Advanced Math 4

Try This First

Try this short activity before reading the lesson.

> Draw seven boxes in a row on paper. Write $4{,}050{,}000$ with one digit in each box. Above each box, write its place name: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions. Now cover every digit except the $5$. What is that $5$ worth all by itself?

The Big Idea

A number is a row of digits. Each place is worth ten times the place to its right. Ten ones make a ten. Ten tens make a hundred. Ten hundreds make a thousand. It keeps going: ten thousands, hundred thousands, millions. A digit's value is the digit times its place. The $7$ in $371{,}502$ is worth $70{,}000$, because it sits in the ten thousands place.

To read a big number, read it in groups of three. $4{,}050{,}000$ is "four million, fifty thousand". The middle group is $050$, which is fifty. The middle group is always thousands. Expanded form writes each digit with its value: $706{,}049 = 700{,}000 + 6{,}000 + 40 + 9$. You do not write the zeros. A zero is worth nothing, but it still holds its place.

Rounding asks: which round number is this closest to? Look at the digit one place to the right of the place you are rounding to. If it is $5$ or more, round up. If it is less than $5$, keep what you have.

Words to Know

• digit: One of the ten symbols $0$ to $9$. The number $371{,}502$ has six digits.
• place value: What a digit is worth because of where it sits. The $7$ in $371{,}502$ is worth $70{,}000$.
• expanded form: The number written as a sum of its place values: $5{,}204 = 5{,}000 + 200 + 4$.
• rounding: Swapping a number for the nearest round one. The nearest ten, hundred or thousand.

Worked Examples and Your Turn

Study each example, then try the paired problem.

Example 1: Read it in groups of three

Problem: Read $2{,}306{,}500$ aloud. Then say what the $3$ is worth.

Solution:

• Split the number into groups of three from the right: $2 \mid 306 \mid 500$. Why: the groups are millions, thousands and ones.
• Read each group, then say its name: "two million, three hundred six thousand, five hundred".
• Find the $3$. It is the first digit of the thousands group. That is the hundred thousands place, so it is worth $300{,}000$.

Answer: Two million, three hundred six thousand, five hundred. The $3$ is worth $300{,}000$.

> Notice: The commas are not decoration. Each comma marks where a group name goes: million, thousand.

Your Turn 1: Read $5{,}204{,}300$ aloud. What is the $2$ worth?

Example 2: Expanded form, then round

Problem: Write $706{,}049$ in expanded form. Then round it to the nearest thousand.

Solution:

• Name each place: $7$ hundred thousands, $0$ ten thousands, $6$ thousands, $0$ hundreds, $4$ tens, $9$ ones.
• Write the non-zero places as a sum: $700{,}000 + 6{,}000 + 40 + 9$. Why: the zeros hold their places but add nothing.
• Find the thousands digit. It is $6$. Look at the digit to its right, the hundreds digit. It is $0$.
• Decide. $0$ is less than $5$, so keep the $6$. Turn everything after it into zeros: $706{,}000$.

Answer: $700{,}000 + 6{,}000 + 40 + 9$. Rounded: $706{,}000$.

> Verification & Check: $706{,}049$ sits between $706{,}000$ and $707{,}000$. It is $49$ from the first and $951$ from the second. So the first is nearer.

Your Turn 2: Write $308{,}027$ in expanded form. Then round it to the nearest thousand.

Example 3: Compare and order

Problem: Put $98{,}760$; $102{,}300$; $99{,}999$; $100{,}020$ in order from greatest to least.

Solution:

• Count the digits first. Why: a six-digit number always beats a five-digit number. So $102{,}300$ and $100{,}020$ are the two biggest.
• Compare the six-digit pair from the left: $1, 0, 2$ against $1, 0, 0$. At the third digit, $2$ beats $0$. So $102{,}300$ is bigger than $100{,}020$.
• Compare the five-digit pair from the left: $9, 9$ against $9, 8$. So $99{,}999$ is bigger than $98{,}760$.

Answer: $102{,}300$; $100{,}020$; $99{,}999$; $98{,}760$

> Notice: More digits means bigger. Same number of digits: compare from the left, and stop at the first place that is different.

Your Turn 3: Put $87{,}650$; $101{,}500$; $89{,}999$; $100{,}090$ in order from greatest to least.

Example 4: Four million fifty, and rounding forty-five thousand

Problem: Read $4{,}050{,}000$ aloud. Then round $45{,}000$ to the nearest ten thousand.

Solution:

• Common mistake: "four million fifty". The word thousand is missing. The fifty has been read as fifty ones, when it is fifty thousands. Another common mistake: $45{,}000$ rounded down to $40{,}000$, because it is exactly halfway.
• Do this instead. Split into groups: $4 \mid 050 \mid 000$. The middle group is thousands. So it is "four million, fifty thousand".
• Round $45{,}000$ to the nearest ten thousand. The ten thousands digit is $4$. The digit to its right is $5$.
• Decide. $5$ or more rounds up. So $50{,}000$. Why: halfway always rounds up. That is the rule everyone agreed on.

Answer: Four million, fifty thousand. Rounded: $50{,}000$.

> Notice: Halfway always rounds up. It is a rule people agreed on so everyone gets the same answer.

Your Turn 4: Read $6{,}020{,}000$ aloud. Then round $75{,}000$ to the nearest ten thousand.

Example 5: Every number that rounds to fifty thousand

Problem: What are the smallest and the largest whole numbers that round to $50{,}000$, when you round to the nearest thousand?

Solution:

• Think about what decides. To land on $50{,}000$, the thousands must round to $50$. The hundreds digit decides.
• Go down. $49{,}500$ has hundreds digit $5$, so it rounds up to $50{,}000$. One less, $49{,}499$, rounds to $49{,}000$. So the smallest is $49{,}500$.
• Go up. $50{,}499$ has hundreds digit $4$, so it stays at $50{,}000$. One more, $50{,}500$, rounds up to $51{,}000$. So the largest is $50{,}499$.

Answer: $49{,}500$ and $50{,}499$

> Notice: A rounded number stands for a whole stretch of numbers. Here it stands for a thousand of them. That is why "about $50{,}000$" is honest.

Your Turn 5: What are the smallest and the largest whole numbers that round to $30{,}000$, to the nearest thousand?

Example 6: Make the biggest and the smallest

Problem: Use each of the digits $5, 0, 8, 2, 7, 1$ once. Make the largest six-digit number. Then make the smallest.

Solution:

• For the largest, put the biggest digit in the biggest place. Then the next biggest, and so on: $875{,}210$.
• For the smallest, you want the smallest digit first. But a number cannot start with $0$. Why: then it would only have five digits. So $1$ goes first, then $0$, then the rest in order: $102{,}578$.

Answer: $875{,}210$ and $102{,}578$

> Notice: A zero at the front does not count as a place. $012{,}578$ is just $12{,}578$.

Your Turn 6: Use each of the digits $3, 0, 9, 4, 6, 2$ once. Make the largest six-digit number and the smallest.

Example 7: How many numbers round there

Problem: How many whole numbers round to $800{,}000$ when you round to the nearest ten thousand?

Solution:

• Start by finding the two ends. Why: every number between the ends counts, so the ends are the whole problem.
• The smallest is $795{,}000$. Why: halfway rounds up. The largest is $804{,}999$. Why: $805{,}000$ would round up to $810{,}000$.
• Count from $795{,}000$ to $804{,}999$. Subtract: $804{,}999 - 795{,}000 = 9{,}999$. Then add $1$. Why: from $795{,}000$ to $795{,}001$ is two numbers, but the difference is only $1$.
• $9{,}999 + 1 = 10{,}000$.

Answer: $10{,}000$ numbers

> Notice: To count a run of numbers, subtract the ends and add one. Finding the ends is the thinking. The counting is one subtraction.

> Verification & Check: Half of them sit below $800{,}000$: from $795{,}000$ to $799{,}999$ is $5{,}000$ numbers. The other $5{,}000$ run from $800{,}000$ to $804{,}999$. Together, $10{,}000$.

Your Turn 7: How many whole numbers round to $60{,}000$ when you round to the nearest thousand?

What If...?

Explore these special cases:

• Case: Ten of the biggest place: $999{,}999 + 1$.
• Resolution: $1{,}000{,}000$. Every place overflows into the next, and a new place appears.
• Case: Rounding a $9$ up: $39{,}700$ to the nearest thousand.
• Resolution: $40{,}000$. The $9$ thousands become $10$ thousands. That carries into the ten thousands.
• Case: Zeros in the middle: $400{,}032$.
• Resolution: Expanded form is $400{,}000 + 30 + 2$. Read it as "four hundred thousand, thirty-two". You say nothing for the zeros.
• Case: Round $45{,}000$ to the nearest thousand.
• Resolution: It already is a thousand: $45{,}000$. Rounding to a place the number already sits on changes nothing.
• Case: Which is bigger, $1{,}000{,}000$ or $999{,}999$?
• Resolution: The one with more digits, always. Even though the other one is all nines.

The chapter continues in the book.