Math Problems: Order of Operations

Posted by David Wetzel

Order of Operations

When there is more than one operation involved in a mathematical problem, it must be solved by using the correct order of operations.

When using a calculator, it will perform operations in the order which you enter them. Therefore, you will need to enter the operations in the correct order for the calculator to give you the right answer.

Order of Operations Acronyms

Use the following Acronyms to help remember order of operations:

Please Excuse My Dear Aunt Sallie (Parenthesis, Exponents, Multiply, Divide, Add, Subtract)

or

Pink Elephants Destroy Mice And Snails (Parenthesis, Exponents, Divide, Multiply, Add, Subtract)

Order of Operations Rules

1. Calculations must be done from left to right.

2. Calculations in parenthesis are done first. When there are more than one set of parenthesis, do the inner parenthesis first.

3. Exponents must be done next.

4. Multiply and divide in the order the operations occur.

5. Add and subtract in the order the operations occur.

Order of Operations Examples

12 ÷ 4 + 3²
12 ÷ 4 + 9
3 + 9
12

Rule 3: Exponent first
Rule 4: Multiply or Divide as they appear
Rule 5: Add or Subtract as they appear

20 ÷ (12 - 2) X 3² - 2
20 ÷ 10 X 3² - 2
20 ÷ 10 X 9 - 2
18 - 2
16

Rule 2: Everything in parenthesis first
Rule 3: Exponents
Rule 4: Multiply and Divide as they appear
Rule 5: Add or Subtract as they appear

Math Skills - Kids Math

Posted by David Wetzel

Math Skills!

A common question when teaching students how to multiply positive and negative numbers is “Why is a Negative times a Negative a Positive?”

It is easy to simply say because that is the way it works; however, it is much better to teach students why this happens as opposed to rote memorization.

The following provides examples of explanations for students.

Why is a negative times a negative positive?

A real-world approach modeling the product of two negatives requires a situation involving two quantities, both of which have two directions. Consider delivering mail by a postal deliverer who makes many mistakes. We can receive (+) a check (+) or a bill (-), and if we receive a check or a bill by mistake, then we may have to return (-) a check (+) or a bill (-).

If we receive (+) checks (+), then we have more (+) money.

If we receive (+) bills (-), then we have less (-) money.

If we were erroneously sent checks then we have to return (-) checks (+), and we have less (-) money.

If we were erroneously sent bills, then we have to return (-) bills (-), and we have more (+)money.

A Pattern Approach

Consider the following table showing the products of a positive number multiplied by a second number that is decreasing in value:

3 x 3 = 9

3 x 2 = 6

3 x 1 = 3

Notice that as the second factor decreases by one, the product decreases by three. It is moving down the number line in steps of three. Continuing this pattern gives us:

3 x 0 = 0

3 x -1 = -3

3 x -2 = -6

3 x -3 = -9

Because your brain detects patterns so well, this is a very reasonable way to see that when positive and negative numbers are multiplied, a negative number results.

Now lets take the last multiplication fact and begin decreasing the lead factor by one.

3 x -3 = -9

2 x -3 = -6

1 x -3 = -3

Now the product is advancing up the number line in increments of three. The product is increasing. If we continue this pattern the following table results:

0 x -3 = 0

-1 x -3 = 3

-2 x -3 = 6

-3 x -3 = 9

Since our brain trusts patterns, the student is confident in the resulting conclusion that a negative multiplied by a negative yields a positive product.

A Number Sense Approach

Let’s consider the problem 2 x 3 which we know to be equal to 6. One way to think about negative numbers is as inverses of positive numbers. In fact, every non-zero number has an inverse, and the sum of a number and it’s inverse is zero. Thus 5 + -5 = 0.

That means that we could read the following problem as, “What is the inverse of the product of two and three?”

-2 x 3 = ?

Since 2 x 3 is six, then the inverse of this must be -6. Thus:

-2 x 3 = -6

Now we have demonstrated by example that when a negative is multiplied by a positive, the product is negative. Since the Commutative Property applies to multiplication, we can rewrite the problem this way:

3 x -2 = -6

Next let’s ask ourselves, “What is the inverse of 3 x -2 = -6?” Clearly it would be:

-3 x -2 = ?

Since we know that the answer to the previous problem was -6, the answer to the inverse problem should be the inverse of -6, which is +6. Thus we have a demonstration that a negative multiplied by a negative is positive.

Source

Adapted from “Answering Your Students Why Questions in Mathematics” by Teacher to Teacher Press

Math Problem - Seven Bridges

Posted by David Wetzel

Problem Solving Strategies

The foundations of topology are often not part of high school math curricula, and thus for many it sounds strange and intimidating.

However, there are some readily graspable ideas at the base of topology that are interesting, fun, and highly applicable to all sorts of situations.

One of these areas is the topology of networks, first developed by Leonhard Euler in 1735. His work in this field was inspired by the following problem.

Seven Bridges Problem

In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts.

Seven bridges were built so that the people of the city could get from one part to another.

The people wondered whether or not someone could walk around the city in a way that would involve crossing each bridge exactly once.

The following is drawing of the seven bridges in Konigsburg, Germany:

Sketch the above map of the city on a sheet of paper and try to ‘plan your journey’ with a pencil in such a way that you trace over each bridge once and only once and you complete the ‘plan’ with one continuous pencil stroke.

The problem now becomes one of drawing this picture without retracing any line and without picking your pencil up off the paper.

Leonhard Euler

A Swiss mathematician (1707 -1783) who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory.

Problem Solving Resources

Math Problem of the Week

Math Problem Solving Game