← 3-1 · Big digits in high places · Build the Largest or Smallest Value from Digit Cards

Big digits in high places · 10 practice problems

2.NBT.A.43.NBT.A.2

Generated variants — 10

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

Variant 1 answer: 421 - 124 = 297

Using the number cards $1$ , $2$ , and $4$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 1, 2, 4 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 1, 2, 4.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 1, 2, 4 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 1, 2, 4 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

To make a difference biggest, make the first number as large as possible and the second as small as possible; the place-value principle (big digits in high places) tells us how to arrange the cards, and a quick systematic check confirms it.

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 4, 2, 1 from hundreds down gives the largest three-digit number 421.

421

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 1, 2, 4 from hundreds down gives the smallest three-digit number 124.

124

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 421 minus 124.

421 - 124 = 297

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 421 - 124 = 297

Review

421 and 124 use each card 1, 2, 4 exactly once, and 421 - 124 = 297 is positive and under 421, which is sensible. Any other pairing of these digits gives a smaller spread, so 297 is the maximum.

List the largest and smallest numbers from 1, 2, 4 and check max minus min: 421 - 124 = 297 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 2 answer: 753 - 357 = 396

Using the number cards $5$ , $3$ , and $7$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 5, 3, 7 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 5, 3, 7.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 5, 3, 7 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 5, 3, 7 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 7, 5, 3 from hundreds down gives the largest three-digit number 753.

753

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 3, 5, 7 from hundreds down gives the smallest three-digit number 357.

357

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 753 minus 357.

753 - 357 = 396

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 753 - 357 = 396

Review

753 and 357 use each card 5, 3, 7 exactly once, and 753 - 357 = 396 is positive and under 753, which is sensible. Any other pairing of these digits gives a smaller spread, so 396 is the maximum.

List the largest and smallest numbers from 5, 3, 7 and check max minus min: 753 - 357 = 396 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 3 answer: 842 - 248 = 594

Using the number cards $2$ , $8$ , and $4$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 2, 8, 4 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 2, 8, 4.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 2, 8, 4 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 2, 8, 4 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 8, 4, 2 from hundreds down gives the largest three-digit number 842.

842

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 2, 4, 8 from hundreds down gives the smallest three-digit number 248.

248

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 842 minus 248.

842 - 248 = 594

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 842 - 248 = 594

Review

842 and 248 use each card 2, 8, 4 exactly once, and 842 - 248 = 594 is positive and under 842, which is sensible. Any other pairing of these digits gives a smaller spread, so 594 is the maximum.

List the largest and smallest numbers from 2, 8, 4 and check max minus min: 842 - 248 = 594 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 4 answer: 962 - 269 = 693

Using the number cards $6$ , $2$ , and $9$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 6, 2, 9 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 6, 2, 9.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 6, 2, 9 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 6, 2, 9 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 9, 6, 2 from hundreds down gives the largest three-digit number 962.

962

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 2, 6, 9 from hundreds down gives the smallest three-digit number 269.

269

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 962 minus 269.

962 - 269 = 693

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 962 - 269 = 693

Review

962 and 269 use each card 6, 2, 9 exactly once, and 962 - 269 = 693 is positive and under 962, which is sensible. Any other pairing of these digits gives a smaller spread, so 693 is the maximum.

List the largest and smallest numbers from 6, 2, 9 and check max minus min: 962 - 269 = 693 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 5 answer: 963 - 369 = 594

Using the number cards $3$ , $6$ , and $9$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 3, 6, 9 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 3, 6, 9.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 3, 6, 9 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 3, 6, 9 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 9, 6, 3 from hundreds down gives the largest three-digit number 963.

963

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 3, 6, 9 from hundreds down gives the smallest three-digit number 369.

369

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 963 minus 369.

963 - 369 = 594

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 963 - 369 = 594

Review

963 and 369 use each card 3, 6, 9 exactly once, and 963 - 369 = 594 is positive and under 963, which is sensible. Any other pairing of these digits gives a smaller spread, so 594 is the maximum.

List the largest and smallest numbers from 3, 6, 9 and check max minus min: 963 - 369 = 594 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 6 answer: 962 - 269 = 693

Using the number cards $9$ , $6$ , and $2$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 9, 6, 2 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 9, 6, 2.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 9, 6, 2 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 9, 6, 2 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 9, 6, 2 from hundreds down gives the largest three-digit number 962.

962

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 2, 6, 9 from hundreds down gives the smallest three-digit number 269.

269

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 962 minus 269.

962 - 269 = 693

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 962 - 269 = 693

Review

962 and 269 use each card 9, 6, 2 exactly once, and 962 - 269 = 693 is positive and under 962, which is sensible. Any other pairing of these digits gives a smaller spread, so 693 is the maximum.

List the largest and smallest numbers from 9, 6, 2 and check max minus min: 962 - 269 = 693 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 7 answer: 731 - 137 = 594

Using the number cards $7$ , $3$ , and $1$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 7, 3, 1 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 7, 3, 1.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 7, 3, 1 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 7, 3, 1 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 7, 3, 1 from hundreds down gives the largest three-digit number 731.

731

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 1, 3, 7 from hundreds down gives the smallest three-digit number 137.

137

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 731 minus 137.

731 - 137 = 594

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 731 - 137 = 594

Review

731 and 137 use each card 7, 3, 1 exactly once, and 731 - 137 = 594 is positive and under 731, which is sensible. Any other pairing of these digits gives a smaller spread, so 594 is the maximum.

List the largest and smallest numbers from 7, 3, 1 and check max minus min: 731 - 137 = 594 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 8 answer: 851 - 158 = 693

Using the number cards $8$ , $1$ , and $5$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 8, 1, 5 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 8, 1, 5.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 8, 1, 5 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 8, 1, 5 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 8, 5, 1 from hundreds down gives the largest three-digit number 851.

851

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 1, 5, 8 from hundreds down gives the smallest three-digit number 158.

158

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 851 minus 158.

851 - 158 = 693

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 851 - 158 = 693

Review

851 and 158 use each card 8, 1, 5 exactly once, and 851 - 158 = 693 is positive and under 851, which is sensible. Any other pairing of these digits gives a smaller spread, so 693 is the maximum.

List the largest and smallest numbers from 8, 1, 5 and check max minus min: 851 - 158 = 693 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 9 answer: 851 - 158 = 693

Using the number cards $1$ , $5$ , and $8$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 1, 5, 8 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 1, 5, 8.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 1, 5, 8 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 1, 5, 8 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 8, 5, 1 from hundreds down gives the largest three-digit number 851.

851

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 1, 5, 8 from hundreds down gives the smallest three-digit number 158.

158

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 851 minus 158.

851 - 158 = 693

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 851 - 158 = 693

Review

851 and 158 use each card 1, 5, 8 exactly once, and 851 - 158 = 693 is positive and under 851, which is sensible. Any other pairing of these digits gives a smaller spread, so 693 is the maximum.

List the largest and smallest numbers from 1, 5, 8 and check max minus min: 851 - 158 = 693 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!

Variant 10 answer: 742 - 247 = 495

Using the number cards $4$ , $7$ , and $2$ , each exactly once, form two three-digit numbers. Write the subtraction that makes the difference of the two numbers as large as possible, and find that difference.

Show solution

Understand

Use the digit cards 4, 7, 2 once each to build two three-digit numbers, then subtract so the difference is as large as possible. Find that largest possible difference.

Givens

The available digit cards are 4, 7, 2.
Each card is used exactly once; to make the difference largest we build the largest and the smallest three-digit number from the same three digits.
Both numbers are three-digit numbers.

Unknowns

Which arrangement of 4, 7, 2 gives the largest number and which gives the smallest.
The largest possible difference of the two numbers.

Constraints

Only the digits 4, 7, 2 may be used.
Each number is a three-digit number (hundreds, tens, units).

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Execute

#2 Make a Systematic List 3.NBT.A.2

A number is largest when the biggest digit sits in the highest place. Ordering 7, 4, 2 from hundreds down gives the largest three-digit number 742.

742

The hundreds place is worth the most, so the largest digit belongs there.

#2 Make a Systematic List 2.NBT.A.4

A number is smallest when the smallest digit sits in the highest place. Ordering 2, 4, 7 from hundreds down gives the smallest three-digit number 247.

247

Putting the smallest digit in the hundreds place makes the whole number as small as it can be.

#6 Guess and Check 3.NBT.A.2

The largest difference comes from the largest number minus the smallest number: 742 minus 247.

742 - 247 = 495

Biggest minus smallest stretches the gap as far apart as the cards allow.

Answer: 742 - 247 = 495

Review

742 and 247 use each card 4, 7, 2 exactly once, and 742 - 247 = 495 is positive and under 742, which is sensible. Any other pairing of these digits gives a smaller spread, so 495 is the maximum.

List the largest and smallest numbers from 4, 7, 2 and check max minus min: 742 - 247 = 495 confirms the answer.

Standards · min grade 3

2.NBT.A.4 Compare two three-digit numbers using symbols — Comparing arrangements to identify the smallest three-digit number.
3.NBT.A.2 Fluently add and subtract within 1000 — Forming the largest number and subtracting to get the difference.

💡 This only needs Grade 3 place-value sense: put big digits up high for the biggest number, small digits up high for the smallest, then subtract!