← 4-1 · A new symbol can define a custom operation · Apply a Newly Defined Operation

A new symbol can define a custom operation · 10 practice problems

4.OA.A.35.OA.A.1

Generated variants — 10

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

Variant 1 answer: 7973

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 35$ . Find the value of $(29 \diamond 14) \diamond 18$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 35. Using this rule, first compute 29-diamond-14, then take that result diamond 18.

Givens

The operation is defined by a diamond b = a x b + 35
We must evaluate (29 diamond 14) diamond 18

Unknowns

The value of (29 diamond 14) diamond 18

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 35

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 29 diamond 14, then feed that result into a second application of the same rule with 18. Each application follows the identical pattern multiply-then-add-35.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 29 and 14: multiply, then add 35. 29 x 14 = 406, and 406 + 35 = 441.

29 \diamond 14 = 29 \times 14 + 35 = 406 + 35 = 441

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 441 and 18: 441 x 18 = 7938, and 7938 + 35 = 7973.

441 \diamond 18 = 441 \times 18 + 35 = 7938 + 35 = 7973

The custom symbol is just a recipe; reusing the same multiply-then-add-35 pattern finishes the job.

Answer: 7973

Review

The answer is dominated by 441 x 18 = 7938; adding 35 nudges it to 7973, which is sensibly just above 7938. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 35, so you could compute both products first (406 and 7938) and add 35 once per step, confirming 7973.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (29 diamond 14) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-35 steps again!

Variant 2 answer: 8697

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 45$ . Find the value of $(52 \diamond 13) \diamond 12$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 45. Using this rule, first compute 52-diamond-13, then take that result diamond 12.

Givens

The operation is defined by a diamond b = a x b + 45
We must evaluate (52 diamond 13) diamond 12

Unknowns

The value of (52 diamond 13) diamond 12

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 45

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 52 diamond 13, then feed that result into a second application of the same rule with 12. Each application follows the identical pattern multiply-then-add-45.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 52 and 13: multiply, then add 45. 52 x 13 = 676, and 676 + 45 = 721.

52 \diamond 13 = 52 \times 13 + 45 = 676 + 45 = 721

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 721 and 12: 721 x 12 = 8652, and 8652 + 45 = 8697.

721 \diamond 12 = 721 \times 12 + 45 = 8652 + 45 = 8697

The custom symbol is just a recipe; reusing the same multiply-then-add-45 pattern finishes the job.

Answer: 8697

Review

The answer is dominated by 721 x 12 = 8652; adding 45 nudges it to 8697, which is sensibly just above 8652. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 45, so you could compute both products first (676 and 8652) and add 45 once per step, confirming 8697.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (52 diamond 13) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-45 steps again!

Variant 3 answer: 5664

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 20$ . Find the value of $(24 \diamond 13) \diamond 17$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 20. Using this rule, first compute 24-diamond-13, then take that result diamond 17.

Givens

The operation is defined by a diamond b = a x b + 20
We must evaluate (24 diamond 13) diamond 17

Unknowns

The value of (24 diamond 13) diamond 17

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 20

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 24 diamond 13, then feed that result into a second application of the same rule with 17. Each application follows the identical pattern multiply-then-add-20.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 24 and 13: multiply, then add 20. 24 x 13 = 312, and 312 + 20 = 332.

24 \diamond 13 = 24 \times 13 + 20 = 312 + 20 = 332

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 332 and 17: 332 x 17 = 5644, and 5644 + 20 = 5664.

332 \diamond 17 = 332 \times 17 + 20 = 5644 + 20 = 5664

The custom symbol is just a recipe; reusing the same multiply-then-add-20 pattern finishes the job.

Answer: 5664

Review

The answer is dominated by 332 x 17 = 5644; adding 20 nudges it to 5664, which is sensibly just above 5644. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 20, so you could compute both products first (312 and 5644) and add 20 once per step, confirming 5664.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (24 diamond 13) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-20 steps again!

Variant 4 answer: 11980

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 10$ . Find the value of $(35 \diamond 16) \diamond 21$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 10. Using this rule, first compute 35-diamond-16, then take that result diamond 21.

Givens

The operation is defined by a diamond b = a x b + 10
We must evaluate (35 diamond 16) diamond 21

Unknowns

The value of (35 diamond 16) diamond 21

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 10

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 35 diamond 16, then feed that result into a second application of the same rule with 21. Each application follows the identical pattern multiply-then-add-10.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 35 and 16: multiply, then add 10. 35 x 16 = 560, and 560 + 10 = 570.

35 \diamond 16 = 35 \times 16 + 10 = 560 + 10 = 570

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 570 and 21: 570 x 21 = 11970, and 11970 + 10 = 11980.

570 \diamond 21 = 570 \times 21 + 10 = 11970 + 10 = 11980

The custom symbol is just a recipe; reusing the same multiply-then-add-10 pattern finishes the job.

Answer: 11980

Review

The answer is dominated by 570 x 21 = 11970; adding 10 nudges it to 11980, which is sensibly just above 11970. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 10, so you could compute both products first (560 and 11970) and add 10 once per step, confirming 11980.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (35 diamond 16) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-10 steps again!

Variant 5 answer: 7369

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 50$ . Find the value of $(27 \diamond 19) \diamond 13$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 50. Using this rule, first compute 27-diamond-19, then take that result diamond 13.

Givens

The operation is defined by a diamond b = a x b + 50
We must evaluate (27 diamond 19) diamond 13

Unknowns

The value of (27 diamond 19) diamond 13

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 50

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 27 diamond 19, then feed that result into a second application of the same rule with 13. Each application follows the identical pattern multiply-then-add-50.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 27 and 19: multiply, then add 50. 27 x 19 = 513, and 513 + 50 = 563.

27 \diamond 19 = 27 \times 19 + 50 = 513 + 50 = 563

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 563 and 13: 563 x 13 = 7319, and 7319 + 50 = 7369.

563 \diamond 13 = 563 \times 13 + 50 = 7319 + 50 = 7369

The custom symbol is just a recipe; reusing the same multiply-then-add-50 pattern finishes the job.

Answer: 7369

Review

The answer is dominated by 563 x 13 = 7319; adding 50 nudges it to 7369, which is sensibly just above 7319. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 50, so you could compute both products first (513 and 7319) and add 50 once per step, confirming 7369.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (27 diamond 19) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-50 steps again!

Variant 6 answer: 11760

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 30$ . Find the value of $(32 \diamond 15) \diamond 23$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 30. Using this rule, first compute 32-diamond-15, then take that result diamond 23.

Givens

The operation is defined by a diamond b = a x b + 30
We must evaluate (32 diamond 15) diamond 23

Unknowns

The value of (32 diamond 15) diamond 23

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 30

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 32 diamond 15, then feed that result into a second application of the same rule with 23. Each application follows the identical pattern multiply-then-add-30.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 32 and 15: multiply, then add 30. 32 x 15 = 480, and 480 + 30 = 510.

32 \diamond 15 = 32 \times 15 + 30 = 480 + 30 = 510

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 510 and 23: 510 x 23 = 11730, and 11730 + 30 = 11760.

510 \diamond 23 = 510 \times 23 + 30 = 11730 + 30 = 11760

The custom symbol is just a recipe; reusing the same multiply-then-add-30 pattern finishes the job.

Answer: 11760

Review

The answer is dominated by 510 x 23 = 11730; adding 30 nudges it to 11760, which is sensibly just above 11730. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 30, so you could compute both products first (480 and 11730) and add 30 once per step, confirming 11760.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (32 diamond 15) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-30 steps again!

Variant 7 answer: 11356

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 60$ . Find the value of $(38 \diamond 17) \diamond 16$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 60. Using this rule, first compute 38-diamond-17, then take that result diamond 16.

Givens

The operation is defined by a diamond b = a x b + 60
We must evaluate (38 diamond 17) diamond 16

Unknowns

The value of (38 diamond 17) diamond 16

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 60

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 38 diamond 17, then feed that result into a second application of the same rule with 16. Each application follows the identical pattern multiply-then-add-60.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 38 and 17: multiply, then add 60. 38 x 17 = 646, and 646 + 60 = 706.

38 \diamond 17 = 38 \times 17 + 60 = 646 + 60 = 706

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 706 and 16: 706 x 16 = 11296, and 11296 + 60 = 11356.

706 \diamond 16 = 706 \times 16 + 60 = 11296 + 60 = 11356

The custom symbol is just a recipe; reusing the same multiply-then-add-60 pattern finishes the job.

Answer: 11356

Review

The answer is dominated by 706 x 16 = 11296; adding 60 nudges it to 11356, which is sensibly just above 11296. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 60, so you could compute both products first (646 and 11296) and add 60 once per step, confirming 11356.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (38 diamond 17) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-60 steps again!

Variant 8 answer: 9848

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 25$ . Find the value of $(41 \diamond 12) \diamond 19$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 25. Using this rule, first compute 41-diamond-12, then take that result diamond 19.

Givens

The operation is defined by a diamond b = a x b + 25
We must evaluate (41 diamond 12) diamond 19

Unknowns

The value of (41 diamond 12) diamond 19

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 25

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 41 diamond 12, then feed that result into a second application of the same rule with 19. Each application follows the identical pattern multiply-then-add-25.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 41 and 12: multiply, then add 25. 41 x 12 = 492, and 492 + 25 = 517.

41 \diamond 12 = 41 \times 12 + 25 = 492 + 25 = 517

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 517 and 19: 517 x 19 = 9823, and 9823 + 25 = 9848.

517 \diamond 19 = 517 \times 19 + 25 = 9823 + 25 = 9848

The custom symbol is just a recipe; reusing the same multiply-then-add-25 pattern finishes the job.

Answer: 9848

Review

The answer is dominated by 517 x 19 = 9823; adding 25 nudges it to 9848, which is sensibly just above 9823. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 25, so you could compute both products first (492 and 9823) and add 25 once per step, confirming 9848.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (41 diamond 12) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-25 steps again!

Variant 9 answer: 6144

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 40$ . Find the value of $(18 \diamond 22) \diamond 14$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 40. Using this rule, first compute 18-diamond-22, then take that result diamond 14.

Givens

The operation is defined by a diamond b = a x b + 40
We must evaluate (18 diamond 22) diamond 14

Unknowns

The value of (18 diamond 22) diamond 14

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 40

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 18 diamond 22, then feed that result into a second application of the same rule with 14. Each application follows the identical pattern multiply-then-add-40.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 18 and 22: multiply, then add 40. 18 x 22 = 396, and 396 + 40 = 436.

18 \diamond 22 = 18 \times 22 + 40 = 396 + 40 = 436

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 436 and 14: 436 x 14 = 6104, and 6104 + 40 = 6144.

436 \diamond 14 = 436 \times 14 + 40 = 6104 + 40 = 6144

The custom symbol is just a recipe; reusing the same multiply-then-add-40 pattern finishes the job.

Answer: 6144

Review

The answer is dominated by 436 x 14 = 6104; adding 40 nudges it to 6144, which is sensibly just above 6104. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 40, so you could compute both products first (396 and 6104) and add 40 once per step, confirming 6144.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (18 diamond 22) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-40 steps again!

Variant 10 answer: 7830

For any two numbers $a$ and $b$ , the operation $\diamond$ is defined by $a \diamond b = a \times b + 15$ . Find the value of $(46 \diamond 11) \diamond 15$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 15. Using this rule, first compute 46-diamond-11, then take that result diamond 15.

Givens

The operation is defined by a diamond b = a x b + 15
We must evaluate (46 diamond 11) diamond 15

Unknowns

The value of (46 diamond 11) diamond 15

Constraints

The inner parentheses must be evaluated first
Each diamond means multiply the two numbers and add 15

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 46 diamond 11, then feed that result into a second application of the same rule with 15. Each application follows the identical pattern multiply-then-add-15.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 46 and 11: multiply, then add 15. 46 x 11 = 506, and 506 + 15 = 521.

46 \diamond 11 = 46 \times 11 + 15 = 506 + 15 = 521

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 521 and 15: 521 x 15 = 7815, and 7815 + 15 = 7830.

521 \diamond 15 = 521 \times 15 + 15 = 7815 + 15 = 7830

The custom symbol is just a recipe; reusing the same multiply-then-add-15 pattern finishes the job.

Answer: 7830

Review

The answer is dominated by 521 x 15 = 7815; adding 15 nudges it to 7830, which is sensibly just above 7815. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 15, so you could compute both products first (506 and 7815) and add 15 once per step, confirming 7830.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (46 diamond 11) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-15 steps again!