# Order of

## Lesson Title:

Arithmetic: Order of operations in mathematical expressions

## Abstract

This lesson is about evaluating numerical expressions, and it is designed for adult learners who are preparing to take their High School Equivalency tests. This course will help the students evaluate numerical expressions correctly by following the correct order of operations, which includes the four basic arithmetical operations and the use of exponents and grouping symbols (Parentheses, brackets, and curly braces).

## Learner Audience / Primary Users

This lesson was designed for adult learners that are preparing to take their High School Equivalency tests as well as math instructors.

## Educational Use

- Curriculum / Instruction

## College & Career Readiness Standards (CCRS) Alignment

**Level:**Adult Education**Grade Level:**Grade Level C**Subject:**Mathematics**Domain:**Operations and Algebraic Thinking**Standard:**Use grouping symbols and evaluate expressions with these symbols. (5.OA.1)**Domain:**Equations and expressions**Standard Description:**Write and evaluate numerical expressions involving whole-number exponents. (6.EE.1)

## Language

English

## Material Type

- Instructional Material
- Homework and Assignments
- Images and Videos
- Diagnostic, formative and final assessments

## Learning Goals

The purpose of this lesson is for learners to:

- Evaluate a numerical expression by using the correct order of operations.
- Apply the correct order when exponents and grouping symbols are used.
- Simplify answers as much as possible.

## Keywords

- Designers for Learning
- Adult Education
- Mathematics
- Arithmetic
- Pre-algebra
- Exponents
- Grouping symbols
- Numerical expressions
- Order of operations
- PEMDAS

## Time Required for Lesson

60 minutes

## Prior Knowledge

Students should be able to:

- Work with the basic arithmetic operations.
- Use parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols.

## Required Resources

Depending on the resources available, this course can be delivered as a fully online course, a face-to-face course, or a hybrid course.

- Access to an electronic device which includes: computers, smartphones and tablets
- Access to the internet
- Whiteboard and whiteboard markers
- Paper
- Pencil
- Eraser

## Lesson Author & License

- Andres Chan

- License: Creative Commons CC BY 4.0 license

## Learning Objectives

By the end of this lesson, the learner should be able to:

- Evaluate a numerical expression by using the correct order of operations.
- Apply the correct order when exponents and grouping symbols are used.
- Simplify answers as much as possible.

## Lesson Topics

Key topics covered in this lesson include:

- Evaluating numerical expressions
- Order of operations
- Grouping symbols and exponents

## Context Summary

This course will engage students to one of the fundamental skills in arithmetic, the order of operations. By learning the correct order in which a numerical expression should be solved, the students will be able to apply this knowledge in future courses in Algebra, Geometry and Pre-calculus.

## Relevance to Practice

When asked to solve a numeric expression, different people get different answers. This is because some of us don't apply the correct order of operations to solve it. There are specific conventions to be applied to solve numerical problems involving more than two operations, such as multiplication, addition, division, etc. We also need to know the order in which to solve operations that contain grouping symbols (brackets, parentheses, and curly braces) and exponents. This lesson aims to provide students with the knowledge to apply these conventions when solving numerical expressions.

## Key Terms and Concepts

- Numerical expression
- Evaluating numerical expressions
- Pre-algebra
- Grouping symbols
- Exponents
- Order of operations
- PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction)
- Simplifying expressions

## Instructional Strategies and Activities

### Warm-Up

**Time: 1 minutes**

The teacher will give the students 20 seconds to solve the following problem:

In the remaining time, the class will reflect on which was the correct answer and why.

The correct answer is **9**. If the students applied the wrong order of operations, they would probably get answers like: 8, 3, etc. Their wrong answers may also include fractions.

This warm-up will serve as a diagnostic test to see how much the students know about the order of operations.

### Introduction

**Time: 1 minute**

The teacher explains the goals of this lesson, which are:

- Identify which operations have priority when evaluating a numerical expression.
- Apply the order of operations correctly when exponents and grouping symbols are used.
- Simplify their answers as much as possible when evaluating numerical expressions.

### Presentation / Modeling / Demonstration

**Time: 17 minutes**

- The students will watch the following video:

- After watching the video, the students will reflect on the order of operations by analyzing the acronym
**PEMDAS**:**P**arentheses/Brackets,**E**xponents,**M**ultiplication,**D**ivision,**A**ddition, and**S**ubtraction. As them to use the mnemonic "**P**lease**e**xcuse**m**y**d**ear**a**unt**S**ally!"

- The teacher should make sure that the students understand that multiplication and division have the same priority, and when they face a problem with both multiplication and division or division and multiplication one next to the other, they should solve the problem from
. For example:__left to right__**12 / 4 x 3 = 12 / 4 x 3 = 3 x 3 = 9** - The same rule applies when there is an addition and subtraction, one next to the other. You should solve the problem from
. For example:__left to right__**16 - 4 + 12 = 16 - 4 + 12 = 12 + 12 = 24** - First, ask them to evaluate expressions with addition, subtraction, multiplication, and division only.
- Now ask them to solve the expression 6+ 8
^{2}÷ 4. The answer will be 20. This is because exponent (square of 8) is evaluated first. - Give them several similar problems with exponents.
- Now introduce the important rule of dealing with grouping symbols, such as parentheses, brackets, and curly braces. Operations inside the groping symbols are calculated first, even before exponents.
- Show them where they might encounter this in real life. An example:
- There are 2 each of 4 species of ducks on a farm and 9 hens. How many animals in total. Answer: 4 X 2 + 9=17. Now consider a situation where each bird lays 3 eggs each. How would you calculate the total number of eggs? Answer: 3 x (4x2 + 9)=51.
- Ask them to solve expressions with grouping symbols with increasing complexity:
- (2 + 5)(3 + 4)
- [3 +(
**15 + 6**) ÷ 7] × 4 - 2{1 + [4(2+1) + 3]

### Guided Practice

**Time: 5 minutes**

The students will play an interactive game to test their mastery of the order of operations. The game has different levels, and if the student has difficulty getting the right answers, there is an option for hints about PEMDAS.

When a student makes a mistake, the game gives an instant feedback on what was wrong.

**Exploring Order of Operations - Use It! (link to the interactive game)**

### Evaluation

**Time: 10 minutes**

On a piece of paper solve the 12 problems presented in the following quiz. To see the answer, just click the 'Answer' link next to each problem, and a complete step-to-step solution of the problem will be shown on screen.

**Order of Operations QUIZ ****(link to the quiz)**

### Application

**1) Time: 6 minutes**

In order to apply all the concepts learned on this lesson, the student will pick two problems from the worksheet presented below. He/she will solve those two problems on a piece of paper, and will check the answers on the answer key at the end of the worksheet. The answer key includes a detailed step-by-step explanation for each problem.

**2) Time: 20 minutes (Group activity)**

Work in pairs where one student comes up with a real-life problem that requires the other student to construct a numerical expression out of it and them to solve it. For example, on a grocery store trip, one would have to know the order of operations to calculate the total discount when buying a number of items each with a different type of discount.

If time permits, after the exercise, the teacher should discuss a couple of these problems in front of the entire class.

## Part 3: Supplementary Resources & References

### Supplementary Resources

**Khan Academy videos:**

- Introduction to Order of Operations, created by Salman Khan, published at: http://www.oercommons.org/courses/pre-algebra-introduction-to-order-of-operations/view, Creative Commons License, NC, SA.
- Arithmetic and Pre-Algebra: Order of Operations Examples, created by Salman Khan, published at: https://www.oercommons.org/courses/arithmetic-and-pre-algebra-order-of-operations-examples, Creative Commons License, NC, SA

### References

- Rob & Jeremy (Math Antics), title of the video: Math Antics - Order Of Operations, URL: https://www.youtube.com/watch?v=dAgfnK528RA, Published on Apr 16, 2012
- Exploring Order of Operations - Use It!, created by LearnAlberta.ca, published at http://www.oercommons.org/courses/exploring-order-of-operations/view
- Interactive math quiz published at http://www.thegreatmartinicompany.com/operations/order-of-operations.html
- Math-Aids.com, published at http://www.math-aids.com/cgi/pdf_viewer_7.cgi?script_name=order_of_operations.pl&skill=1&type=5&language=0&memo=&answer=1&x=149&y=42

### Attribution Statements

- Exploring Order of Operations - Use It!, created by LearnAlberta.ca, published at http://www.oercommons.org/courses/exploring-order-of-operations/view, NC
- Order of Operations Quiz, created by The Shodor Education Foundation, Inc., published at http://www.oercommons.org/courses/order-of-operations-quiz/view, CC no restrictions.
- Interactive math quiz published at http://www.thegreatmartinicompany.com/operations/order-of-operations.html
- Order of Operations PDF worksheet, generated by Math-Aids.com, published at http://www.math-aids.com/cgi/pdf_viewer_7.cgi?script_name=order_of_operations.pl&skill=1&type=5&language=0&memo=&answer=1&x=149&y=42