Method 1: Rounding

Rounding Rules:

• If the digit is 0, 1, 2, 3, or 4 → round DOWN
• If the digit is 5, 6, 7, 8, or 9 → round UP
• When adding/subtracting, round to the highest place value[web:45]

Example 1: Round to Nearest Ten

Problem: \( 47 + 82 \)

Round: \( 47 \rightarrow 50 \) and \( 82 \rightarrow 80 \)

Estimate: \( 50 + 80 = 130 \)

(Actual answer: 129)

Example 2: Round to Nearest Hundred

Problem: \( 4{,}567 - 1{,}234 \)

Round: \( 4{,}567 \rightarrow 4{,}600 \) and \( 1{,}234 \rightarrow 1{,}200 \)

Estimate: \( 4{,}600 - 1{,}200 = 3{,}400 \)

(Actual answer: 3,333)

Method 2: Front-End Estimation

Steps:

1. Identify the number with the least digits
2. Keep the first digit of all numbers
3. Change all other digits to 0
4. Add or subtract

Example:

Problem: \( 5{,}467 + 326 \)

Front-End: \( 5{,}000 + 300 = 5{,}300 \)

Method 3: Compatible Numbers

Example:

Problem: \( 48 + 52 + 37 \)

Compatible: \( 48 + 52 = 100 \), then \( 100 + 37 = 137 \)

Round → Calculate → Check Reasonableness

Example 1: Addition Word Problem

Problem: A store sold 4,892 toys in January and 3,156 toys in February. About how many toys were sold in both months?

Solution:

Round to nearest thousand: \( 4{,}892 \rightarrow 5{,}000 \) and \( 3{,}156 \rightarrow 3{,}000 \)

Estimate: \( 5{,}000 + 3{,}000 = 8{,}000 \) toys

Example 2: Subtraction Word Problem

Problem: There are 8,765 students in a school district. If 3,248 students are in elementary school, about how many are not?

Solution:

Round to nearest thousand: \( 8{,}765 \rightarrow 9{,}000 \) and \( 3{,}248 \rightarrow 3{,}000 \)

Estimate: \( 9{,}000 - 3{,}000 = 6{,}000 \) students

Steps for Addition:

1. Line up numbers by place value (ones under ones, tens under tens, etc.)
2. Start with the ones place
3. Add digits in each column
4. If sum is 10 or more, regroup (carry) to the next place value
5. Continue adding left through all place values

Example: Adding with Regrouping

Step 1: \( 7 + 5 = 12 \) (write 2, carry 1)

Step 2: \( 6 + 9 + 1 = 16 \) (write 6, carry 1)

Step 3: \( 5 + 8 + 1 = 14 \) (write 4, carry 1)

Step 4: \( 4 + 3 + 1 = 8 \)

Addition Formula:

Steps for Subtraction:

1. Line up numbers by place value
2. Start with the ones place
3. If top digit is smaller than bottom digit, regroup (borrow) from next place value
4. Subtract digits in each column
5. Continue subtracting left through all place values

Example: Subtracting with Regrouping

Step 1: Can't do \( 2 - 8 \), borrow: \( 12 - 8 = 4 \)

Step 2: \( 2 - 7 \), borrow: \( 12 - 7 = 5 \)

Step 3: \( 4 - 6 \), borrow: \( 14 - 6 = 8 \)

Step 4: \( 6 - 2 = 4 \)

Subtraction Formula:

Addition Key Words:

add, plus, sum, total, altogether, combined, in all, both, increased by, more than

Subtraction Key Words:

subtract, minus, difference, take away, left, remain, fewer, less than, decreased by, how many more

Example 1: Two-Step Problem

Problem: A library had 12,456 books. They bought 3,789 new books and donated 1,234 old books. How many books does the library have now?

Solution:

Step 1: Add new books: \( 12{,}456 + 3{,}789 = 16{,}245 \)

Step 2: Subtract donated books: \( 16{,}245 - 1{,}234 = 15{,}011 \)

Answer: 15,011 books

Example 2: Comparison Problem

Problem: City A has a population of 456,789. City B has a population of 234,567. How many more people live in City A?

Solution:

Subtract to find the difference: \( 456{,}789 - 234{,}567 = 222{,}222 \)

Answer: 222,222 more people

Type 1: Missing Addend

Strategy: Subtract to find missing addend

Example: \( 345 + ? = 892 \)

Solution: \( 892 - 345 = 547 \)

Answer: \( 345 + 547 = 892 \)

Type 2: Missing Sum

Strategy: Add the numbers together

Example: \( 4{,}567 + 2{,}345 = ? \)

Solution: \( 4{,}567 + 2{,}345 = 6{,}912 \)

Type 3: Missing Minuend or Subtrahend

Strategy for \( ? - b = c \): Add to find minuend

Example: \( ? - 348 = 1{,}797 \)

Solution: \( 1{,}797 + 348 = 2{,}145 \)

Answer: \( 2{,}145 - 348 = 1{,}797 \)

Strategy for \( a - ? = c \): Subtract to find subtrahend

Example: \( 5{,}000 - ? = 3{,}456 \)

Solution: \( 5{,}000 - 3{,}456 = 1{,}544 \)

Answer: \( 5{,}000 - 1{,}544 = 3{,}456 \)

If \( a + b = c \), then \( c - b = a \) and \( c - a = b \)

If \( a - b = c \), then \( a = b + c \) and \( b = a - c \)

Property 1: Commutative Property

Meaning: The order of addends doesn't change the sum[web:46][web:49]

Examples:

\( 2 + 4 = 4 + 2 = 6 \)

\( 567 + 234 = 234 + 567 = 801 \)

Property 2: Associative Property

Meaning: The grouping of addends doesn't change the sum[web:46][web:49]

Examples:

\( (4 + 2) + 3 = 4 + (2 + 3) = 9 \)

\( (100 + 50) + 25 = 100 + (50 + 25) = 175 \)

Property 3: Identity Property (Zero Property)

Meaning: Adding zero to any number gives the same number[web:46][web:52]

Examples:

\( 14 + 0 = 14 \)

\( 9{,}876 + 0 = 9{,}876 \)

Property 4: Closure Property

Meaning: When you add any two whole numbers, the result is always a whole number[web:46][web:49]

Examples:

\( 4 + 3 = 7 \) (all whole numbers)

\( 125 + 678 = 803 \) (all whole numbers)

Property 5: Additive Inverse Property

Meaning: A number and its opposite (negative) add to zero[web:49]

Examples:

\( 8 + (-8) = 0 \)

\( 250 + (-250) = 0 \)

Strategy 1: Make Friendly Numbers (Using Commutative Property)

Goal: Rearrange addends to create easier sums

Example: \( 78 + 25 + 22 \)

Rearrange: \( 78 + 22 + 25 = 100 + 25 = 125 \)

(We paired 78 and 22 to make 100)

Strategy 2: Group for Easier Addition (Using Associative Property)

Goal: Group numbers that are easy to add mentally

Example: \( 17 + 36 + 3 \)

Regroup: \( 17 + (36 + 3) = 17 + 39 = 56 \)

Or: \( (17 + 3) + 36 = 20 + 36 = 56 \)

Strategy 3: Break Apart Numbers (Decomposition)

Goal: Break numbers into parts that are easier to work with

Example: \( 247 + 198 \)

Break 198 into \( 200 - 2 \)

Calculate: \( 247 + 200 - 2 = 447 - 2 = 445 \)

Strategy 4: Compensation Method

Goal: Round one number, then adjust the answer

Example: \( 456 + 299 \)

Think: \( 299 \) is close to \( 300 \)

Calculate: \( 456 + 300 = 756 \)

Adjust: \( 756 - 1 = 755 \)

Strategy 5: Use Doubles

Goal: Look for doubles or near-doubles

Example: \( 48 + 49 \)

Think: This is close to \( 48 + 48 \)

Calculate: \( 48 + 48 = 96 \), then \( 96 + 1 = 97 \)

Complex Example:

Problem: \( 125 + 78 + 75 + 22 \)

Solution:

Step 1: Use Commutative Property to rearrange

\( (125 + 75) + (78 + 22) \)

Step 2: Use Associative Property to group

\( 200 + 100 = 300 \)

Answer: 300