Variant 1 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $0$ , $1$ , $2$ , $5$ , $6$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $2 \to 5$ , $5 \to 2$ , $6 \to 9$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 0, 1, 2, 5, 6, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 0, 1, 2, 5, 6.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 6, 5, 2, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 652

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 652 each 180 degrees changes their shapes.

6 \to 9,\quad 5 \to 2,\quad 2 \to 5

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 5, 2, 9 gives the rotated number.

652 \xrightarrow{180^\circ} 529

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

652 - 529 = 123

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 652; rotated = 529; difference = 652 - 529 = 123

4 · Reviewdoes it hold up?

652 is the biggest three-digit number from the cards, and 529 uses only valid turned digits. 652 - 529 = 123 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 652, then turn it upside-down (digits change AND flip order) to get 529 - Grade 4 place-value and subtraction!

Variant 2 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $6$ , $1$ , $5$ , $0$ , $2$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $2 \to 5$ , $5 \to 2$ , $6 \to 9$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 6, 1, 5, 0, 2, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 6, 1, 5, 0, 2.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 6, 5, 2, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 652

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 652 each 180 degrees changes their shapes.

6 \to 9,\quad 5 \to 2,\quad 2 \to 5

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 5, 2, 9 gives the rotated number.

652 \xrightarrow{180^\circ} 529

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

652 - 529 = 123

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 652; rotated = 529; difference = 652 - 529 = 123

4 · Reviewdoes it hold up?

652 is the biggest three-digit number from the cards, and 529 uses only valid turned digits. 652 - 529 = 123 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 652, then turn it upside-down (digits change AND flip order) to get 529 - Grade 4 place-value and subtraction!

Variant 3 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $8$ , $5$ , $9$ , $1$ , $0$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $5 \to 2$ , $8 \to 8$ , $9 \to 6$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 8, 5, 9, 1, 0, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 8, 5, 9, 1, 0.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 9, 8, 5, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 985

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 985 each 180 degrees changes their shapes.

9 \to 6,\quad 8 \to 8,\quad 5 \to 2

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 2, 8, 6 gives the rotated number.

985 \xrightarrow{180^\circ} 286

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

985 - 286 = 699

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 985; rotated = 286; difference = 985 - 286 = 699

4 · Reviewdoes it hold up?

985 is the biggest three-digit number from the cards, and 286 uses only valid turned digits. 985 - 286 = 699 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 985, then turn it upside-down (digits change AND flip order) to get 286 - Grade 4 place-value and subtraction!

Variant 4 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $9$ , $1$ , $2$ , $0$ , $5$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $2 \to 5$ , $5 \to 2$ , $9 \to 6$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 9, 1, 2, 0, 5, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 9, 1, 2, 0, 5.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 9, 5, 2, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 952

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 952 each 180 degrees changes their shapes.

9 \to 6,\quad 5 \to 2,\quad 2 \to 5

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 5, 2, 6 gives the rotated number.

952 \xrightarrow{180^\circ} 526

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

952 - 526 = 426

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 952; rotated = 526; difference = 952 - 526 = 426

4 · Reviewdoes it hold up?

952 is the biggest three-digit number from the cards, and 526 uses only valid turned digits. 952 - 526 = 426 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 952, then turn it upside-down (digits change AND flip order) to get 526 - Grade 4 place-value and subtraction!

Variant 5 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $5$ , $6$ , $1$ , $8$ , $0$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $5 \to 2$ , $6 \to 9$ , $8 \to 8$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 5, 6, 1, 8, 0, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 5, 6, 1, 8, 0.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 8, 6, 5, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 865

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 865 each 180 degrees changes their shapes.

8 \to 8,\quad 6 \to 9,\quad 5 \to 2

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 2, 9, 8 gives the rotated number.

865 \xrightarrow{180^\circ} 298

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

865 - 298 = 567

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 865; rotated = 298; difference = 865 - 298 = 567

4 · Reviewdoes it hold up?

865 is the biggest three-digit number from the cards, and 298 uses only valid turned digits. 865 - 298 = 567 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 865, then turn it upside-down (digits change AND flip order) to get 298 - Grade 4 place-value and subtraction!

Variant 6 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $8$ , $1$ , $0$ , $6$ , $2$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $2 \to 5$ , $6 \to 9$ , $8 \to 8$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 8, 1, 0, 6, 2, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 8, 1, 0, 6, 2.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 8, 6, 2, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 862

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 862 each 180 degrees changes their shapes.

8 \to 8,\quad 6 \to 9,\quad 2 \to 5

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 5, 9, 8 gives the rotated number.

862 \xrightarrow{180^\circ} 598

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

862 - 598 = 264

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 862; rotated = 598; difference = 862 - 598 = 264

4 · Reviewdoes it hold up?

862 is the biggest three-digit number from the cards, and 598 uses only valid turned digits. 862 - 598 = 264 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 862, then turn it upside-down (digits change AND flip order) to get 598 - Grade 4 place-value and subtraction!

Variant 7 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $5$ , $1$ , $0$ , $8$ , $2$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $2 \to 5$ , $5 \to 2$ , $8 \to 8$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 5, 1, 0, 8, 2, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 5, 1, 0, 8, 2.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 8, 5, 2, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 852

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 852 each 180 degrees changes their shapes.

8 \to 8,\quad 5 \to 2,\quad 2 \to 5

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 5, 2, 8 gives the rotated number.

852 \xrightarrow{180^\circ} 528

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

852 - 528 = 324

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 852; rotated = 528; difference = 852 - 528 = 324

4 · Reviewdoes it hold up?

852 is the biggest three-digit number from the cards, and 528 uses only valid turned digits. 852 - 528 = 324 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 852, then turn it upside-down (digits change AND flip order) to get 528 - Grade 4 place-value and subtraction!

Variant 8 medium

Representative Problem

From 5 number cards, choose 3 different cards to form the largest possible three-digit number. Then rotate that number $180\degree$ clockwise (turn all three digits at once, as one block) to read a new number, and find the difference between the original number and the rotated number.

Number cards: $8$ , $2$ , $1$ , $0$ , $9$

When a card is turned $180\degree$ , each digit looks like another digit: $0 \to 0$ , $1 \to 1$ , $2 \to 5$ , $8 \to 8$ , $9 \to 6$ . Remember the whole number is flipped at once, so the order of the digits also reverses.

The largest three-digit number you can make is [ ].
Rotating that number $180\degree$ clockwise gives [ ].
Therefore the difference of the two numbers is [ ] $-$ [ ] $=$ [ ].

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

From the number cards 8, 2, 1, 0, 9, pick 3 different cards to build the largest three-digit number. Then turn that number 180 degrees as one block (each digit changes shape and the whole order reverses) to read a new number, and subtract to find the difference.

Givens

Number cards available: 8, 2, 1, 0, 9.
When a card is turned 180 degrees each digit maps to its upside-down shape (for example 9 to 6, 6 to 9, 5 to 5).
The whole number is turned at once, so the digit order also reverses.
Exactly 3 different cards are used to form a three-digit number.

Unknowns

The largest three-digit number.
The number read after turning it 180 degrees.
The difference between the two numbers.

Constraints

Three distinct cards (no repeats).
The first digit cannot be 0 (it must be a three-digit number).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram

To get the largest number I order the usable digits from greatest to least and place them left to right. Then I diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

3 · Execute4 carry out the plan

1Build the largest three-digit number

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

Order the digits from greatest to least: 9, 8, 2, 1, 0. The largest three-digit number puts the three biggest in the hundreds, tens, and ones places.

\text{largest} = 982

Bigger digits in higher place values make a bigger number.

2Turn each digit 180 degrees

#1 Draw a Diagram 4.G.A.3

Turning the digits of 982 each 180 degrees changes their shapes.

9 \to 6,\quad 8 \to 8,\quad 2 \to 5

A half-turn changes each digit into the shape it looks like upside down.

3Reverse the order to read the rotated number

#1 Draw a Diagram 4.G.A.3

Because the whole number is turned at once, the order reverses too. Reading the turned digits 5, 8, 6 gives the rotated number.

982 \xrightarrow{180^\circ} 586

Spinning a strip of digits halfway flips left-right as well as turning each symbol.

4Subtract to find the difference

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

Find the difference between the original number and the rotated number.

982 - 586 = 396

Straightforward multi-digit subtraction once both numbers are known.

Answer: Largest = 982; rotated = 586; difference = 982 - 586 = 396

4 · Reviewdoes it hold up?

982 is the biggest three-digit number from the cards, and 586 uses only valid turned digits. 982 - 586 = 396 is a difference consistent with the two numbers.

Another way: Guess and check: test a few large candidates, confirm the greatest, then verify the rotation by turning the written number upside down.

Standardsmin grade 4

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers — Forming the largest number using place value and subtracting to get the difference.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding how each digit's shape changes and the order reverses under a 180-degree turn.

💡Takeaway. Put the biggest digits first to make 982, then turn it upside-down (digits change AND flip order) to get 586 - Grade 4 place-value and subtraction!

Flipping or turning digits forms new digits · 8 practice problems

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