English:Whole Numbers and Place Value
Whole Numbers and Place Value
Whole Numbers and Place Value
Grades 5–6
Numbers are everywhere: on sports scores, maps, clocks, building signs, prices, population tables, game statistics, and science data. In this course, you will learn how to read, write, compare, round, and explain large whole numbers by using place value.
Introduction
A whole number is 0 or a positive counting number such as 1, 2, 3, 10, 245, or 6,000,000. Whole numbers do not include negative numbers, fractions, or numbers with a fractional decimal part. In this course, the term whole numbers always includes zero.
Our usual number-writing system is a base-ten positional system. It uses the ten digits 0 through 9. The value of a digit depends on both the digit itself and the position where it appears. That idea is called place value.
Learning Goals
By the end of this aiMOOC, you should be able to explain the difference between a digit and a number, identify the value of any digit in a multi-digit whole number, write numbers in standard, word, expanded, and unit forms, compare and order whole numbers, round whole numbers to a chosen place, and use place value to solve real-world problems.
Digits, Numbers, and the Base-Ten System
A digit is one written symbol. The decimal system uses 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. A number is a mathematical idea, while a numeral is a written representation of a number. For example, the numeral 507 uses three digits to represent the number five hundred seven.
The word base ten tells you that groups are built in tens. Ten ones make one ten, ten tens make one hundred, and ten hundreds make one thousand. This pattern continues without ending.
Base-ten blocks make this pattern visible. A small unit can stand for one. Ten units can be grouped into a ten, ten tens into a hundred, and ten hundreds into a thousand. The same multiplicative pattern continues for much larger place values.
Whole Numbers on a Number Line
Whole numbers can be placed in order on a number line. Zero is the starting whole number. Moving to the right gives larger whole numbers: 0, 1, 2, 3, 4, and so on. Negative numbers may appear to the left of zero on a full integer number line, but they are not whole numbers.
When two whole numbers are written without leading zeros and have different numbers of digits, the number with more digits is greater. For example, 9,999 has four digits, while 10,000 has five digits, so 10,000 is greater.
Understanding Place Value
Each place in a whole number has a value that is ten times the value of the place immediately to its right. Moving one place left multiplies the place value by ten. Moving one place right makes the place value one tenth as large, until you reach the ones place.
A place-value chart helps you see the structure of a large number.
| Billions | Hundred millions | Ten millions | Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|---|---|---|
| 2 | 4 | 7 | 5 | 0 | 8 | 3 | 6 | 1 | 9 |
The chart represents 2,475,083,619. The digit 5 is in the millions place, so its value is 5,000,000. The digit 8 is in the ten-thousands place, so its value is 80,000. The 0 in the hundred-thousands place shows that there are no hundred-thousands in this number, but the zero still holds that place.
A Digit Is Not the Same as Its Value
Look at the digit 7 in these numbers:
7,204 has a 7 worth 7,000.
70,204 has a 7 worth 70,000.
700,204 has a 7 worth 700,000.
The digit stays the same, but its value changes because its position changes. This is the key idea of positional notation.
Powers of Ten
Place values follow powers of ten:
1 = 10 to the power zero.
10 = 10 to the power one.
100 = 10 to the power two.
1,000 = 10 to the power three.
10,000 = 10 to the power four.
You do not need advanced exponent rules to use this pattern. Just notice that each move one place to the left makes the place value ten times as large.
Writing Whole Numbers in Different Forms
The same whole number can be represented in several useful ways. Switching between forms helps you understand what each digit contributes.
Standard Form
Standard form writes a number using digits. Example:
604,052
Commas separate groups of three digits in many English-language number-writing conventions. From right to left, the groups are ones, thousands, millions, billions, and so on.
Word Form
Word form writes the number using words:
six hundred four thousand fifty-two
English number-name conventions vary slightly by region. Some speakers include the word and in whole-number names. Follow the convention used by your teacher or school.
Expanded Form
Expanded form shows the value contributed by each nonzero digit:
604,052 = 600,000 + 4,000 + 50 + 2
The zero digits are not written as separate addends because they contribute zero.
Unit Form
Unit form names the amount in each place:
604,052 = 6 hundred thousands + 4 thousands + 5 tens + 2 ones
Unit form can help you explain why the same digit has different values in different positions.
Zero as a Placeholder
Zero is a number, and it is also an essential digit in our place-value system. Inside a multi-digit numeral, zero can act as a placeholder. It tells you that a particular place has no units while keeping the other digits in their correct positions.
For example, in 507,204, the first zero is in the ten-thousands place and the second zero is in the tens place. If you removed either zero, you would create a different numeral with a different value.
Leading zeros are different. Writing 00042 as a number still represents 42. However, leading zeros can matter in codes, room numbers, product numbers, or identification labels because those are labels rather than quantities.
An abacus is another way to make place value visible. Columns or rods can represent different place values, so the position of a bead or counter changes what it represents.
Comparing and Ordering Whole Numbers
To compare two whole numbers, begin at the greatest place value.
If the numbers have different numbers of digits, the number with more digits is greater. If they have the same number of digits, compare digits from left to right until you find the first place where the digits differ.
Example:
805,472 is greater than 795,999 because both numbers have six digits, but 8 hundred-thousands is greater than 7 hundred-thousands.
You can use the symbols >, <, and = when comparing numbers. Read them as greater than, less than, and equal to.
Ordering Several Numbers
Suppose you need to order these numbers from least to greatest:
480,205; 408,250; 480,025; 408,520
Compare the hundred-thousands places first. The numbers beginning with 4 are tied there, so continue place by place. The correct order is:
408,250; 408,520; 480,025; 480,205
A place-value chart can reduce mistakes because it lines up matching places.
Rounding Whole Numbers
Rounding gives a nearby number that is easier to use for estimating. First choose the place to which you are rounding. Then look at the digit immediately to its right.
If that digit is 0, 1, 2, 3, or 4, keep the rounding-place digit the same. If it is 5, 6, 7, 8, or 9, increase the rounding-place digit by one. Replace all digits to the right with zeros.
Example: Round 368,421 to the nearest ten thousand. The ten-thousands digit is 6, and the thousands digit is 8, so round up. The result is 370,000.
Round the same number to the nearest hundred thousand. The hundred-thousands digit is 3, and the ten-thousands digit is 6, so round up. The result is 400,000.
Rounding is useful when an exact number is unnecessary, but you should not replace exact data with rounded data when precision matters.
Place Value in Real Life
Large whole numbers appear in many subjects. In geography, you may compare population counts or distances. In science, you may record measurements or counts. In economics, you may read budgets or sales totals. In computer science, you may work with file sizes, counts, or other numerical data.
Place value helps you answer questions such as: Which value is larger? Which digit has the greatest effect on the total? How much does a number change if one digit moves one place left? Which rounded value is useful for a quick estimate?
A good habit is to connect every digit to its place and its value. For example, in 3,482,615, the digit 8 is in the ten-thousands place and represents 80,000. Saying only “the 8 is in the ten-thousands place” tells the position; saying “the 8 has a value of 80,000” tells the value.
Common Mistakes and How to Avoid Them
Mistake: Reading digits separately instead of reading the whole number. Fix it by grouping digits into periods of three from the right.
Mistake: Confusing the digit with its value. Fix it by naming both the place and the value.
Mistake: Dropping a zero that holds a place. Fix it by checking the number in a place-value chart.
Mistake: Comparing only the last digits of large numbers. Fix it by comparing from the greatest place value first.
Mistake: Rounding from the wrong digit. Fix it by underlining the rounding place and looking only at the digit immediately to its right.
Interactive Tasks
Quiz: Test Your Knowledge
Which choice is a whole number? (406) (!three point five) (!negative eight) (!one half)
Which digit is in the hundred-thousands place in 7,462,319? (4) (!7) (!6) (!2)
What is the value of the digit 6 in 2,681,450? (600000) (!60000) (!6000) (!600)
Which expanded form matches 305,070? (300000 plus 5000 plus 70) (!300000 plus 500 plus 70) (!30000 plus 5000 plus 70) (!300000 plus 5000 plus 7)
Which standard form matches nine million forty thousand six? (9040006) (!9400006) (!9004006) (!9040060)
How does the value of 4 in 45,210 compare with the value of 4 in 4,521? (The first value is ten times the second) (!The first value is one hundred times the second) (!The values are equal) (!The first value is one tenth of the second)
Which statement correctly compares 708,090 and 708,900? (Seven hundred eight thousand nine hundred is greater) (!Seven hundred eight thousand ninety is greater) (!The two numbers are equal) (!Both numbers are less than seven hundred thousand)
What is 642,781 rounded to the nearest ten thousand? (640000) (!650000) (!643000) (!600000)
In 507,204, what place is held by the zero between 5 and 7? (Ten thousands place) (!Thousands place) (!Hundreds place) (!Tens place)
What is the greatest number that can be made using 4, 0, 8, 2, and 9 exactly once? (98420) (!94820) (!98240) (!89420)
Memory Game
| Whole number | Zero or a positive counting number with no fractional part |
| Digit | A single symbol used to write numerals in base ten |
| Place value | The value a digit has because of its position |
| Expanded form | A way to show a number as the sum of its nonzero place values |
| Placeholder | A zero that keeps other digits in their correct positions |
| Rounding | Replacing a number with a nearby value that is useful for estimating |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Ones place | Position for single units |
| Tens place | Position for groups of ten |
| Hundreds place | Position for groups of one hundred |
| Thousands place | Position for groups of one thousand |
| Millions place | Position for groups of one million |
...
Crossword Puzzle
| Digit | What is one symbol used to write a numeral called? |
| Numeral | What is a written representation of a number called? |
| Expanded | Which form writes a number as a sum of place values? |
| Standard | Which form writes a number using digits? |
| Rounding | What process replaces a number with a nearby useful estimate? |
| Compare | What verb means to decide which of two numbers is greater or smaller? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Place Value Poster: Create a colorful place-value chart from ones to millions and add two example numbers with every digit correctly labeled.
- Digit Detective: Find five large whole numbers in books, newspapers, websites, signs, or class materials and explain the place and value of one chosen digit in each number.
- Number Forms: Make a set of cards that shows the same six-digit number in standard form, word form, expanded form, and unit form, then trade cards with a partner and check each other.
- Human Number Line: Work with classmates to arrange number cards from least to greatest and explain which place value helped you decide the order.
Standard
- Base Ten Model: Build a paper, block, or digital model showing how ten units of one place regroup into one unit of the next place, then photograph or draw your model and explain the pattern.
- Large Number Interview: Interview an adult about a job or hobby that uses large whole numbers and write a short report explaining where place value, comparison, or rounding is useful.
- Comparison Challenge: Design a card or board game in which players compare and order multi-digit whole numbers, write clear rules, and test the game with classmates.
- Rounding Field Study: Visit a suitable place such as a shop, library, sports facility, or school office and collect examples of exact and rounded whole numbers, then explain why rounding is useful in each case.
Advanced
- Place Value Video: Produce a two- to four-minute teaching video that explains digit value, zero as a placeholder, and one common place-value mistake using your own examples.
- Data Investigation: Use a trustworthy public dataset containing large whole numbers, select at least eight values, order them, round them to useful places, and explain what information is lost or kept by rounding.
- Numeral System Comparison: Compare base-ten positional notation with another numeral system such as Roman numerals and explain why position and zero play different roles in the two systems.
- Place Value Teaching Project: Create and teach a short mini-lesson for younger learners using a visual model, a practice problem, and an exit question, then improve your lesson after collecting feedback.
Learning Assessment
- Place Value Reasoning: Explain why moving a nonzero digit one place to the left makes its value ten times as large, and support your explanation with two different numerical examples.
- Error Analysis: A student says that the 6 in 462,915 has a value of 6,000; identify the error, correct it, and describe a method that would prevent the mistake.
- Number Reconstruction: Create a whole number that has 8 in the millions place, 3 in the ten-thousands place, and zeros in at least two other places, then write it in standard and expanded form and justify every digit.
- Comparison Strategy: Compare two seven-digit numbers that differ in at least three places and explain why only the first differing place is needed to decide which number is greater.
- Rounding Decision: Describe a real situation in which rounding to the nearest thousand is useful and another situation in which the exact number should be kept, then explain the difference.
- Transfer Challenge: Choose a large number from science, geography, economics, or another subject and show how place value helps you interpret, compare, and communicate that number accurately.
Evidence of Learning
- Knowledge: You can identify whole numbers, digits, place-value positions, powers-of-ten patterns, and the role of zero.
- Skills: You can read, write, compare, order, decompose, and round multi-digit whole numbers and explain your reasoning.
- Products: Your posters, models, games, reports, datasets, videos, or mini-lessons show accurate use of place value.
- Reasoning: You can explain why a digit changes value when its position changes and diagnose common place-value errors.
- Transfer: You can use place value to interpret large numbers in other school subjects and in real-life situations.
OERs on the Topic
The following English Wikipedia article gives background on positional notation and the mathematical idea that a digit's value depends on its position.
Linked Learning Areas
Place value connects strongly with arithmetic, estimation, number sense, measurement, data, and later work with decimals. A strong understanding of whole-number place value makes multi-digit addition, subtraction, multiplication, division, and decimal place value easier to understand.