Grade 3 Math

Try This First

Try this short activity before reading the lesson.

> Draw a number line from $340$ to $350$ with a tick for every number. Put a dot on $347$. Which end is it closer to? Now draw a line from $300$ to $400$ with a tick every ten. Put a dot on $347$ again. Which end is it closer to now? The dot did not move. The question did.

The Big Idea

Rounding means swapping a number for a nearby round one. A round number ends in zero. You round when the exact number does not matter right now. "About $350$ people came" is honest even if $347$ came.

The rule: decide which place you are rounding to. Look at the digit just to the right of it. If that digit is $5$ or more, round up. If it is less than $5$, keep the digit and turn the rest into zeros. $347$ to the nearest ten: the ones digit is $7$, so round up to $350$. To the nearest hundred: the tens digit is $4$, so round down to $300$.

Adding and subtracting big numbers works in columns, one place at a time. When a column gets more than $9$, carry a ten to the next place. When you need to take away more than you have, borrow a ten from the next place. Rounding first gives you an estimate. The estimate tells you if your exact answer makes sense.

Words to Know

• round: Swap a number for the nearest number that ends in zero. $347$ rounds to $350$ or to $300$, depending on the place you choose.
• estimate: A quick answer that is close, made from rounded numbers. $482 + 319$ is about $500 + 300 = 800$.
• regroup: Trade ten of one place for one of the next place, or the other way. $10$ ones become $1$ ten when you carry.

Worked Examples and Your Turn

Study each example, then try the paired problem.

Example 1: Round to the nearest ten and hundred

Problem: Round $683$ to the nearest ten. Then round it to the nearest hundred.

Solution:

• Nearest ten: the tens digit is $8$. Look one place to the right, at the ones digit. It is $3$.
• $3$ is less than $5$, so keep the $8$ and make the ones a zero: $680$. Why: $683$ is closer to $680$ than to $690$.

<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 360 76" width="360" height="76" role="img" aria-label="Number line from 680 to 690 with a tick for every number. A dot on 683 is 3 steps from 680 and 7 steps from 690, so it is closer to 680."><line x1="22" y1="40" x2="338" y2="40" stroke="#333" stroke-width="1.5"/><line x1="30" y1="31" x2="30" y2="49" stroke="#333" stroke-width="1.5"/><text x="30" y="64" font-family="Source Sans 3" font-size="13" text-anchor="middle" fill="#333">680</text><line x1="60" y1="34" x2="60" y2="46" stroke="#333" stroke-width="1.5"/><line x1="90" y1="34" x2="90" y2="46" stroke="#333" stroke-width="1.5"/><line x1="120" y1="34" x2="120" y2="46" stroke="#333" stroke-width="1.5"/><line x1="150" y1="34" x2="150" y2="46" stroke="#333" stroke-width="1.5"/><line x1="180" y1="31" x2="180" y2="49" stroke="#333" stroke-width="1.5"/><text x="180" y="64" font-family="Source Sans 3" font-size="13" text-anchor="middle" fill="#333">685</text><line x1="210" y1="34" x2="210" y2="46" stroke="#333" stroke-width="1.5"/><line x1="240" y1="34" x2="240" y2="46" stroke="#333" stroke-width="1.5"/><line x1="270" y1="34" x2="270" y2="46" stroke="#333" stroke-width="1.5"/><line x1="300" y1="34" x2="300" y2="46" stroke="#333" stroke-width="1.5"/><line x1="330" y1="31" x2="330" y2="49" stroke="#333" stroke-width="1.5"/><text x="330" y="64" font-family="Source Sans 3" font-size="13" text-anchor="middle" fill="#333">690</text><circle cx="120" cy="40" r="6" fill="#e06666" stroke="#333" stroke-width="1"/><text x="120" y="26" font-family="Source Sans 3" font-size="13" text-anchor="middle" fill="#333" font-weight="700">683</text></svg>

• Nearest hundred: the hundreds digit is $6$. Look at the tens digit. It is $8$.
• $8$ is $5$ or more, so the $6$ becomes $7$: $700$. Why: $683$ is closer to $700$ than to $600$.

Answer: $680$ and $700$

> Notice: The same number rounds two different ways. What changes is the place you are rounding to.

Your Turn 1: Round $256$ to the nearest ten. Then round it to the nearest hundred.

Example 2: Estimate, then add

Problem: Estimate $482 + 319$ by rounding to the nearest hundred. Then find the exact sum. Say why the estimate is a little low.

Solution:

• Round each number: $482$ is about $500$, and $319$ is about $300$. Estimate: $500 + 300 = 800$.
• Now add exactly, ones first: $2 + 9 = 11$. Write $1$, carry $1$ ten.
• Tens: $8 + 1 + 1 = 10$. Write $0$, carry $1$ hundred.
• Hundreds: $4 + 3 + 1 = 8$. The sum is $801$.
• Compare. $801$ is close to $800$, so the sum makes sense. The estimate is a little low because $482$ went up by $18$ but $319$ went down by $19$. The rounding lost one.

Answer: Estimate $800$; exact $801$.

> Verification & Check: Take $319$ away from $801$: $801 - 319 = 482$. Back where you started.

Your Turn 2: Estimate $563 + 228$ to the nearest hundred, then find the exact sum.

Example 3: Subtract across a zero

Problem: Find $704 - 356$.

Solution:

• Ones: $4 - 6$ cannot be done. Look at the tens for a ten to borrow. The tens digit is $0$. Nothing there.
• Go one more place. Borrow $1$ hundred from the $7$: it becomes $6$. That hundred is $10$ tens. Now the tens digit is $10$.
• Borrow $1$ ten from those $10$ tens: the tens become $9$, and the ones become $14$.
• Now subtract each column: $14 - 6 = 8$, $9 - 5 = 4$, $6 - 3 = 3$. The answer is $348$.

Answer: $348$

> Notice: A zero cannot lend. You go one more place to the left, and the zero becomes a $9$ after it lends one ten on.

> Verification & Check: Add it back: $348 + 356 = 704$. Yes.

Your Turn 3: Find $503 - 247$.

Example 4: Three rounds down, and the wrong borrow

Problem: Round $347$ to the nearest ten. Then find $603 - 278$.

Solution:

• Common mistake: "the $3$ rounds down, so $340$". The $3$ is the hundreds digit. It has nothing to do with rounding to the nearest ten. You look at the digit to the right of the tens place. That is the ones digit, $7$.
• $7$ is $5$ or more, so round up: $350$.
• Now the subtraction. Common mistake: borrowing from the $6$ straight into the ones, so the ones become $13$ and the $6$ becomes $5$, leaving the tens at $0$. Then $0 - 7$ gets stuck, and a child writes $7$ anyway. That gives $375$, and it is wrong.
• Do it in order. The ones need a ten. The tens have $0$, so borrow a hundred first: $6$ becomes $5$, tens become $10$. Now lend one of those tens to the ones: tens become $9$, ones become $13$.
• Subtract: $13 - 8 = 5$, $9 - 7 = 2$, $5 - 2 = 3$. The answer is $325$.

Answer: $350$ and $325$

> Verification & Check: Estimate first: $600 - 300 = 300$. $325$ is near $300$. $375$ is further off, but the estimate alone cannot be sure. The add-back is sure: $325 + 278 = 603$.

Your Turn 4: Round $362$ to the nearest ten. Then find $402 - 165$.

Example 5: A story with a missing part

Problem: A school has $815$ children. $468$ of them are girls. About how many boys are there? Then find the exact number.

Solution:

• Draw a bar. The whole bar is $815$. One part is $468$ girls. The other part is the boys, and it is the unknown.

<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 380 90" width="380" height="90" role="img" aria-label="Bar model. A whole bar of 815 children is split into two parts drawn to scale: 468 girls and an unknown part for the boys."><line x1="20" y1="34" x2="20" y2="26" stroke="#333" stroke-width="1.2"/><line x1="20" y1="26" x2="360" y2="26" stroke="#333" stroke-width="1.2"/><line x1="360" y1="26" x2="360" y2="34" stroke="#333" stroke-width="1.2"/><text x="190" y="21" font-family="Source Sans 3" font-size="13" text-anchor="middle" fill="#333">815 children</text><rect x="20" y="38" width="195.2" height="40" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><text x="117.6" y="63" font-family="Source Sans 3" font-size="13" text-anchor="middle" fill="#333">468 girls</text><rect x="215.2" y="38" width="144.8" height="40" fill="#ffffff" stroke="#333" stroke-width="1.5"/><text x="287.6" y="63" font-family="Source Sans 3" font-size="13" text-anchor="middle" fill="#333">? boys</text></svg>

• Estimate: $800 - 500 = 300$. So there are about $300$ boys.
• Subtract exactly. Ones: $5 - 8$ needs a borrow. Tens: $1$ becomes $0$, ones become $15$. $15 - 8 = 7$.
• Tens: $0 - 6$ needs a borrow. Hundreds: $8$ becomes $7$, tens become $10$. $10 - 6 = 4$. Hundreds: $7 - 4 = 3$. That is $347$ boys.

Answer: About $300$; exactly $347$ boys.

> Verification & Check: $347 + 468 = 815$. The two parts make the whole.

Your Turn 5: A jar holds $920$ beads. $573$ are blue and the rest are red. Estimate the red ones, then find the exact number.

Example 6: Every number that rounds to 500

Problem: A number rounds to $500$ when rounded to the nearest ten. It also rounds to $500$ when rounded to the nearest hundred. What is the smallest it could be? What is the largest?

Solution:

• Start with the tighter rule. Nearest ten gives $500$ only for $495$ up to $504$. Why: $495$ has ones digit $5$, so it goes up to $500$. $505$ would go up to $510$.
• Now check the nearest hundred for those numbers. $495$: the tens digit is $9$, so it rounds up to $500$. Good. $504$: the tens digit is $0$, so it rounds down to $500$. Good.
• Every number from $495$ to $504$ passes both rules. The smallest is $495$ and the largest is $504$.

Answer: $495$ and $504$

> Notice: When two rules must both be true, start with the stricter one. It leaves fewer numbers to check.

Your Turn 6: A number rounds to $800$ to the nearest ten and to the nearest hundred. What are the smallest and largest it could be?

The chapter continues in the book.