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.

• 1st Year
• Foundations of Mathematical Thinking
• 4 activities
• 15 min–30 min

Learning Objectives

1. Compare the results of joining two sets of objects with the results of partitioning a single set into two parts.
2. Analyze how changing the order of addends affects the sum using concrete objects.
3. Explain the relationship between addition and subtraction as inverse operations using number sentences.
4. Calculate the total when combining sets of objects up to 20.
5. Identify the two parts that make up a whole number up to 20.

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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.

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Quick Check

After the Pairs Activity: Join the Sets, provide each pair with 12 counters. Ask them to combine a group of 5 and 7, then write the total on a mini-whiteboard. Next, ask them to partition the 12 counters into two groups and record both parts to assess their understanding of both operations.

Discussion Prompt

During the Fact Family Circle, present the sentence 7 + 3 = 10 and ask students: 'How can this help us solve 10 - 3?' Listen for explanations that connect the subtraction to the partition of the total set, noting who can articulate the inverse relationship.

Exit Ticket

After the Number Bond Mats activity, give each student a card with the number 15. Ask them to write one combining sentence (e.g., 9 + 6 = 15) and one partitioning sentence (e.g., 15 = 7 + 8) to assess their ability to represent both operations for the same total.

Extensions & Scaffolding

• Challenge pairs to find all possible ways to combine two numbers to make a given total (e.g., 10) using counters, then record their findings on a grid.
• Scaffolding for Partitioning Baskets: Provide pre-partitioned sets in a tray so students can focus on counting the parts rather than dividing.
• Deeper exploration: Introduce missing addend problems during the Number Bond Mats activity, using phrases like 'What goes with 4 to make 9?' to encourage flexible thinking.

Key Vocabulary

Combine	To join two or more groups of objects together to find a total amount.
Partition	To separate a whole group of objects into smaller parts.
Addend	A number that is added to another number in an addition problem.
Sum	The result when two or more numbers are added together.
Difference	The result when one number is subtracted from another number.