Integers
Warm-up
Post a vertical thermometer and call out a winter forecast: "It was 3° at sunset and dropped 8° overnight." Ask students to show the overnight low with a finger on the thermometer before anyone speaks. Most hands land at −5° — collect the strategies: some counted down 8, some went 3-to-zero then 5 more.
The warm-up's job is to surface the two mental models students already own — counting through zero, and splitting the drop at zero — because today's lesson formalizes both.
Explore
Give partners a number line from −10 to 10 and a set of "journey cards": start at 4, move 6 left; start at −3, move 5 right; start at −2, move 4 left. For each card, students record the journey as a number sentence (4 − 6 = −2) and mark the landing point.
After eight journeys, ask pairs to sort their number sentences into "crossed zero" and "stayed on one side" piles, and to write one sentence describing what crossing zero felt like arithmetically. Circulate and listen for the split-at-zero strategy — it's the one that generalizes.
Formalize
Formalize the number line as the home of the integers: whole numbers and their opposites, with zero as the centre of symmetry. Every integer has an opposite the same distance from zero on the other side, and that distance is its absolute value:
Comparing integers is a position question, not a size-of-digits question: on the number line, left means less, so even though 7 beats 4 as a digit. Anchor the two vocabulary pairs — opposite (mirror across zero) and absolute value (distance from zero, always non-negative) — before practice.
Practice
Practice mixes representation and comparison: order sets like ; write the integer for "12 m below sea level"; find opposites and absolute values; and answer two crossing-zero journey problems.
Exit ticket: "Which is greater, or ? Draw the number line evidence in one sketch."
Exit ticket
Practice mixes representation and comparison: order sets like ; write the integer for "12 m below sea level"; find opposites and absolute values; and answer two crossing-zero journey problems.
Exit ticket: "Which is greater, or ? Draw the number line evidence in one sketch."
Step 1: Place each on the number line: farthest left, then , then , then , then .
Step 2: Read left to right: .
Step 3: Spot-check the danger pair: vs . Digits say 8 beats 6; the line says sits farther LEFT, so it's less. Position wins.
Step 4: Say one comparison in temperature language to anchor it: " is colder than ."
A submarine sits at m (45 m below sea level). A helicopter hovers at m. Which is farther from sea level, and how far apart are they?
Step 1: Farther from sea level = larger absolute value: and . The submarine is farther from sea level.
Step 2: Distance between them: from up to is 45; from up to is 30 more.
Step 3: Total separation: m.
Step 4: Note what we did NOT do: we never computed mechanically — we split the trip at zero, the same move as the temperature warm-up. Position can be negative; distance never is.
Step 1: Read the expression as language, not decoration: means "the OPPOSITE of ."
Step 2: Locate on the line, then mirror it across zero: land at . So .
Step 3: Test the pattern the student can trust: the opposite of 5 is ; the opposite of is 5; opposite-of-opposite returns home. Two mirror flips face you forward again.
Step 4: Preview why this matters: next year, subtracting a negative will use exactly this idea — will mean "8 minus the opposite of 3's opposite"… better: will unwrap via opposites. Plant the mirror image now; the rule arrives later with a reason attached.