Addition of Integers and Rational Numbers

Active learning works for this topic because combining and partitioning numbers are physical actions. When students manipulate objects, they see that addition and subtraction are not just symbols on a page but actions that change quantities. Moving from counting all objects to breaking totals apart builds the flexibility students need to solve problems in multiple ways.

• 1st Class
• Foundations of Mathematical Thinking
• 3 activities
• 15 min–30 min

Learning Objectives

1. Calculate the sum of two positive integers up to 20 using manipulatives and number lines.
2. Compare the results of adding numbers in different orders to demonstrate the commutative property of addition.
3. Identify and explain the part-part-whole relationship in addition problems involving numbers up to 20.
4. Represent addition problems using concrete objects and pictorial models.

Inquiry Circle

⏱ 30 min·Small Groups

Inquiry Circle: The Part-Part-Whole Lab

Groups are given a 'whole' number (e.g., 12) and a pile of cubes. They must find as many ways as possible to split the 12 cubes into two hoops. They record each combination as a number sentence to see the patterns.

Prepare & details

What happens to the total when you add more objects to a group?

Facilitation Tip: During The Part-Part-Whole Lab, circulate and ask guiding questions like, 'How did you decide which group to split first?' to encourage deeper thinking.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Role Play

⏱ 20 min·Small Groups

Role Play: The Number Shop

One student is the 'Shopkeeper' with a set number of items (the whole). A 'Customer' buys some (a part), and the class must figure out how many are left (the other part). This makes the abstract concept of subtraction a concrete social interaction.

Prepare & details

How can you use a number line to help you add two numbers together?

Facilitation Tip: In The Number Shop role play, model how to use language such as 'I have 8 apples. I sell 3. How many are left?' to connect actions to number sentences.

Setup: Open space or rearranged desks for scenario staging

Materials: Character cards with backstory and goals, Scenario briefing sheet

• Apply
• Analyze
• Evaluate
• Social Awareness
• Self-Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: Story Sums

The teacher provides a simple equation like 5 + 3 = 8. Partners must come up with a real-life story to match it (e.g., 5 birds on a fence, 3 more fly down). They share their stories to see how the same math applies to different situations.

Prepare & details

Can you show that adding 5 and 3 gives the same answer as adding 3 and 5?

Facilitation Tip: For Story Sums, provide sentence starters to support students who struggle, such as 'There were ___ birds on the fence. ___ more landed. Now there are ___.'

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Experienced teachers know that students need repeated exposure to the idea that addition and subtraction are related. Avoid teaching subtraction as only 'taking away.' Instead, use stories and objects to show subtraction as 'finding the difference' or 'comparing.' Research suggests that students who can explain their thinking using multiple representations develop stronger number sense and problem-solving skills.

What to Expect

Students should show confidence in using objects, drawings, or symbols to explain how numbers can be combined and separated. They should recognize that problems can be solved in different ways and explain why their methods make sense. Participation in discussions and role play indicates engagement with the concept of inverse operations.

Watch Out for These Misconceptions

Common Misconception

During The Part-Part-Whole Lab, watch for students assuming subtraction only means 'taking away.'

What to Teach Instead

Use the tower of blocks in the lab to show how to find the difference between two groups. Ask, 'How many more blocks are in this tower?' This redirects students to see subtraction as a comparison, not just removal.

Common Misconception

During The Number Shop role play, watch for students ignoring the order of numbers in subtraction.

What to Teach Instead

Give students a scenario like 'You have 2 apples and give away 5.' Ask them to act it out with props. The physical impossibility of giving away more than you have makes the need for correct order clear.

Assessment Ideas

Quick Check

After The Part-Part-Whole Lab, present students with 15 counters and ask them to show two different partitions and record the addition sentences. Collect these to check their understanding of flexible grouping.

Exit Ticket

After The Number Shop role play, give each student a card with a number sentence like '8 + 4 = ?'. Ask them to solve it using the number line on the back and write a sentence explaining what happens when you add more to a group.

Discussion Prompt

During Story Sums, pose the question, 'Can you show that adding 6 and 2 gives the same answer as adding 2 and 6?' Allow students to use manipulatives or drawings to demonstrate their thinking, then facilitate a class discussion about the commutative property.

Extensions & Scaffolding

• Challenge students to create their own story problems using a set of counters or drawings, then trade with a partner to solve.
• Scaffolding: Provide number lines or ten-frames for students who struggle to visualize the actions of adding or subtracting.
• Deeper exploration: Ask students to explain why 5 + 3 equals 3 + 5 using drawings or objects, then discuss how this relates to the commutative property.

Key Vocabulary

Addend	A number that is added to another number in an addition problem. For example, in 5 + 3 = 8, both 5 and 3 are addends.
Sum	The result when two or more numbers are added together. In 5 + 3 = 8, the sum is 8.
Number Line	A straight line marked with a series of numbers at intervals, used to visualize mathematical operations like addition.
Part-Part-Whole	A model that shows how two smaller numbers (parts) combine to make a larger number (whole) through addition.