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Teacher Guide: Order of Operations

Learning Objectives

Students will…

• Evaluate numerical expressions using the order of operations.

• Understand how parentheses can be used to change the order of operations.

Vocabulary

expression, order of operations

Lesson Overview

In the Order of Operations Gizmo™, students choose the correct order in which to evaluate numerical expressions. The Gizmo itself provides feedback for incorrect choices.

The first two problems in the Gizmo will always be the same. After those, the problems are randomized.

The Student Exploration sheet contains one activity.

Suggested Lesson Sequence

1. Pre-Gizmo activity ([pic] 5 – 10 minutes)

Review the steps in the order of operations. Then ask students to use the order of operations to evaluate the following expressions:

2 + 3 ( 5 [17] 7 – 2 ( 3 [1] 8 + 4 ÷ 2 [10] 2 ( 5 + 4 ÷ 2 [12] (10 – 4) ÷ 3 [2]

2. Prior to using the Gizmo ([pic] 10 – 15 minutes)

Before students are at the computers, pass out the Student Exploration sheets and ask students to complete the Prior Knowledge Questions. Discuss student answers as a class. Afterwards, if possible, use a projector to introduce the Gizmo and demonstrate its basic operations. Show students how to take snapshots in the Gizmo and paste the images into a blank document.

3. Gizmo activity ([pic] 15 – 20 minutes)

Assign students to computers, individually or in pairs. Have students work through the Student Exploration (SE) sheet, using the Gizmo. (Or, you can use a projector and do the SE as a teacher-led activity.) Either way, we recommend doing page 1 of the SE (Prior Knowledge Questions and Gizmo Warm-up) plus one or both pages of the activity.

*ELL Adaptation* – If you have ELL students in your class, you may need to focus more explicitly on vocabulary. Focus in particular on multiple-meaning words used in this lesson, such as “operation,” “order,” “sign,” “higher,” and “perform.”

4. Follow-up activity ([pic] 5 – 10 minutes)

Have students use calculators and the expression 5 ( 3 – 8 ÷ 4 + 22 to solve the following problems:

• Type in the whole expression, and hit the ENTER key. What is the answer? [17]

• Now, use the order of operations to evaluate the expression. How does this answer compare to the calculator answer? [Both answers are the same.]

• What do you think will happen if you type in one operation at a time and then hit the ENTER key? [The answer won’t be the same.] Why? [Hitting ENTER makes the calculator perform that operation right away. It’s essentially like putting parentheses around each part.] Verify this with your calculators.

Mathematical Background

The order of operations is a set of rules used to clarify which operations to perform first. The acronym PEMDAS describes the four rules:

1. Parentheses – Evaluate expressions inside parentheses or grouping symbols.

2. Exponents – Evaluate powers.

3. Multiply, Divide – Multiply and divide in order from left to right.

4. Add, Subtract – Add and subtract in order from left to right.

Without these rules, many expressions would have more than one possible value. For example, the expression 20 – 15 ÷ 5 equals 17. The order of operations makes it so because you do the division first.

But, if you do the subtraction first and then evaluate the expression from left to right, you’ll get an incorrect answer of 1. However, parentheses or grouping symbols can be used to “change” the order. If parentheses are placed around 20 – 15 in the expression above, then the answer is 1.

Let’s say you buy a pair of jeans for $25 and a sweater for $13. The sales tax is 8%. If you use the expression $25 + $13 ( 0.08 to find the amount of sales tax, the tax will be $26.04! But, if you use parentheses, then the tax is ($25 + $13) ( 0.08 = $3.04, which is correct.

Other commonly used grouping symbols are brackets and the fraction bar. Brackets like [ ] are used when grouping symbols are nested as in ([3 – 2] ( 4) + 5. Start with the innermost set of brackets to evaluate this expression.

For fractions such as [pic](which essentially means (5 + 32) ÷ (9 – 2)), evaluate the numerator and then the denominator. Then, simplify the fraction if possible.

Selected Web Resources

Order of operations video (Khan Academy):

Order of operations game:

Order of operations lesson:

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[pic]

[pic]

[pic]

[pic]

[pic]

20 – 15 ÷ 5 = 20 – 3

= 17

(20 – 15) ÷ 5 = 5 ÷ 5

= 1

([3 – 2] • 4) + 5 = (1 • 4) + 5

= 4 + 5

= 9

[pic] = [pic]

= 2

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