Practice · Evaluate a newly defined operation symbol · Sensim Math

Variant 1 easy answer: 225

Suppose a new operation $\odot$ is defined as shown below. Find the value of $25 \odot (6 \odot 4)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 6 and 4 first, then apply it again with 25.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $25 \odot (6 \odot 4)$ .

Unknowns

The value of $25 \odot (6 \odot 4)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $6 \odot 4$

#7 Identify Subproblems 5.OA.A.1

Substitute 6 for the first number and 4 for the second.

(6 + 4) \times (6 - 4) = 10 \times 2 = 20

The parentheses are now worth 20.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 20 leaves a single use of the symbol.

25 \odot (6 \odot 4) = 25 \odot 20

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $25 \odot 20$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 25 and 20.

(25 + 20) \times (25 - 20) = 45 \times 5 = 225

The value is 225.

Answer: 225

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 25 + 20 = 45 and 25 - 20 = 5, and their product is 225.

Another way: The rule is the difference-of-squares pattern, so the answer is also $25^2 - 20^2 = 625 - 400 = 225$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 45 by 5.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 2 easy answer: 528

Suppose a new operation $\odot$ is defined as shown below. Find the value of $28 \odot (5 \odot 3)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 5 and 3 first, then apply it again with 28.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $28 \odot (5 \odot 3)$ .

Unknowns

The value of $28 \odot (5 \odot 3)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $5 \odot 3$

#7 Identify Subproblems 5.OA.A.1

Substitute 5 for the first number and 3 for the second.

(5 + 3) \times (5 - 3) = 8 \times 2 = 16

The parentheses are now worth 16.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 16 leaves a single use of the symbol.

28 \odot (5 \odot 3) = 28 \odot 16

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $28 \odot 16$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 28 and 16.

(28 + 16) \times (28 - 16) = 44 \times 12 = 528

The value is 528.

Answer: 528

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 28 + 16 = 44 and 28 - 16 = 12, and their product is 528.

Another way: The rule is the difference-of-squares pattern, so the answer is also $28^2 - 16^2 = 784 - 256 = 528$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 44 by 12.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 3 easy answer: 116

Suppose a new operation $\odot$ is defined as shown below. Find the value of $30 \odot (8 \odot 6)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 8 and 6 first, then apply it again with 30.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $30 \odot (8 \odot 6)$ .

Unknowns

The value of $30 \odot (8 \odot 6)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $8 \odot 6$

#7 Identify Subproblems 5.OA.A.1

Substitute 8 for the first number and 6 for the second.

(8 + 6) \times (8 - 6) = 14 \times 2 = 28

The parentheses are now worth 28.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 28 leaves a single use of the symbol.

30 \odot (8 \odot 6) = 30 \odot 28

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $30 \odot 28$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 30 and 28.

(30 + 28) \times (30 - 28) = 58 \times 2 = 116

The value is 116.

Answer: 116

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 30 + 28 = 58 and 30 - 28 = 2, and their product is 116.

Another way: The rule is the difference-of-squares pattern, so the answer is also $30^2 - 28^2 = 900 - 784 = 116$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 58 by 2.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 4 easy answer: 448

Suppose a new operation $\odot$ is defined as shown below. Find the value of $32 \odot (7 \odot 5)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 7 and 5 first, then apply it again with 32.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $32 \odot (7 \odot 5)$ .

Unknowns

The value of $32 \odot (7 \odot 5)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $7 \odot 5$

#7 Identify Subproblems 5.OA.A.1

Substitute 7 for the first number and 5 for the second.

(7 + 5) \times (7 - 5) = 12 \times 2 = 24

The parentheses are now worth 24.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 24 leaves a single use of the symbol.

32 \odot (7 \odot 5) = 32 \odot 24

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $32 \odot 24$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 32 and 24.

(32 + 24) \times (32 - 24) = 56 \times 8 = 448

The value is 448.

Answer: 448

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 32 + 24 = 56 and 32 - 24 = 8, and their product is 448.

Another way: The rule is the difference-of-squares pattern, so the answer is also $32^2 - 24^2 = 1024 - 576 = 448$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 56 by 8.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 5 medium answer: 496

Suppose a new operation $\odot$ is defined as shown below. Find the value of $35 \odot (6 \odot 3)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 6 and 3 first, then apply it again with 35.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $35 \odot (6 \odot 3)$ .

Unknowns

The value of $35 \odot (6 \odot 3)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $6 \odot 3$

#7 Identify Subproblems 5.OA.A.1

Substitute 6 for the first number and 3 for the second.

(6 + 3) \times (6 - 3) = 9 \times 3 = 27

The parentheses are now worth 27.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 27 leaves a single use of the symbol.

35 \odot (6 \odot 3) = 35 \odot 27

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $35 \odot 27$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 35 and 27.

(35 + 27) \times (35 - 27) = 62 \times 8 = 496

The value is 496.

Answer: 496

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 35 + 27 = 62 and 35 - 27 = 8, and their product is 496.

Another way: The rule is the difference-of-squares pattern, so the answer is also $35^2 - 27^2 = 1225 - 729 = 496$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 62 by 8.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 6 medium answer: 576

Suppose a new operation $\odot$ is defined as shown below. Find the value of $40 \odot (9 \odot 7)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 9 and 7 first, then apply it again with 40.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $40 \odot (9 \odot 7)$ .

Unknowns

The value of $40 \odot (9 \odot 7)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $9 \odot 7$

#7 Identify Subproblems 5.OA.A.1

Substitute 9 for the first number and 7 for the second.

(9 + 7) \times (9 - 7) = 16 \times 2 = 32

The parentheses are now worth 32.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 32 leaves a single use of the symbol.

40 \odot (9 \odot 7) = 40 \odot 32

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $40 \odot 32$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 40 and 32.

(40 + 32) \times (40 - 32) = 72 \times 8 = 576

The value is 576.

Answer: 576

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 40 + 32 = 72 and 40 - 32 = 8, and their product is 576.

Another way: The rule is the difference-of-squares pattern, so the answer is also $40^2 - 32^2 = 1600 - 1024 = 576$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 72 by 8.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 7 medium answer: 847

Suppose a new operation $\odot$ is defined as shown below. Find the value of $44 \odot (7 \odot 4)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 7 and 4 first, then apply it again with 44.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $44 \odot (7 \odot 4)$ .

Unknowns

The value of $44 \odot (7 \odot 4)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $7 \odot 4$

#7 Identify Subproblems 5.OA.A.1

Substitute 7 for the first number and 4 for the second.

(7 + 4) \times (7 - 4) = 11 \times 3 = 33

The parentheses are now worth 33.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 33 leaves a single use of the symbol.

44 \odot (7 \odot 4) = 44 \odot 33

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $44 \odot 33$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 44 and 33.

(44 + 33) \times (44 - 33) = 77 \times 11 = 847

The value is 847.

Answer: 847

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 44 + 33 = 77 and 44 - 33 = 11, and their product is 847.

Another way: The rule is the difference-of-squares pattern, so the answer is also $44^2 - 33^2 = 1936 - 1089 = 847$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 77 by 11.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 8 medium answer: 729

Suppose a new operation $\odot$ is defined as shown below. Find the value of $45 \odot (10 \odot 8)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 10 and 8 first, then apply it again with 45.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $45 \odot (10 \odot 8)$ .

Unknowns

The value of $45 \odot (10 \odot 8)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $10 \odot 8$

#7 Identify Subproblems 5.OA.A.1

Substitute 10 for the first number and 8 for the second.

(10 + 8) \times (10 - 8) = 18 \times 2 = 36

The parentheses are now worth 36.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 36 leaves a single use of the symbol.

45 \odot (10 \odot 8) = 45 \odot 36

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $45 \odot 36$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 45 and 36.

(45 + 36) \times (45 - 36) = 81 \times 9 = 729

The value is 729.

Answer: 729

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 45 + 36 = 81 and 45 - 36 = 9, and their product is 729.

Another way: The rule is the difference-of-squares pattern, so the answer is also $45^2 - 36^2 = 2025 - 1296 = 729$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 81 by 9.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 9 hard answer: 900

Suppose a new operation $\odot$ is defined as shown below. Find the value of $50 \odot (11 \odot 9)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 11 and 9 first, then apply it again with 50.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $50 \odot (11 \odot 9)$ .

Unknowns

The value of $50 \odot (11 \odot 9)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $11 \odot 9$

#7 Identify Subproblems 5.OA.A.1

Substitute 11 for the first number and 9 for the second.

(11 + 9) \times (11 - 9) = 20 \times 2 = 40

The parentheses are now worth 40.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 40 leaves a single use of the symbol.

50 \odot (11 \odot 9) = 50 \odot 40

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $50 \odot 40$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 50 and 40.

(50 + 40) \times (50 - 40) = 90 \times 10 = 900

The value is 900.

Answer: 900

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 50 + 40 = 90 and 50 - 40 = 10, and their product is 900.

Another way: The rule is the difference-of-squares pattern, so the answer is also $50^2 - 40^2 = 2500 - 1600 = 900$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 90 by 10.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 10 hard answer: 464

Suppose a new operation $\odot$ is defined as shown below. Find the value of $60 \odot (9 \odot 5)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 9 and 5 first, then apply it again with 60.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $60 \odot (9 \odot 5)$ .

Unknowns

The value of $60 \odot (9 \odot 5)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $9 \odot 5$

#7 Identify Subproblems 5.OA.A.1

Substitute 9 for the first number and 5 for the second.

(9 + 5) \times (9 - 5) = 14 \times 4 = 56

The parentheses are now worth 56.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 56 leaves a single use of the symbol.

60 \odot (9 \odot 5) = 60 \odot 56

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $60 \odot 56$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 60 and 56.

(60 + 56) \times (60 - 56) = 116 \times 4 = 464

The value is 464.

Answer: 464

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 60 + 56 = 116 and 60 - 56 = 4, and their product is 464.

Another way: The rule is the difference-of-squares pattern, so the answer is also $60^2 - 56^2 = 3600 - 3136 = 464$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 116 by 4.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 11 hard answer: 2964

Suppose a new operation $\odot$ is defined as shown below. Find the value of $70 \odot (12 \odot 10)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 12 and 10 first, then apply it again with 70.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $70 \odot (12 \odot 10)$ .

Unknowns

The value of $70 \odot (12 \odot 10)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $12 \odot 10$

#7 Identify Subproblems 5.OA.A.1

Substitute 12 for the first number and 10 for the second.

(12 + 10) \times (12 - 10) = 22 \times 2 = 44

The parentheses are now worth 44.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 44 leaves a single use of the symbol.

70 \odot (12 \odot 10) = 70 \odot 44

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $70 \odot 44$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 70 and 44.

(70 + 44) \times (70 - 44) = 114 \times 26 = 2964

The value is 2964.

Answer: 2964

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 70 + 44 = 114 and 70 - 44 = 26, and their product is 2964.

Another way: The rule is the difference-of-squares pattern, so the answer is also $70^2 - 44^2 = 4900 - 1936 = 2964$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 114 by 26.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Variant 12 hard answer: 1216

Suppose a new operation $\odot$ is defined as shown below. Find the value of $80 \odot (11 \odot 7)$ .

$a \odot b = (a + b) \times (a - b)$

Show solution

1 · Understandwhat's really being asked

A new symbol is defined: putting two numbers on either side of it means add them, subtract them, and multiply those two results. We must apply it to 11 and 7 first, then apply it again with 80.

Givens

The rule is: the first number plus the second, times the first minus the second.
The expression to evaluate is $80 \odot (11 \odot 7)$ .

Unknowns

The value of $80 \odot (11 \odot 7)$ .

Constraints

The parentheses say the inner operation happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #9 Solve an Easier Related Problem

The symbol is only a recipe. Do the parenthesised part on its own, replace it with the number it produces, then run the same recipe once more. Two small problems instead of one strange-looking one.

3 · Execute4 carry out the plan

1Read the recipe the symbol stands for

#9 Solve an Easier Related Problem 5.OA.A.2

The rule says: add the two numbers, subtract the second from the first, then multiply those two results together.

a \odot b = (a + b) \times (a - b)

A new symbol is just a shorthand for steps we already know.

2Do the inner subproblem first: $11 \odot 7$

#7 Identify Subproblems 5.OA.A.1

Substitute 11 for the first number and 7 for the second.

(11 + 7) \times (11 - 7) = 18 \times 4 = 72

The parentheses are now worth 72.

3Rewrite the whole problem with the inner value

#7 Identify Subproblems 5.OA.A.1

Replacing the parenthesised part by 72 leaves a single use of the symbol.

80 \odot (11 \odot 7) = 80 \odot 72

The nested expression has flattened into one step.

4Apply the recipe to the outer operation: $80 \odot 72$

#7 Identify Subproblems 5.NBT.B.5

Same rule again, now with 80 and 72.

(80 + 72) \times (80 - 72) = 152 \times 8 = 1216

The value is 1216.

Answer: 1216

4 · Reviewdoes it hold up?

Both factors are checkable on their own: 80 + 72 = 152 and 80 - 72 = 8, and their product is 1216.

Another way: The rule is the difference-of-squares pattern, so the answer is also $80^2 - 72^2 = 6400 - 5184 = 1216$ .

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the parenthesised operation before the outer one.
5.OA.A.2 Write simple expressions that record calculations with numbers — Reading the defined symbol as an expression to record and evaluate.
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm — Multiplying 152 by 8.

💡Takeaway. An unfamiliar symbol is just a recipe. Do the innermost one first and replace it with its value; then it is arithmetic you know.

Evaluate a newly defined operation symbol · 12 practice problems

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