Algebra Backpack
533571113
Number Factories 01
Bob Albrecht & George Firedrake (MathBackpacks@q.com)
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 license.
http://creativecommons.org/licenses/by-nc/3.0/
Read or download our Numbers units at www.curriki.org. Search for albrecht numbers.
Contents as of 2008-11-02

Page 2: The Digit Factory

Page 4: The Million Factory
Half-baked ideas for another number factory
Use the number 1 as many times as you wish, parentheses, and the operations +, , , and / to
manufacture a target number in several different ways. Possible constraints:






No constraints. For example, OK to use 1 two or more times in one number: 11, 111, et cetera.
Cannot use 1 more than once in a number. Can't use 11, 111, et cetera.
Use each operation at least once.
Use each operation once and only once.
Use each operation once and only once in this order: +, , , 
Use each operation once and only once in this order: , , , +
Possible special target numbers:








Odd numbers less than 100.
Even numbers less than 100.
Prime numbers less than 100.
Composite numbers less than 100.
Square numbers less than 100.
Triangular numbers less than 100.
Fibonacci numbers less than 100.
Et cetera, et cetera.
Onward to The Digit Factory and The Million Factory!
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The Digit Factory
In the Digit Factory, your task is to use raw materials to construct numerical expressions that have
values equal to decimal digits.
The raw materials are:



These natural numbers: 1, 2, 3, 4, and 5.
Operations: addition, subtraction, multiplication, and division.
Parentheses: ( ).
The finished products are numerical expressions whose values are decimal digits.

Decimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
You are the Digit Maker. For each decimal digit 0 through 9, write a numerical expression whose value
is equal to the digit. In constructing your expression, please follow these rules:



Use each raw material number (1, 2, 3, 4, and 5) once and only once.
Use each operation (addition, subtraction, multiplication, and division) once and only once. No
other operations may be used.
Use as many parentheses as you want.
We use a TI-84 Plus Graphing Calculator to check each expression and make sure
that its value is a decimal digit. Here's a handy list of the TI-84's addition,
subtraction, multiplication, and division keys and the symbols that appear on the
calculator's display screen.
Operation
Key
Displayed
as
+
Addition
+
Subtraction
–
–
Multiplication

*
Division

/
Examples: Three numerical expressions. The value of each expression is exactly 5. To be sure, we
entered each expression into our TI-84 Plus and got 5 every time. We show each expression using
operation symbols for TI-84 keys (+, –, , ) and the corresponding symbols displayed on the TI-84's
display screen (+, –, *, /).
Expression using TI-84 key symbols
Expression as displayed
231+4–5
2*3/1+4–5
5
(3 + 2 – 1)  5  4
(3 + 2 – 1) * 5 / 4
5
(3 + 2)  (1  5 – 4)
(3 + 2) / (1 * 5 – 4)
5
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Your Turn. Write numerical expressions for each decimal digit 0 through 9. For extra elegance,
construct expressions in which one or more of the following is true:




The numbers 1, 2, 3, 4, and 5 are in increasing order from left to right.
The numbers are in decreasing order from left to right, like this: 5, 4, 3, 2, 1.
The operations appear in the order: +, –, , . (Displayed in the order +, –, *, /.)
The operations appear in the order: , , –, +. (Displayed in the order /, *, –, +.)
Numerical Expressions
A numerical expression is an algebraic alakazam consisting of numbers, operation symbols, and grouping
symbols such as parentheses. In the Digit Factory, the operations are addition (+), subtraction (–),
multiplication ( or *), and division ( or /).
To construct a numerical expression, you can't just throw numbers and operation symbols together in any
old willy-nilly harum-scarum way. For example:
These are numerical expressions
These are not numerical expressions
7+5
7+–+5
7–5
7–+–5
75
75
75
75
23+45
23+45
(2 + 3)  (4 + 5)
(2 + + + 3)   (4 + + + 5)
To evaluate a numerical expression, you use an order of operations. This order of operations is built into
the TI-84 Plus, most other calculators, and number-crunching computer tools such as Microsoft Excel. If
the expression has no parentheses:
1. First do all multiplications and divisions from left to right.
2. Then do all additions and subtractions from left to right.
If there are sub-expressions enclosed in parentheses, do them first. If there are parentheses enclosed in
parentheses, start on the inside and work your way out.
If you don't understand everything you know about numerical expressions, order of operations, and
related stuff, trundle on over to these helpful Internet sites:




Order of Operations (http://www.aaamath.com/pro73c-order-of-operations.html)
Amby's Math Resources: Order of Operations (http://amby.com/educate/ord-op/)
Funbrain.com's Operation Order Algebra Game (http://www.funbrain.com/algebra/)
Puzzle game uses arithmetic functions, order of operations
(http://www.theproblemsite.com/games/onetoten.asp)
May dragons of good fortune be with you as you numericalally express yourself.
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The Million Factory
In the Million Factory, your task is to use raw materials to construct numerical expressions that have a
value equal to exactly 1,000,000 (one million).
The raw materials are:



Decimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Operations: addition, subtraction, multiplication, division, and exponentiation (raising a number
to a power).
Parentheses: ( ).
The finished products are numerical expressions that have a value exactly equal to:

1,000,000 (one million).
You are the Million Maker. Write numerical expressions that each have the value of exactly 1,000,000
(one million). In constructing your expressions, please follow these rules:



Use each decimal digit, 0 through 9, once and only once.
Use the operations addition, subtraction, multiplication, division, and exponentiation as many
times as you want. No other operations may be used.
Use as many parentheses as you want.
We use a TI-84 Plus Graphing Calculator to check each expression and make sure that
its value is exactly 1,000,000. Here's a handy list of the TI-84's operation keys and the
corresponding symbols that appear on the calculator's display screen.
Operation
Key
Displayed as
Addition
+
+
Subtraction
–
–
Multiplication

*
Division

/
Exponentiation
^
^
Down yonder are three numerical expressions. The value of each expression is exactly 1,000,000. To be
sure, we entered each expression into our TI-84 Plus and got 1,000,000 every time. We show each
expression using operation symbols for keys (+, –, , , ^) and symbols displayed on the calculator's
display (+, –, *, /, ^).
Expression using TI-84 key symbols
Expression as displayed
10^6 + (2 + 3 – 5)  (8  4 + 9 – 7)
10^6 + (2 + 3 – 5) * (8 / 4 + 9 – 7)
1,000.000
10^7  (2  5) + (9 – 6 – 3)  (4 + 8)
10^7 / (2 * 5) + (9 – 6 – 3) * (4 + 8)
1,000,000
((9 + 1)  (8 + 2)  (7 + 3))^(6 – 4 + 0  5)
((9 + 1) * (8 + 2) * (7 + 3))^(6 – 4 + 0 / 5)
1,000,000
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Your Turn. Write numerical expressions that each have a value of exactly 1,000,000 (one million). For
extra elegance, construct expressions in which:




Each operation (+, –, , , ^) appears at least once.
Each operation (+, –, , , ^) appears exactly once.
The digits appear in descending order from left to right: 9, 8, 7, 6, 5, 4, 3, 2, 1, 0
The digits appear in ascending order from left to right: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Numerical Expressions
A numerical expression is an algebraic alakazam consisting of numbers, operation symbols, and grouping
symbols such as parentheses. In the Million Factory, the operations are addition (+), subtraction (–),
multiplication ( or *), division ( or /), and exponentiation (^).
To construct a numerical expression, you can't just throw numbers and operation symbols together in any
old willy-nilly harum-scarum way. For example:
These are numerical expressions
These are not numerical expressions
7+5
7+–+5
7–5
7–+–5
75
75
75
75
(2 + 3)  (4 + 5)
(2 + + + 3)   (4 + + + 5)
3^2 + 4^2
3^^2 + 4^^2
To evaluate a numerical expression, you use an order of operations. This order of operations is built into
the TI-84 Plus, most other calculators, and number-crunching computer tools such as Microsoft Excel. If
the expression has no parentheses:
1. First do all exponentiations.
2. Next do all multiplications and divisions from left to right.
3. Then do all additions and subtractions from left to right.
If there are sub-expressions enclosed in parentheses, do them first. If there are parentheses enclosed in
parentheses, start on the inside and work your way out.
If you don't understand everything you know about numericalal expressions, order of operations, and
related stuff, trundle on over to these helpful Internet sites:


The Language of Algebra - Order of operations - First Glance
(http://www.math.com/school/subject2/lessons/S2U1L2GL.html)
Amby's Math Resources: Order of Operations (http://amby.com/educate/ord-op/)
May dragons of good fortune be with you as you make millions.
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