A 1-Derful Post for 1/1

My sons have refrigerator magnets with digits and binary operators, which they use to create expressions, equations, and dates. Recently, they created the following equation:

1 + 1 ÷ 1 – 1 × 1 = 1

They asked if it was correct. Oh, no, that’s not how things work in this house. “You tell me,” I said.

Eli said, “One plus one is two, divided by one is two, minus one is one, times one is one. It’s true.”

Anyone who teaches middle school has seen students make this type of order of operations error. The equation is true if operations are performed left-to-right but not if the conventional order of operations is applied.

On a calculator, I entered

1 + 2 ÷ 2

and asked, “What is the value of this expression?” Sure enough, they thought it would be three-halves, and they were surprised to see two displayed when the ENTER key was pressed.

Eli looked puzzled, and Alex looked cross. “Oh, right,” said Alex. “We have to do multiplication and division first.”

They then concluded that their equation was indeed true, and this time for the right reasons.

But it made me wonder:

What is the probability that the following equation will be true, if the four binary operators are randomly placed in the blanks with each operator used only once?

1 __ 1 __ 1 __ 1 __ 1 = 1

I’ll tell you that (a) I was surprised by the results and (b) I didn’t have to check every possible equation to arrive at the answer; in fact, I didn’t even have to check a quarter of them.

Surgeon: I have so many patients to see today! Who should I do surgery on first?
Nurse: Follow the order of operations.

How many calculus teachers does it take to screw in a light bulb?
0.99999999…

Smooth Operators

I recently learned about a cool problem that involves the four binary operations.

“My life is all arithmetic,” the young businesswoman explained. “I try to add to my income, subtract from my weight, divide my time, and avoid multiplying.”

A fifth-grader teacher, who is spending a week at NCTM Headquarters for the Illuminations Summer Institute, shared the problem with me. He uses it to practice basic operations and to develop an understanding of place value with his students.

Choose any positive integer as the starting value, and choose a different positive integer as the ending value. Then, perform any of the following moves on the starting number:

• Add 1.
• Subtract 1.
• Multiply by 10.
• Divide by 10.

Continue to perform moves until you’ve reached the ending number.

For example, you can get from 8 to 71 with the following sequence of moves:

• Subtract 1: 8 – 1 = 7
• Multiply by 10: 7 × 10 = 70
• Add 1: 70 + 1 = 71

For any given starting and ending number, what is the fewest number of moves required?

In general, can you find an algorithm to predict the minimum number of moves for any given starting and ending numbers?

The teacher who shared the problem with me said that students have fun with 777 as the starting number and 888 as the ending number. It is a fine problem for kids to explore, but it proved to be a red herring for me — it led me to make false assumptions for determining a general solution.

Enjoy.