Combining Opposite Changes

Overview

Students model addition of positive and negative rational numbers through a real-world temperature story. Working as partners, they use counters, draw a matching representation, and write an equation, then compare strategies and justify how the parts combine to find the total.

Learning intentions

• Students will be able to represent positive and negative quantities with counters.
• Students will be able to connect a counter model, drawing, and equation.
• Students will be able to add rational numbers in a real-world context.
• Students will be able to explain why opposite quantities combine or cancel.

Success criteria

• I can use positive and negative counters to model a situation.
• I can draw a representation that matches my counters.
• I can write an addition equation with the correct sign.
• I can explain how the parts combine to find the total.

Curriculum links

• Addition of rational numbers: interpret a sum as movement in a positive or negative direction and recognize additive inverses.
• Expressions and Equations: solve multi-step real-life problems involving positive and negative rational numbers and assess whether an answer is reasonable.
• Expressions and Equations: use variables and equations to represent quantities in a real-world problem.
• Mathematical practice: use tools strategically and explain reasoning with precision.

Lesson structure (15 minutes)

1. 0–2 min · Hook and predict. Teacher opens the temperature story introduction and displays: “At 6 a.m., the temperature was −4°F. By noon, it rose 7°F. What is the temperature at noon?” Students make a quick prediction and explain whether the final temperature should be positive or negative.

2. 2–5 min · Model with counters. Teacher gives each student two-color integer counters and demonstrates that one color represents +1 and the other represents −1; teacher models −4 + 7 by placing four negative counters and seven positive counters together. Students build the same model, pair opposite counters, and identify any zero pairs.

3. 5–8 min · Connect representations. Teacher draws a number line or counter sketch on the model-and-equation slide, emphasizing that each positive and negative pair sums to zero, then writes −4 + 7 = 3. Students complete the matching counter drawing and equation on the rational-number modeling sheet and label the answer with units.

4. 8–11 min · Partner problem. Teacher presents a second situation on the partner challenge slide: “A hiker is 12 feet below sea level and climbs 5.5 feet. What is the hiker’s elevation now?” Students work together with counters, draw a representation, and write an equation such as −12 + 5.5 = −6.5. One student models and the other records, then partners switch roles if time allows.

5. 11–13 min · Compare and explain. Teacher asks partners to compare their counter model and drawing, using the prompts “What combined?” and “What canceled?” Students explain why the answer remains negative and check whether −6.5 feet is reasonable because climbing 5.5 feet from −12 feet does not reach sea level.

6. 13–15 min · Exit check. Teacher displays the exit question: “A bank account is $6.25 below zero. A deposit of $10 is made. Write and solve an equation, then explain the result.” Students complete the final item on the rational-number modeling sheet independently and briefly share their reasoning.

Resources

• the rational-number addition slide deck
• the rational-number modeling sheet
• Two-color integer counters, at least 20 per student
• Optional number-line strips
• Pencils and colored pencils
• Whiteboard or document camera
• Timer

Assessment

• Observe whether students correctly assign meanings to positive and negative counters and create zero pairs.
• Check that each student connects the counter model, drawing, equation, and units; ask, “How does your model prove your answer?”
• Use the exit response to identify whether students can add rational numbers and explain the sign and reasonableness of the result.

Differentiation

• Support students with a labeled key showing one color for positive and one for negative, a horizontal number line, and sentence starters: “I started at ___,” “I moved ___,” and “The total is ___ because ___.”
• Use whole-number counters first, then replace one positive counter with a half-unit or decimal label when modeling 5.5.
• For students who need reduced cognitive load, provide the equation frame ___ + ___ = ___ and allow them to explain orally while the partner records.
• Challenge students to solve the hiker problem using a number line and counters, then compare which representation makes the answer easiest to justify. Pair discussion should give both students a speaking and modeling role.