Ordering 3-digit numbers

Task

1. Arrange the following numbers from least to greatest:

$$ 476 \qquad \qquad 647 \qquad \qquad 74 \qquad \qquad 674 \qquad \qquad 467 $$

______ ______ ______ ______ ______

2. Arrange the following numbers from greatest to least:

$$ 326 \qquad \qquad 362 \qquad \qquad 63 \qquad \qquad 623 \qquad \qquad 632 $$

______ ______ ______ ______ ______

IM Commentary

Each number has at most 3 digits so that students have the opportunity to think about how digit placement affects the size of the number. Each group also contains a two-digit number so that students have to do more than just compare the first digit, the second digit, etc.

Solution

1. Least to greatest:

Arranging the numbers from least to greatest means writing the numbers in an ordered list according to their values. The smallest number should be written on the left, with the next smallest number written to its immediate right. The process continues for all the given numbers, the largest of which should be on the right.

First we look for the smallest number given. $74$ is the smallest number given because it is the only number with no 100s. (There is an implied zero in the hundred’s place.)

We now look for the second smallest number. There are two numbers, $476$ and $467$, with four $100$s (fours in the hundred’s place). We must now consider the ten’s place. $467$ is the next smallest number because it only has six $10s$ (a six in the ten’s place), while $476$ has seven $10$s (a seven in the ten’s place).

The next smallest number is $476$ because it only has four $100$s (a four in the hundred’s place), while all the other remaining numbers have six $100$s (sixes in the hundred’s place).

We now have two numbers remaining, $647$ and $674$. Both numbers have six $100$s (sixes in the hundred’s place), so we must compare their $10$s. $647$ is smaller than $674$ because in has just four $10$s (a four in the ten’s place), rather than seven $100$s (a seven in the hundred’s place).

This leaves $674$ as the largest number.

$$ 74 \qquad \qquad 467 \qquad \qquad 479 \qquad \qquad 647 \qquad \qquad 674 $$

2. Greatest to least

Arranging the numbers from greatest to least means writing the numbers in an ordered list according to their values. The largest number should be written on the left, with the next largest number written to its immediate right. The process continues for all the given numbers, the smallest of which should be on the right.

First we look for the largest number given. There are three numbers that begin with six: $63$, $623$ and $632$. However, the $6$ in $63$ is in the tens place, not in the hundreds place. There are zero hundreds, making $63$ the smallest number, the only number that is less than $100$.

$623$ and $632$ both have six $100$s (sixes in the hundred’s place), so it is necessary to compare the digits in the $10$’s place. $623$ has two $10$s (a two in the ten’s place), while $632$ has three 10s (a tens in the ten’s place. Since the value of the three in the $10$'s place is larger than two in the $10$'s place, $632$ is the largest number and $623$ is the second largest number.

The two remaining numbers both start with three. Comparing $326$ and $362$, which both have $3$ hundreds (a three in the hundred’s place), means comparing the numbers in the ten’s place. Six tens (a six in the ten’s place) is larger than three tens (a three in the ten’s place), so $326$ is smaller than $362$.

$$ 632 \qquad \qquad 623 \qquad \qquad 362 \qquad \qquad 326 \qquad \qquad 63 $$