Mathematics · 2nd Grade · Algebraic Thinking: Patterns and Equations · Weeks 19-27

Writing Equations for Even and Odd

Students write an equation to express an even number as a sum of two equal addends.

• Common Core State Standards
• CCSS.Math.Content.2.OA.C.3

About This Topic

Foundations of multiplication begin with the study of rectangular arrays. In second grade, students use arrays with up to 5 rows and 5 columns to visualize repeated addition. They learn that the total number of objects can be found by adding the number in each row or the number in each column. This spatial arrangement helps students move away from counting by ones toward more efficient group-counting strategies.

This topic aligns with CCSS standards for using addition to find the total number of objects arranged in rectangular arrays and writing equations to express the total as a sum of equal addends. It bridges the gap between simple addition and the concept of 'groups of' that defines multiplication. Students grasp this concept faster through structured discussion and peer explanation as they compare how they 'see' the same array in different ways.

Key Questions

1. Construct an equation to demonstrate that any even number can be formed by adding two identical numbers.
2. Analyze the relationship between an even number and its two equal addends.
3. Predict what happens when you try to express an odd number as a sum of two equal addends.

Learning Objectives

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

Before You Start

Addition Facts to 20

Why: Students need to be fluent with basic addition facts to easily identify pairs of addends that form a sum.

Identifying Even and Odd Numbers

Why: Understanding the definition of even and odd numbers is fundamental to writing equations that represent them.

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.

Watch Out for These Misconceptions

Common Misconception

Confusing rows (horizontal) with columns (vertical).

What to Teach Instead

This is a common vocabulary error. Use physical cues: 'Rows go across like a rowboat' and 'Columns go up and down like the columns on a tall building.' Have students trace the lines with their fingers while saying the words.

Common Misconception

Thinking that rotating an array changes the total number of objects.

What to Teach Instead

Students may think 3 rows of 5 is different from 5 rows of 3. Use a 'Turn and Talk' activity with physical tiles where they rotate their own array to see that the quantity is conserved even when the orientation changes.

Active Learning Ideas

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.

30 min·Small Groups

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.

25 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).

15 min·Pairs

Real-World Connections

• Bakers often divide batches of cookies into two equal trays, writing an equation like 12 cookies + 12 cookies = 24 cookies to represent the total.
• When setting up chairs for a school assembly, organizers might arrange them in rows with an equal number in each row, such as 10 chairs + 10 chairs = 20 chairs, to ensure fair spacing.

Assessment Ideas

Exit Ticket

Provide students with the number 14. Ask them to write an equation showing 14 as the sum of two equal addends. Then, ask them to write one sentence explaining why 15 cannot be written as the sum of two equal addends.

Quick Check

Display a collection of 10 objects. Ask students to draw two equal groups and write the equation that represents the total. Then, display 7 objects and ask them to explain why they cannot make two equal groups.

Discussion Prompt

Pose the question: 'How can we use addition to show if a number is even or odd?' Guide students to share their equations for even numbers and their reasoning for odd numbers, focusing on the concept of equal addends.

Frequently Asked Questions

What are the best hands-on strategies for teaching arrays?

Using physical manipulatives like counters, square tiles, or even snacks (like crackers) allows students to build and break apart arrays. Active learning strategies like 'Array Architects' encourage them to experiment with different configurations, which builds a deeper understanding of factors and products.

Why do we teach arrays before multiplication facts?

Arrays provide a visual model for what multiplication actually represents: equal groups. Without this foundation, multiplication becomes a matter of rote memorization rather than logical reasoning.

How do arrays help with repeated addition?

An array organizes objects so that the 'groups' are obvious. Instead of seeing 12 random dots, a student sees 3 groups of 4. This makes it much easier to write and solve the equation 4 + 4 + 4 = 12.

What is a 'rectangular' array?

A rectangular array is an arrangement of objects in rows and columns where every row has the same number of objects and every column has the same number of objects, forming a rectangle shape.