Order of operations and "bubbles"

Or you could call it a "cloud" or "balloon", instead of a "bubble". Anyhow, the idea is simple: in a given math problem with many operations, have the child encircle in a bubble/cloud/balloon the operation to be done first. He/she can use colored pencils or crayons for more fun.

I just had this idea when my 2nd daughter was studying order of operations in addition and subtraction... that is, problems with many additions and subtractions, including parenthesis, things like

100 + 20 − (50 + 10)

or
20 − 8 − 6    vs.   20 − (8 − 6)

(Those are found in Math Mammoth grade 3 curriculum.)


So what we did was write some of the problems from the book on the whiteboard and she "bubbled" or drew bubbles around the operation to be done first. Well, her bubbles look like ovals, but she was thinking of them as bubbles--and that made it fun for her!

Click to enlarge

Then we did another fun thing, which is that she made me a difficult math problem with lots of additions and subtractions. It's seen in the bottom on the board. Then I solved it...  and the answer turned out negative! Making problems for mom is always fun for little ones.

The same idea will of course work if you are dealing with multiplication and division as well.

Order of operations and "bubbles"

Or you could call it a "cloud" or "balloon", instead of a "bubble". Anyhow, the idea is simple: in a given math problem with many operations, have the child encircle in a bubble/cloud/balloon the operation to be done first. He/she can use colored pencils or crayons for more fun.

I just had this idea when my 2nd daughter was studying order of operations in addition and subtraction... that is, problems with many additions and subtractions, including parenthesis, things like

100 + 20 − (50 + 10)

or
20 − 8 − 6    vs.   20 − (8 − 6)

(Those are found in Math Mammoth grade 3 curriculum.)


So what we did was write some of the problems from the book on the whiteboard and she "bubbled" or drew bubbles around the operation to be done first. Well, her bubbles look like ovals, but she was thinking of them as bubbles--and that made it fun for her!

Click to enlarge

Then we did another fun thing, which is that she made me a difficult math problem with lots of additions and subtractions. It's seen in the bottom on the board. Then I solved it...  and the answer turned out negative! Making problems for mom is always fun for little ones.

The same idea will of course work if you are dealing with multiplication and division as well.

Games for order of operations

Photo courtesy of minibe09

If your student or child needs more practice with order of operations, here are some online games for that. These shouldn't include integers, but just the four operations (addition, subtraction, multiplication, division) and parenthesis.

Most kids just love using games and interactive quizzes for practice instead of worksheets. Have fun!


Choose Math Operation
Choose the mathematical operation(s) so that the number sentence is true. Practice the role of zero and one in basic operations or operations with negative numbers. This helps develop number sense and logical thinking.
www.homeschoolmath.net/operation-game.php


Division and Order of operations and

Division and Addition - Order of Operations
Two mystery picture games.
www.dositey.com/2008/math/m/mystery2MD.htm
www.dositey.com/2008/math/m/mystery2AD.htm


Order of Operations Quiz
A 10-question online quiz that includes two different operations and possibly parenthesis in each question. You can also modify the quiz parameters yourself.
www.thatquiz.org/tq-1/?-j8f-la



The Order of Operations Millionaire
Answer multiple-choice questions that have to do with the order of operations, and win a million. It can be played alone or in two teams.
www.math-play.com/Order-of-Operations-Millionaire/order-of-operations-millionaire.html


Exploring Order of Operations (Object Interactive)
The program shows an expression, and you click on the correct operation (either +, —, ×, ÷ or exponent) to be done first. The program then solves that operation, and you click on the next operation to be performed, etc., until it is solved. Lastly, the resource includes a game where you click on the falling blocks in the order that order of operations would dictate.
www.learnalberta.ca/content/mejhm/html/object_interactives/order_of_operations/use_it.html


Order of Operations Practice
A simple online quiz of 10 questions that use parenthesis and the four operations.
www.onlinemathlearning.com/order-of-operations-practice.html


Quick Calculate
Practice your arithmetic of all four operations plus order of operations.
themathgames.com/arithmetic-games/addition-subtraction-multiplication-division/quick-calculate-game.php

Games for order of operations

Photo courtesy of minibe09

If your student or child needs more practice with order of operations, here are some online games for that. These shouldn't include integers, but just the four operations (addition, subtraction, multiplication, division) and parenthesis.

Most kids just love using games and interactive quizzes for practice instead of worksheets. Have fun!


Choose Math Operation
Choose the mathematical operation(s) so that the number sentence is true. Practice the role of zero and one in basic operations or operations with negative numbers. This helps develop number sense and logical thinking.
www.homeschoolmath.net/operation-game.php


Division and Order of operations and

Division and Addition - Order of Operations
Two mystery picture games.
www.dositey.com/2008/math/m/mystery2MD.htm
www.dositey.com/2008/math/m/mystery2AD.htm


Order of Operations Quiz
A 10-question online quiz that includes two different operations and possibly parenthesis in each question. You can also modify the quiz parameters yourself.
www.thatquiz.org/tq-1/?-j8f-la



The Order of Operations Millionaire
Answer multiple-choice questions that have to do with the order of operations, and win a million. It can be played alone or in two teams.
www.math-play.com/Order-of-Operations-Millionaire/order-of-operations-millionaire.html


Exploring Order of Operations (Object Interactive)
The program shows an expression, and you click on the correct operation (either +, —, ×, ÷ or exponent) to be done first. The program then solves that operation, and you click on the next operation to be performed, etc., until it is solved. Lastly, the resource includes a game where you click on the falling blocks in the order that order of operations would dictate.
www.learnalberta.ca/content/mejhm/html/object_interactives/order_of_operations/use_it.html


Order of Operations Practice
A simple online quiz of 10 questions that use parenthesis and the four operations.
www.onlinemathlearning.com/order-of-operations-practice.html


Quick Calculate
Practice your arithmetic of all four operations plus order of operations.
themathgames.com/arithmetic-games/addition-subtraction-multiplication-division/quick-calculate-game.php

Order of operations / PEMDAS

PEMDAS does not cover matrices...
Photo courtesy of Stuartpilbrow

Someone asked me recently...

Could you please share with me your opinion of the "Please Excuse My Dear Aunt Sally" simplifying expressions. Any feedback could you give will be appreciated. Thank you.

This "PEMDAS" rule is a mnemonic for order of operations:

Please = Parenthesis
Excuse = Exponents
My = Multiplication
Dear = Division
Aunt = Addition
Sally = Subtraction

There's nothing wrong with using a mnemonic to remember the order of operations. However, one has to bear in mind that

1. This rule is not all-inclusive. It omits for example square roots. But the rule is good for all elementary grades. (Square roots would be on the same level or rank with exponents, by the way.)
    
2. The rule doesn't spell out the fact that in reality multiplication and division are "on the same level" or rank. This means that if you have several multiplications and divisions, you do them from left to right, and not "multiplication first, then division".
   
   For example: 60 ÷ 5 × 4. You go from left to right, and first do 60 ÷ 5 = 12. Then you multiply 12 × 4 = 48.
   
   If you want 5 × 4 to be done first, it needs to be in parenthesis: 60 ÷ (5 × 4). Here, first do 5 × 4 = 20, and then 60 ÷ 20 = 3.
   
   Similarly, addition and subtraction are on the same level: if both exist in an expression, they are to be done from left to right.
   
   An example: Simplify the expression 2 × 5 − 6 + 8.
   
   1: Multiply 2 × 5 = 10.
       The expression is now 10 − 6 + 8.
   
   2. Subtract. 10 − 6 = 4.
       The expression is now 4 + 8.
   
   3. Add. 4 + 8 = 12.
   
   So, perhaps it's more illustrative to lay out the PEMDAS rule like this:
   
   Please = Parenthesis
   Excuse = Exponents
   My Dear = Multiplication & Division
   Aunt Sally = Addition & Subtraction
   
   ...and say it with little pauses at the commas: Please, Excuse, My Dear, Aunt Sally.
   
   
   
3. Fraction line as a division symbol generates implicit groupings or parenthesis in the fact that anything written on top or bottom of the fraction line is to be done before the division. It's as if the whole numerator and denominator were inside parenthesis.
   
   So...
   
   
   
   

   4 × 32
   --------
   7 − 22
   

   
   means the same as (4 × 3²) ÷ (7 − 2²). Simplifying this is left as an exercise for the reader. The answer is 12.
   

See also a detailed lesson on order of operations at Purplemath and Wikipedia's note on mnemonics.

Order of operations / PEMDAS

PEMDAS does not cover matrices...
Photo courtesy of Stuartpilbrow

Someone asked me recently...

Could you please share with me your opinion of the "Please Excuse My Dear Aunt Sally" simplifying expressions. Any feedback could you give will be appreciated. Thank you.

This "PEMDAS" rule is a mnemonic for order of operations:

Please = Parenthesis
Excuse = Exponents
My = Multiplication
Dear = Division
Aunt = Addition
Sally = Subtraction

There's nothing wrong with using a mnemonic to remember the order of operations. However, one has to bear in mind that

1. This rule is not all-inclusive. It omits for example square roots. But the rule is good for all elementary grades. (Square roots would be on the same level or rank with exponents, by the way.)
    
2. The rule doesn't spell out the fact that in reality multiplication and division are "on the same level" or rank. This means that if you have several multiplications and divisions, you do them from left to right, and not "multiplication first, then division".
   
   For example: 60 ÷ 5 × 4. You go from left to right, and first do 60 ÷ 5 = 12. Then you multiply 12 × 4 = 48.
   
   If you want 5 × 4 to be done first, it needs to be in parenthesis: 60 ÷ (5 × 4). Here, first do 5 × 4 = 20, and then 60 ÷ 20 = 3.
   
   Similarly, addition and subtraction are on the same level: if both exist in an expression, they are to be done from left to right.
   
   An example: Simplify the expression 2 × 5 − 6 + 8.
   
   1: Multiply 2 × 5 = 10.
       The expression is now 10 − 6 + 8.
   
   2. Subtract. 10 − 6 = 4.
       The expression is now 4 + 8.
   
   3. Add. 4 + 8 = 12.
   
   So, perhaps it's more illustrative to lay out the PEMDAS rule like this:
   
   Please = Parenthesis
   Excuse = Exponents
   My Dear = Multiplication & Division
   Aunt Sally = Addition & Subtraction
   
   ...and say it with little pauses at the commas: Please, Excuse, My Dear, Aunt Sally.
   
   
   
3. Fraction line as a division symbol generates implicit groupings or parenthesis in the fact that anything written on top or bottom of the fraction line is to be done before the division. It's as if the whole numerator and denominator were inside parenthesis.
   
   So...
   
   
   
   

   4 × 32
   --------
   7 − 22
   

   
   means the same as (4 × 3²) ÷ (7 − 2²). Simplifying this is left as an exercise for the reader. The answer is 12.
   

See also a detailed lesson on order of operations at Purplemath and Wikipedia's note on mnemonics.