Foundations of Mathematical Thinking · 1st Class

Active learning ideas

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.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Strand 3: Number - N.1.1
• NCCA: Junior Cycle - Strand 3: Number - N.1.2

• 15–30 min
• Pairs → Whole Class
• 3 activities

Activity 01

• 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.

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.

What to look for

Present students with a collection of 15 counters. Ask them to show two different ways to partition the 15 counters into two groups and record the addition sentence for each. For example, 10 + 5 = 15 and 8 + 7 = 15.

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

Activity 02

• 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.

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.

What to look for

Give each student a card with a number sentence, such as '6 + 4 = ?'. Ask them to solve the problem using a number line drawn on the back of the card and write the sum. Then, ask them to write one sentence explaining what happens to the total when you add more objects to a group.

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

Activity 03

• 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.

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 ___.'

What to look for

Pose the question: 'Can you show that adding 5 and 3 gives the same answer as adding 3 and 5?' Allow students to use manipulatives or draw pictures to demonstrate their thinking. Facilitate a class discussion where students share their findings and explain the concept of the commutative property.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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Methods used in this brief

• Inquiry Circle
• Role Play
• Think-Pair-Share