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+++ title = "Integer foundations" description = "Model directed magnitude on number lines; use the zero principle and tiles to add and subtract integers, and see subtraction as adding an opposite." template = "chapter.html" weight = 8 [extra] subject = "math" chapter_id = "integer-foundations" strand = "Number & Algebra" +++
Integers extend whole numbers to include values below zero. They are useful for temperature, elevation, and money balances.
Directed numbers on a number line
Positive numbers are to the right of zero, negative numbers are to the left. Moving right increases value; moving left decreases value.
Compare quickly
\(a > b\) means \(a\) is to the right of \(b\) on the number line.
Absolute value and opposites
Absolute value is distance from zero: \(|-7|=7\). Opposites have equal distance from zero and opposite signs, such as \(+5\) and \(-5\).
Adding and subtracting integers
You can think of addition and subtraction as movement on the number line.
- \(a+b\): start at \(a\), move by \(b\).
- \(a-b = a+(-b)\): subtracting is adding the opposite.
Example: \(6-(-4)=6+4=10\).
Zero principle with tiles
A positive tile and a negative tile form a zero pair. Add or remove zero pairs to simplify without changing value.
Checkpoint: evaluate \(-3 + 8\)
\(-3 + 8 = 5\). Start at \(-3\), move 8 units right.
Checkpoint: evaluate \(4 - 9\)
\(4 - 9 = -5\). Equivalent to \(4 + (-9)\).
Checkpoint: explain why \(2-(-2)=4\)
Subtracting \(-2\) means adding its opposite \(+2\), so \(2-(-2)=2+2=4\).