Using Algebra to Explain Arithmetic Using Arithmetic to Explain Algebra

1
Using Algebra to Explain ArithmeticUsing
Arithmetic to Explain Algebra

• Joe Hill
• Director of Math and Technology
• Rockingham County Public Schools

2
What is Arithmetic?

• Main Entry arithmetic
• Pronunciation -'rith-m-"tik
• Function noun
• Date 15th century
• 1 a a branch of mathematics that deals usually
  with the nonnegative real numbers including
  sometimes the transfinite cardinals and with the
  application of the operations of addition,
  subtraction, multiplication, and division to them
• Source Merriam-Webster online
  http//www.m-w.com/

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What is Algebra?

• Main Entry algebra
• Pronunciation 'al-j-br
• Function noun
• Etymology Medieval Latin, from Arabic al-jabr,
  literally, the reduction
• Date 1551
• 1 a generalization of arithmetic in which
  letters representing numbers are combined
  according to the rules of arithmetic
• Source Merriam-Webster online
  http//www.m-w.com/

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Purpose of this presentation

• To give you several examples of how to use
  Arithmetic when teaching Algebra
• To reinforce an Algebraic skill
• To explain why a trick works
• To show why a step in an Algebraic process is or
  is not justified

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Virginia Standards of Learning

• A.3 The student will justify steps used in
  simplifying expressions and solving equations and
  inequalities. Justifications will include the
  use of concrete objects pictorial
  representations and the properties of real
  numbers, equality, and inequality.
• AII.1 The student will identify field properties,
  axioms of equality and inequality, and properties
  of order that are valid for the set of real
  numbers and its subsets, complex numbers, and
  matrices

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An arithmetic example of this SOL

• Add 1 2 3 98 99
• (1 99) (2 98) (3 97) (48 52)
  (49 51) 50 Commutative and Associative
  Properties of Addition
• (100 100 100 100 100) 50
  Substitution
• 100 (1 1 1 1 1) 50 Distributive
  Property
• 100 (49) 50 Substitution
• 4950 Substitution

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A Trick from David Copperfield

• The next slide has a layout of various rooms in a
  school.
• You are allowed to move from one room to another
  one HORIZONTALLY and VERTICALLY only.
• Ill gradually take away rooms youre not
  currently in.

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Move horizontally or vertically only. As you
move, Ill take away rooms I know youre not in.
You cant move through a room I remove.
Start in Math, Science, English, or Social Studies
Youre not here!
Youre not here!
Youre not here!
Youre not here!
Youre not here!
Move 2
Youre not here!
Youre not here!
Youre not here!
Move 3
Move 5
Move 3
Youre in Math, arent you?
Move 1
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How did that work?
Start in Math, Science, English, or Social Studies
Youre not here!
Youre not here!
Youre not here!
Youre not here!
Youre not here!
Move 2
Youre not here!
Youre not here!
Youre not here!
Move 3
Move 5
Move 3
Youre in Math, arent you?
Move 1
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Virginia Standards of Learning

• A.2 The student will represent verbal
  quantitative situations algebraically and
  evaluate these expressions for given replacement
  values of the variables. Students will choose an
  appropriate computational technique, such as
  mental mathematics, calculator, or paper and
  pencil.

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An addition trick for this SOL

• Ill use the chalkboard for this
• Ill take three numbers in the range 1,000 to
  9,999 from the audience.
• Ill add a couple more to make the problem
  tougher.
• Then well add up the five huge numbersyou with
  a calculator and me in my head.
• Ill add faster than you can.

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Example

• 2,419 (from audience)
• 3,892 (from audience)
• 4,535 (from audience)
• 7,580 (from me)
• 6,107 (from me)
• 24,533 (from me--instantly)

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How did that work?

• First number from you A
• Second number from you B
• Third number from you C
• First number from me 9999 - A
• Second number from me 9999 - B
• Sum 20000 C - 2

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A Calendar Trick for this SOL

• Ill show you a calendar.
• You pick a 3 x 3 matrix of numbers and find their
  sum.
• Tell me the sum and Ill tell you the matrix you
  chose.
• then well see how Algebra can explain this.

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Sample Month
Pick any 3 x 3 block of numbers and find their
sum.
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Sample Month
If the sum is 189 then the block is...
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Sample Month
If the sum is 108 then the block chosen is...
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How did that work?
x - 7 (one week ago)
x - 6 (six days ago)
x - 8 (8 days ago)
x - 1 (yesterday)
x 1 (tomorrow)
x (today)
x 7 (one week from today)
x 8 (eight days from today)
x 6 (six days from today)
When you add up the nine values, you get 9x where
x is the middle number.
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Virginia Standards of Learning

• A.3 The student will justify steps used in
  simplifying expressions and solving equations and
  inequalities. Justifications will include the
  use of concrete objects pictorial
  representations and the properties of real
  numbers, equality, and inequality.

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Heres an arithmetic trick involving this SOL

• Whats your Favorite Activity?

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On this chart are some of your favorite activities
Sleeping
Doing Math
Watching TV
Golfing
Swimming
Shopping
Walking/Exercising
Reading
Eating
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Favorite Activity trick

• Enter a three digit number into your calculator
• Add 2
• Multiply by 2
• Subtract your original number
• Add 5
• Subtract your original number again
• Trace around the wheel starting at the top.
• Doing Math is your favorite, huh?

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How did that work?

• Choose a three digit number x
• Add 2 (x 2)
• Multiply by 2 2(x 2)
• Subtract original number 2(x 2) - x
• Add 5 2(x 2) - x 5
• Subtract original number
• 2(x 2) - x 5 - x
• Trace around the wheel 9

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Another Favorite Number trick

• Choose a three digit number
• Mix up its digits to form another three digit
  number
• Subtract the smaller from the larger
• Add the digits of the result
• Trace around the wheel again.

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How did that work?

• A three digit number can be written 100x 10y
  z
• Suppose you mixed up the digits to form 100z
  10x y
• Subtracting, you get 90x 9y - 99z
• 9(10x y - 11z)
• a multiple of nine

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Virginia Standards of Learning

• A.12 The student will factor completely first-
  and second-degree binomials and trinomials in one
  or two variables. The graphing calculator will
  be used as a tool for factoring and for
  confirming algebraic factorizations.
• AII.5 The student will identify and factor
  completely polynomials representing the
  difference of squares, perfect square trinomials,
  the sum and difference of cubes, and general
  trinomials.

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An arithmetic example of this SOL

• Factor 391
• (400 - 9)
• 202 - 32
• (20 - 3)(20 3)
• (17)(23)
• Dont believe me, ask your calculator!

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Another example..

• Factor 899
• (900 - 1)
• 302 - 12
• (30 - 1)(30 1)
• (29)(31)

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Another example..

• Factor 589
• 625 - 36
• 252 - 62
• (25 - 6)(25 6)
• (19)(31)

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Another example

• Factor 133
• 169 - 36
• 132 - 62
• (13 - 6)(13 6)
• (7)(19)

• Factor 133
• 125 8
• 53 23
• (5 2)(52 - (5)(2) 22)
• (7)(19)
• OR

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Another one

• Factor 973
• 1000 - 27
• 103 - 33
• (10 - 3)(102 (10)(3) 32)
• (7)(139)
• OR

• Factor 973
• 5329 - 4356
• 732 - 662
• (73 - 66)(73 66)
• (7)(139)

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A Math Trick for these SOLs

• Enter any three digit number into your
  calculator. For example, 725
• Repeat the same three digits to make a six digit
  number. For example, 725725
• Divide by 13
• Surprise, no remainder!
• Divide the answer by 11
• Surprise, no remainder!
• Divide the answer by 7
• Double surprise this time.

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Why did that work?

• When you converted the three digit number abc
  into abcabc, you multiplied it by 1001
• Is 1001 prime? 1000 1
• 103 13
• (10 1)(102 - (10)(1) 12)
• (11)(91)
• (11)(102 - 32)
• (11)(10 - 3)(10 3)
• (11)(7)(13)

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Virginia Standards of Learning

• A.5 The student will create and use tabular,
  symbolic, graphical, verbal, and physical
  representations to analyze a given set of data
  for the existence of a pattern, determine the
  domain and range of relations, and identify the
  relations that are functions.

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Some arithmetic examples for this SOL
Study these examples. See a pattern?
25 x 25
55 x 55
225
3025
35 x 35
75 x 75
1225
5625
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Your turn....
85 x 85
7225
995 x 995
99025
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How did that work?

• Suppose a number that ends in a 5 and is to be
  multiplied by itself. If this number is of the
  form x5 it can be written as 10x 5. For
  example 75 10(7) 5. This arithmetic trick
  asks us to multiply x5 times itself.
• (10x 5)2
• (100x2 100x 25)
• 100(x2 x) 25
• 100x(x 1) 25

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Virginia Standards of Learning

• A.11 The student will add, subtract, and multiply
  polynomials and divide polynomials with monomial
  divisors, using concrete objects, pictorial and
  area representations, and algebraic manipulations.

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Some arithmetic examples for this SOL using (a
b)(a - b)

• Multiply (59)(61)
• (60 - 1)(60 1)
• 3599

• Multiply (76)(84)
• (80 - 4)(80 4)
• 6384

• Multiply (67)(73)
• (70 - 3)(70 3)
• 4891

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Examples using (a b)2

• Find 512
• (50 1)2
• 502 2(50) 12
• 2601

• Find 492
• (50 - 1)2
• 502 - 2(50) 12
• 2401

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Examples using FOIL

• Multiply (12)(15)
• (10 2)(10 5)
• 100 20 50 10
• 180

• Multiply (13)(19)
• (10 3)(10 9)
• 100 30 90 27
• 247

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An arithmetic trick for this SOL

• Can you multiply large numbers such as 93 x 98
  mentally?
• Youll soon be able to!
• This technique comes from Scott Flansburg, The
  Human Calculator

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Can you do mental multiplications like this?
98 x 88
98 x 93
95 x 97
8624
9114
9215
83 x 99
92 x 89
91 x 94
8217
8188
8554
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How do you do that?

• 96
• x 93

• 4 from 100
• 7 from 100

8928
28 (4 x 7) 89 93 - 4
92 x 94
8 from 100 6 from 100
8648
48 (8 x 6) 86 92 - 6
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Why does that work? Algebra!

• Suppose you want to multiply A x B.
• Let A 100 - x, B 100 - y. That is, x and y
  are the differences between 100 and the original
  numbers.
• Note If A lt B then x gt y. If A is the smaller
  original number, y is the smaller of the other
  numbers.
• So AB (100 - x)(100 - y)
• (10,000 - 100x - 100y xy)
• 100(100 - x - y) xy
• 100(A - y) xy

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Virginia Standards of Learning

• A.13 The student will express the square root of
  a whole number in simplest radical form and
  approximate square roots to the nearest tenth.
• AII.7 The student will solve equations containing
  rational expressions and equations containing
  radical expressions algebraically and
  graphically. Graphing calculators will be used
  for solving and for confirming the algebraic
  solutions.

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Some arithmetic examples of this SOL

• Add v20 v80
• v20 v80 2v5 4v5
• 6v5
• v180 so v20 v80 v180
• Subtract v192 - v75
• v192 - v75 8v3 - 5v3
• 3v3
• v27 so v192 - v75 v27
• Dont believe me--ask your calculator!

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Virginia Standards of Learning

• A.10 The student will apply the laws of exponents
  to perform operations on expressions with
  integral exponents, using scientific notation
  when appropriate.

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An arithmetic example of this SOL

• Multiply and express in exponential form (450
  )(820)
• (22)50 (23)20
• 2100 260
• 2160
• 480
• 1640

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Another arithmetic example of this SOL

• Add, expressing your answer in exponential form
• 212 212
• 2(212)
• 21(212)
• 213

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Another arithmetic example of this SOL

• What is one-third of 999?
• 999/3
• (32)99 /3
• 3198 /31
• 3197

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Another example

• Add and express your answer in scientific
  notation (4 x 108)(2 x 109)
• (4 x 108)(2 x 109)
• (4 x 108)(20 x 108)
• (4 20)108
• (24)108
• 2.4 x 109

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Virginia Standards of Learning

• AII.2 The student will add, subtract, multiply,
  divide, and simplify rational expressions,
  including complex fractions.

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Use arithmetic to explain this SOL

• Can you cancel the b's?
• 10a b
• 10b c

Can you cancel the 8's? 28 83
Can you cancel the 6's? 16 64
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In conclusion

• Use arithmetic whenever you can to reinforce
  Algebraic ideas
• Students will be amazed that what you show them
  actually works!

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My contact information

• Joe Hill
• Director of Math and Technology
• Rockingham County Public Schools
• 2 S. Main Street
• Harrisonburg VA 22801
• Phone [phone]
• Fax [phone]
• E-mail jhill_at_rockingham.k12.va.us