Writing Equations for Even and Odd

Active learning works for this topic because students need to physically manipulate objects and visualize groups to move beyond rote counting. Arrays provide a spatial bridge between addition and multiplication, making abstract ideas concrete through hands-on exploration.

• 2nd Grade
• Mathematics
• 3 activities
• 15 min–30 min

Learning Objectives

1. Write an equation demonstrating that any even number can be expressed as the sum of two equal addends.
2. Analyze the relationship between an even number and its two equal addends by comparing different equations.
3. Predict and explain why an odd number cannot be expressed as the sum of two equal addends.
4. Identify pairs of equal addends that sum to a given even number.

Ready-to-Use Activities

Inquiry Circle

⏱ 30 min·Small Groups

Inquiry Circle: Array Architects

Groups are given 20 sticky notes and asked to create as many different rectangular arrays as possible. They must write the repeated addition equation for both the rows and the columns for every array they build.

Prepare & details

Construct an equation to demonstrate that any even number can be formed by adding two identical numbers.

Facilitation Tip: During Array Architects, circulate with a tray of tiles and model how to arrange them into rows while naming each part aloud, e.g., 'I have 3 rows of 5 tiles.'

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

Gallery Walk

⏱ 25 min·Pairs

Gallery Walk: Real World Arrays

The teacher displays photos of real-world arrays (an egg carton, a muffin tin, a window pane). Students walk around in pairs, writing the 'row equation' and 'column equation' for each image on a recording sheet.

Prepare & details

Analyze the relationship between an even number and its two equal addends.

Facilitation Tip: During the Gallery Walk, ask students to photograph one array they find and write its equation on an index card before moving to the next display.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: The Array Flip

The teacher shows an array of 3 rows and 4 columns. Students solve for the total. Then, the teacher rotates the array 90 degrees. Pairs discuss whether the total changed and why the new equation (4+4+4) still equals the old one (3+3+3+3).

Prepare & details

Predict what happens when you try to express an odd number as a sum of two equal addends.

Facilitation Tip: During the Think-Pair-Share for The Array Flip, give each pair two sticky notes to rotate their array and label both orientations with equations before sharing with the class.

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

Teachers should introduce arrays as a way to see equal groups, not just as a grid. Avoid rushing to multiplication; focus on repeated addition first. Research shows that students who physically build and manipulate arrays develop stronger number sense and retain the concept longer. Use consistent language: 'rows go left to right' and 'columns go top to bottom' to prevent confusion.

What to Expect

Successful learning looks like students confidently describing arrays with precise vocabulary, writing correct equations for even and odd totals, and explaining why rotating an array doesn’t change its total. They should also articulate why odd numbers cannot be split into two equal groups.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Array Architects, watch for students mixing up the terms rows and columns.

What to Teach Instead

Pause the group work and have students trace the rows with their fingers while saying, 'Rows go across like a rowboat.' Then, have them trace the columns while saying, 'Columns go up and down like the columns on a tall building.' Repeat this cue before they continue building.

Common Misconception

During Think-Pair-Share: The Array Flip, watch for students thinking that rotating an array changes the total number of objects.

What to Teach Instead

Hand each pair their array and ask them to rotate it 90 degrees. Then, prompt them to write both equations (e.g., 3 rows of 5 and 5 rows of 3) and ask, 'Did the total change?' Guide them to recognize that the quantity stays the same even when the orientation changes.

Assessment Ideas

Exit Ticket

After Collaborative Investigation: Array Architects, provide students with the number 18. Ask them to write two different array equations that represent 18 as the sum of equal addends and explain why 19 cannot be written this way.

Quick Check

During Gallery Walk: Real World Arrays, display a poster with two arrays: one with 8 objects arranged as 2 rows of 4 and another with 9 objects arranged as 3 rows of 3. Ask students to circle the even total and write the equation for it.

Discussion Prompt

After Think-Pair-Share: The Array Flip, pose the question, 'How can we use addition to decide if a number is even or odd?' Have students share their equations for even numbers and explain why odd numbers cannot be split into two equal groups, using their array examples as evidence.

Extensions & Scaffolding

• Challenge: Provide students with 24 tiles and ask them to create as many different array configurations as possible, recording each equation and identifying which configurations represent even or odd totals.
• Scaffolding: For students struggling with vocabulary, provide a sentence frame: 'This array has ____ rows of ____ tiles. The total is ____ because ____ + ____ = ____.'
• Deeper: Invite students to research and present on how arrays are used in real-world design, such as seating charts or garden layouts.

Key Vocabulary

even number	A whole number that can be divided exactly by 2, meaning it has no remainder. Even numbers end in 0, 2, 4, 6, or 8.
odd number	A whole number that cannot be divided exactly by 2, meaning it has a remainder of 1. Odd numbers end in 1, 3, 5, 7, or 9.
addend	One of the numbers that is added together in an addition problem. For example, in 3 + 3 = 6, both 3s are addends.
sum	The result of adding two or more numbers together. In 3 + 3 = 6, 6 is the sum.
equation	A mathematical sentence that shows two expressions are equal, using an equals sign (=). For example, 4 + 4 = 8 is an equation.