Practice · Define a custom operator and evaluate it · Sensim Math

Variant 1 easy answer: 30

Using the rules defined below, find the value of $[\,36 \bigstar 60\,] \blacklozenge 10$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 36 star 60.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,36 \bigstar 60\,] \blacklozenge 10$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 36 star 60

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 36 and 60.

\gcd(36, 60) = 12

The largest number dividing both is 12.

2Evaluate the bracket: count the factors of 12

#2 Make a Systematic List 4.OA.B.4

List every number that divides 12 exactly, then count them.

1, 2, 3, 4, 6, 12

There are 6 of them, so the bracket is worth 6.

3Evaluate the outer expression: 6 diamond 10

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 6 and 10.

\operatorname{lcm}(6, 10) = 30

The whole expression is worth 30.

Answer: 30

4 · Reviewdoes it hold up?

Check the last step both ways: 30 divided by 6 is 5 and 30 divided by 10 is 3, both whole, and no smaller number does that.

Another way: Since 12 is the greatest common factor, its factors are exactly the common factors of 36 and 60; counting those directly gives the same 6.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 36 and 60, and listing the factors of 12.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 2 easy answer: 44

Using the rules defined below, find the value of $[\,45 \bigstar 60\,] \blacklozenge 22$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 45 star 60.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,45 \bigstar 60\,] \blacklozenge 22$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 45 star 60

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 45 and 60.

\gcd(45, 60) = 15

The largest number dividing both is 15.

2Evaluate the bracket: count the factors of 15

#2 Make a Systematic List 4.OA.B.4

List every number that divides 15 exactly, then count them.

1, 3, 5, 15

There are 4 of them, so the bracket is worth 4.

3Evaluate the outer expression: 4 diamond 22

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 4 and 22.

\operatorname{lcm}(4, 22) = 44

The whole expression is worth 44.

Answer: 44

4 · Reviewdoes it hold up?

Check the last step both ways: 44 divided by 4 is 11 and 44 divided by 22 is 2, both whole, and no smaller number does that.

Another way: Since 15 is the greatest common factor, its factors are exactly the common factors of 45 and 60; counting those directly gives the same 4.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 45 and 60, and listing the factors of 15.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 3 easy answer: 60

Using the rules defined below, find the value of $[\,48 \bigstar 64\,] \blacklozenge 12$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 48 star 64.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,48 \bigstar 64\,] \blacklozenge 12$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 48 star 64

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 48 and 64.

\gcd(48, 64) = 16

The largest number dividing both is 16.

2Evaluate the bracket: count the factors of 16

#2 Make a Systematic List 4.OA.B.4

List every number that divides 16 exactly, then count them.

1, 2, 4, 8, 16

There are 5 of them, so the bracket is worth 5.

3Evaluate the outer expression: 5 diamond 12

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 5 and 12.

\operatorname{lcm}(5, 12) = 60

The whole expression is worth 60.

Answer: 60

4 · Reviewdoes it hold up?

Check the last step both ways: 60 divided by 5 is 12 and 60 divided by 12 is 5, both whole, and no smaller number does that.

Another way: Since 16 is the greatest common factor, its factors are exactly the common factors of 48 and 64; counting those directly gives the same 5.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 48 and 64, and listing the factors of 16.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 4 easy answer: 108

Using the rules defined below, find the value of $[\,56 \bigstar 70\,] \blacklozenge 27$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 56 star 70.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,56 \bigstar 70\,] \blacklozenge 27$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 56 star 70

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 56 and 70.

\gcd(56, 70) = 14

The largest number dividing both is 14.

2Evaluate the bracket: count the factors of 14

#2 Make a Systematic List 4.OA.B.4

List every number that divides 14 exactly, then count them.

1, 2, 7, 14

There are 4 of them, so the bracket is worth 4.

3Evaluate the outer expression: 4 diamond 27

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 4 and 27.

\operatorname{lcm}(4, 27) = 108

The whole expression is worth 108.

Answer: 108

4 · Reviewdoes it hold up?

Check the last step both ways: 108 divided by 4 is 27 and 108 divided by 27 is 4, both whole, and no smaller number does that.

Another way: Since 14 is the greatest common factor, its factors are exactly the common factors of 56 and 70; counting those directly gives the same 4.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 56 and 70, and listing the factors of 14.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 5 medium answer: 24

Using the rules defined below, find the value of $[\,50 \bigstar 75\,] \blacklozenge 8$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 50 star 75.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,50 \bigstar 75\,] \blacklozenge 8$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 50 star 75

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 50 and 75.

\gcd(50, 75) = 25

The largest number dividing both is 25.

2Evaluate the bracket: count the factors of 25

#2 Make a Systematic List 4.OA.B.4

List every number that divides 25 exactly, then count them.

1, 5, 25

There are 3 of them, so the bracket is worth 3.

3Evaluate the outer expression: 3 diamond 8

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 3 and 8.

\operatorname{lcm}(3, 8) = 24

The whole expression is worth 24.

Answer: 24

4 · Reviewdoes it hold up?

Check the last step both ways: 24 divided by 3 is 8 and 24 divided by 8 is 3, both whole, and no smaller number does that.

Another way: Since 25 is the greatest common factor, its factors are exactly the common factors of 50 and 75; counting those directly gives the same 3.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 50 and 75, and listing the factors of 25.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 6 medium answer: 90

Using the rules defined below, find the value of $[\,32 \bigstar 80\,] \blacklozenge 18$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 32 star 80.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,32 \bigstar 80\,] \blacklozenge 18$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 32 star 80

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 32 and 80.

\gcd(32, 80) = 16

The largest number dividing both is 16.

2Evaluate the bracket: count the factors of 16

#2 Make a Systematic List 4.OA.B.4

List every number that divides 16 exactly, then count them.

1, 2, 4, 8, 16

There are 5 of them, so the bracket is worth 5.

3Evaluate the outer expression: 5 diamond 18

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 5 and 18.

\operatorname{lcm}(5, 18) = 90

The whole expression is worth 90.

Answer: 90

4 · Reviewdoes it hold up?

Check the last step both ways: 90 divided by 5 is 18 and 90 divided by 18 is 5, both whole, and no smaller number does that.

Another way: Since 16 is the greatest common factor, its factors are exactly the common factors of 32 and 80; counting those directly gives the same 5.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 32 and 80, and listing the factors of 16.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 7 medium answer: 28

Using the rules defined below, find the value of $[\,54 \bigstar 81\,] \blacklozenge 14$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 54 star 81.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,54 \bigstar 81\,] \blacklozenge 14$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 54 star 81

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 54 and 81.

\gcd(54, 81) = 27

The largest number dividing both is 27.

2Evaluate the bracket: count the factors of 27

#2 Make a Systematic List 4.OA.B.4

List every number that divides 27 exactly, then count them.

1, 3, 9, 27

There are 4 of them, so the bracket is worth 4.

3Evaluate the outer expression: 4 diamond 14

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 4 and 14.

\operatorname{lcm}(4, 14) = 28

The whole expression is worth 28.

Answer: 28

4 · Reviewdoes it hold up?

Check the last step both ways: 28 divided by 4 is 7 and 28 divided by 14 is 2, both whole, and no smaller number does that.

Another way: Since 27 is the greatest common factor, its factors are exactly the common factors of 54 and 81; counting those directly gives the same 4.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 54 and 81, and listing the factors of 27.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 8 medium answer: 84

Using the rules defined below, find the value of $[\,24 \bigstar 90\,] \blacklozenge 21$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 24 star 90.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,24 \bigstar 90\,] \blacklozenge 21$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 24 star 90

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 24 and 90.

\gcd(24, 90) = 6

The largest number dividing both is 6.

2Evaluate the bracket: count the factors of 6

#2 Make a Systematic List 4.OA.B.4

List every number that divides 6 exactly, then count them.

1, 2, 3, 6

There are 4 of them, so the bracket is worth 4.

3Evaluate the outer expression: 4 diamond 21

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 4 and 21.

\operatorname{lcm}(4, 21) = 84

The whole expression is worth 84.

Answer: 84

4 · Reviewdoes it hold up?

Check the last step both ways: 84 divided by 4 is 21 and 84 divided by 21 is 4, both whole, and no smaller number does that.

Another way: Since 6 is the greatest common factor, its factors are exactly the common factors of 24 and 90; counting those directly gives the same 4.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 24 and 90, and listing the factors of 6.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 9 hard answer: 120

Using the rules defined below, find the value of $[\,72 \bigstar 96\,] \blacklozenge 15$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 72 star 96.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,72 \bigstar 96\,] \blacklozenge 15$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 72 star 96

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 72 and 96.

\gcd(72, 96) = 24

The largest number dividing both is 24.

2Evaluate the bracket: count the factors of 24

#2 Make a Systematic List 4.OA.B.4

List every number that divides 24 exactly, then count them.

1, 2, 3, 4, 6, 8, 12, 24

There are 8 of them, so the bracket is worth 8.

3Evaluate the outer expression: 8 diamond 15

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 8 and 15.

\operatorname{lcm}(8, 15) = 120

The whole expression is worth 120.

Answer: 120

4 · Reviewdoes it hold up?

Check the last step both ways: 120 divided by 8 is 15 and 120 divided by 15 is 8, both whole, and no smaller number does that.

Another way: Since 24 is the greatest common factor, its factors are exactly the common factors of 72 and 96; counting those directly gives the same 8.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 72 and 96, and listing the factors of 24.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 10 hard answer: 100

Using the rules defined below, find the value of $[\,66 \bigstar 99\,] \blacklozenge 25$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 66 star 99.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,66 \bigstar 99\,] \blacklozenge 25$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 66 star 99

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 66 and 99.

\gcd(66, 99) = 33

The largest number dividing both is 33.

2Evaluate the bracket: count the factors of 33

#2 Make a Systematic List 4.OA.B.4

List every number that divides 33 exactly, then count them.

1, 3, 11, 33

There are 4 of them, so the bracket is worth 4.

3Evaluate the outer expression: 4 diamond 25

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 4 and 25.

\operatorname{lcm}(4, 25) = 100

The whole expression is worth 100.

Answer: 100

4 · Reviewdoes it hold up?

Check the last step both ways: 100 divided by 4 is 25 and 100 divided by 25 is 4, both whole, and no smaller number does that.

Another way: Since 33 is the greatest common factor, its factors are exactly the common factors of 66 and 99; counting those directly gives the same 4.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 66 and 99, and listing the factors of 33.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 11 hard answer: 18

Using the rules defined below, find the value of $[\,40 \bigstar 100\,] \blacklozenge 9$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 40 star 100.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,40 \bigstar 100\,] \blacklozenge 9$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 40 star 100

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 40 and 100.

\gcd(40, 100) = 20

The largest number dividing both is 20.

2Evaluate the bracket: count the factors of 20

#2 Make a Systematic List 4.OA.B.4

List every number that divides 20 exactly, then count them.

1, 2, 4, 5, 10, 20

There are 6 of them, so the bracket is worth 6.

3Evaluate the outer expression: 6 diamond 9

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 6 and 9.

\operatorname{lcm}(6, 9) = 18

The whole expression is worth 18.

Answer: 18

4 · Reviewdoes it hold up?

Check the last step both ways: 18 divided by 6 is 3 and 18 divided by 9 is 2, both whole, and no smaller number does that.

Another way: Since 20 is the greatest common factor, its factors are exactly the common factors of 40 and 100; counting those directly gives the same 6.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 40 and 100, and listing the factors of 20.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Variant 12 hard answer: 40

Using the rules defined below, find the value of $[\,84 \bigstar 126\,] \blacklozenge 20$ .

$a \bigstar b$ : the greatest common factor of $a$ and $b$
$a \blacklozenge b$ : the least common multiple of $a$ and $b$
$[\,a\,]$ : the number of factors of $a$

Show solution

1 · Understandwhat's really being asked

Three invented symbols are defined: a star means greatest common factor, a diamond means least common multiple, and brackets mean 'count the factors'. We must evaluate them in the order the notation sets out, starting with 84 star 126.

Givens

The star gives the greatest common factor of the two numbers.
The diamond gives the least common multiple of the two numbers.
The brackets give how many factors the number inside has.

Unknowns

The value of $[\,84 \bigstar 126\,] \blacklozenge 20$ .

Constraints

The brackets group the star operation, so it happens first.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List

None of the three symbols is new mathematics -- each renames something familiar. Peel them from the inside out, one small answer at a time, and list the factors rather than trying to count them in your head.

3 · Execute3 carry out the plan

1Evaluate the innermost subproblem: 84 star 126

#2 Make a Systematic List 4.OA.B.4

The star asks for the greatest common factor of 84 and 126.

\gcd(84, 126) = 42

The largest number dividing both is 42.

2Evaluate the bracket: count the factors of 42

#2 Make a Systematic List 4.OA.B.4

List every number that divides 42 exactly, then count them.

1, 2, 3, 6, 7, 14, 21, 42

There are 8 of them, so the bracket is worth 8.

3Evaluate the outer expression: 8 diamond 20

#7 Identify Subproblems 5.OA.A.1

The diamond asks for the least common multiple of 8 and 20.

\operatorname{lcm}(8, 20) = 40

The whole expression is worth 40.

Answer: 40

4 · Reviewdoes it hold up?

Check the last step both ways: 40 divided by 8 is 5 and 40 divided by 20 is 2, both whole, and no smaller number does that.

Another way: Since 42 is the greatest common factor, its factors are exactly the common factors of 84 and 126; counting those directly gives the same 8.

Standardsmin grade 5

4.OA.B.4 Find all factor pairs and recognize multiples; determine whether a number is a multiple — Finding the greatest common factor of 84 and 126, and listing the factors of 42.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate them — Working the bracketed part before the outer operation.

💡Takeaway. Invented symbols are labels for things you already know. Work from the innermost bracket outwards and each one turns into a number.

Define a custom operator and evaluate it · 12 practice problems

Generated variants — 12

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