Practice · Evaluate a custom-defined operation symbol · Sensim Math

Variant 1 easy

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

$a \star b = a \times 9 + a \times b$

$\square \star 3 = 60$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 9 + a \times b$
$\square \star 3 = 60$

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 $\square \star 3$ 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 \star b = a \times 9 + a \times b

.

\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

.

\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.

\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: $5 \star 3$ = 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

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

$a \star b = a \times 5 + a \times b$

$\square \star 5 = 60$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 5 + a \times b$
$\square \star 5 = 60$

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 $\square \star 5$ 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 \star b = a \times 5 + a \times b

.

\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

.

\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.

\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: $6 \star 5$ = 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

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

$a \star b = a \times 5 + a \times b$

$\square \star 4 = 72$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 5 + a \times b$
$\square \star 4 = 72$

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 $\square \star 4$ 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 \star b = a \times 5 + a \times b

.

\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

.

\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.

\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: $8 \star 4$ = 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

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

$a \star b = a \times 3 + a \times b$

$\square \star 9 = 72$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 3 + a \times b$
$\square \star 9 = 72$

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 $\square \star 9$ 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 \star b = a \times 3 + a \times b

.

\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

.

\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.

\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: $6 \star 9$ = 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

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

$a \star b = a \times 2 + a \times b$

$\square \star 6 = 72$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 2 + a \times b$
$\square \star 6 = 72$

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 $\square \star 6$ 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 \star b = a \times 2 + a \times b

.

\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

.

\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.

\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: $9 \star 6$ = 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

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

$a \star b = a \times 7 + a \times b$

$\square \star 4 = 77$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 7 + a \times b$
$\square \star 4 = 77$

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 $\square \star 4$ 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 \star b = a \times 7 + a \times b

.

\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

.

\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.

\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: $7 \star 4$ = 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

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

$a \star b = a \times 6 + a \times b$

$\square \star 6 = 84$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 6 + a \times b$
$\square \star 6 = 84$

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 $\square \star 6$ 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 \star b = a \times 6 + a \times b

.

\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

.

\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.

\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: $7 \star 6$ = 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

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

$a \star b = a \times 4 + a \times b$

$\square \star 7 = 88$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 4 + a \times b$
$\square \star 7 = 88$

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 $\square \star 7$ 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 \star b = a \times 4 + a \times b

.

\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

.

\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.

\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: $8 \star 7$ = 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

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

$a \star b = a \times 8 + a \times b$

$\square \star 2 = 90$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 8 + a \times b$
$\square \star 2 = 90$

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 $\square \star 2$ 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 \star b = a \times 8 + a \times b

.

\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

.

\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.

\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: $9 \star 2$ = 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

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

$a \star b = a \times 7 + a \times b$

$\square \star 8 = 135$

Your answer

Show solution

1 · Understandwhat's really being asked

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

Givens

The rule: $a \star b = a \times 7 + a \times b$
$\square \star 8 = 135$

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 $\square \star 8$ 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 \star b = a \times 7 + a \times b

.

\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

.

\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.

\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: $9 \star 8$ = 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!

Evaluate a custom-defined operation symbol · 10 practice problems

Generated variants — 10

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