Addition: combine and count on

Key idea

Addition means putting parts together to make a total. When you see \(a+b\), you can think: start at \(a\) and add \(b\) more. On a number line, addition is like hopping forward.

Important property

Addition is commutative, which means you can switch the order and the sum stays the same: \(\,a+b=b+a\).

Worked example

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Summary

• \(a+b\) means combine parts to make a total (or count on).
• You can switch addends: \(a+b=b+a\).

Subtraction: take away and find the difference

Key idea

Subtraction can mean take away (remove from a set) or difference (compare two numbers). When you see \(a-b\), you can think: start at \(a\) and remove \(b\). On a number line, subtraction is like hopping backward.

Inverse relationship

Addition and subtraction are inverse operations. If \(a-b=c\), then you can check with addition: \(c+b=a\).

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Summary

• \(a-b\) can mean take away or find the difference.
• Check subtraction with addition: if \(a-b=c\), then \(c+b=a\).

Fast strategies for addition and subtraction facts

Key strategies

• +0 / −0: the number stays the same
• Make a ten: move to 10 (or the next ten) first
• Doubles: \(6+6\), \(7+7\), \(8+8\)
• Near doubles: \(7+8\) is one more than \(7+7\)
• Bridge tens: for example, \(28+7 = 28+2+5 = 30+5\)
• Compensation: adjust, then fix (for example, \(52-19 = 52-20+1\))

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Summary

• Use strategies like make-a-ten, doubles, and compensation.
• Good mental math is step-by-step, not guessing.

Multi-digit addition and subtraction with regrouping

Key idea

Line up digits by place value (ones under ones, tens under tens). For addition, if the ones add to 10 or more, regroup 10 ones as 1 ten. For subtraction, if you can’t subtract in the ones place, regroup 1 ten as 10 ones.

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Summary

• Use place value alignment to avoid mistakes.
• Regroup when needed (carry in addition, borrow in subtraction).

Multi-step addition and subtraction (left to right)

Key idea

Addition and subtraction have the same priority. When an expression has only \(+\) and \(−\), you work from left to right (after you do anything inside parentheses).

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Summary

• With only \(+\) and \(−\), work left to right.
• Show steps to avoid common sign mistakes.

Check your work and solve missing-number equations

Key idea

If a number is missing, use the inverse operation:

• If \(x+b=a\), then \(x=a-b\).
• If \(a-x=b\), then \(x=a-b\).

This is also a great way to check subtraction with addition (and addition with subtraction).

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Summary

• Use inverse operations to find missing numbers.
• Always check: a quick check prevents small mistakes from becoming big ones.

Why addition and subtraction matter

Where you use addition and subtraction

• Money: totals and change (paying and getting change back).
• Distance and time: how far, how long, and how much remains.
• Temperature: increases and decreases (differences between readings).
• Data: comparing values to find the difference.
• Geometry: perimeter is found by adding side lengths.

Worked example: making change

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Fun facts (a little history)

• Symbols: The plus sign \(+\) and minus sign \(−\) are used worldwide to show addition and subtraction.
• More meanings: The minus sign can also show a negative number (for example, \(-3\)).
• Mental math: Before calculators, people relied on smart strategies (like making tens) to add and subtract quickly.

Final recap

• Addition combines parts to make a total: \(a+b\).
• Subtraction removes or compares to find a difference: \(a-b\).
• Use strategies (make ten, doubles, compensation) to be fast and accurate.
• Use place value and regrouping for multi-digit problems.
• Check your answers using inverse operations.