Place value to a million and rounding

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课程笔记

Place Value to a Million

Place value is the value of a digit based on its position in a number.
In the number 1,234,567, the digit 1 is in the millions place, so it means 1,000,000.
The digit 2 is in the hundred thousands place, so it means 200,000.
The digit 3 is in the ten thousands place, so it means 30,000.
The digit 4 is in the thousands place, so it means 4,000.
The digit 5 is in the hundreds place, so it means 500.
The digit 6 is in the tens place, so it means 60.
The digit 7 is in the ones place, so it means 7.

Reading and Writing Numbers to a Million

To read a large number, split it into groups of three digits from the right: 1,234,567.
Read each group and say its name: one million, two hundred thirty-four thousand, five hundred sixty-seven.
When writing numbers in words, use a hyphen for numbers from 21 to 99 (like twenty-one).
Use commas to separate groups of three digits in numbers of 1,000 or more.
The number 1,000,000 is one million, and 999,999 is nine hundred ninety-nine thousand, nine hundred ninety-nine.

Comparing and Ordering Numbers

To compare numbers, look at the digits from left to right.
The number with more digits is larger (e.g., 1,000,000 > 999,999).
If two numbers have the same number of digits, compare the first digit that is different.
Use the symbols < (less than), > (greater than), and = (equal to) to show comparison.
To order numbers, write them from smallest to largest (ascending) or largest to smallest (descending).

Decimals to Thousandths

A decimal is a number with a decimal point, like 0.5 or 3.142.
The first digit after the decimal point is the tenths place.
The second digit after the decimal point is the hundredths place.
The third digit after the decimal point is the thousandths place.
In 0.375, the digit 3 is in the tenths place, 7 is in the hundredths place, and 5 is in the thousandths place.

Writing Decimals in Expanded Form

Expanded form shows the value of each digit in a number.
For a decimal like 0.375, expanded form is 0.3 + 0.07 + 0.005.
For a whole number like 4,302, expanded form is 4,000 + 300 + 2.
You can also write expanded form using fractions: 0.375 = 3/10 + 7/100 + 5/1000.
Expanded form helps you understand the value of each digit.

Comparing Decimals

To compare decimals, line up the decimal points and compare digits from left to right.
If one decimal has fewer digits, you can add zeros to the right to make them the same length.
For example, 0.5 is greater than 0.45 because 0.50 > 0.45.
Use <, >, or = to compare decimals.
Ordering decimals is like ordering whole numbers, but you compare the decimal parts.

Rounding Whole Numbers

Rounding means making a number simpler but close to its original value.
To round to the nearest ten, look at the ones digit. If it is 5 or more, round up; if it is 4 or less, round down.
Example: 47 rounded to the nearest ten is 50, because the ones digit is 7 (5 or more).
To round to the nearest hundred, look at the tens digit.
Example: 1,234 rounded to the nearest hundred is 1,200, because the tens digit is 3 (less than 5).

Rounding Decimals

To round a decimal to the nearest tenth, look at the hundredths digit.
If the hundredths digit is 5 or more, round up the tenths digit; if it is 4 or less, keep the tenths digit the same.
Example: 3.47 rounded to the nearest tenth is 3.5, because the hundredths digit is 7.
To round to the nearest hundredth, look at the thousandths digit.
Example: 0.375 rounded to the nearest hundredth is 0.38, because the thousandths digit is 5.

Counting in Powers of 10

Powers of 10 are numbers like 10, 100, 1,000, 10,000, 100,000, and 1,000,000.
Counting forwards in powers of 10 means adding 10, 100, 1,000, etc.
Counting backwards means subtracting those amounts.
Example: Counting backwards by 100 from 1,000: 1,000, 900, 800, 700, 600.
You can also count through zero into negative numbers, like -10, -20, -30.

Roman Numerals to 1,000

Roman numerals are an old way of writing numbers using letters: I, V, X, L, C, D, M.
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000.
To write a number, combine these symbols. If a smaller number comes before a larger one, subtract it (like IV = 4).
If a smaller number comes after a larger one, add it (like VI = 6).
Example: 49 is written as XLIX (40 + 9).

幻灯片

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练习题

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1.What is the value of the digit 7 in the number 3,742,105?
Easy
- A700,000
- B70,000
- C7,000
- D700
2.Which of these numbers is the largest?
Easy
- A456,789
- B456,798
- C456,897
- D456,978
3.The number 1,000,000 has 7 digits.
Easy
True or false?
4.Which number is equal to 5,000,000 + 40,000 + 300 + 2?
Medium
- A5,040,302
- B5,400,302
- C5,004,302
- D5,040,032
5.Which of the following numbers are less than 1,000,000? (select all that apply)
Easy
- A999,999
- B1,000,001
- C999,999.9
- D1,000,000.0
6.Put these numbers in order from smallest to largest.
Medium
- 345,678
- 345,768
- 345,687
- 345,876
7.When rounding 2,345,678 to the nearest hundred thousand, you get 2,300,000.
Easy
True or false?
8.A school has 1,234,567 books in its library. If you round this number to the nearest ten thousand, what do you get?
Medium
- A1,230,000
- B1,240,000
- C1,200,000
- D1,300,000

复习笔记

## Place Value to a Million and Rounding ### Place Value to a Million - **Place value** tells you the value of each digit in a number. For example, in 345,678, the digit 4 is in the ten-thousands place, so it means 40,000. - **Digits** are the symbols 0-9. A number like 1,000,000 has 7 digits. - **Expanded form** shows the value of each digit: 345,678 = 300,000 + 40,000 + 5,000 + 600 + 70 + 8. - **Comparing numbers**: Use < (less than), > (greater than), or = (equal to). Compare digits from left to right. - **Ordering**: Put numbers in order from smallest to largest or largest to smallest. - **Decimals**: Read and write decimals to thousandths. For example, 0.375 is three hundred seventy-five thousandths. ### Rounding - **Rounding** means changing a number to a nearby, simpler number. For example, 23.4476 rounded to two decimal places is 23.45. - **Rounding whole numbers**: Look at the digit to the right of the place you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down. For example, round 123,456 to the nearest thousand: look at the hundreds digit (4), so 123,000. - **Rounding decimals**: Same rule. Round 3.14159 to the nearest hundredth: look at the thousandths digit (1), so 3.14. - **Rounding to a given place**: You can round to any place, like tens, hundreds, or tenths. - **Wavy equals sign (≈)** means "is approximately equal to". For example, 9.98 ≈ 10. - **Why round?** To make numbers easier to report or understand. For example, if a number is 123,456 but only accurate to the nearest hundred, it is better to say "about 123,500". - **Rounding errors**: When you round, you lose some exactness. In long calculations, errors can add up. ### Key Tips - Always identify the place value you are rounding to. - Look at the digit immediately to the right. - If that digit is 5 or more, round up; if 4 or less, round down. - Practice with big numbers and decimals!

Place value: The value of a digit based on its position in a number. For example, in 345,678, the 4 is in the ten-thousands place, so it means 40,000.
Digit: A single symbol used to make numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.
Expanded form: A way to write a number that shows the value of each digit. For example, 345,678 = 300,000 + 40,000 + 5,000 + 600 + 70 + 8.
Rounding: Changing a number to a nearby, simpler number. For example, 23.4476 rounded to two decimal places is 23.45.
Approximately equal to (≈): A symbol that means 'is about'. For example, 9.98 ≈ 10.
Round up: To increase the digit in the place you are rounding to by 1, when the digit to the right is 5 or more.
Round down: To keep the digit in the place you are rounding to the same, when the digit to the right is 4 or less.

闪卡

What is the value of the digit 7 in the number 3,475,209?

The digit 7 is in the ten-thousands place, so its value is 70,000.

Write the number 2,304,567 in expanded form.

2,000,000 + 300,000 + 0 + 4,000 + 500 + 60 + 7.

Compare: 456,789 ___ 456,798. Use <, >, or =.

456,789 < 456,798 because 789 is less than 798.

What is the value of the digit 5 in the number 0.25?

The digit 5 is in the hundredths place, so its value is 5 hundredths (0.05).

Write the decimal 0.307 in words.

Three hundred seven thousandths.

Round 3,456 to the nearest hundred.

3,500. The tens digit is 5, so round up.

Round 12.345 to the nearest tenth.

12.3. The hundredths digit is 4, so round down.

What does the symbol ≈ mean?

It means 'approximately equal to'. For example, 9.98 ≈ 10.

Round 999,999 to the nearest ten thousand.

1,000,000. The thousands digit is 9, so round up.

Write the number 1,000,000 in words.

One million.

What is the place value of the digit 4 in 0.004?

The digit 4 is in the thousandths place, so its value is 4 thousandths (0.004).

Order these numbers from least to greatest: 234,567; 234,576; 234,567.

234,567; 234,567; 234,576. (If duplicates, list both.)

Round 0.999 to the nearest hundredth.

1.00. The thousandths digit is 9, so round up.

What is the value of the digit 0 in 5,000,000?

Each zero is in a place with value 0, so the digit 0 has a value of 0.

Write 0.5 as a fraction in simplest form.

1/2.

Compare: 0.7 ___ 0.70. Use <, >, or =.

They are equal, so 0.7 = 0.70.

Round 12,345 to the nearest thousand.

12,000. The hundreds digit is 3, so round down.

What is the place value of the digit 8 in 8,000,000?

It is in the millions place, so its value is 8,000,000.

Write 0.25 as a fraction in simplest form.

1/4.

Round 3.14159 to the nearest thousandth.

3.142. The ten-thousandths digit is 5, so round up.

What is the value of the digit 6 in 0.06?

The digit 6 is in the hundredths place, so its value is 6 hundredths (0.06).

Round 1,234,567 to the nearest million.

1,000,000. The hundred-thousands digit is 2, so round down.

Write 0.125 as a fraction in simplest form.

1/8.

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