NUMBER TRICKS

Number Trick #1

• Choose a number

• Add 3

• Multiply by 2

• Add 8

• Divide by 2

• Subtract the original number

• What is your result

• Show why this trick works using algebra.

Number Trick #1

• Choose a number—x

• Add 3—x + 3

• Multiply by 2—2(x+3) = 2x + 6

• Add 8—2x + 14

• Divide by 2—x + 7

• Subtract the original number—x

• What is your result—7

Multiple Representations

Multiple Representation

• Can also be done by drawing pictures

• A number =

• Add 3 =

Multiple Representation

• Can also be done by drawing pictures

• A number =

• Add 3 =

• Multiple by 2

• Add 8

• Divide by 2

• Subtract number

Questions

• Does the initial number selected have to be a single digit? – No, any number will work with this trick.

• What happens at the last step if instead of subtracting the original number you subtract 7? – The answer will always be the original number

selected instead of 7.

• Does the order of the commands make a difference

• Students are problem solving.

• Students enhance their fluency in mental calculations

• Problems are personalized which provides motivation to find a solution.

• Students formulate and reformulate generalized solution patterns from specific cases.

• Students manipulate symbolic expressions.

Considerations

• Begin by doing the trick.

• Calculations may be done mentally or with calculator.

• Have students pick another number and try the trick.

• Have the students do the trick with parents or someone in a different class.

• Challenge the students to find out why the trick works.

Notes

• Once the trick is demonstrated have the students try to figure out why it works.

• Introduce the visual and algebraic columns at an appropriate time.

• Provide students visual or algebraic and have them create other columns

• Multiplying 2n + 4 by 3 = 3(2n + 4) can be used to show distributive property.

• Have students create their own number tricks• A blank table and other tricks is provided with the word

document.

Phone number

• Enter the first 3 digits of your phone number, not the area code.

• Multiply by 80• Add 1• Multiply by 250• Add the last 4 digits of your phone number• Add the last 4 digits of your phone number again• Subtract 250• Divide by 2• What is the result?

Why does this work?

• Multiplying by 80 and 250 is the same as multiplying by 10,000. This creates a number which is the first three numbers of your phone followed by four zeros. Adding 1 and them multiplying by 250 is canceled by subtracting 250 in a latter step. The last 4 digits of the number is added twice so that it can be divided by 2.

Birthday Trick• Start with the number of your birth month

• Add 2

• Multiply by 200

• Subtract 400

• Add the number for the day of your birth

• Add the day again

• Multiply by 5000

• Add the 4 digit year of your birth

• What is your result?

Why does this work?

Write down a two-digit number

• Add 20

• Multiply by 4

• Add 200

• divide by 4

• subtract your number

• What do you get?

Why it works

• N

• N + 20

• 4(N + 20)

• 4N +80 + 200

• (4N + 280)/4

• N + 70 – N

• 70

• Pick a number

• Add 20

• Multiply by 4

• Add 200

• divide by 4

• subtract your number

• What do you get.

Eating Chocolate

• How many times a week do you want to eat chocolate? (Pick more than one but less than 10 times).

• Multiply that number by 2.• Add 5.• Multiply that result by 50.• If your birthday has occurred this year, add 1760 to that

result. If your birthday is still to come this year, add 1759.• Subtract the year (all four digits) in which you were born.• The result will be a three-digit number. The first digit will

be the number of times a week that you want to eat chocolate, and the last two digits will be your age.

Extension

• Extension: Suppose you wanted chocolate 10 or more times a week, or no chocolate at all. Could you still make the puzzle work?

• What if you are over 100 years old?

• Will the puzzle need to be adjusted when the current year becomes 2012?

Two Sums

• Mathematics Teaching in the Middle School, September 2010, p 68-71.

• Challenge: Place a 3X3 grid anywhere on a 100’s board (1-10 on first row, 11-20 on second, etc).

• Find the sums of the two diagonals of thee 3X3 grid.

• How to the two sums compare? Will this comparison always hold true? Why?

Solution

• The sum of the diagonals will always be equal.

• The sum will always be 3 times the middle number.

• Also the number in a cross (from the middle above and below, from the middle left and right) will also be equal and 3 times the middle number.

N - 11 N - 10 N - 9

N - 1 N N + 1N + 9 N + 10 N +11

The sum of the diagonals is(N-11) + N + (N+11) = 3N

(N-9) + N + (N+9) = 3N 

The sum of the cross (the middle rows and middle column)

(N-10) + N (N +10) = 3N(N-1) + N + (N +1) = 3N

Challenges & Questions

• Begin with N being the number in the top left corner and fill in the rest of the numbers)

• Will this work with a 4X4 grid or a 5X5 grid?

• What if the 100’s board is an 8X8 grid instead of a 10X10 grid?

Number Card Trick

• Take a deck of cards and take out the face cards leaving the numbers Ace (1) through 10.

• Pick any card

• Add the next number

• Multiply by 5

• Add 6-clubs, 7 diamond, 8-heart, 9 spade

To find the card:

• Take the answer subtract 5, the first digit is the card and the second is the suit

• Example– Card is 8 of diamonds, add 8+9 = 17, multiply

by 5 = 85 add 7 for diamonds get 92– Person subtracts 5 (92-5) = 87, card was 8 of

(7)diamonds

Why

• n Pick any card

• n + (n+1) Add the next number

• (n + (n+1))*5 = 10n + 5 Multiply by 5

• Add 6-clubs, 7 diamond, 8-heart, 9 spade

• Solution

• Subtract 5

Questions

• Does it work for 10?

• Would it work for higher numbers?

• Why is the add 5 necessary?

• Could the numbers for suits be different?

Your Favorite Person

• Pick your favorite number between 1-9

• Multiply by 3 then

• Add 3

• Then again Multiply by 3 (I'll wait while you get the calculator....)

• You'll get a 2 or 3 digit number....

• Add the digits together

Use Your Number to Find Your Favorite Person

1. Albert Einstein

2. Oprah Winfrey

3. Snoopy

4. Bill Clinton

5. Bill Gates

6. George W. Bush

7. Barack Obama

8. Babe Ruth

9. Lenny VerMaas

10.10. John Fitzgerald Kennedy

Mathematical Private Eye

• Think of your favorite number

• Multiply it by 3

• Add your favorite number and 1 to it.

• Add 11 to it

• Divide by 4

• Add 2 to your answer

• What is your answer?

• To get the favorite number subtract 5

Mathematical Private Eye

• I know your Favorite Number!

• (Alternate) I know your favorite number!

• I Know What Month You Were Born!

• I Know Your Mother’s Age!

• I Know Your Favorite Day of the Week.

• I Know What Numbers You Are Thinking Of! Three at a Time!

Solving Simultaneous Equations

• I know Your Birthdate

Web Examples

• Magic Goffer– http://www.learnenglish.org.uk/games/magic-g

opher-central.swf

Visa Number: 1234 5678 9012 3456

1 3 5 7 9 1 3 5

2 4 6 8 0 2 4 6

ADD 3 NUMBERS:a. 2 x Row 1 Sumb. Row 2 Sumc. # of numbers in Row 1 great than or equal to 5

Look at the Units Digit of this sum.

Visa Number: 1234 5678 9012 3456

1 3 5 7 9 1 3 5

2 4 6 8 0 2 4 6

ADD 3 NUMBERS:a. 2 x Row 1 Sumb. Row 2 Sumc. # of numbers in Row 1 great than or equal to 5

Look at the Units Digit of this sum. Is it 0?

• Write down 4-digit number

• Write down 4-digit number

• Scramble the digits to form a new 4-digit number

• Write down 4-digit number

• Scramble the digits to form a new 4-digit number

• Subtract smaller # from larger #

• Write down 4-digit number

• Scramble the digits to form a new 4-digit number

• Subtract smaller # from larger #

• Draw circle around 1 digit of this difference (but not a 0)

• Write down 4-digit number

• Scramble the digits to form a new 4-digit number

• Subtract smaller # from larger #

• Draw circle around 1 digit of this difference (but not a 0)

• Jumble remaining digits to form new number.

• Write down 4-digit number

• Scramble the digits to form a new 4-digit number

• Subtract smaller # from larger #

• Draw circle around 1 digit of this difference (but not a 0)

• Jumble remaining digits to form new number.

• Share this number.

• Write down 4-digit number• Scramble the digits to form a new 4-

digit number• Subtract smaller # from larger #• Draw circle around 1 digit of this

difference (but not a 0)• Jumble remaining digits to form new

number.• Share this number.• I can tell you the digit you circled.