Understanding Place Value

🔢 Understanding Place Value

Part 1 of 5 — What Place Value Means

Topics in This Part

🔑 Key Concept: In our number system, the position of a digit tells you its value. The same digit $7$ can mean $7$, or $70$, or $700$ — it all depends on which place it sits in.

Digits vs. Numbers

A digit is one of the ten symbols we write numbers with:

$0,\; 1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9$

A number is built by putting digits together. The number $352$ is made of the digits $3$, $5$, and $2$.

But those digits are not worth $3$, $5$, and $2$. Their value depends on where they sit:

Add the values: $300 + 50 + 2 = 352$. ✓

💡 This is why our system is called base ten — each place is worth ten times as much as the place just to its right.

The Place-Value Columns

Reading from the right, the columns grow by ten each step:

The ones place is always farthest to the right. As you move left, the value of each place gets bigger.

Example: the number 4,286

So $4{,}286 = 4{,}000 + 200 + 80 + 6$.

🔑 Remember: A comma is placed every three digits from the right. It helps you see the groups: ones, then thousands, then millions.

Concept Check 🎯

Finding a Digit's Value

To find what a digit is worth, multiply the digit by the value of its place.

$\text{value} = \text{digit} \times \text{place value}$

Example: the value of $6$ in $3{,}640$

The $6$ sits in the hundreds place, so its value is:

$6 \times 100 = 600$

The $6$ is worth six hundred, not six.

Example: the value of $4$ in $3{,}640$

The $4$ sits in the tens place, so its value is:

$4 \times 10 = 40$

⚠️ Watch out: Don't confuse the digit with its value. In $3{,}640$ the digit is $6$, but its value is $600$.

Find the Value 🧮

Write the value of the underlined digit (not just the digit).

1) In $2{,}\underline{5}03$, the $5$ is worth $\,?$ 2) In $\underline{7}{,}219$, the $7$ is worth $\,?$ 3) In $4{,}9\underline{8}0$, the $8$ is worth $\,?$

Putting It Together

For any digit, ask two questions:

1. Which place is it in? (Count from the ones place on the right.)
2. What is one of that place worth? ($1$, $10$, $100$, $1{,}000$, …)

Multiply, and you have the digit's value.

🔑 One last reminder: The places, right to left, are ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions. Each is ten times the one before it.

Name That Place 🔽

For the number $\textbf{6,253}$, choose what each digit means.

🔢 Understanding Place Value

Part 2 of 5 — Three Ways to Write a Number

🔑 The Idea: Every whole number can be written three ways — as digits (standard form), as a sum of place values (expanded form), and in words (word form). They all name the same number.

The Three Forms

Notice the expanded form of $3{,}408$ has no tens term, because the tens digit is $0$. A zero holds the place but adds nothing.

💡 Expanded form is just place value written as a sum. Each digit becomes (digit $\times$ place value), and you add them up.

Writing Expanded Form

Break the number into one piece per non-zero digit.

Example: $6{,}752$

$6{,}752 = 6{,}000 + 700 + 50 + 2$

Example with a zero: $5{,}069$

The thousands digit is $5$, hundreds is $0$, tens is $6$, ones is $9$:

$5{,}069 = 5{,}000 + 60 + 9$

⚠️ Skip the $0$ place in expanded form — writing "$+\,0$" is allowed but usually left out. Just be sure the other digits land in the right place.

Concept Check 🎯

Word Form

To write a number in words, read each group of three (each comma section) and say its group name.

Example: $52{,}170$

• The thousands group is $52$ → "fifty-two thousand"
• The last group is $170$ → "one hundred seventy"

$52{,}170 = \text{fifty-two thousand, one hundred seventy}$

🔑 Rules to remember:

• Put a comma in word form where the comma goes in the number.
• Hyphenate two-word numbers like twenty-one and fifty-two.
• We do not say the word "and" in a whole number — say "two hundred five," not "two hundred and five."

Match the Forms 🔽

Fill in the missing form for each number.

Going Backward to Standard Form

You can also work the other way: start with expanded or word form and rebuild the standard number. Just place each digit in its correct column, and use a $0$ wherever a place is missing.

Example: $5{,}000 + 70 + 8$

There is no hundreds term, so the hundreds digit is $0$:

$5{,}000 + 0 + 70 + 8 = 5{,}078$

💡 The trickiest part is remembering the zeros. If a place isn't named, its digit is $0$.

Back to Standard Form 🧮

Write each as a standard-form number (digits only — you may include the comma or not).

1) $7{,}000 + 500 + 60 + 3 = \,?$ 2) $9{,}000 + 40 + 2 = \,?$ 3) "three thousand, eight hundred" $= \,?$

🔢 Understanding Place Value

Part 3 of 5 — The "Ten Times" Rule

🔑 Why it works: Each place is worth ten times the place on its right. So the same digit is worth ten times more each step you move it to the left.

A Digit Is 10 Times Bigger to the Left

Look at the digit $4$ as it slides one place to the left:

Each step to the left, the value is multiplied by $10$:

$4 \xrightarrow{\times 10} 40 \xrightarrow{\times 10} 400 \xrightarrow{\times 10} 4{,}000$

💡 And going the other way: each step to the right the value is divided by $10$ (it becomes one tenth as big). $400 \to 40 \to 4$.

Multiply and Divide by 10 🔽

Use the "ten times to the left, one tenth to the right" rule.

Comparing the Same Digit in Two Places

Example: the two $7$s in $7{,}700$

• The left $7$ is in the thousands place → worth $7{,}000$.
• The right $7$ is in the hundreds place → worth $700$.

How many times bigger is the left $7$?

$7{,}000 \div 700 = 10$

The left $7$ is worth $10$ times as much as the right $7$. That is always true for neighboring places.

🔑 The key fact: A digit in any place is worth $10$ times the same digit one place to its right, and $\dfrac{1}{10}$ as much one place to its left.

Concept Check 🎯

A Quick Pattern for Multiplying by 10

Because each place is ten times the one to its right, multiplying a whole number by $10$ just shifts every digit one place left — which looks like adding a zero:

$50 \times 10 = 500 \qquad 7 \times 1000 = 7000$

Dividing by $10$ does the opposite for numbers that end in zero — it removes a zero: $800 \div 10 = 80$.

💡 This shortcut only "adds a zero" for whole numbers. The real reason it works is the place-value pattern, not a trick.

Ten Times Practice 🧮

1) $50 \times 10 = \,?$ 2) $7 \times 1000 = \,?$ 3) The $9$ in $9{,}000$ is worth how many times the $9$ in $900$? $\,?$

🔢 Understanding Place Value

Part 4 of 5 — Comparing, Ordering & Rounding

🔑 Big Payoff: Place value lets you compare any two numbers, put a list in order, and round to a friendly number — all by looking at digits one place at a time.

Comparing Numbers

To compare two whole numbers, line them up and compare from the left:

1. The number with more digits is larger.
2. If they have the same number of digits, compare the leftmost place. The bigger digit wins.
3. If those digits tie, move one place right and compare again.

We write comparisons with these symbols:

Example: compare $3{,}471$ and $3{,}419$

Both have $3$ in thousands and $4$ in hundreds — a tie so far. Move to the tens:

$7 > 1 \quad\Rightarrow\quad 3{,}471 > 3{,}419$

💡 Tip: The $>$ and $<$ symbols always open toward the bigger number, like a hungry mouth eating the larger amount.

Greater, Less, or Equal? 🔽

Choose the symbol that makes each statement true.

Rounding to a Place

Rounding replaces a number with a nearby "friendly" number. To round to a given place:

1. Underline the digit in the rounding place.
2. Look at the digit just to its right (the "decider").
3. If the decider is $5$ or more, round up (add $1$ to the underlined digit). If it is $4$ or less, round down (keep the underlined digit).
4. Change every digit to the right to $0$.

Example: round $3{,}472$ to the nearest hundred

The hundreds digit is $4$; the digit to its right (tens) is $7$.

$7 \ge 5 \quad\Rightarrow\quad \text{round up}: \; 3{,}472 \approx 3{,}500$

Example: round $3{,}472$ to the nearest thousand

The thousands digit is $3$; the digit to its right (hundreds) is $4$.

$4 < 5 \quad\Rightarrow\quad \text{round down}: \; 3{,}472 \approx 3{,}000$

⚠️ Only the decider digit matters — the one immediately to the right of the rounding place. Don't peek further right than that.

Concept Check 🎯

Ordering a List

To put several numbers in order, compare them two at a time using place value, just like before. To find the largest, scan for the biggest digit in the highest place where they differ.

Example: order $4{,}109$, $4{,}901$, $4{,}190$ from least to greatest

All share $4$ in the thousands place, so compare hundreds: $1$, $9$, and $1$. The $9$ ($4{,}901$) is biggest. The other two tie at hundreds ($1$), so compare tens: $0$ vs $9$ — so $4{,}109 < 4{,}190$.

$4{,}109 \;<\; 4{,}190 \;<\; 4{,}901$

💡 Always start at the highest place (farthest left) and move right only when there's a tie.

Order & Round 🧮

1) Which is the largest of $4{,}109$, $4{,}901$, and $4{,}190$? $\,?$ 2) Round $7{,}350$ to the nearest hundred. $\,?$ 3) Round $7{,}350$ to the nearest thousand. $\,?$

🔢 Understanding Place Value

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) name a digit's place and value, (2) write numbers in standard, expanded, and word form, (3) use the "ten times" rule, and (4) compare, order, and round. Let's put it all together.

Quick Reference

⚠️ Top reminders: a $0$ still holds a place (so $4{,}205$ has a tens digit of $0$); when rounding, only the single decider digit matters; the symbols $>$ and $<$ open toward the larger number.

Warm-Up 🔽

A quick check across everything you've learned, using the number $\textbf{8,406}$.

Nicely done. Now try a few mixed questions that pull from every part of the lesson.

Mixed Practice 🎯

One Last Check

You've reached the end. The Exit Quiz below has three questions that touch the biggest ideas of the lesson: place value, the ten-times rule, and rounding. Take your time and use what you've learned.

Exit Quiz ✅

Answer all three to finish the lesson.