Basic Formula:

\(\text{Addend} + \text{Addend} = \text{Sum}\)

OR

\(\text{Part} + \text{Part} = \text{Whole}\)

Vocabulary:

• • Addend: The numbers being added
• • Sum: The answer when we add
• • Plus (+): The addition symbol

Example: \(7 + 5 = 12\)
• Addends: \(7\) and \(5\)
• Sum: \(12\)

Basic Formula:

\(\text{Minuend} − \text{Subtrahend} = \text{Difference}\)

OR

\(\text{Whole} − \text{Part} = \text{Part}\)

Vocabulary:

• • Minuend: The number we start with
• • Subtrahend: The number we take away
• • Difference: The answer when we subtract
• • Minus (−): The subtraction symbol

Example: \(15 − 8 = 7\)
• Minuend: \(15\)
• Subtrahend: \(8\)
• Difference: \(7\)

Strategy 1: Counting On (Addition)

Start with the bigger number and count up!

Example: \(8 + 5 = ?\)
Start at \(8\), then count: \(9, 10, 11, 12, 13\)
Answer: \(8 + 5 = 13\) ✓

Strategy 2: Counting Back (Subtraction)

Start with the first number and count backwards!

Example: \(14 − 6 = ?\)
Start at \(14\), count back \(6\): \(13, 12, 11, 10, 9, 8\)
Answer: \(14 − 6 = 8\) ✓

Strategy 3: Making 10

Break numbers to make \(10\) first, then add the rest!

Example: \(7 + 5 = ?\)
Break it: \(7 + 3 + 2 = 10 + 2 = 12\)
Answer: \(7 + 5 = 12\) ✓

Strategy 4: Using Doubles

If you know doubles, use them to help!

Example: \(6 + 7 = ?\)
I know \(6 + 6 = 12\), so \(6 + 7 = 12 + 1 = 13\)
Answer: \(6 + 7 = 13\) ✓

Strategy 1: Adding Tens and Ones Separately

Add the tens first, then add the ones!

Example: \(34 + 25 = ?\)
Tens: \(30 + 20 = 50\)
Ones: \(4 + 5 = 9\)
Total: \(50 + 9 = 59\)
Answer: \(34 + 25 = 59\) ✓

Strategy 2: Using Place Value

Break numbers into tens and ones, then combine!

Example: \(67 − 32 = ?\)
\(67 = 60 + 7\)
\(32 = 30 + 2\)
Tens: \(60 − 30 = 30\)
Ones: \(7 − 2 = 5\)
Total: \(30 + 5 = 35\)
Answer: \(67 − 32 = 35\) ✓

Strategy 3: Jumping by 10s on a Number Line

Jump forward or backward by tens, then by ones!

Example: \(43 + 24 = ?\)
Start at \(43\)
Jump \(+10\) → \(53\)
Jump \(+10\) → \(63\)
Jump \(+4\) → \(67\)
Answer: \(43 + 24 = 67\) ✓

Ways to Make 10:

• • \(0 + 10 = 10\)
• • \(1 + 9 = 10\)
• • \(2 + 8 = 10\)
• • \(3 + 7 = 10\)
• • \(4 + 6 = 10\)
• • \(5 + 5 = 10\)
• • \(20 − 10 = 10\)
• • \(15 − 5 = 10\)

Ways to Make 50:

• • \(25 + 25 = 50\)
• • \(30 + 20 = 50\)
• • \(40 + 10 = 50\)
• • \(45 + 5 = 50\)
• • \(100 − 50 = 50\)
• • \(75 − 25 = 50\)
• • \(60 − 10 = 50\)

Important Formula:

If \(a + b = n\), then \(b + a = n\) (Commutative Property)
And \(n − a = b\) and \(n − b = a\)

The Golden Rule:

\(\text{Left Side} = \text{Right Side}\)

Whatever is on the left MUST equal what is on the right!

How to Solve Balance Equations:

1. Solve the side with known numbers
2. That answer is what the other side must equal
3. Find the missing number to make both sides equal
4. Check your work! Both sides should be the same

Example 1:

Problem: \(8 + 4 = ? + 5\)

Step 1: Solve left side: \(8 + 4 = 12\)
Step 2: Now we have: \(12 = ? + 5\)
Step 3: What plus \(5\) equals \(12\)? Answer: \(7\)
Step 4: Check: \(8 + 4 = 12\) and \(7 + 5 = 12\) ✓
Answer: \(?\) \(= 7\)

Example 2:

Problem: \(15 − 6 = 20 − ?\)

Step 1: Solve left side: \(15 − 6 = 9\)
Step 2: Now we have: \(9 = 20 − ?\)
Step 3: What minus from \(20\) equals \(9\)? Answer: \(11\)
Step 4: Check: \(15 − 6 = 9\) and \(20 − 11 = 9\) ✓
Answer: \(?\) \(= 11\)

Basic Formula:

\(\text{Input} + \text{Rule} = \text{Output}\)

OR

\(\text{Input} − \text{Rule} = \text{Output}\)

How to Find the Rule:

1. Compare input to output: Is output bigger or smaller?
2. If output is bigger → use addition (+)
3. If output is smaller → use subtraction (−)
4. Find the difference: How much was added or subtracted?
5. Test your rule on all the numbers!

Example - Addition Rule:

Finding the Rule:
• \(3 → 8\): difference is \(5\) (output is bigger, so add)
• \(5 → 10\): difference is \(5\) ✓
• \(7 → 12\): difference is \(5\) ✓
Rule: Add \(5\) or \(\text{Input} + 5 = \text{Output}\) ✓

Example - Subtraction Rule:

Finding the Rule:
• \(15 → 8\): difference is \(7\) (output is smaller, so subtract)
• \(20 → 13\): difference is \(7\) ✓
• \(18 → 11\): difference is \(7\) ✓
Rule: Subtract \(7\) or \(\text{Input} − 7 = \text{Output}\) ✓

Quick Rules:

• If the answer is BIGGER → use + (addition)
• If the answer is SMALLER → use − (subtraction)

Example 1:

Problem: \(12\) ___ \(5 = 17\)

Step 1: Look at the answer: \(17\) is BIGGER than \(12\)
Step 2: When the answer is bigger, we add!
Step 3: Check: \(12 + 5 = 17\) ✓
Answer: The sign is +

Example 2:

Problem: \(20\) ___ \(8 = 12\)

Step 1: Look at the answer: \(12\) is SMALLER than \(20\)
Step 2: When the answer is smaller, we subtract!
Step 3: Check: \(20 − 8 = 12\) ✓
Answer: The sign is −

Addition Key Words (Use +):

• • Add, Plus, Sum, Total
• • Altogether, Combined, Together
• • In all, More, Increase
• • Join, Put together

Subtraction Key Words (Use −):

• • Subtract, Minus, Difference
• • Take away, Remove, Left
• • Less, Fewer, Decrease
• • Gave away, Lost, Spent
• • How many more?

Addition Problem:

Problem: Sarah has \(15\) stickers. Her friend gives her \(8\) more stickers. How many stickers does Sarah have now?

Solution:
• Important numbers: \(15\) and \(8\)
• Key word: "gives more" → Addition!
• Number sentence: \(15 + 8 = ?\)
• Solve: \(15 + 8 = 23\)
Answer: Sarah has \(23\) stickers. ✓

Subtraction Problem:

Problem: There are \(42\) apples in a basket. \(17\) apples are eaten. How many apples are left?

Solution:
• Important numbers: \(42\) and \(17\)
• Key words: "eaten" and "left" → Subtraction!
• Number sentence: \(42 − 17 = ?\)
• Solve: \(42 − 17 = 25\)
Answer: There are \(25\) apples left. ✓

Comparison Problem:

Problem: Jake has \(35\) marbles. Emma has \(58\) marbles. How many more marbles does Emma have than Jake?

Solution:
• Important numbers: \(35\) and \(58\)
• Key words: "how many more" → Subtraction!
• Number sentence: \(58 − 35 = ?\)
• Solve: \(58 − 35 = 23\)
Answer: Emma has \(23\) more marbles. ✓

Parts of a Number Sentence:

\(\text{Number}\) \(\text{Operation Sign}\) \(\text{Number}\) \(=\) \(\text{Answer}\)

Examples:
\(7 + 3 = 10\) (Addition sentence)
\(15 − 6 = 9\) (Subtraction sentence)

How to Write from Pictures or Words:

1. Count the first group (write the first number)
2. Decide: adding or taking away? (choose + or −)
3. Count the second group (write the second number)
4. Write the equal sign (=)
5. Find the answer

Example 1: Write from a Picture

Situation: You see \(5\) red apples and \(4\) green apples.

Number Sentence: \(5 + 4 = 9\)
(We're putting groups together, so we add!)

Example 2: Write from Words

Situation: There were \(18\) birds. \(7\) birds flew away.

Number Sentence: \(18 − 7 = 11\)
(Birds are leaving, so we subtract!)