Advanced Math 5

Chapter at a Glance

• Chapter: 1
• Strand: Foundations
• Difficulty: Level 1 of 5
• Builds on: Numbers to 1,000,000, Decimals

Try This First

Try this short activity before reading the lesson.

> Write the digits of $4.2$ on small squares of paper, one digit each. Put a coin between them for the decimal point. Now multiply by $10$: slide every digit one place to the left, and drop a $0$ into the empty place. What number is on the table now? Slide once more for another $\times 10$. Did the digits change, or only where they sit?

The Big Idea

A number is a row of digits. Each place is worth ten times the place to its right. That does not change when a number gets past a billion. There are just more places. To read $2{,}340{,}506{,}018$, read three digits at a time: two billion, three hundred forty million, five hundred six thousand, eighteen.

Multiplying by $10$ moves every digit one place to the left. Multiplying by $100$ moves it two places. By $10^3$, three places. Dividing moves the digits the other way. "Add a zero" is only part of this rule. $56{,}000 \times 10$ does end in a new zero. But $4.2 \times 100$ is $420$, not $4.200$. The digits have to move through the point to make the number bigger.

Rounding uses the same places. Find the place you were asked for. Look at the digit to its right. Then decide.

Words to Know

• place value: What a digit is worth because of where it sits. The $4$ in $40{,}000{,}000$ is worth forty million.
• power of ten: $10^3 = 10 \times 10 \times 10 = 1000$. The small number counts the zeros, and the places the digits move.
• round: Swap a number for the nearest one at a chosen place. $7{,}485{,}392{,}610$ to the nearest million is $7{,}485{,}000{,}000$.
• billion: $1{,}000{,}000{,}000$. A thousand million. Nine zeros.

Outcomes and Milestones

• Students can: read and round through $1{,}000{,}000{,}000$; multiply and divide by $10, 100, 1{,}000$ and $10^k$.
• Watch for: $4.2 \times 100 = 4.200$; $56{,}000 \div 100 = 56{,}0$.
• Ready when: $3.07 \times 10^3$ and $4{,}800 \div 10^2$ are both automatic.

Worked Examples and Your Turn

Study each example, then try the paired problem.

Example 1: Read it in groups of three

Problem: Write $2{,}340{,}506{,}018$ in words. What is the $4$ worth?

Solution:

• Split the number at the commas: $2$ | $340$ | $506$ | $018$. Why: the groups are billions, millions, thousands and ones.
• Read each group, then say its name: two billion, three hundred forty million, five hundred six thousand, eighteen.
• Find the $4$. It is in the millions group, in the tens place of that group. So it is worth ten millions, four times: $40{,}000{,}000$.

Answer: Two billion, three hundred forty million, five hundred six thousand, eighteen. The $4$ is worth forty million

> Notice: The group $018$ is read as "eighteen". The zero at the front holds the hundreds place and says nothing. Leave out a group and every group to its left gets the wrong name.

Your Turn:

• Prompt: Write $5{,}207{,}013{,}400$ in words. What is the $7$ worth?
• Answer: Five billion, two hundred seven million, thirteen thousand, four hundred. The $7$ is in the ones place of the millions group, so it is worth $7{,}000{,}000$.

Example 2: Round at two different places

Problem: Round $7{,}485{,}392{,}610$ to the nearest million. Then round it to the nearest hundred million.

Solution:

• Find the millions digit. It is the $5$ in $7{,}48\underline{5}$. Look one place to its right: $3$. That is less than $5$, so the $5$ stays. Turn every digit after it into $0$: $7{,}485{,}000{,}000$.
• Find the hundred-millions digit. It is the $4$ in $7{,}\underline{4}85$. Look one place to its right: $8$. That is $5$ or more, so the $4$ goes up to $5$. The rest become zeros: $7{,}500{,}000{,}000$.

Answer: $7{,}485{,}000{,}000$ and $7{,}500{,}000{,}000$

> Verification & Check: Both answers are between $7$ billion and $8$ billion, like the number itself. The second one is rougher than the first. Rounding to a bigger place always loses more detail.

Your Turn:

• Prompt: Round $6{,}372{,}845{,}190$ to the nearest million. Then to the nearest ten million.
• Answer: Millions digit $2$, next digit $8$, so round up: $6{,}373{,}000{,}000$. Ten-millions digit $7$, next digit $2$, so it stays: $6{,}370{,}000{,}000$.

Example 3: Powers of ten, both ways

Problem: Work out $3.07 \times 10^3$ and $4{,}800 \div 10^2$.

Solution:

• Count the places for $10^3$: three. Move the point three places to the right, filling with a zero: $3.07 \to 30.7 \to 307 \to 3070$.
• Count the places for $10^2$: two. Dividing moves the point to the left: $4800 \to 480 \to 48$.

Answer: $3070$ and $48$

> Notice: The small number on the $10$ is the number of places. Say it as you move: "one, two, three".

> Verification & Check: $3 \times 1000 = 3000$, and a little more. $3070$ fits. And $48 \times 100 = 4800$.

Your Turn:

• Prompt: Work out $5.09 \times 10^2$ and $67{,}000 \div 10^3$.
• Answer: Two places right: $509$. Three places left: $67$.

Example 4: Adding zeros

Problem: Work out $4.2 \times 100$ and $56{,}000 \div 100$.

Solution:

• Common mistake: $4.2 \times 100 = 4.200$, with two zeros stuck on the end. People do this because it works for whole numbers. But $4.200$ is the same as $4.2$. Nothing got bigger.
• Do this instead. Move the point two places to the right: $4.2 \to 42 \to 420$. Why: the $2$ moved from the tenths place to the tens place, and that is what makes the number a hundred times bigger.
• For $56{,}000 \div 100$, move the point two places to the left: $56000 \to 5600 \to 560$.

Answer: $420$ and $560$

> Notice: "Add a zero" only works when there is nothing after the point to move through. It is one case of the rule, not the rule.

> Verification & Check: $4 \times 100 = 400$ and $0.2 \times 100 = 20$. Together, $420$. And $560 \times 100 = 56{,}000$.

Your Turn:

• Prompt: Work out $3.6 \times 100$ and $48{,}000 \div 100$.
• Answer: $3.6 \to 36 \to 360$. And $48000 \to 4800 \to 480$.

Example 5: A power of ten in a story

Problem: A crate holds $10^4$ bolts. Each bolt has a mass of $0.65$ g. What is the mass of all the bolts in grams? In kilograms?

Solution:

• Move the point four places to the right: $0.65 \to 6.5 \to 65 \to 650 \to 6500$ g.
• Change grams to kilograms. A kilogram is $1000$ g, so divide by $10^3$. Move the point three places left: $6500 \to 6.5$ kg.

Answer: $6500$ g $= 6.5$ kg

> Verification & Check: Ten thousand bolts at a bit more than half a gram each. That is a bit more than $5$ kg. Yes.

Your Turn:

• Prompt: A box holds $10^3$ screws. Each screw has a mass of $2.4$ g. What is the total mass in grams, and in kilograms?
• Answer: $2.4 \times 1000 = 2400$ g. $2400 \div 1000 = 2.4$ kg.

Example 6: Compare two shifted numbers

Problem: Which is larger, $3.9 \times 10^5$ or $41 \times 10^4$?

Solution:

• Write the first one out: $3.9 \times 10^5 = 390{,}000$.
• Write the second one out: $41 \times 10^4 = 410{,}000$.
• Compare them: $410{,}000$ is bigger than $390{,}000$.

Answer: $41 \times 10^4$

> Notice: The bigger power of ten does not always win. Write both numbers out, then compare.

Your Turn:

• Prompt: Which is larger, $2.8 \times 10^4$ or $27 \times 10^3$?
• Answer: $28{,}000$ against $27{,}000$. So $2.8 \times 10^4$.

Example 7: One hundred copies minus five copies

Problem: A number multiplied by $100$ is $4{,}750$ more than the same number multiplied by $5$. What is the number?

Solution:

• Start by saying what the two products are. Times $100$ is one hundred copies of the number. Times $5$ is five copies. Why: you do not know the number, but you can still count copies of it.
• Take the five copies away from the hundred copies. That leaves $95$ copies. The story says that difference is $4{,}750$.
• Divide to find one copy: $4750 \div 95 = 50$. Why: $95$ equal copies make $4{,}750$, so one copy is $4{,}750$ shared $95$ ways.

Answer: $50$

> Notice: A number you do not know can still be counted in copies. Ninety-five copies is one thing, and dividing finds one copy. That is the first step of algebra without any letter.

> Verification & Check: $50 \times 100 = 5000$ and $50 \times 5 = 250$. The difference is $4750$.

Your Turn:

• Prompt: A number multiplied by $10$ is $360$ more than the same number multiplied by $4$. What is the number?
• Answer: $10 - 4 = 6$ copies make $360$. One copy is $360 \div 6 = 60$. Check: $600 - 240 = 360$.

What If...?

Explore these special cases:

• Case: $560 \times 10$
• Resolution: $5600$. Here "add a zero" is right. There is nothing after the point, so the empty place gets a $0$.
• Case: $45 \div 100$
• Resolution: $0.45$. Dividing can push digits past the point. The answer is less than $1$.
• Case: $10^0$ and $10^1$
• Resolution: $10^1 = 10$. And $10^0 = 1$: zero places moved, so the number does not change.
• Case: Round $999{,}600{,}000$ to the nearest million.
• Resolution: $1{,}000{,}000{,}000$. The $6$ rounds the $9$ up, and that carries through every $9$ to its left.
• Case: $0.5 \times 10$
• Resolution: $5$. The point moves past the only digit. Not $0.50$.

The chapter continues in the book.