← List two-digit numbers from digit conditions · Pin Down a Number from Digit and Range Conditions

List two-digit numbers from digit conditions · 12 practice problems

1.NBT.B.2K.MD.B.3

Generated variants — 12

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

Variant 1 easy answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $1$ and less than $4$ , and whose ones digit is greater than $3$ and at most $5$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 1 or more but under 4; the ones digit must be above 3, up to and including 5. We must count every number that fits.

Givens

Tens digit: at least 1, less than 4.
Ones digit: greater than 3, at most 5.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 1' includes 1 itself; 'less than 4' stops before 4.

1, 2, 3

3 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 3' starts after 3; 'at most 5' includes 5.

4, 5

2 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

14, 15, 24, 25, 34, 35

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 3 tens digits and 2 ones digits, and every combination appears exactly once.

3 \times 2 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 14 and 35 both fit, while 44 fails the tens rule and 13 fails the ones rule.

Another way: Multiplying the two counts, 3 x 2, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 2 easy answer: 9 numbers

We want to make two-digit whole numbers whose tens digit is at least $3$ and less than $6$ , and whose ones digit is greater than $0$ and at most $3$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 3 or more but under 6; the ones digit must be above 0, up to and including 3. We must count every number that fits.

Givens

Tens digit: at least 3, less than 6.
Ones digit: greater than 0, at most 3.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 3' includes 3 itself; 'less than 6' stops before 6.

3, 4, 5

3 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 0' starts after 0; 'at most 3' includes 3.

1, 2, 3

3 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

31, 32, 33, 41, 42, 43, 51, 52, 53

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 3 tens digits and 3 ones digits, and every combination appears exactly once.

3 \times 3 = 9

There are 9 such numbers.

Answer: 9 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 31 and 53 both fit, while 61 fails the tens rule and 30 fails the ones rule.

Another way: Multiplying the two counts, 3 x 3, gives 9 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 3 medium answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $5$ and less than $7$ , and whose ones digit is greater than $3$ and at most $6$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 5 or more but under 7; the ones digit must be above 3, up to and including 6. We must count every number that fits.

Givens

Tens digit: at least 5, less than 7.
Ones digit: greater than 3, at most 6.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 5' includes 5 itself; 'less than 7' stops before 7.

5, 6

2 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 3' starts after 3; 'at most 6' includes 6.

4, 5, 6

3 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

54, 55, 56, 64, 65, 66

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 2 tens digits and 3 ones digits, and every combination appears exactly once.

2 \times 3 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 54 and 66 both fit, while 74 fails the tens rule and 53 fails the ones rule.

Another way: Multiplying the two counts, 2 x 3, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 4 easy answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $4$ and less than $6$ , and whose ones digit is greater than $4$ and at most $7$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 4 or more but under 6; the ones digit must be above 4, up to and including 7. We must count every number that fits.

Givens

Tens digit: at least 4, less than 6.
Ones digit: greater than 4, at most 7.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 4' includes 4 itself; 'less than 6' stops before 6.

4, 5

2 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 4' starts after 4; 'at most 7' includes 7.

5, 6, 7

3 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

45, 46, 47, 55, 56, 57

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 2 tens digits and 3 ones digits, and every combination appears exactly once.

2 \times 3 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 45 and 57 both fit, while 65 fails the tens rule and 44 fails the ones rule.

Another way: Multiplying the two counts, 2 x 3, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 5 easy answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $2$ and less than $5$ , and whose ones digit is greater than $5$ and at most $7$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 2 or more but under 5; the ones digit must be above 5, up to and including 7. We must count every number that fits.

Givens

Tens digit: at least 2, less than 5.
Ones digit: greater than 5, at most 7.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 2' includes 2 itself; 'less than 5' stops before 5.

2, 3, 4

3 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 5' starts after 5; 'at most 7' includes 7.

6, 7

2 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

26, 27, 36, 37, 46, 47

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 3 tens digits and 2 ones digits, and every combination appears exactly once.

3 \times 2 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 26 and 47 both fit, while 56 fails the tens rule and 25 fails the ones rule.

Another way: Multiplying the two counts, 3 x 2, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 6 medium answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $5$ and less than $8$ , and whose ones digit is greater than $0$ and at most $2$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 5 or more but under 8; the ones digit must be above 0, up to and including 2. We must count every number that fits.

Givens

Tens digit: at least 5, less than 8.
Ones digit: greater than 0, at most 2.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 5' includes 5 itself; 'less than 8' stops before 8.

5, 6, 7

3 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 0' starts after 0; 'at most 2' includes 2.

1, 2

2 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

51, 52, 61, 62, 71, 72

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 3 tens digits and 2 ones digits, and every combination appears exactly once.

3 \times 2 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 51 and 72 both fit, while 81 fails the tens rule and 50 fails the ones rule.

Another way: Multiplying the two counts, 3 x 2, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 7 medium answer: 4 numbers

We want to make two-digit whole numbers whose tens digit is at least $2$ and less than $4$ , and whose ones digit is greater than $6$ and at most $8$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 2 or more but under 4; the ones digit must be above 6, up to and including 8. We must count every number that fits.

Givens

Tens digit: at least 2, less than 4.
Ones digit: greater than 6, at most 8.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 2' includes 2 itself; 'less than 4' stops before 4.

2, 3

2 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 6' starts after 6; 'at most 8' includes 8.

7, 8

2 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

27, 28, 37, 38

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 2 tens digits and 2 ones digits, and every combination appears exactly once.

2 \times 2 = 4

There are 4 such numbers.

Answer: 4 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 27 and 38 both fit, while 47 fails the tens rule and 26 fails the ones rule.

Another way: Multiplying the two counts, 2 x 2, gives 4 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 8 hard answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $7$ and less than $9$ , and whose ones digit is greater than $1$ and at most $4$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 7 or more but under 9; the ones digit must be above 1, up to and including 4. We must count every number that fits.

Givens

Tens digit: at least 7, less than 9.
Ones digit: greater than 1, at most 4.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 7' includes 7 itself; 'less than 9' stops before 9.

7, 8

2 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 1' starts after 1; 'at most 4' includes 4.

2, 3, 4

3 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

72, 73, 74, 82, 83, 84

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 2 tens digits and 3 ones digits, and every combination appears exactly once.

2 \times 3 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 72 and 84 both fit, while 92 fails the tens rule and 71 fails the ones rule.

Another way: Multiplying the two counts, 2 x 3, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 9 medium answer: 4 numbers

We want to make two-digit whole numbers whose tens digit is at least $3$ and less than $5$ , and whose ones digit is greater than $7$ and at most $9$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 3 or more but under 5; the ones digit must be above 7, up to and including 9. We must count every number that fits.

Givens

Tens digit: at least 3, less than 5.
Ones digit: greater than 7, at most 9.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 3' includes 3 itself; 'less than 5' stops before 5.

3, 4

2 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 7' starts after 7; 'at most 9' includes 9.

8, 9

2 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

38, 39, 48, 49

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 2 tens digits and 2 ones digits, and every combination appears exactly once.

2 \times 2 = 4

There are 4 such numbers.

Answer: 4 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 38 and 49 both fit, while 58 fails the tens rule and 37 fails the ones rule.

Another way: Multiplying the two counts, 2 x 2, gives 4 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 10 hard answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $1$ and less than $3$ , and whose ones digit is greater than $6$ and at most $9$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 1 or more but under 3; the ones digit must be above 6, up to and including 9. We must count every number that fits.

Givens

Tens digit: at least 1, less than 3.
Ones digit: greater than 6, at most 9.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 1' includes 1 itself; 'less than 3' stops before 3.

1, 2

2 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 6' starts after 6; 'at most 9' includes 9.

7, 8, 9

3 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

17, 18, 19, 27, 28, 29

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 2 tens digits and 3 ones digits, and every combination appears exactly once.

2 \times 3 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 17 and 29 both fit, while 37 fails the tens rule and 16 fails the ones rule.

Another way: Multiplying the two counts, 2 x 3, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 11 hard answer: 9 numbers

We want to make two-digit whole numbers whose tens digit is at least $6$ and less than $9$ , and whose ones digit is greater than $2$ and at most $5$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 6 or more but under 9; the ones digit must be above 2, up to and including 5. We must count every number that fits.

Givens

Tens digit: at least 6, less than 9.
Ones digit: greater than 2, at most 5.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 6' includes 6 itself; 'less than 9' stops before 9.

6, 7, 8

3 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 2' starts after 2; 'at most 5' includes 5.

3, 4, 5

3 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

63, 64, 65, 73, 74, 75, 83, 84, 85

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 3 tens digits and 3 ones digits, and every combination appears exactly once.

3 \times 3 = 9

There are 9 such numbers.

Answer: 9 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 63 and 85 both fit, while 93 fails the tens rule and 62 fails the ones rule.

Another way: Multiplying the two counts, 3 x 3, gives 9 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.

Variant 12 hard answer: 6 numbers

We want to make two-digit whole numbers whose tens digit is at least $8$ and less than $10$ , and whose ones digit is greater than $5$ and at most $8$ .

How many such whole numbers can be made in all?

Show solution

1 · Understandwhat's really being asked

A two-digit number has a tens digit and a ones digit. The tens digit must be 8 or more but under 10; the ones digit must be above 5, up to and including 8. We must count every number that fits.

Givens

Tens digit: at least 8, less than 10.
Ones digit: greater than 5, at most 8.
The number has exactly two digits.

Unknowns

How many such numbers there are.

Constraints

'At least' and 'at most' include their endpoint; 'less than' and 'greater than' do not.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #5 Look for a Pattern

The two digits are chosen independently, so work out the allowed values for each place separately and then pair them up. Listing in order shows nothing is missed or repeated.

3 · Execute4 carry out the plan

1Find the allowed tens digits

#2 Make a Systematic List 1.NBT.B.2

'At least 8' includes 8 itself; 'less than 10' stops before 10.

8, 9

2 choices for the tens place.

2Find the allowed ones digits

#2 Make a Systematic List 1.NBT.B.2

'Greater than 5' starts after 5; 'at most 8' includes 8.

6, 7, 8

3 choices for the ones place.

3List every number by pairing the digits

#2 Make a Systematic List 1.NBT.B.2

Take each allowed tens digit with each allowed ones digit, in order, so the list is complete.

86, 87, 88, 96, 97, 98

Every pairing gives a different number.

4Count the list and notice the pattern

#5 Look for a Pattern K.MD.B.3

There are 2 tens digits and 3 ones digits, and every combination appears exactly once.

2 \times 3 = 6

There are 6 such numbers.

Answer: 6 numbers

4 · Reviewdoes it hold up?

Check the endpoints: 86 and 98 both fit, while 106 fails the tens rule and 85 fails the ones rule.

Another way: Multiplying the two counts, 2 x 3, gives 6 without writing the list -- but the list is what shows the multiplication is the right move.

Standardsmin grade 1

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones — Treating the tens and ones places as separate choices.
K.MD.B.3 Classify objects into given categories and count the numbers in each — Counting the finished list of numbers.

💡Takeaway. Read the endpoint words carefully: 'at least' and 'at most' let a digit in, 'less than' and 'greater than' keep it out.