Read, compare and round whole numbers through one million

Learning goal: Use the tenfold relationship between neighboring places to read, write, compare and round whole numbers up to one million. In the decimal place-value system, a digit one place to the left represents ten times as much. The same written digit can therefore have very different values in different positions.

In 444,444, the leftmost four represents four hundred thousand, while the next represents forty thousand. Four hundred thousand is ten times forty thousand. Continuing right, the values are four thousand, four hundred, forty and four. Each value is ten times the next. The digit stays four; its place determines the represented amount.

Standard form writes a number with digits, such as 406,082. Word form is four hundred six thousand eighty-two. Expanded form is 400,000 + 6,000 + 80 + 2. Zero placeholders show that the ten-thousands and hundreds places have no units of those sizes. Omitting those placeholders from standard form would change the number.

Read the groups around the comma carefully. In 406,082, the left group names thousands and the right group names the remaining ones, tens and hundreds. One million is written 1,000,000. These examples stay within the fourth-grade whole-number range through one million.

To compare two whole numbers, first check the greatest place each number reaches. A number that reaches a greater place is the greater number: 320,000 > 48,000 because 320,000 has three hundred-thousands and 48,000 has none, even though the first digit 3 is less than the first digit 4. When both numbers reach the same greatest place, compare their digits starting from that place. To compare 406,082 and 460,028, both have four hundred-thousands. At the ten-thousands place, zero is less than six, so 406,082 < 460,028. Later digits cannot overcome that difference. Comparing only the final digit would ignore most of each number's value.

Rounding replaces a number with a nearby multiple of a requested place value. To round 406,082 to the nearest ten thousand, locate 400,000 and 410,000. The midpoint is 405,000. Since 406,082 is above the midpoint, it is nearer 410,000. To the nearest hundred thousand, the neighbors are 400,000 and 500,000; now the number is nearer 400,000.

The rounding place matters. Rounding 406,082 to the nearest thousand gives 406,000, while the nearest hundred gives 406,100. At an exact halfway point, this whole-number convention chooses the greater neighboring multiple. For example, 450,000 rounds to 500,000 to the nearest hundred thousand. Explain the neighboring multiples before applying a digit shortcut.

Rounding can reach a new place. The nearest hundred thousand to 960,000 is 1,000,000, not 900,000. The next multiple after nine hundred thousand is one million. A result ending in zeros is not automatically correct; check that it is a multiple of the requested unit and nearer under the stated convention.

Worked representation: Write 703,050 in expanded form: 700,000 + 3,000 + 50. Its word form is seven hundred three thousand fifty. The omitted terms in expanded form have zero value, but standard form still needs zeros to preserve the place positions.

Worked comparison and rounding: 703,050 > 703,005 because the tens digits are five and zero after all greater places match. Rounded to the nearest hundred, 703,050 lies halfway between 703,000 and 703,100, so choose 703,100 using the stated convention.

Guided practice: Use a place-value chart to represent numbers with internal zeros and repeated digits. Move one digit one place left and explain the tenfold change. Compare pairs that first differ in different places, then round the same number to several requested places.

Independent evidence: Read and write an unfamiliar number in standard, word and expanded forms; justify a comparison with >, < or =; and round using neighboring multiples. Include a midpoint and a case that rounds to one million. If a digit rule produces a result outside the neighboring multiples, return to the number line and represented values.