Transcript number tricks - lenny

NUMBER TRICKS
Number Trick #1
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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
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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
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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
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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
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Add 20
Multiply by 4
Add 200
divide by 4
subtract your
number
• What do you get?
Why it works
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Pick a number
Add 20
Multiply by 4
Add 200
divide by 4
subtract your
number
• What do you get.
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N
N + 20
4(N + 20)
4N +80 + 200
(4N + 280)/4
N + 70 – N
70
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+9
N
N + 10
N+1
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
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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
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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
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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
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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
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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/magicgopher-central.swf
Visa Number: 1234 5678 9012 3456
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ADD 3 NUMBERS:
a. 2 x Row 1 Sum
b. Row 2 Sum
c. # 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
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ADD 3 NUMBERS:
a. 2 x Row 1 Sum
b. Row 2 Sum
c. # 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 4digit number
• Write down 4-digit number
• Scramble the digits to form a new 4digit number
• Subtract smaller # from larger #
• Write down 4-digit number
• Scramble the digits to form a new 4digit 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 4digit 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 4digit 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 4digit 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.