Practice · Make result largest or smallest · Sensim Math

Variant 1 easy

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$6 \times 7 + 3 - 12 \div 3$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 6 times 7 plus 3 minus 12 divided by 3 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $6 \times 7 + 3 - 12 \div 3$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 6 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

6 \times 7 + 3 - 12 \div 3 = 41

41 is the baseline to beat.

2Notice 6 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 6 gets multiplied by 6, so adding to that group is worth 6 times as much as adding outside it.

6 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $56

56 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 41,\quad a \times (b + c - d) \div e \to -4,\quad a \times b + (c - d) \div e \to 39,\quad (a \times b + c - d) \div e \to 11

The largest result is 56.

Answer: 56

4 · Reviewdoes it hold up?

The winner beats the unbracketed 41, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 2 easy

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$4 \times 13 + 5 - 10 \div 2$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 4 times 13 plus 5 minus 10 divided by 2 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $4 \times 13 + 5 - 10 \div 2$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 4 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

4 \times 13 + 5 - 10 \div 2 = 52

52 is the baseline to beat.

2Notice 4 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 4 gets multiplied by 4, so adding to that group is worth 4 times as much as adding outside it.

4 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $67

67 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 52,\quad a \times (b + c - d) \div e \to 16

The largest result is 67.

Answer: 67

4 · Reviewdoes it hold up?

The winner beats the unbracketed 52, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 3 easy

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$13 \times 5 + 3 - 14 \div 7$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 13 times 5 plus 3 minus 14 divided by 7 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $13 \times 5 + 3 - 14 \div 7$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 13 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

13 \times 5 + 3 - 14 \div 7 = 66

66 is the baseline to beat.

2Notice 13 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 13 gets multiplied by 13, so adding to that group is worth 13 times as much as adding outside it.

13 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $102

102 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 66

The largest result is 102.

Answer: 102

4 · Reviewdoes it hold up?

The winner beats the unbracketed 66, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 4 easy

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$7 \times 9 + 2 - 15 \div 5$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 7 times 9 plus 2 minus 15 divided by 5 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $7 \times 9 + 2 - 15 \div 5$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 7 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

7 \times 9 + 2 - 15 \div 5 = 62

62 is the baseline to beat.

2Notice 7 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 7 gets multiplied by 7, so adding to that group is worth 7 times as much as adding outside it.

7 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $74

74 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 62,\quad (a \times b + c - d) \div e \to 10

The largest result is 74.

Answer: 74

4 · Reviewdoes it hold up?

The winner beats the unbracketed 62, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 5 medium

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$10 \times 8 + 5 - 16 \div 4$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 10 times 8 plus 5 minus 16 divided by 4 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $10 \times 8 + 5 - 16 \div 4$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 10 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

10 \times 8 + 5 - 16 \div 4 = 81

81 is the baseline to beat.

2Notice 10 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 10 gets multiplied by 10, so adding to that group is worth 10 times as much as adding outside it.

10 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $126

126 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 81

The largest result is 126.

Answer: 126

4 · Reviewdoes it hold up?

The winner beats the unbracketed 81, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 6 medium

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$8 \times 6 + 7 - 18 \div 6$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 8 times 6 plus 7 minus 18 divided by 6 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $8 \times 6 + 7 - 18 \div 6$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 8 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

8 \times 6 + 7 - 18 \div 6 = 52

52 is the baseline to beat.

2Notice 8 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 8 gets multiplied by 8, so adding to that group is worth 8 times as much as adding outside it.

8 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $101

101 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 52

The largest result is 101.

Answer: 101

4 · Reviewdoes it hold up?

The winner beats the unbracketed 52, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 7 medium

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$9 \times 5 + 4 - 20 \div 5$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 9 times 5 plus 4 minus 20 divided by 5 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $9 \times 5 + 4 - 20 \div 5$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 9 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

9 \times 5 + 4 - 20 \div 5 = 45

45 is the baseline to beat.

2Notice 9 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 9 gets multiplied by 9, so adding to that group is worth 9 times as much as adding outside it.

9 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $77

77 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 45

The largest result is 77.

Answer: 77

4 · Reviewdoes it hold up?

The winner beats the unbracketed 45, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 8 medium

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$5 \times 11 + 6 - 21 \div 7$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 5 times 11 plus 6 minus 21 divided by 7 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $5 \times 11 + 6 - 21 \div 7$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 5 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

5 \times 11 + 6 - 21 \div 7 = 58

58 is the baseline to beat.

2Notice 5 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 5 gets multiplied by 5, so adding to that group is worth 5 times as much as adding outside it.

5 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $82

82 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 58

The largest result is 82.

Answer: 82

4 · Reviewdoes it hold up?

The winner beats the unbracketed 58, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 9 hard

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$12 \times 4 + 9 - 24 \div 8$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 12 times 4 plus 9 minus 24 divided by 8 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $12 \times 4 + 9 - 24 \div 8$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 12 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

12 \times 4 + 9 - 24 \div 8 = 54

54 is the baseline to beat.

2Notice 12 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 12 gets multiplied by 12, so adding to that group is worth 12 times as much as adding outside it.

12 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $153

153 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 54

The largest result is 153.

Answer: 153

4 · Reviewdoes it hold up?

The winner beats the unbracketed 54, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 10 hard

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$11 \times 3 + 8 - 27 \div 9$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 11 times 3 plus 8 minus 27 divided by 9 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $11 \times 3 + 8 - 27 \div 9$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 11 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

11 \times 3 + 8 - 27 \div 9 = 38

38 is the baseline to beat.

2Notice 11 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 11 gets multiplied by 11, so adding to that group is worth 11 times as much as adding outside it.

11 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $118

118 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 38

The largest result is 118.

Answer: 118

4 · Reviewdoes it hold up?

The winner beats the unbracketed 38, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 11 hard

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$6 \times 12 + 4 - 30 \div 6$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 6 times 12 plus 4 minus 30 divided by 6 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $6 \times 12 + 4 - 30 \div 6$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 6 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

6 \times 12 + 4 - 30 \div 6 = 71

71 is the baseline to beat.

2Notice 6 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 6 gets multiplied by 6, so adding to that group is worth 6 times as much as adding outside it.

6 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $91

91 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 71,\quad a \times (b + c - d) \div e \to -14

The largest result is 91.

Answer: 91

4 · Reviewdoes it hold up?

The winner beats the unbracketed 71, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Variant 12 hard

Insert one pair of parentheses ( ) into the expression below so that the result is as large as possible, then find that largest result. (The result must be a whole number.)

$9 \times 7 + 6 - 32 \div 8$

Your answer

Show solution

1 · Understandwhat's really being asked

The expression 9 times 7 plus 6 minus 32 divided by 8 is given. One pair of brackets may go anywhere. We want the placement that makes the value largest, with a whole-number result.

Givens

The expression is $9 \times 7 + 6 - 32 \div 8$ .
Exactly one pair of parentheses may be inserted.
The result must be a whole number.

Unknowns

The largest result obtainable.

Constraints

Parentheses may only group a run of the expression as written; the numbers cannot be reordered.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List#5 Look for a Pattern

There are only a handful of places brackets can go, so they can all be tried. But thinking first narrows it: 9 multiplies whatever follows it, so brackets that make that partner bigger have the most leverage.

3 · Execute4 carry out the plan

1See what happens with no parentheses

#5 Look for a Pattern 5.OA.A.1

Ordinary order of operations: the multiplication and division first, then the addition and subtraction.

9 \times 7 + 6 - 32 \div 8 = 65

65 is the baseline to beat.

2Notice 9 is a multiplier -- grow what it multiplies

#5 Look for a Pattern 5.OA.A.2

Everything inside brackets right after 9 gets multiplied by 9, so adding to that group is worth 9 times as much as adding outside it.

9 \times (\ldots)

Leverage sits with the multiplier.

3Finish that candidate placement

#6 Guess and Check 5.OA.A.1

Working out the most promising bracketing gives its value.

a \times (b + c) - d \div e$ gives $113

113 is the candidate to beat.

4Check the other reasonable placements

#2 Make a Systematic List 5.OA.A.1

Each remaining bracketing that still leaves a whole number comes out smaller.

(a \times b + c) - d \div e \to 65

The largest result is 113.

Answer: 113

4 · Reviewdoes it hold up?

The winner beats the unbracketed 65, which it must if the brackets did anything useful.

Another way: Evaluating every placement without thinking first gives the same answer; the multiplier argument just says which one to try first.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. — Evaluating each bracketed version in the right order.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. — Reading which grouping the multiplier amplifies, before computing.

💡Takeaway. Brackets next to a multiplier are worth the most -- whatever you put inside gets multiplied along with everything else.

Make result largest or smallest · 12 practice problems

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