Representing Difference as Subtraction

Introduction

In this lesson plan, your learner will explore the concept of the "difference as subtraction." This lesson will demonstrate how the difference between two numbers can be represented as subtraction through real-world contexts, building on their prior understanding of subtraction in part-whole models and first-then-now stories.

Before beginning the lesson, your learner should be familiar with part-part-whole bar models and writing subtraction equations.

Key Concepts for Representing Difference as Subtraction

Subtraction can be used to represent different kinds of scenarios. Here is a brief summary of the three types of scenarios your learner will work with.

• Subtraction in Partitioning: Finding the unknown part in a part-part-whole scenario. For example, "If there are 8 apples and 3 are taken away, how many are left? 8 - 3 = 5."
• Subtraction as Reduction: Using subtraction in word problems that follow a sequence such as first-then-now where some of a quantity is taken away. For example, "There were 5 swans on the lake, then 3 flew away. How many swans are left? 5 - 3 = 2."
• Subtraction as Difference: Using subtraction in comparison scenarios. For example, "There are 5 ducks on the pond and 3 ducks on the lake. How many more ducks are on the pond? The difference is 5 - 3 = 2."

Teaching Plan

The following activities will help your learner develop their understanding of difference as subtraction.

Examples and visuals to support the lesson:

Introducing Difference as Subtraction

• Use real-world contexts to show that the difference between two numbers can be represented as subtraction. This is conceptually more complex than partitioning or reduction because the difference refers to an absence.
• For example: "There are eight children and only three pencils. How many more pencils does the teacher need so each child has one pencil?"
• Use pictures and part-part-whole models (such as a bar model) to represent the problem. Explain to your learner that we are partitioning the children into those who have pencils and those who do not.
• Have your learner describe what each number represents: "The 8 represents the number of children. The 3 represents the number of pencils. The 5 represents the difference; it is how many more pencils the teacher needs."
• Link difference to subtraction calculations, and encourage your learner to describe what the equation is showing. For example, "The difference between eight and three can be written as 8 - 3 = 5."

Skill Check
I can use bar models and subtraction equations to show the difference between numbers.

2. Exploring Same Difference with Manipulatives

• Next, reinforce your learner's understanding of difference by identifying situations where the difference is the same.
• Start with concrete manipulatives like Cuisenaire rods. Ask your learner to find pairs of rods with a length difference equal to a smaller rod. For example, "Green and white rods have a length difference of a red rod."
• Move on to exploring other "same difference" rods without assigning numerical values initially. Instead, describe the differences using the unit of a particular rod color.

Skill Check
I can find pairs of rods that have the same length difference.

Extending Same Difference to Numerical Values

• Use multilink cubes to support exploring same differences numerically. For example: Start with four red cubes alongside one blue cube (the difference is three).
• Add one cube of the corresponding color to each set (five red cubes, two blue cubes). Highlight that the "gap" remains the same (the difference is still three).
• Continue adding or subtracting an equal number of cubes from each set, noting the difference remains three.
• Introduce equations alongside the concrete/pictorial representations, emphasizing "same differences." Here are some examples that all have a difference of three: 4 - 1 = 3; 5 - 2 = 3; 6 - 3 = 3.
• Draw attention to the non-commutative property of subtraction. For example, "The difference between six and three is equivalent to the difference between three and six, but 6 - 3 is not equal to 3 - 6."

Skill Check
I can recognize subtraction equations that have the same difference.

Practicing with Subtraction Stories

When your learner is secure in understanding difference as a representation of subtraction, explore subtraction stories representing various structures. You can also have your learner write their own difference stories based on given subtraction calculations. Here are some examples:

• "Five children are playing in the playground. Three are on the swing. How many are on the roundabout?" (partitioning)
• "There were five swans on a lake, then three flew away. How many swans are left?" (reduction/take away)
• "There are five ducks on the pond and three on the lake. How many more ducks are on the pond?" (difference)

Skill Check
I can solve different kinds of subtraction story problems including ones that ask me to find the difference.

Summary

With the activities in this lesson plan, your learner will understand the concept of "difference as subtraction" using real-world contexts, practice with manipulatives, and explore abstract concepts through subtraction stories and symbolic notation. They will build a strong foundation in recognizing and describing the difference between two numbers.

Teaching Plan adapted from NCETM under OGL license v3.