← Evaluate a custom-defined operation symbol · Apply a Newly Defined Operation

Evaluate a custom-defined operation symbol · 10 practice problems

3.OA.D.83.OA.B.5

Generated variants — 10

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

Variant 1 easy answer: 5

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×9+a×ba \star b = a \times 9 + a \times b

□⋆3=60\square \star 3 = 60

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 9, then add a times b.' We are told that □⋆3\square \star 3 equals 60, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×9+a×ba \star b = a \times 9 + a \times b
  • □⋆3=60\square \star 3 = 60
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆3\square \star 3 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 3 in the rule a⋆b=a×9+a×ba \star b = a \times 9 + a \times b.
□⋆3=□×9+□×3\square \star 3 = \square \times 9 + \square \times 3
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 9 of the □\square and 3 of the □\square gives 12 of the □\square.
□×9+□×3=□×12\square \times 9 + \square \times 3 = \square \times 12
9 groups plus 3 groups of the same size is 12 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 12 = 60. Since 5 times 12 = 60, the □\square is 5.
□×12=60⇒□=5\square \times 12 = 60 \Rightarrow \square = 5
Testing the factor 5 (because 5 times 12 = 60) confirms the answer at once.
Answer: 5
4 · Reviewdoes it hold up?

Check: 5⋆35 \star 3 = 5 times 9 + 5 times 3 = 45 + 15 = 60, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 9 plus □\square times 3 = 60, then divide 60 by 12 directly (work backwards, tool 11) to get 5.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 9 and $\square$ times 3 into $\square$ times 12 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 12 = 60
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 9 groups plus 3 groups is simply 12 groups!
Variant 2 easy answer: 6

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×5+a×ba \star b = a \times 5 + a \times b

□⋆5=60\square \star 5 = 60

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 5, then add a times b.' We are told that □⋆5\square \star 5 equals 60, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×5+a×ba \star b = a \times 5 + a \times b
  • □⋆5=60\square \star 5 = 60
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆5\square \star 5 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 5 in the rule a⋆b=a×5+a×ba \star b = a \times 5 + a \times b.
□⋆5=□×5+□×5\square \star 5 = \square \times 5 + \square \times 5
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 5 of the □\square and 5 of the □\square gives 10 of the □\square.
□×5+□×5=□×10\square \times 5 + \square \times 5 = \square \times 10
5 groups plus 5 groups of the same size is 10 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 10 = 60. Since 6 times 10 = 60, the □\square is 6.
□×10=60⇒□=6\square \times 10 = 60 \Rightarrow \square = 6
Testing the factor 6 (because 6 times 10 = 60) confirms the answer at once.
Answer: 6
4 · Reviewdoes it hold up?

Check: 6⋆56 \star 5 = 6 times 5 + 6 times 5 = 30 + 30 = 60, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 5 plus □\square times 5 = 60, then divide 60 by 10 directly (work backwards, tool 11) to get 6.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 5 and $\square$ times 5 into $\square$ times 10 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 10 = 60
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 5 groups plus 5 groups is simply 10 groups!
Variant 3 easy answer: 8

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×5+a×ba \star b = a \times 5 + a \times b

□⋆4=72\square \star 4 = 72

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 5, then add a times b.' We are told that □⋆4\square \star 4 equals 72, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×5+a×ba \star b = a \times 5 + a \times b
  • □⋆4=72\square \star 4 = 72
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆4\square \star 4 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 4 in the rule a⋆b=a×5+a×ba \star b = a \times 5 + a \times b.
□⋆4=□×5+□×4\square \star 4 = \square \times 5 + \square \times 4
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 5 of the □\square and 4 of the □\square gives 9 of the □\square.
□×5+□×4=□×9\square \times 5 + \square \times 4 = \square \times 9
5 groups plus 4 groups of the same size is 9 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 9 = 72. Since 8 times 9 = 72, the □\square is 8.
□×9=72⇒□=8\square \times 9 = 72 \Rightarrow \square = 8
Testing the factor 8 (because 8 times 9 = 72) confirms the answer at once.
Answer: 8
4 · Reviewdoes it hold up?

Check: 8⋆48 \star 4 = 8 times 5 + 8 times 4 = 40 + 32 = 72, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 5 plus □\square times 4 = 72, then divide 72 by 9 directly (work backwards, tool 11) to get 8.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 5 and $\square$ times 4 into $\square$ times 9 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 9 = 72
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 5 groups plus 4 groups is simply 9 groups!
Variant 4 medium answer: 6

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×3+a×ba \star b = a \times 3 + a \times b

□⋆9=72\square \star 9 = 72

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 3, then add a times b.' We are told that □⋆9\square \star 9 equals 72, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×3+a×ba \star b = a \times 3 + a \times b
  • □⋆9=72\square \star 9 = 72
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆9\square \star 9 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 9 in the rule a⋆b=a×3+a×ba \star b = a \times 3 + a \times b.
□⋆9=□×3+□×9\square \star 9 = \square \times 3 + \square \times 9
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 3 of the □\square and 9 of the □\square gives 12 of the □\square.
□×3+□×9=□×12\square \times 3 + \square \times 9 = \square \times 12
3 groups plus 9 groups of the same size is 12 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 12 = 72. Since 6 times 12 = 72, the □\square is 6.
□×12=72⇒□=6\square \times 12 = 72 \Rightarrow \square = 6
Testing the factor 6 (because 6 times 12 = 72) confirms the answer at once.
Answer: 6
4 · Reviewdoes it hold up?

Check: 6⋆96 \star 9 = 6 times 3 + 6 times 9 = 18 + 54 = 72, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 3 plus □\square times 9 = 72, then divide 72 by 12 directly (work backwards, tool 11) to get 6.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 3 and $\square$ times 9 into $\square$ times 12 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 12 = 72
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 3 groups plus 9 groups is simply 12 groups!
Variant 5 medium answer: 9

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×2+a×ba \star b = a \times 2 + a \times b

□⋆6=72\square \star 6 = 72

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 2, then add a times b.' We are told that □⋆6\square \star 6 equals 72, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×2+a×ba \star b = a \times 2 + a \times b
  • □⋆6=72\square \star 6 = 72
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆6\square \star 6 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 6 in the rule a⋆b=a×2+a×ba \star b = a \times 2 + a \times b.
□⋆6=□×2+□×6\square \star 6 = \square \times 2 + \square \times 6
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 2 of the □\square and 6 of the □\square gives 8 of the □\square.
□×2+□×6=□×8\square \times 2 + \square \times 6 = \square \times 8
2 groups plus 6 groups of the same size is 8 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 8 = 72. Since 9 times 8 = 72, the □\square is 9.
□×8=72⇒□=9\square \times 8 = 72 \Rightarrow \square = 9
Testing the factor 9 (because 9 times 8 = 72) confirms the answer at once.
Answer: 9
4 · Reviewdoes it hold up?

Check: 9⋆69 \star 6 = 9 times 2 + 9 times 6 = 18 + 54 = 72, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 2 plus □\square times 6 = 72, then divide 72 by 8 directly (work backwards, tool 11) to get 9.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 2 and $\square$ times 6 into $\square$ times 8 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 8 = 72
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 2 groups plus 6 groups is simply 8 groups!
Variant 6 medium answer: 7

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×7+a×ba \star b = a \times 7 + a \times b

□⋆4=77\square \star 4 = 77

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 7, then add a times b.' We are told that □⋆4\square \star 4 equals 77, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×7+a×ba \star b = a \times 7 + a \times b
  • □⋆4=77\square \star 4 = 77
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆4\square \star 4 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 4 in the rule a⋆b=a×7+a×ba \star b = a \times 7 + a \times b.
□⋆4=□×7+□×4\square \star 4 = \square \times 7 + \square \times 4
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 7 of the □\square and 4 of the □\square gives 11 of the □\square.
□×7+□×4=□×11\square \times 7 + \square \times 4 = \square \times 11
7 groups plus 4 groups of the same size is 11 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 11 = 77. Since 7 times 11 = 77, the □\square is 7.
□×11=77⇒□=7\square \times 11 = 77 \Rightarrow \square = 7
Testing the factor 7 (because 7 times 11 = 77) confirms the answer at once.
Answer: 7
4 · Reviewdoes it hold up?

Check: 7⋆47 \star 4 = 7 times 7 + 7 times 4 = 49 + 28 = 77, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 7 plus □\square times 4 = 77, then divide 77 by 11 directly (work backwards, tool 11) to get 7.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 7 and $\square$ times 4 into $\square$ times 11 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 11 = 77
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 7 groups plus 4 groups is simply 11 groups!
Variant 7 medium answer: 7

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×6+a×ba \star b = a \times 6 + a \times b

□⋆6=84\square \star 6 = 84

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 6, then add a times b.' We are told that □⋆6\square \star 6 equals 84, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×6+a×ba \star b = a \times 6 + a \times b
  • □⋆6=84\square \star 6 = 84
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆6\square \star 6 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 6 in the rule a⋆b=a×6+a×ba \star b = a \times 6 + a \times b.
□⋆6=□×6+□×6\square \star 6 = \square \times 6 + \square \times 6
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 6 of the □\square and 6 of the □\square gives 12 of the □\square.
□×6+□×6=□×12\square \times 6 + \square \times 6 = \square \times 12
6 groups plus 6 groups of the same size is 12 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 12 = 84. Since 7 times 12 = 84, the □\square is 7.
□×12=84⇒□=7\square \times 12 = 84 \Rightarrow \square = 7
Testing the factor 7 (because 7 times 12 = 84) confirms the answer at once.
Answer: 7
4 · Reviewdoes it hold up?

Check: 7⋆67 \star 6 = 7 times 6 + 7 times 6 = 42 + 42 = 84, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 6 plus □\square times 6 = 84, then divide 84 by 12 directly (work backwards, tool 11) to get 7.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 6 and $\square$ times 6 into $\square$ times 12 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 12 = 84
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 6 groups plus 6 groups is simply 12 groups!
Variant 8 hard answer: 8

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×4+a×ba \star b = a \times 4 + a \times b

□⋆7=88\square \star 7 = 88

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 4, then add a times b.' We are told that □⋆7\square \star 7 equals 88, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×4+a×ba \star b = a \times 4 + a \times b
  • □⋆7=88\square \star 7 = 88
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆7\square \star 7 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 7 in the rule a⋆b=a×4+a×ba \star b = a \times 4 + a \times b.
□⋆7=□×4+□×7\square \star 7 = \square \times 4 + \square \times 7
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 4 of the □\square and 7 of the □\square gives 11 of the □\square.
□×4+□×7=□×11\square \times 4 + \square \times 7 = \square \times 11
4 groups plus 7 groups of the same size is 11 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 11 = 88. Since 8 times 11 = 88, the □\square is 8.
□×11=88⇒□=8\square \times 11 = 88 \Rightarrow \square = 8
Testing the factor 8 (because 8 times 11 = 88) confirms the answer at once.
Answer: 8
4 · Reviewdoes it hold up?

Check: 8⋆78 \star 7 = 8 times 4 + 8 times 7 = 32 + 56 = 88, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 4 plus □\square times 7 = 88, then divide 88 by 11 directly (work backwards, tool 11) to get 8.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 4 and $\square$ times 7 into $\square$ times 11 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 11 = 88
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 4 groups plus 7 groups is simply 11 groups!
Variant 9 hard answer: 9

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×8+a×ba \star b = a \times 8 + a \times b

□⋆2=90\square \star 2 = 90

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 8, then add a times b.' We are told that □⋆2\square \star 2 equals 90, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×8+a×ba \star b = a \times 8 + a \times b
  • □⋆2=90\square \star 2 = 90
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆2\square \star 2 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 2 in the rule a⋆b=a×8+a×ba \star b = a \times 8 + a \times b.
□⋆2=□×8+□×2\square \star 2 = \square \times 8 + \square \times 2
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 8 of the □\square and 2 of the □\square gives 10 of the □\square.
□×8+□×2=□×10\square \times 8 + \square \times 2 = \square \times 10
8 groups plus 2 groups of the same size is 10 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 10 = 90. Since 9 times 10 = 90, the □\square is 9.
□×10=90⇒□=9\square \times 10 = 90 \Rightarrow \square = 9
Testing the factor 9 (because 9 times 10 = 90) confirms the answer at once.
Answer: 9
4 · Reviewdoes it hold up?

Check: 9⋆29 \star 2 = 9 times 8 + 9 times 2 = 72 + 18 = 90, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 8 plus □\square times 2 = 90, then divide 90 by 10 directly (work backwards, tool 11) to get 9.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 8 and $\square$ times 2 into $\square$ times 10 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 10 = 90
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 8 groups plus 2 groups is simply 10 groups!
Variant 10 hard answer: 9

The symbol ⋆\star is defined as follows. Find the number that belongs in □\square.

a⋆b=a×7+a×ba \star b = a \times 7 + a \times b

□⋆8=135\square \star 8 = 135

Show solution
1 · Understandwhat's really being asked

A new symbol ⋆\star is defined so that a⋆ba \star b means 'multiply a by 7, then add a times b.' We are told that □⋆8\square \star 8 equals 135, and we must find the number that goes in the □\square.

Givens
  • The rule: a⋆b=a×7+a×ba \star b = a \times 7 + a \times b
  • □⋆8=135\square \star 8 = 135
Unknowns
  • The number that belongs in the □\square
Constraints
  • The □\square stands for one whole number
  • Both products use the same first number (the □\square)
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns □⋆8\square \star 8 into a single multiplication once we notice both terms share the □\square as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the □\square.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8
Replace a with the □\square and b with 8 in the rule a⋆b=a×7+a×ba \star b = a \times 7 + a \times b.
□⋆8=□×7+□×8\square \star 8 = \square \times 7 + \square \times 8
Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5
Both products multiply the same □\square. Adding 7 of the □\square and 8 of the □\square gives 15 of the □\square.
□×7+□×8=□×15\square \times 7 + \square \times 8 = \square \times 15
7 groups plus 8 groups of the same size is 15 groups; this is the distributive property kids already use.

3Find the □\square

#6 Guess and Check 3.OA.A.3
We need □\square times 15 = 135. Since 9 times 15 = 135, the □\square is 9.
□×15=135⇒□=9\square \times 15 = 135 \Rightarrow \square = 9
Testing the factor 9 (because 9 times 15 = 135) confirms the answer at once.
Answer: 9
4 · Reviewdoes it hold up?

Check: 9⋆89 \star 8 = 9 times 7 + 9 times 8 = 63 + 72 = 135, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: □\square times 7 plus □\square times 8 = 135, then divide 135 by 15 directly (work backwards, tool 11) to get 9.

Standardsmin grade 3
  • 3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
  • 3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining $\square$ times 7 and $\square$ times 8 into $\square$ times 15 using the distributive property
  • 3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the $\square$ from $\square$ times 15 = 135
💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 7 groups plus 8 groups is simply 15 groups!