Introduction

Exponents and Order of Operations is an important Grade 6 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with exponents and order of operations.

What Is Exponents and Order of Operations?

Exponents and Order of Operations means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Exponents and Order of Operations

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: Which correctly describes the first step in evaluating \(5 + 4^2 \times 2\)?

Visual Model 1

  • A. Add 5 and 4.
  • B. Multiply \(5 \times 2\).
  • C. Multiply \(4 \times 2\).
  • D. Evaluate \(4^2 = 16\).

Why it works: PEMDAS requires exponents before multiplication. The diagram shows the first step is evaluating \(4^2 = 16\). Then multiply: \(16 \times 2 = 32\). Finally add: \(5 + 32 = 37\).

Answer: \(37\)

Visual Model 2

Question: What is the first step in evaluating \(3^2 + 4 \times 2\)?

Visual Model 2

  • A. Add 3 and 4.
  • B. Evaluate the exponent: \(3^2 = 9\).
  • C. Multiply \(4 \times 2\).
  • D. Multiply \(3 \times 2\).

Why it works: Following PEMDAS, exponents come before multiplication and addition. The diagram shows Step 1 is evaluating \(3^2 = 9\).

Answer: \(9\)

Worked Examples

Example 1

Question: What is the value after Step 1 (parentheses)?

Example 1

  • A. \(5^2 \times 2\)
  • B. \(4 \times 1 \times 2\)
  • C. \(5 + 1 \times 2\)
  • D. \(16 \times 2\)
  1. Inside the parentheses: \(4 + 1 = 5\).
  2. After Step 1, the expression becomes \(5^2 \times 2\), which is now ready for the exponent in Step 2.

Answer: \(5^2 \times 2\)

Example 2

Question: Evaluate: \(50 - 2^3 \times 5\)

Example 2

  • A. \(10\)
  • B. \(20\)
  • C. \(30\)
  • D. \(40\)
  1. Following the diagram: Step 1: Exponent \(2^3 = 8\).
  2. Step 2: Multiply \(8 \times 5 = 40\).
  3. Step 3: Subtract \(50 - 40 = 10\).

Answer: \(10\)

Example 3

Question: What is the correct value of \((3 \times 2)^2\)?

Example 3

  • A. \(12\)
  • B. \(18\)
  • C. \(36\)
  • D. \(72\)
  1. Order of operations starts inside the parentheses: \(3 \times 2 = 6\).
  2. Then apply the exponent to that single number: \(6^2 = 36\).
  3. (Note: writing it as \(3^2 \times 2^2\) first happens to give the same answer for multiplication inside parentheses, but the diagram models the correct PEMDAS path: simplify the parentheses to one number, then square.)

Answer: \(36\)

Real-World Word Problems

Problem 1

Question: A student wrote: "\(2 + 3^2 = 5^2 = 25\)." What is the mistake?

  • A. The exponent should be applied only to the base 3, not the sum.
  • B. Exponents cannot be used with addition.
  • C. The result must be negative.
  • D. There is no mistake.

Why it works: Apply the exponent first: \(3^2 = 9\). Then add: \(2 + 9 = 11\). The student incorrectly added before applying the exponent, treating it as \((2+3)^2\).

Answer: The exponent should be applied only to the base \(3\), not the sum.

Problem 2

Question: A student evaluated \(2 + 3 \times 2^2\) and got \(20\). What was the student's error?

  • A. The student added first instead of applying the exponent first.
  • B. The student multiplied by 2 twice instead of once.
  • C. The student computed \(5^2 = 25\) and forgot to subtract 5.
  • D. The student's answer is correct.

Why it works: Correct order: exponent \(2^2 = 4\), then multiply \(3 \times 4 = 12\), then add \(2 + 12 = 14\). The student likely computed \((2+3) \times 2^2 = 5 \times 4 = 20\), adding before the exponent.

Answer: The student added first instead of applying the exponent first.

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

Evaluate: \(3^2+2\times(5-1)\)

  • A. \(9\)
  • B. \(17\)
  • C. \(25\)
  • D. \(40\)

Question 2

What is \(2^4\) in expanded form?

  • A. \(2+2+2+2\)
  • B. \(2\times4\)
  • C. \(4\times4\)
  • D. \(2\times2\times2\times2\)

Question 3

Evaluate: \(5^3\)

  • A. \(15\)
  • B. \(25\)
  • C. \(125\)
  • D. \(243\)

Question 4

What is \(6^2\)?

  • A. \(12\)
  • B. \(36\)
  • C. \(42\)
  • D. \(62\)

Question 5

Evaluate: \(10^2 \div 5 + 3\)

  • A. \(20\)
  • B. \(27\)
  • C. \(30\)
  • D. \(23\)

Question 6

Which expression equals \(24\)?

  • A. \(2^3 + 4 \times 2\)
  • B. \(3^2 + 6 + 3\)
  • C. \(4^2 + 8\)
  • D. \(5^2 - 2\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(17\)

Work inside the parentheses first: \(5-1=4\). Then exponent: \(3^2=9\). Then multiply: \(2\times4=8\). Finally add: \(9+8=17\).

Question 2

Answer: \(2\times2\times2\times2\)

An exponent shows how many times to multiply the base by itself. \(2^4\) means \(2\) multiplied \(4\) times, which is \(2\times2\times2\times2=16\).

Question 3

Answer: \(125\)

\(5^3 = 5 \times 5 \times 5 = 125\). The exponent 3 means we multiply 5 by itself 3 times (a cube).

Question 4

Answer: \(36\)

\(6^2 = 6 \times 6 = 36\). A square exponent (power of 2) means multiply the base by itself once.

Question 5

Answer: \(23\)

First, compute the exponent: \(10^2 = 100\). Then divide: \(100 \div 5 = 20\). Finally add: \(20 + 3 = 23\).

Question 6

Answer: \(24\)

Evaluate each expression using order of operations. Choice C gives \(4^2+8=16+8=24\).

Timed practice quizzes

Related Quizzes

After studying Exponents and Order of Operations, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 6 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Exponents and Order of Operations becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.