Foundations of Mathematical Thinking · Junior Infants · Number Systems and Operations · Autumn Term

Operations with Integers: Addition & Subtraction

Students will perform addition and subtraction of integers, using various models and understanding the concept of absolute value.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Strand 3: Number - N.1.2

About This Topic

Operations with integers focus on addition and subtraction of positive and negative numbers, using models such as number lines, two-color counters, and vertical number lines. Students explore how adding opposites results in zero, and practice rules like subtracting a negative equals adding a positive. Absolute value emerges as the distance from zero on the number line, regardless of direction. These skills address NCCA Junior Cycle Strand 3 Number standard N.1.2, supporting predictions about sums and differences.

This topic extends whole number operations into the integers, preparing students for rational numbers and algebra. Key questions guide learning: predict adding positive and negative integers, justify equivalence of subtraction rules, and distinguish sums from differences. Real-world contexts like temperature changes, bank balances, or sea-level elevations make concepts relevant and build number sense.

Active learning shines here because integer operations are abstract without visuals. Manipulatives let students physically model additions and subtractions, revealing patterns through trial and error. Group tasks with number lines encourage justification of results, turning rules into discovered truths that stick.

Key Questions

1. Predict the outcome of adding a positive and a negative integer.
2. Justify why subtracting a negative number is equivalent to adding a positive number.
3. Differentiate between the sum and the difference of two integers.

Learning Objectives

• Calculate the sum of two integers, including positive and negative values, using a number line model.
• Calculate the difference between two integers, including positive and negative values, using two-color counters.
• Explain the concept of absolute value as the distance from zero on a number line.
• Predict the sign of the sum when adding a positive and a negative integer, justifying the prediction with examples.
• Justify why subtracting a negative integer is equivalent to adding its positive counterpart.

Before You Start

Addition and Subtraction of Whole Numbers

Why: Students need a solid foundation in adding and subtracting non-negative numbers before extending these operations to include negative integers.

Introduction to Number Lines

Why: Familiarity with representing numbers on a number line is essential for understanding integer operations and the concept of absolute value.

Key Vocabulary

Integer	A whole number (not a fraction or decimal) that can be positive, negative, or zero. Examples include -3, 0, and 5.
Absolute Value	The distance of a number from zero on the number line, always a non-negative value. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.
Opposite Integers	Two integers that are the same distance from zero on the number line but in opposite directions. For example, 4 and -4 are opposite integers.
Sum	The result of adding two or more numbers together. For example, the sum of -2 and 3 is 1.
Difference	The result of subtracting one number from another. For example, the difference between 5 and -3 is 8.

Watch Out for These Misconceptions

Common Misconception

Subtracting a negative number means subtracting a positive.

What to Teach Instead

Students often apply whole number rules rigidly. Hands-on number line work shows moving right for the negative's opposite, building intuition. Pair discussions help articulate why -5 - (-3) equals -2.

Common Misconception

Adding a larger positive and smaller negative always yields positive.

What to Teach Instead

This overlooks magnitude comparison. Two-color counters reveal pairing to zero first, clarifying direction. Group modeling exposes errors through peer checks and repeated practice.

Common Misconception

Absolute value changes the sign of negative numbers.

What to Teach Instead

Learners confuse it with negation. Measuring distances on shared number lines corrects this visually. Collaborative sketches reinforce absolute value as non-negative distance only.

Active Learning Ideas

Number Line Walks: Integer Journeys

Mark a floor number line from -10 to 10 with tape. Students start at zero and follow cards with instructions like '+3' or '-2'. Pairs discuss and predict endpoints before moving, then record results on worksheets. End with sharing one justification.

30 min·Pairs

Two-Color Counters: Zero Pairs Game

Provide red (negative) and yellow (positive) counters. Students model problems like -3 + 5 by pairing opposites to make zeros, then count leftovers. Switch to subtraction by removing pairs. Groups compete to solve 10 problems fastest with explanations.

25 min·Small Groups

Elevator Challenges: Real-World Integers

Print elevator floor cards from -5 to 15. Students draw sequences like 'down 4, up 7' and track position on personal number lines. Pairs create their own problems, trade, and solve while noting absolute values of floors.

35 min·Pairs

Integer War Card Game

Use cards labeled -10 to 10. Players flip two cards per round, add or subtract based on color rules, and compare results. Highest absolute value wins the round. Debrief misconceptions as a class.

20 min·Pairs

Real-World Connections

• Temperature changes in Dublin during winter months often involve adding or subtracting degrees. For example, if the temperature is -2°C and drops by 3°C, students can calculate the new temperature as -5°C.
• Bank account balances can be modeled using integers. A deposit of €50 into an account with a balance of -€20 results in a new balance of €30, demonstrating addition of integers.

Assessment Ideas

Quick Check

Present students with a number line. Ask them to model the problem -3 + 5 by moving their finger or a marker. Then, ask them to write the final answer and explain their steps in one sentence.

Exit Ticket

Give each student a card with a subtraction problem involving a negative number, such as 7 - (-2). Ask them to rewrite the problem as an addition problem and then solve it, showing their work.

Discussion Prompt

Pose the question: 'If you have €10 in your pocket and you owe your friend €5, how would you represent this using integers? What happens to your 'money' if you pay them back?' Guide students to discuss the meaning of negative numbers and subtraction in this context.

Frequently Asked Questions

How do you introduce integer addition and subtraction?

Start with concrete models like number lines and counters before rules. Pose real contexts such as temperature or debt to hook interest. Guide discovery through key questions, ensuring students predict, test, and justify outcomes for deep understanding.

What are common errors with subtracting negatives?

Many treat subtraction of negative as subtracting positive, yielding wrong signs. Address via manipulatives showing the rule as adding opposite. Repeated pair practice with explanations solidifies equivalence, aligning with NCCA emphasis on justification.

How does absolute value fit into integer operations?

Absolute value measures distance from zero, aiding comparisons in addition and subtraction. Teach it alongside number lines to show | -4 | = 4. Applications like speed or debt reinforce its role without signs, building toward rational numbers.

Why use active learning for integer operations?

Active approaches make abstract signs concrete through movement and manipulatives. Students physically act out additions on floors or pair counters, discovering rules independently. Group justifications during games address misconceptions immediately, boosting retention and confidence per NCCA active methodologies.