UK K12 GCSE/A-Level: Year 7 KS3 Mathematics - Algebraic Expressions, Collecting Like Terms

Learning Objectives

By the end of this topic, students will be able to:

• Identify and collect like terms in algebraic expressions
• Simplify expressions by combining like terms
• Understand the concept of coefficients and constants in algebraic expressions
• Apply the concept of collecting like terms to solve problems in various contexts
• Explain the reasoning behind collecting like terms in algebraic expressions

Core Concepts

Algebraic expressions are a combination of variables, numbers, and mathematical operations. Like terms are terms that have the same variable(s) raised to the same power. For example, 2x and 5x are like terms because they both contain the variable x raised to the power of 1. Collecting like terms involves combining these terms to simplify the expression.

Imagine you have a collection of boxes, each containing a different number of identical toys. If you have two boxes with 2x toys and one box with 5x toys, you can combine these boxes to get a total of (2x + 5x) = 7x toys. This is similar to collecting like terms in algebraic expressions, where you combine terms with the same variable(s) raised to the same power.

Worked Examples

Example 1: Collecting Like Terms

Simplify the expression: 2x + 3x + 4

To collect like terms, we need to identify the terms with the same variable(s) raised to the same power. In this case, the terms 2x and 3x are like terms because they both contain the variable x raised to the power of 1. We can combine these terms to get:

2x + 3x = 5x

Now we have the expression 5x + 4. Since 4 is a constant term, we cannot combine it with the term 5x. Therefore, the simplified expression is:

5x + 4

Example 2: Collecting Like Terms with Coefficients

Simplify the expression: 2x + 5x - 3x

In this example, we have three terms with the same variable x raised to the power of 1. We can collect these terms by combining their coefficients:

2x + 5x - 3x = (2 + 5 - 3)x = 4x

The simplified expression is 4x.

Common Misconceptions

• Students may think that collecting like terms only involves adding or subtracting the coefficients, without considering the variable(s) raised to the same power.
• Students may forget to check if the terms have the same variable(s) raised to the same power before collecting like terms.
• Students may not understand the concept of coefficients and how they relate to collecting like terms.

Exam Tips

• Make sure to identify the like terms in the expression before collecting them.
• Check if the terms have the same variable(s) raised to the same power before collecting like terms.
• Simplify the expression by combining the coefficients of the like terms.
• Be careful when dealing with negative coefficients and constants.

MCQs with Explanations

MCQ 1: [F]

What is the simplified form of the expression: 2x + 3x + 4?

A) 5x + 4 B) 2x + 3x C) 5x - 4 D) x + 4

Correct answer: A) 5x + 4 Why the distractors fail: * B) 2x + 3x is incorrect because it does not combine the like terms. * C) 5x - 4 is incorrect because it incorrectly combines the constant term with the like terms. * D) x + 4 is incorrect because it does not combine the like terms.

MCQ 2: [H]

Simplify the expression: 2x + 5x - 3x

A) 4x B) 2x + 5x C) 5x - 3x D) x - 3x

Correct answer: A) 4x Why the distractors fail: * B) 2x + 5x is incorrect because it does not combine the like terms. * C) 5x - 3x is incorrect because it does not combine the like terms. * D) x - 3x is incorrect because it incorrectly combines the constant term with the like terms.

MCQ 3: [F]

What is the simplified form of the expression: x + 2x + 3?

A) 3x + 3 B) x + 2x C) 3x + 2 D) x + 3

Correct answer: A) 3x + 3 Why the distractors fail: * B) x + 2x is incorrect because it does not combine the like terms. * C) 3x + 2 is incorrect because it incorrectly combines the constant term with the like terms. * D) x + 3 is incorrect because it does not combine the like terms.

MCQ 4: [H]

Simplify the expression: 2x^2 + 3x^2 - 4x^2

A) x^2 B) 2x^2 + 3x^2 C) 3x^2 - 4x^2 D) x^2 - 4x^2

Correct answer: A) x^2 Why the distractors fail: * B) 2x^2 + 3x^2 is incorrect because it does not combine the like terms. * C) 3x^2 - 4x^2 is incorrect because it does not combine the like terms. * D) x^2 - 4x^2 is incorrect because it incorrectly combines the constant term with the like terms.

MCQ 5: [F]

What is the simplified form of the expression: 2x + 3x - 2

A) 5x - 2 B) 2x + 3x C) 5x + 2 D) x - 2

Correct answer: A) 5x - 2 Why the distractors fail: * B) 2x + 3x is incorrect because it does not combine the like terms. * C) 5x + 2 is incorrect because it incorrectly combines the constant term with the like terms. * D) x - 2 is incorrect because it does not combine the like terms.

Short-answer Questions

1. Simplify the expression: 2x + 3x + 4

(Answer should be 5x + 4)

1. Simplify the expression: 2x^2 + 3x^2 - 4x^2

(Answer should be x^2)

1. Simplify the expression: x + 2x + 3

(Answer should be 3x + 3)

1. Simplify the expression: 2x + 5x - 3x

(Answer should be 4x)

1. Simplify the expression: 2x^2 + 3x^2 - 4x^2

(Answer should be x^2)