Flipping or turning digits forms new digits

4.G.A.3 · adapt · grade 4

Archetype: Transformations Preserve Measures · step in a 8-type progression

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: $0 \to 0$ , $1 \to 1$ , $8 \to 8$ , $9 \to 6$ , $5 \to 2$ . 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 [ ] $-$ [ ] $=$ [ ].

Show solution

Understand

From the five 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: 0 to 0, 1 to 1, 8 to 8, 9 to 6, 5 to 2.
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).

Plan

#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 carefully diagram the 180-degree turn (each digit's new shape, and the reversed order) to read the rotated number, and finally subtract.

Execute

#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, ones places: 9, then 8, then 5.

\text{largest} = 985

Bigger digits in higher place values make a bigger number - basic place-value sense.

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

Turning the digits of 985 each 180 degrees: 9 becomes 6, 8 becomes 8, 5 becomes 2.

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

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

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

Because the whole number is turned at once, the order reverses too. The original ones digit ends up on the left. So 985 (turned) reads as the reverse of 6, 8, 2, which is 2, 8, 6 = 286.

985 \xrightarrow{180^\circ} 286

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

#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

Review

985 is the biggest three-digit number from {9,8,5,1,0}, and 286 uses only valid turned digits (6,8,2). 985 - 286 = 699 is a positive three-digit difference, which is sensible since the original is larger.

Guess and check (tool 6): test a few large candidates like 985, 981, 980 and confirm 985 is greatest, then verify the rotation by physically turning the written number upside down.

Standards · min 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.

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