Subtraction Magic of Nines — Longevity Calculator

Last week we explored the concept of difference by subtracting birth year from death year to calculate the longevity of scientists and mathematicians. The K-1st graders subtracted using standard algorithm with no borrowing and the older students mastered both borrowing and subtracting with adding. Some of the students were inspired create lists of female scientists and their short biographies. I have attached this pdf here.

 

We continued our exploration of subtraction by looking at the magic of nines. With all students we learned a mathemagician “trick” with three-digit numbers. The students soon learned that there is no magic at all — it is the properties of the number nine. 

 

Subtraction Magic of Nines: 3-Digit Numbers

 

STEP 1: THE MATHEMAGICIAN WILL ASK YOU TO WRITE DOWN A 3-DIGIT NUMBER WITH 3 DISTINCT (DIFFERENT) DIGITS

STEP 2:  THE MATHEMAGICIAN WILL ASK YOU TO REVERSE THE DIGITS

STEP 3:  THE MATHEMAGICIAN WILL ASK YOU TO SUBTRACT THE LOWER NUMBER FROM THE LARGER NUMBER

STEP 4:  THE MATHEMAGICIAN WILL ASK YOU TO TELL THEM THE FIRST OR LAST DIGIT OF YOUR ANSWER AND THE MATHEMAGICIAN WILL TELL YOU YOUR ANSWER

STEP 5: FINALLY,  THE MATHEMAGICIAN WILL ASK YOU TO REVERSE THE DIGITS OF YOUR ANSWER AND ADD IT TO YOUR ANSWER AND WILL TELL YOU THE SUM

 

THE ONLY POSSIBLE SOLUTIONS ARE:

099 198 297

396 495 594

693 792 891

 

Since there are only nine possible solutions, asking the audience for the first or last digit of their answer, will allow you to solve their answer. The sum of the first and last number always will be 9 and the middle digit will always be 9. It gets tricky if the audience says that their first number is 9; in this case, their first number is the middle number of 099 or just 99.

 

A bonus trick here is having the audience reverse the digits of their answer and add their answer to that number. The sum will always be 1,089. This is a really cool number to explore. It has factors of 1x1,089; 3x363; 9x121; 11x99; 33x33.

 

Subtraction Magic of Nines: Any-Digit Numbers

 

The older children (3rd grade and up) explored another amazing “trick” with nines as follows:

 

STEP 1: THE MATHEMAGICIAN WILL ASK YOU TO WRITE DOWN ANY-DIGIT NUMBER 

STEP 2: THE MATHEMAGICIAN WILL ASK YOU TO JUMBLE THE DIGITS IN ANY ORDER

STEP 3: THE MATHEMAGICIAN WILL ASK YOU TO SUBTRACT THE LOWER NUMBER FROM THE LARGER NUMBER

STEP 4: THE MATHEMAGICIAN WILL ASK YOU TO CIRCLE ANY NON-ZERO NUMBER IN THE ANSWER AND TELL THEM THE OTHER  DIGITS OF THE ANSWER IN ANY ORDER

STEP 5: THE MATHEMAGICIAN WILL TELL YOU THAT THEY WILL BE ABLE TO DETERMINE WHAT NUMBER YOU CIRCLED

 

SOLUTION: FIND THE DIGITAL ROOT OF THE REMAINING DIGITS OF THE ANSWER; TO FIND THE DIGITAL ROOT, ADD THE DIGITS AND CONTINUE TO ADD THEM UNTIL YOU HAVE A ONE DIGIT NUMBER; SUBTRACT THE DIGITAL ROOT FROM 9 AND THE DIFFERENCE IS THE CIRCLED NUMBER

Note: If your digital root is 9, the circled number is "9"; also if you are not able to determine the circled number, either your digital root sum and subtraction from 9 is incorrect or their subtraction is in error.

 

These lessons reinforce the following objectives:

  • subtraction with borrowing
  • reversing digits
  • the definition of “distinct” digits (different)
  • placing the larger number on the top of a subtraction question
  • adding the reverse of your solution
  • factors of 1,089
  • finding the digital root of a number until you reach one digit.

Longevity Calculator

 

A life insurance company came up with a longevity calculator which determines your average life expectancy given 12 factors such as age, weight, height, diet, blood pressure, exercise, stress, seatbelt use, driving record, use of recreational drugs, alcohol use, smoking, and family history. I was able to increase my life expectancy from 77 years to 95 by my family history and making certain decisions about lifestyle.

 

Try this with your children, parents, friends to see how your decisions affect your life expectancy; and remember, also “don’t get hit by a truck.” A metaphor for being safe. The link is https://www.northwesternmutual.com/learning-center/tools/the-longevity-game and the pdf of my test is attached.

 

 

 

 

 

 

AttachmentSize
Subtraction_Magic_of_Nines_with_3-Digit_Numbers.pdf182.87 KB
Subtraction_Magic_of_Nines_with_Any-Digit_Number.pdf102.76 KB
Longevity_Calculator-LIfe_Expectancy.pdf546.81 KB