Lesson 5 — Operating with Integers

Summary

Learners perform all four operations with integers, using a number line and zero pairs to see each move, then stating the sign rules: add/subtract by moving up or down, and multiply/divide where same signs give positive and different signs give negative. They explain each step in a real context such as temperature or debt.

Objectives

• Add, subtract, multiply, and divide integers, and explain each step of the reasoning. (D05.S1.07.02)

Connection

The temperature is −5 °C at dawn and rises 8 degrees by noon: −5 + 8 = +3 °C. A farmer digs 5 metres into debt and then earns 5: the two cancel to zero. Negative numbers name what is below zero — cold below freezing, depth below sea level, money owed. Adding a negative moves down; subtracting a negative turns around and moves up. Negative numbers were used in China more than two thousand years ago — the Nine Chapters on the Mathematical Art (c. 200 BCE) counted with red rods for positive and black rods for negative — and the Indian mathematician Brahmagupta wrote rules for them in the 7th century. They are a shared human tool, not one culture’s idea.

Materials

• Vertical number line (temperature)
• Integer operation cards
• Two-colour counters

Preparation

• Copy number lines and integer operation cards.
• Have worked examples ready: −5 + 8 = 3; 3 − 5 = −2; 3 − (−2) = 5; −3 × −2 = 6.
• Recall from Grade 6: positive and negative numbers on a number line.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) add a positive moves up the number line; add a negative moves down; (2) subtract means move the opposite way, so subtracting a negative moves up (3 − (−2) = 5); and (3) multiply/divide sign rules — same signs give positive, different signs give negative. Because integer operations are a foundational skill, use explicit instruction, worked examples, and guided practice before independent work (Kirschner, Sweller & Clark, 2006). Watch for the slips of subtracting a negative as if it were adding, and of mixing up the multiply/divide sign rules. Retrieval: ask learners to recall the number line and ordering of integers from Grade 6 — operations are moves on that same line. The global lens: negative numbers appear in the Chinese Nine Chapters on the Mathematical Art (c. 200 BCE, red and black counting rods) and in Brahmagupta’s Indian rules (7th century CE); the ideas are shared human achievements. The egalitarian lens (docs/philosophy.md §4): negative numbers let anyone name debt, cold, and depth precisely — they are tools for honest accounting, not gatekeeping.

Procedure

1. Gather (5 min). Last time you operated on fractions and decimals. Today you operate on numbers below zero — the integers — and explain each sign.
2. Meet the moves (10 min). On your number line, adding a positive moves up, adding a negative moves down. Subtracting moves the opposite way: subtracting a negative moves up. Build zero pairs with counters: a +1 and a −1 cancel to zero.
3. Worked example — add and subtract (12 min). −5 + 8: start at −5, move up 8, land on +3. 3 − 5: start at 3, move down 5, land on −2. 3 − (−2): subtract −2 means move up 2, land on +5 (subtracting a debt raises you, as if the debt were forgiven).
4. Worked example — multiply and divide (10 min). −3 × 2: three groups of −2 is −6. −3 × −2: the negative flips the direction twice, landing back on +6. Rule: same signs → positive, different signs → negative. The same rule holds for division: −6 ÷ 2 = −3.
5. Practice with a partner (10 min). Take your integer operation cards. For each, move on the number line, write the operation, and say the sign rule. Trade and check each other’s signs.
6. Close (5 min). Add moves up or down; subtracting a negative moves up; multiply/divide with same signs → positive, different signs → negative. Explain each step — that is honest math.

Differentiation

• Support: Use only add/subtract with a ready number line and step the moves physically; describe each move aloud so learners with low vision can follow by listening.
• Extension: Solve multi-step integer expressions (e.g., −3 + (−4) − (−5)) and explain the sign at each step; connect to a real debt-and-credit story.

Assessment

• Formative (observation/self): Can the learner perform each of the four integer operations and explain the sign rule and number-line move behind each step?
• Self-check: The learner asks, “Did I move the right way (up for add-positive, down for add-negative, opposite for subtract)? For multiply/divide, did same signs give positive and different signs give negative, and does my move match my written answer?”

Home connection

At home, use a real below-zero situation — temperature, a depth, or money owed and paid back — and write and explain one integer addition and one subtraction, checking the signs.

Resources

• Integer operations are standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The notes that negative numbers appear in the Chinese Nine Chapters on the Mathematical Art (c. 200 BCE, red and black counting rods) (MacTutor, “Nine Chapters on the Mathematical Art,” S-289) and in Brahmagupta’s rules (India, 7th century CE) (MacTutor, “Brahmagupta,” S-290) are documented history; treating all traditions as worthy is a value (docs/philosophy.md §4). The claim that negative numbers make honest accounting possible is a value commitment.