Foundations of Mathematical Thinking · 1st Year

Active learning ideas

Combining and Partitioning Numbers

Active learning works because combining and partitioning numbers require physical and visual engagement to build number sense. When students manipulate objects, they connect abstract symbols to concrete actions, making relationships between numbers visible and memorable. This hands-on approach helps solidify foundational concepts before moving to symbolic representations.

• NCCA Curriculum Specifications
• NCCA: Primary - Number
• NCCA: Primary - Algebra

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

Activity 01

• Problem-Based Learning
• 20 min · Pairs

Pairs Activity: Join the Sets

Provide each pair with two bowls of counters in different colors and number cards from 1 to 5. Partners select cards, join the sets, and count the total together before recording it on a mat. Switch roles and repeat with new cards to explore different totals.

Compare how addition and subtraction are like two sides of the same coin.

Facilitation Tip

During the Pairs Activity: Join the Sets, circulate to ensure pairs are verbalizing the total aloud while moving the counters together.

What to look for

Provide students with a set of 12 counters. Ask them to first combine a group of 5 counters with a group of 7 counters and write the total. Then, ask them to partition the set of 12 counters into two groups and write the two parts.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 02

• Problem-Based Learning
• 30 min · Small Groups

Small Groups: Partitioning Baskets

Give groups baskets with 6-10 objects like blocks or buttons. One student partitions into two groups based on a target part, such as make four, while others verify by recombining and counting. Groups share strategies and draw their partitions.

Analyze what happens to the total when we change the order of the numbers we are adding.

Facilitation Tip

In Partitioning Baskets, ask each child to demonstrate their partition before recording it on the whiteboard.

What to look for

Present the number sentence 7 + 3 = 10. Ask students: 'If we know 7 + 3 = 10, how can we use this to figure out 10 - 3?' Guide them to explain that subtraction takes apart what addition joined.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 03

• Problem-Based Learning
• 25 min · Whole Class

Whole Class: Fact Family Circle

Form a circle with students holding number cards for a fact family, like 5, 5, 10. Call out equations such as five plus five, and students arrange to show it; reverse for subtraction. Rotate cards to test order changes.

Explain how we can use a known fact like 5 plus 5 to solve 5 plus 6.

Facilitation Tip

For the Fact Family Circle, model how to rotate the numbers to show all four sentences while keeping the total constant.

What to look for

Give each student a card with a number (e.g., 15). Ask them to write two number sentences on the card: one showing how to combine two numbers to make 15, and one showing how to partition 15 into two parts.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Activity 04

• Problem-Based Learning
• 15 min · Individual

Individual: Number Bond Mats

Each student gets a mat with wholes from 5 to 10 and dry-erase markers. They draw bonds for given parts or find missing parts using counters, then write matching sentences. Collect mats to discuss patterns.

Compare how addition and subtraction are like two sides of the same coin.

What to look for

Provide students with a set of 12 counters. Ask them to first combine a group of 5 counters with a group of 7 counters and write the total. Then, ask them to partition the set of 12 counters into two groups and write the two parts.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Teach this topic by prioritizing concrete experiences before abstract symbols. Use consistent language like 'join' and 'partition' to reinforce actions. Avoid rushing to number sentences; instead, let students describe their actions first. Research suggests that frequent opportunities to manipulate materials and discuss relationships strengthens relational understanding and reduces reliance on counting by ones.

Successful learning looks like students confidently joining and separating groups of objects, explaining their actions with clear language, and recognizing the inverse relationship between addition and subtraction. They should use strategies like counting on or bridging to ten, and articulate their thinking with examples.

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Watch Out for These Misconceptions

• During the Pairs Activity: Join the Sets, watch for students who recount the entire group each time they join sets instead of counting on from the first set.

  Prompt them to say the first set aloud, then 'count on' the second set by touching each counter while saying the next numbers. Model this with your own set to reinforce the strategy.

• During the Fact Family Circle, watch for students who rearrange the numbers without recognizing the constant total.

  Have them place the total in the center and physically rotate the addends around it, saying each sentence aloud to connect the actions to the symbols.

• During the Number Bond Mats activity, watch for students who write only one possible partition for a number.

  Ask them to find another way to split the number and draw it on the mat, reinforcing that multiple partitions exist for the same total.

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Methods used in this brief

• Problem-Based Learning