Core Curriculum Objective(s): The student will evaluate, demonstrate, and apply order of operations (including the use of grouping symbols). 6NR3-2,
II.A.2.
Overview: The student will complete activities that will demonstrate the various
mnemonic devices that can be used to remember the correct order of operations when interpreting a mathematical expression.
Focus/Essential Questions:
Time Frame: One to three 50-minute class periods
Resources/Materials:
Textbook, Mathematics: Applications and Connections, Course 1 by Glencoe
Websites:
http://forum.swarthmore.edu/dr.math/faq/faq.order.operations.html
http://www.blarg.net/~math/order.html
http://www.funbrain.com/algebra/index.html (Simple game to show order of operations)
http://www.aplusmath.com/cgi-bin/Worksheets/Order_Of_Operations
http://amby.com/educate/ord-op/ (Tutorial and Pretest for Order of Operations)
http://www.math.utep.edu/sosmath/
http://www.gcse.com/Maths/Easy/bd1.htm (BODMAS - Tutorial) Excellent for final review or pretest (allows the students to self test)
http://www.eduplace.com/math/mathsteps/4/a/index.html Basic information presented on Grade 4 Level on Order of Operations
http://www.algebrahelp.com/simpoopspg2.htm Good instruction done step by step for order of operations
Culminating Assessment: The student will complete activities that will demonstrate the various pneumonic devices that can be used to remember the correct order of operations when interpreting a mathematical expression.
| Order of Operations Rubric |
| Descriptor |
Met Requirement |
Partially Met Requirement |
Did Not Meet
requirement |
| Student participates in all activities. |
20 |
10 |
0 |
| Computations are completed neatly and accurately. |
20 |
10 |
0 |
| Student completes pneumonic device with all steps included. |
20 |
10 |
0 |
| Work is completed in a timely manner. |
20 |
10 |
0 |
|
Total points earned |
|
Instructional Activities:
1. The teacher will begin the lesson by asking the FOCUS/ESSENTIAL QUESTIONS to focus the students' thinking on the objective.
2. Students should be introduced to pneumonic devices that will help them remember the steps taken when they are interpreting an expression. An optional step in this process is to divide students up into groups of three students and let them come up with a pneumonic device to show to the class. The teacher would monitor the groups making sure the students included all the steps in their pneumonic device. The groups then could then present their pneumonic device to the class. Appendices A & B should be used here to help introduce students to pneumonic devices.
3. Next, the teacher will utilize EXAMPLES A and B to introduce Order of Operations lesson to the students. These can be made into transparencies, or the teacher may choose her own examples.
4. Next, have students complete Order of Operations Worksheet 1. Review the answers by asking for volunteers to work the problems on the overhead. The teacher should monitor all responses making corrections wherever needed.
5. The teacher can download the worksheet from the following site to use with the class as an additional practice if more than one day is to be used to work on this objective.
http://www.aplusmath.com/cgi-in/Worksheets/Order_Of_Operations
The lesson can be extended with some of the Websites listed under Materials/Resources.
http://www.gcse.com/Maths/Easy/bd1.htm is good for pretest or posttest.
http://www.funbrain.com/algebra/index.html is a game students can use to review order of operations.
|
Order
of Operations
1.
Parenthesis ( ) and Brackets[
] from
the inside out.
2.
Exponents of numbers or parenthesis.
3.
Multiplication and Division in the order they
appear.
4.
Addition and Subtraction in the order they
appear.
|
EXAMPLE A
Below is the first expression that we will be simplifying:
4
+ (6 - 4) * 22
To simplify this expression, use the Order of Operations reference above. The first step in the Order of Operations is to simplify parenthesis and brackets from the inside out. Remember to use the Order of Operations when simplifying the inside of the parenthesis. In this problem there is only one operation inside the parenthesis 6 - 4. All that has to be done is subtract 4 from 6. The answer is shown below.
4
+ (2) * 22
The next step to simplify exponents. 22 becomes 4. This is shown below.
2
+ (2) * 4
Next, simplify multiplication and division in the order that they appear. There is no division, only multiplication. Multiply (2) and 4:
2
+ 8
The final step is to simplify addition and subtraction (combine like terms). The answer is...
10
EXAMPLE 2
Let's continue with the lesson. Look at the following mathematical expression:
6 + [5 (6 - 1) / 5]
Again this expression contains an operation (6 - 1) within parenthesis, but this time they are enclosed in brackets as well. We have to subtract 1 from 6. The expression now looks like this...
6 + [5 (5) / 5]
We now have to complete the operations within the brackets. According to the table from the previous example, we must complete multiplication and division from left to right as they appear. That means we have to multiple 5 (5). The expression now looks like this:
6 + [(25) / 5]
We will continue with the operation within the brackets. We have to divide 25 by 5 with the following results:
6 + [5]
Now, we can complete the order of operations by adding 6 and 5.
The answer is...
11
|
Order of Operations Worksheet 1
1. Parenthesis ( ) and Brackets[ ] from the inside out.
2. Exponents of numbers or parenthesis.
3. Multiplication and Division in the order they appear.
4. Addition and Subtraction in the order they appear. |
Complete the following set of expressions using the information from above to complete your work.
| 1. 2 * 3 + (5 -2) = |
2. 4 + 2 (3 - 1) = |
| 3. 6 - 2 + 3 (4 + 2) = |
4. 5 * 3 + 4 (3 - 1) = |
| 5. 7 + (6 - 1) - (4 -1) = |
6. 8 - 2 + (7 - 2) + 6 = |
| 7. 6 + 2 - (4 * 1) = |
8. 5 - 3 + 6 - (2 + 1) = |
Order of Operations Worksheet 1 Key
Complete the following set of expressions using the information from above to complete your work.
1. 2 * 3 + (5 -2) =
2 * 3 + (3) =
6 + 3 =
9 |
2. 4 + 2 (3 - 1) =
4 + 2 (2) =
4 + 4 =
8 |
3. 6 - 2 + 3 (4 + 2) =
6 - 2 + 3 (6) =
6 - 2 + 18 =
4 + 18 =
22 |
4. 5 * 3 + 4 (3 - 1) =
5 * 3 + 4 (2) =
15 + 8 =
23 |
5. 7 + (6 - 1) - (4 -1) =
7 + (5) - (3) =
12 - 3 =
9 |
6. 8 - 2 + (7 - 2) + 6 =
8 - 2 + (5) + 6 =
6 + 5 + 6 =
11 + 6 =
17 |
7. 6 + 2 - (4 * 1) =
6 + 2 - (4) =
8 - (4) =
4 |
8. 5 - 3 + 6 - (2 + 1) =
5 - 3 + 6 - (3) =
2 + 6 - (3) =
8 - (3) =
5 |
Appendix A
PNEUMONIC DEVICES
PEMDAS
(You might remember this as "Please excuse my dear Aunt Sally.")
1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction
This means that you should do what is possible within parentheses first, then exponents, then multiplication and division (from left to right), and then addition and subtraction (from left to right).
Some people are taught to remember BEDMAS:
1. Brackets
Exponents
Division and Multiplication, left to right
Addition and Subtraction, left to right
Here are two examples:
3 + 5 x 7 = ?
3 + 5 x 7 = 3 + 35 = 38
(1 + 3) x (8 - 4) = ?
(1 + 3) x (8 - 4) = 4 x 4 = 16
Calculations based on the Order of Operations
The order of operations are rules that govern which mathematical operations are done first.
-
Do operations in parentheses and other grouping symbols first. If there are grouping symbols within other grouping symbols do the innermost first.
-
Do multiplication and division operations from left to right.
-
Do addition and subtraction operations from left to right
Example:
2+3*(4+(6*3-8))*2
2+3*(4+(18-8))*2
2+3*(4+10)*2
2+3*14*2
2+42*2
2+84
86
APPENDIX B - PNEUMONIC DEVICE CHART
|
Pleasant
Dreams My Dear Aunt Susie
Another Pneumonic Device for Order of Operations
A Variation on PEMDAS |
| 1'st |
PARENTHESES
( )
[ ] |
Grouping symbols are always done from the innermost set outward. |
| 2'nd |
EXPONENTS |
Remember applying an exponent is different than just multiplying.
EX. 34 means
3 x 3 x 3 x 3 = 81 |
| 3'rd |
MULTIPLY |
These operations are done together in the order they appear from left to right because they have the SAME IMPORTANCE. |
| DIVIDE |
| 4'th |
ADD |
These two operations are of the SAME IMPORTANCE. These are done in the order that they appear from left to right. |
| SUBTRACT |
|
