
Variant 2 answer: A = 10, B = 36, C = 64

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, 82,\ 4,\ 6,\ 8 from the left, and the leftmost column (the numbers being added to) reads 2, 4, 62,\ 4,\ 6 from the top. Inside the grid the entries 44,\ 66,\ 88,\ AA / 66,\ 88,\ 1010,\ 1212 / 88,\ 1010,\ 1212,\ 1414 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, 86,\ 7,\ 8, and the leftmost column reads 6, 7, 86,\ 7,\ 8 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 2 4 6 8 2 4 6 4 6 8 A 6 8 10 12 8 10 12 14 Multiplication table × 6 7 8 6 7 8 B 42 48 42 49 56 48 56 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, 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.

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 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=102 + 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.
#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.
6×6=36, 8×8=646 \times 6 = 36,\ 8 \times 8 = 64
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
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.
6×6=36,8×8=646 \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

Review

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.

Instead of the sum/product rule, extend each row and column by its constant step 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!
Variant 3 answer: A = 40, B = 25, C = 36

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, 3010,\ 20,\ 30 from the left, and the leftmost column (the numbers being added to) reads 10, 20, 3010,\ 20,\ 30 from the top. Inside the grid the entries 2020,\ 3030,\ 4040 / 3030,\ AA,\ 5050 / 4040,\ 5050,\ 6060 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, 64,\ 5,\ 6, and the leftmost column reads 4, 5, 64,\ 5,\ 6 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 10 20 30 10 20 30 20 30 40 30 A 50 40 50 60 Multiplication table × 4 5 6 4 5 6 16 20 24 20 B 30 24 30 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): 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.

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 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=4020 + 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.
#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.
5×5=25, 6×6=365 \times 5 = 25,\ 6 \times 6 = 36
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
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.
5×5=25,6×6=365 \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

Review

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.

Instead of the sum/product rule, extend each row and column by its constant step 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!
Variant 4 answer: A = 25, B = 9, C = 25

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, 155,\ 10,\ 15 from the left, and the leftmost column (the numbers being added to) reads 5, 10, 15, 205,\ 10,\ 15,\ 20 from the top. Inside the grid the entries 1010,\ 1515,\ 2020 / 1515,\ 2020,\ 2525 / 2020,\ 2525,\ 3030 / AA,\ 3030,\ 3535 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, 73,\ 5,\ 7, and the leftmost column reads 3, 5, 73,\ 5,\ 7 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 5 10 15 5 10 15 20 10 15 20 15 20 25 20 25 30 A 30 35 Multiplication table × 3 5 7 3 5 7 B 15 21 15 C 35 21 35 49
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): 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.

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 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=2520 + 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.
#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.
3×3=9, 5×5=253 \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
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.
3×3=9,5×5=253 \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

Review

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.

Instead of the sum/product rule, extend each row and column by its constant step 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!
Variant 5 answer: A = 7, B = 16, C = 64

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, 51,\ 3,\ 5 from the left, and the leftmost column (the numbers being added to) reads 2, 4, 62,\ 4,\ 6 from the top. Inside the grid the entries 33,\ 55,\ 77 / 55,\ 77,\ 99 / AA,\ 99,\ 1111 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, 82,\ 4,\ 6,\ 8, and the leftmost column reads 2, 4, 6, 82,\ 4,\ 6,\ 8 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 1 3 5 2 4 6 3 5 7 5 7 9 A 9 11 Multiplication table × 2 4 6 8 2 4 6 8 4 8 12 16 8 B 24 32 12 24 36 48 16 32 48 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): 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.

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 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=76 + 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.
#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.
4×4=16, 8×8=644 \times 4 = 16,\ 8 \times 8 = 64
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
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.
4×4=16,8×8=644 \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

Review

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.

Instead of the sum/product rule, extend each row and column by its constant step 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!
Variant 6 answer: A = 10, B = 9, C = 25

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, 62,\ 4,\ 6 from the left, and the leftmost column (the numbers being added to) reads 2, 4, 6, 82,\ 4,\ 6,\ 8 from the top. Inside the grid the entries 44,\ 66,\ 88 / 66,\ 88,\ AA / 88,\ 1010,\ 1212 / 1010,\ 1212,\ 1414 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, 51,\ 3,\ 5, and the leftmost column reads 1, 3, 51,\ 3,\ 5 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 2 4 6 2 4 6 8 4 6 8 6 8 A 8 10 12 10 12 14 Multiplication table × 1 3 5 1 3 5 1 3 5 3 B 15 5 15 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.
  • 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, so A = 4 + 6 = 10. (Across a row the values go up by 2 each step.)
4+6=104 + 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 down, every cell is the product of its two heads.
3×3=9, 5×5=253 \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
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.
3×3=9,5×5=253 \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 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.

Instead of the sum/product rule, extend each row and column by its constant step 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!
Variant 7 answer: A = 12, B = 9, C = 16

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, 93,\ 6,\ 9 from the left, and the leftmost column (the numbers being added to) reads 3, 6, 93,\ 6,\ 9 from the top. Inside the grid the entries 66,\ 99,\ AA / 99,\ 1212,\ 1515 / 1212,\ 1515,\ 1818 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, 42,\ 3,\ 4, and the leftmost column reads 2, 3, 42,\ 3,\ 4 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 3 6 9 3 6 9 6 9 A 9 12 15 12 15 18 Multiplication table × 2 3 4 2 3 4 4 6 8 6 B 12 8 12 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): 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.

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 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=123 + 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.
#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.
3×3=9, 4×4=163 \times 3 = 9,\ 4 \times 4 = 16
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
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.
3×3=9,4×4=163 \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

Review

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.

Instead of the sum/product rule, extend each row and column by its constant step 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!
Variant 8 answer: A = 5, B = 4, C = 36

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, 31,\ 2,\ 3 from the left, and the leftmost column (the numbers being added to) reads 1, 2, 3, 41,\ 2,\ 3,\ 4 from the top. Inside the grid the entries 22,\ 33,\ 44 / 33,\ 44,\ 55 / 44,\ AA,\ 66 / 55,\ 66,\ 77 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, 62,\ 4,\ 6, and the leftmost column reads 2, 4, 62,\ 4,\ 6 from the top. In this part of the multiplication table, two cells are left blank as B and C.

Addition table + 1 2 3 1 2 3 4 2 3 4 3 4 5 4 A 6 5 6 7 Multiplication table × 2 4 6 2 4 6 B 8 12 8 16 24 12 24 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): 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.

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 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=53 + 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.
#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.
2×2=4, 6×6=362 \times 2 = 4,\ 6 \times 6 = 36
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
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.
2×2=4,6×6=362 \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

Review

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.

Instead of the sum/product rule, extend each row and column by its constant step 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!