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Multiplication table folds to equal cells · 8 practice problems

3.OA.B.53.OA.C.7

Generated variants — 8

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

Variant 1 answer: 90

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $5$ , $6$ , $7$ , $8$ written across the top and $5$ , $6$ , $7$ , $8$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $5{\times}5$ , $6{\times}6$ , $7{\times}7$ , $8{\times}8$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $8$ and top number $6$ .
B is the cell with left number $6$ and top number $7$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 5, 6, 7, 8 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 8, top 6.
B is the cell with left 6, top 7.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Folding along the diagonal is a spatial mirror that swaps the row and column heads of a cell, so I picture the mirror move, use the commutative pattern a x b = b x a to find each meeting value, then add.

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 8, top 6, so its mirror across the diagonal is at left 6, top 8. That meeting cell holds 6 x 8 = 48 (and A itself is 8 x 6 = 48, the same number).

6 \times 8 = 48

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 6, top 7, so its mirror is at left 7, top 6. That meeting cell holds 7 x 6 = 42.

7 \times 6 = 42

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 48 and the cell meeting B holds 42, so their sum is 48 + 42 = 90.

48 + 42 = 90

Once both meeting values are known, the answer is a simple addition.

Answer: 90

Review

Both meeting cells lie in the 5-to-8 multiplication range, where products run from 25 to 64; 48 and 42 sit in that range, and their sum 90 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!

Variant 2 answer: 58

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $4$ , $5$ , $6$ , $7$ written across the top and $4$ , $5$ , $6$ , $7$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $4{\times}4$ , $5{\times}5$ , $6{\times}6$ , $7{\times}7$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $7$ and top number $4$ .
B is the cell with left number $6$ and top number $5$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 4, 5, 6, 7 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 7, top 4.
B is the cell with left 6, top 5.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 7, top 4, so its mirror across the diagonal is at left 4, top 7. That meeting cell holds 4 x 7 = 28 (and A itself is 7 x 4 = 28, the same number).

4 \times 7 = 28

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 6, top 5, so its mirror is at left 5, top 6. That meeting cell holds 5 x 6 = 30.

5 \times 6 = 30

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 28 and the cell meeting B holds 30, so their sum is 28 + 30 = 58.

28 + 30 = 58

Once both meeting values are known, the answer is a simple addition.

Answer: 58

Review

Both meeting cells lie in the 4-to-7 multiplication range, where products run from 16 to 49; 28 and 30 sit in that range, and their sum 58 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!

Variant 3 answer: 77

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $3 \times 3$ table with $5$ , $6$ , $7$ written across the top and $5$ , $6$ , $7$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $5{\times}5$ , $6{\times}6$ , $7{\times}7$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $7$ and top number $5$ .
B is the cell with left number $6$ and top number $7$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 3x3 multiplication table with heads 5, 6, 7 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 7, top 5.
B is the cell with left 6, top 7.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 7, top 5, so its mirror across the diagonal is at left 5, top 7. That meeting cell holds 5 x 7 = 35 (and A itself is 7 x 5 = 35, the same number).

5 \times 7 = 35

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 6, top 7, so its mirror is at left 7, top 6. That meeting cell holds 7 x 6 = 42.

7 \times 6 = 42

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 35 and the cell meeting B holds 42, so their sum is 35 + 42 = 77.

35 + 42 = 77

Once both meeting values are known, the answer is a simple addition.

Answer: 77

Review

Both meeting cells lie in the 5-to-7 multiplication range, where products run from 25 to 49; 35 and 42 sit in that range, and their sum 77 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!

Variant 4 answer: 40

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $2$ , $4$ , $6$ , $8$ written across the top and $2$ , $4$ , $6$ , $8$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $2{\times}2$ , $4{\times}4$ , $6{\times}6$ , $8{\times}8$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $8$ and top number $2$ .
B is the cell with left number $6$ and top number $4$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 2, 4, 6, 8 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 8, top 2.
B is the cell with left 6, top 4.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 8, top 2, so its mirror across the diagonal is at left 2, top 8. That meeting cell holds 2 x 8 = 16 (and A itself is 8 x 2 = 16, the same number).

2 \times 8 = 16

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 6, top 4, so its mirror is at left 4, top 6. That meeting cell holds 4 x 6 = 24.

4 \times 6 = 24

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 16 and the cell meeting B holds 24, so their sum is 16 + 24 = 40.

16 + 24 = 40

Once both meeting values are known, the answer is a simple addition.

Answer: 40

Review

Both meeting cells lie in the 2-to-8 multiplication range, where products run from 4 to 64; 16 and 24 sit in that range, and their sum 40 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!

Variant 5 answer: 110

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $6$ , $7$ , $8$ , $9$ written across the top and $6$ , $7$ , $8$ , $9$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $6{\times}6$ , $7{\times}7$ , $8{\times}8$ , $9{\times}9$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $9$ and top number $6$ .
B is the cell with left number $8$ and top number $7$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 6, 7, 8, 9 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 9, top 6.
B is the cell with left 8, top 7.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 9, top 6, so its mirror across the diagonal is at left 6, top 9. That meeting cell holds 6 x 9 = 54 (and A itself is 9 x 6 = 54, the same number).

6 \times 9 = 54

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 8, top 7, so its mirror is at left 7, top 8. That meeting cell holds 7 x 8 = 56.

7 \times 8 = 56

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 54 and the cell meeting B holds 56, so their sum is 54 + 56 = 110.

54 + 56 = 110

Once both meeting values are known, the answer is a simple addition.

Answer: 110

Review

Both meeting cells lie in the 6-to-9 multiplication range, where products run from 36 to 81; 54 and 56 sit in that range, and their sum 110 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!

Variant 6 answer: 38

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $3$ , $4$ , $5$ , $6$ written across the top and $3$ , $4$ , $5$ , $6$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $3{\times}3$ , $4{\times}4$ , $5{\times}5$ , $6{\times}6$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $6$ and top number $3$ .
B is the cell with left number $5$ and top number $4$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 3, 4, 5, 6 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 6, top 3.
B is the cell with left 5, top 4.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 6, top 3, so its mirror across the diagonal is at left 3, top 6. That meeting cell holds 3 x 6 = 18 (and A itself is 6 x 3 = 18, the same number).

3 \times 6 = 18

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 5, top 4, so its mirror is at left 4, top 5. That meeting cell holds 4 x 5 = 20.

4 \times 5 = 20

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 18 and the cell meeting B holds 20, so their sum is 18 + 20 = 38.

18 + 20 = 38

Once both meeting values are known, the answer is a simple addition.

Answer: 38

Review

Both meeting cells lie in the 3-to-6 multiplication range, where products run from 9 to 36; 18 and 20 sit in that range, and their sum 38 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!

Variant 7 answer: 22

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $2$ , $3$ , $4$ , $5$ written across the top and $2$ , $3$ , $4$ , $5$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $2{\times}2$ , $3{\times}3$ , $4{\times}4$ , $5{\times}5$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $5$ and top number $2$ .
B is the cell with left number $4$ and top number $3$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 2, 3, 4, 5 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 5, top 2.
B is the cell with left 4, top 3.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 5, top 2, so its mirror across the diagonal is at left 2, top 5. That meeting cell holds 2 x 5 = 10 (and A itself is 5 x 2 = 10, the same number).

2 \times 5 = 10

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 4, top 3, so its mirror is at left 3, top 4. That meeting cell holds 3 x 4 = 12.

3 \times 4 = 12

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 10 and the cell meeting B holds 12, so their sum is 10 + 12 = 22.

10 + 12 = 22

Once both meeting values are known, the answer is a simple addition.

Answer: 22

Review

Both meeting cells lie in the 2-to-5 multiplication range, where products run from 4 to 25; 10 and 12 sit in that range, and their sum 22 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!

Variant 8 answer: 62

In the multiplication table, fold along the green dashed line. Find the two numbers that meet cell A and cell B, then find the sum of those two numbers.

The multiplication table is a $4 \times 4$ table with $3$ , $5$ , $7$ , $9$ written across the top and $3$ , $5$ , $7$ , $9$ written down the left side. Each cell holds the product of its top number and its left number.

The green dashed line is the diagonal through the cells where the left number equals the top number ( $3{\times}3$ , $5{\times}5$ , $7{\times}7$ , $9{\times}9$ ). When the table is folded along this line, the two cells that land on each other hold the same number.

A is the cell with left number $9$ and top number $3$ .
B is the cell with left number $7$ and top number $5$ .

Fold along the dashed line, find the number in the cell that meets A and the number in the cell that meets B, and add those two numbers together.

Show solution

Understand

In a 4x4 multiplication table with heads 3, 5, 7, 9 across and down, fold along the diagonal where left equals top. Find the numbers in the cells that land on A and B after folding, then add those two numbers.

Givens

Each cell holds (left number) x (top number).
The fold line is the diagonal through the equal-head cells.
When folded, two mirror cells land on each other and hold the same number.
A is the cell with left 9, top 3.
B is the cell with left 7, top 5.

Unknowns

The number in the cell that meets A after folding.
The number in the cell that meets B after folding.
The sum of those two numbers.

Constraints

Folding along the left=top diagonal swaps a cell at (left a, top b) with the cell at (left b, top a).
Because a x b = b x a, the meeting cell holds the same number.

Plan

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

Execute

#1 Draw a Diagram 3.OA.B.5

A is at left 9, top 3, so its mirror across the diagonal is at left 3, top 9. That meeting cell holds 3 x 9 = 27 (and A itself is 9 x 3 = 27, the same number).

3 \times 9 = 27

Folding swaps the two heads of a cell, and a x b equals b x a, so the meeting cell shows the same product.

#5 Look for a Pattern 3.OA.B.5

B is at left 7, top 5, so its mirror is at left 5, top 7. That meeting cell holds 5 x 7 = 35.

5 \times 7 = 35

Same idea: swapping the heads keeps the product, so the mirror cell is also the same value.

#1 Draw a Diagram 3.OA.C.7

The cell meeting A holds 27 and the cell meeting B holds 35, so their sum is 27 + 35 = 62.

27 + 35 = 62

Once both meeting values are known, the answer is a simple addition.

Answer: 62

Review

Both meeting cells lie in the 3-to-9 multiplication range, where products run from 9 to 81; 27 and 35 sit in that range, and their sum 62 is reasonable.

You could fill in the whole product table, mark the diagonal, and physically check that the cells across from A and B read the same values before adding.

Standards · min grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the commutative property a x b = b x a to find the mirror cells across the fold.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the two products and adding the results.

💡 Folding the table just swaps the row and column numbers, and since a x b = b x a, the mirror cell shows the very same product!