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Mathler Order of Operations. When they look at a mathematical equation, it should mean exactly what they do, or they will not be able to discuss it, collaborate efficiently, or even agree on the puzzle’s solutions! They must decide on the best way to carry out operations to have a common understanding.

“4 + 6” is easy! Everybody is familiar with the meaning of numbers; in the first place, “+” means addition, so everyone will realize that adding 4 and 6 means adding 4 and 6, and the result will be 10. There’s only one option to choose from, and there’s no choice as to which one to start with. If an expression is composed of more than one operation, such as in Mathler and HardMathler, it may determine which one is performed first.

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**Example 1**: 20 – 4 + 3

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A. If you subtract the first 20 4 – 20 = 16 after that, add 16 and 3, you’ll get 19 but

B. If you add first 4+3 = 7, then subtract 20 – 7, you’ll get 13!

**Example 2**: 12 + 4 *6

A. If you perform the addition first with 12 + 4 =16, multiply 16 times 6, and you’ll get the number 96.

B. If you perform the multiplication first 4 * 6 = 24 and add 24 + 12, you receive 36.

**Example 3**: 6 / 3 * 2

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A. If you perform the division first 6 3 x 6 = 2, multiply 2 * 2 and get 4. BUT

B. If you multiply 3 times 2 to get 6 and then divide 6 by 6, you’ll get 1.

If you are familiar with the basic procedure, you can have fun and choose whether you prefer A or B as the proper method to assess each situation. (Answers in the discussion below.)

**The “Order of Operations”** for Mathler Puzzles

**In the beginning,**if there is an expression in parentheses, check that.**Second**, examine divisions and multiplications starting from the left.**Third**, examine subtractions and additions by comparing left and right.

Do you feel you have to alter one of your options of what you can do with the examples you have seen? (Last chance before finding the solution!)

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For **Example 2,** the best answer to this is: there are no parentheses. The multiplication occurs at the next grade of precedence, so you must do the multiplication first, and then the addition

12 + 4 * 6

= 12 + 24

= 36

Your author is a part of the team at Arundel Mobile Professionals. Taught adults prealgebra for several years. She observed that many adults had forgotten or were allowed to be misled or even been taught incorrectly (!) about the sequence of operations. This led them to believe that they assumed that all multiplications must be completed before any divisions and additions before any subtractions. This could get out of the way of locating a Mathler solution and make solvers believe that Mathler is causing errors.

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The confusion was created or exacerbated by teaching the sequence of operations with “PEMDAS” (and it’s all very memorable memory aid “Please Excuse My Dear Aunt Sally.”) PEMDAS is described and taught in classes as well as on the internet (including by people you’d expect to be more knowledgeable) as explaining six levels of prioritization that correspond to each letter of “PEMDAS,” as follows:

**INCORRECT Mathler ORDER OF OPERATIONS 6 different levels of prioritization:**

- Then, you must evaluate the expressions using
**the P**arentheses, - Second, analyze
**your E**xponents (not employed for Mahler), ~~Third, evaluate~~~~multiplications,~~~~Fourth, review Divisions~~~~Fifth, review Additions,~~~~Sixth, analyze Subtractions.~~

This misinterpretation of the sequence of operations is commonplace. It is a massive obstacle to students’ success in math courses and STEM-related subjects (perhaps more significant than making it harder for students to figure out Mathler!) If you were taught/taught/remembered it this way, forgive my beloved Aunt Sally and others who have made fun of her name and learn it according to the following.

**Correct Order of OPERATIONS: Four levels of precedence:**

- First, look at expressions using
**The P**arentheses, - Second, analyze
**the E**xponents (not employed by Mahler), - Third, explore
**Third, evaluate**multiplications and**D**ivisions from left to right. - Fourth, examine
**the A**dditions or**S**ubtractions from left to the right.

In the past, before when Aunt Sally came into the picture, it was known as “the order of prioritization,” or the hierarchy of operations. j. If you must use PEMDAS to remember this, you might try writing it as P E MD AS and read it as PLEASE- Excuse-M’Dear-AuntSally.

In **example 1**, you can answer the question with A. There aren’t any parentheses, and there are no divisions or multiplications. Subtractions and additions should be performed from the direction of left to right. Subtractions are left after the acquisition; therefore, do the subtraction before the addition:

20 – 4 + 3

= 16 + 3

= 19

About the title, the reason “Why 3 – 2 + 1 = 2, not 0” is that subtraction and addition are performed from left to right. In other words, the subtraction occurs in the opposite direction of the addition. Therefore, subtraction should be done first! If you believe that adding should be performed first, and the sum of 3 – 2 + 1 is 0, You have the Aunt Sally issue! It’s not your fault. You might have been taught incorrectly due to an insufficient amount of training in textbooks or due to poor explanations by teachers. However, this is your time to make it right!

3 – 2 + 1

= 1 + 1

= 2

In **Example 3.**, the right answer would be A. There are no parentheses. the division and multiplication are on the next stage of precedence and are carried out by alternating left and right. The division follows the multiplication, and so is the division before the multiplication

6 / 3 * 2

= 2 * 2

= 4

## NOTES:

## 1 . Examining the operation order

An excellent way to determine whether you’ve applied an Order of Operations correctly is to evaluate your results against a “scientific calculator” like the TI-83 or the scientific version of the online calculator. The “four-function calculator” (hardware or online) that doesn’t let users input the entire expression before evaluating it won’t be able to work.

Students who aren’t aware of the proper order of operations can pass STEM and math classes using a scientific calculator. TI-83 is the most commonly used calculator that is used for introductory algebra.

- Operation order at the same level as precedence

Subtraction and addition are assessed from left to right at the same time. If you are learning about negative numbers, you are taught that you can write addition or subtraction in the same way as the other by changing the operators and then replacing the operator’s number using its negative. Similar to learning about fractions, you will learn that you can write a multiplication or division in the other by altering the operator and then replacing the number that follows the operator with its “reciprocal.” Examples: Applying brackets around negative numbers and fractions to make them clearer:

- 8 – 3 = 8 + [-3] = 5
- 11 + 7 = 11 – [-7] = 18
- 16 / 4 = 16 * [1/4] = 4
- 6 * 5 = 6 / [1/5] = 30

So :

- 8 3 +. is compared from left to right and has the same amount as 8 + [-3] + 10. Both 15!
- 16/4 * 2 is evaluated from left to right and has the same amount of 16 * 1/4 2

- Mathler Order of Operations for advanced math

Advanced math requires other operations that must be included in the sequence of operations. Mathematics classes typically follow the textbook; however, calculators (for instance) must be precise regarding this. It might be interesting to know the complete “order of operations” for the TI-83 model, the model Texas Instruments calls the EOS TM (Equation Operating System.) If you’ve not taken a lot of algebra, you may not be able to recognize all the symbols — don’t be concerned about that. You will notice that the Mathler operations are in the correct order.

**For the very Mathematically Thinking:** A lot more could be said about the subject of notation and the order of operations. Other Wikipedia entries:

Mathematical Notation (and its ” See also” hyperlinks)

Ordnance of Operations (and its ” See also” links, specifically reverse Polish notation (really!) )