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Algebra Backpack
Algebra-Numbers-01
Algebra: Numbers 01
Bob Albrecht & George Firedrake
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 license.
http://creativecommons.org/licenses/by-nc/3.0/
Edit date: 2011-12-25
This booklet is a slow, very slow teach yourself algebra unit
that introduces beginners to natural numbers and whole numbers.
TABLE OF CONTENTS
Natural Numbers
Successors and Predecessors
Number Lines
Odd Numbers and Even Numbers
Whole Numbers
Zero (0) and One (1)
Square Numbers
Triangular Numbers
Mathemagical Black Hole 123
Self-Test Exercises
Self-Test Answers
Ahoy Reader:
This is a teach yourself algebra unit. In easily
digestible nibbles, we present tutorials, examples, and
things for you to do called Your Turn with
Answers.
If you are reading this on your computer, you can
click on a section heading listed under TABLE OF
CONTENTS to go to that section.
The section called A Mathemagical Black Hole is
a math recreation. Use it to amaze and amuse your
friends.
To Teachers and Tutors: Below is a list of key words and phrases in this teach yourself unit.
To Students: You can skip this list and begin your algebra exploration on the next page. After
you have completed this unit, you will know much about the words and phrases in the list.
Key words and phrases in order of appearance: natural numbers, counting numbers,
little squares, base-10-block unit cube, set, ellipsis, postulate, successor, predecessor, number
line, location, tick mark, graph, odd number, even number, model an even number, model an odd
number, quotient, remainder, zero (0), one (1), property, square number, model a square number,
palindrome, consecutive odd numbers, triangular number, consecutive numbers, model a
triangular number, mathemagical black hole, catenate
1
Algebra Backpack
Algebra-Numbers-01
Natural Numbers | TOC
All aboard! We begin with the natural numbers: 1, 2, 3, 4, 5, and so on. You can use natural
numbers to count objects, so they are also called counting numbers.
Let's count little squares (■).
1
2
3
■
■■
■■■
4
5
6
■■■■ ■■■■■ ■■■■■■
and so on
et cetera
Our little squares ( ■■■■■■ ) are like base-10 block unit cubes. We
found a pile of unit cubes online at Nasco.
http://www.enasco.com/product/TB17412T
Product #TB17412T  100 unit cubes for $2.55
Natural numbers and counting numbers are names for the same set (bunch; collection) of
numbers.

Natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, et cetera.

Counting numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on.
A set of three dots ( ... ) called an ellipsis means "et cetera" or "and so on."

Natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

Counting numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
Natural numbers keep on going and going and going. You pick a natural number and we can pick
a greater one. Easy – we pick the number that is one more than your number. You pick 1; we
pick 2. You grab 2; we glom onto 3. You choose 99; we select 100. You slyly think of a secret
number; we admire your cleverness and say, "Our number is one more than your secret number."
About natural numbers
The first natural number is 1.
There is no last natural number.
The least natural number is 1.
There is no greatest natural number.
2
Algebra Backpack
Algebra-Numbers-01
Successors and Predecessors | TOC
A postulate is a statement that a bunch of people agree is true. Algebra books are loaded with
postulates that mathematicians agree are true. Postulates provide the foundation on which we can
build algebraic structures. Here are our first postulates for natural numbers:
Postulate: The first natural number is 1.
Postulate: Every natural number a has a successor a + 1 that is a natural number.
Postulate: Every natural number a, except 1, has a predecessor a – 1 that is a natural
number.
The second postulate tells you that natural numbers are never ending.
There is no greatest natural number.
Imagine a greatest natural number. Your number has a successor that is
one more than your number and is greater than your number.
Every natural number has a successor that is 1 more than that natural number. Every natural
number except 1 has a predecessor that is 1 less than that natural number.
 The number 1 is the first natural number and does not have a predecessor. 
Successors and predecessors of natural numbers 1 to 10
Number
Successor
Predecessor
Number
Successor
Predecessor
1
2
none
6
7
5
2
3
1
7
8
6
3
4
2
8
9
7
4
5
3
9
10
8
5
6
4
10
11
9
The least natural number is 1. It does not have a predecessor.
The successor of 1 is 2. The predecessor of 2 is 1.
There is no greatest natural number.
3
Algebra Backpack
Algebra-Numbers-01
Your Turn. Answer the first part of questions 1 through 5 with Y for yes or N for no.
1. Is there a first natural number? _____
If yes, what is it? ______________________
2. Is there a last natural number? _____
If yes, what is it? ______________________
3. Is there a least natural number? _____
If yes, what is it? ______________________
4. Is there a greatest natural number? _____
If yes, what is it? ______________________
5. Complete each sentence by writing the successor or predecessor of each natural number.
a. The successor of 1 is _____.
b. The predecessor of 1 is _____.
c. The successor of 4 is _____.
d. The predecessor of 4 is _____.
e. The successor of 20 is _____.
f. The predecessor of 20 is _____.
g. The successor of 999,999 is ____________.
h. The predecessor of 999,999 is __________.
Answers
1. Is there a first natural number? Y
If yes, what is it? 1
2. Is there a last natural number? N
If yes, what is it? No last natural number.
3. Is there a least natural number? Y
If yes, what is it? 1
4. Is there a greatest natural number? N
If yes, what is it? No greatest natural number.
3. Complete each sentence by writing the successor or predecessor of each natural number.
a. The successor of 1 is 2.
b. The predecessor of 1 is none.
c. The successor of 4 is 5.
d. The predecessor of 4 is 3.
e. The successor of 20 is 21.
f. The predecessor of 20 is 19.
g. The successor of 999,999 is 1,000,000.
h. The predecessor of 999,999 is 999,998.
Every natural number has a successor that is 1 more than the number.
Every natural number except 1 has a predecessor that is 1 less than the number.
4
Algebra Backpack
Algebra-Numbers-01
Number Lines | TOC
A number line is a line, usually horizontal, that you can use to graphically display numbers. The
number line shown below has nine locations labeled 1 through 9.
┼──┼──┼──┼──┼──┼──┼──┼──┼───►
1
2
3
4
5
6
7
8
9 ...
The natural numbers on this number line are equally spaced. The distance between adjacent
numbers is 1. The first natural number (1) is at the left end of the number line. At the right end,
the arrow (──► ) and ellipsis (...) remind you that the natural numbers are never-ending. A
tick mark ( ┼ ) on the number line marks the location of each natural number.
You can graph a particular natural number on the number line by drawing a dot ( ● ) at the
natural number's location. Here is a number line graph of the natural numbers 1, 3, and 6.
●──┼──●──┼──┼──●──┼──┼──┼───►
1 2 3 4 5 6 7 8 9 ...
Your Turn
1. What numbers are graphed on the number line below? _____, _____, and _____.
┼──┼──┼──●──┼──┼──●──┼──●───►
1 2 3 4 5 6 7 8 9 ...
2. Graph the natural numbers 2, 5, and 8 on the number line.
┼──┼──┼──┼──┼──┼──┼──┼──┼───►
1 2 3 4 5 6 7 8 9 ...
3. On the number line below, graph the successor of 4 and the predecessor of 9.
┼──┼──┼──┼──┼──┼──┼──┼──┼───►
1 2 3 4 5 6 7 8 9 ...
Answers
1. 4, 7, and 9.
2. ┼──●──┼──┼──●──┼──┼──●──┼───►
1 2 3 4 5 6 7 8 9 ...
3. ┼──┼──┼──┼──●──┼──┼──●──┼───►
1 2 3 4 5 6 7 8 9 ...
5
Algebra Backpack
Algebra-Numbers-01
Odd Numbers and Even Numbers | TOC
Every natural number is either an odd number or an even number.

Natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...

Odd numbers:
1,

Even numbers:
3,
2,
5,
4,
7,
6,
9,
8,
11,
10,
...
12, ...
Abracadabra! You can split an even number into two equal natural numbers.
Numbers
Little squares
Split 2 into 1 and 1.
Split ■■ into ■ and ■
Split 6 into 3 and 3.
Split ■■■■■■ into ■■■ and ■■■
Split 10 into 5 and 5
Split ■■■■■■■■■■ into ■■■■■ and ■■■■■
Your Turn Split each even number into two equal natural numbers.
1. Split 4 into _____ and _____.
2. Split ■■■■ into ______ and ______.
3. Split ■■■■■■■■ into ________ and ________.
4. Split 8 into _____ and _____.
5. Split 18 into _____ and _____.
6. Split 20 into ____ and ____.
7. Split 30 into ____ and ____.
8. Split 98 into ____ and ____.
9. Split 124 into _____ and _____.
10. Split 246 into ______ and ______.
Answers
1. Split 4 into 2 and 2.
2. Split ■■■■ into ■■ and ■■.
3. Split ■■■■■■■■ into ■■■■ and ■■■■.
4. Split 8 into 4 and 4.
5. Split 18 into 9 and 9.
6. Split 20 into 10 and 10.
7. Split 30 into 15 and 15.
8. Split 98 into 49 and 49.
9. Split 124 into 62 and 62.
10. Split 246 into 123 and 123.
Remember: You can split any even number into
two equal natural numbers.
6
Algebra Backpack
Algebra-Numbers-01
You can write any even number as 2 times a natural number. Below we show examples using
both the word 'times' and the multiplication symbol ''.
2 = 2 times 1
4 = 2 times 2
6 = 2 times 3
8 = 2 times 4
10 = 2 times 5
2=21
4=22
6=23
8=24
10 = 2  5
Your Turn Write each even number as a) 2 times a natural number and b) 2  a natural number.
1a. 12 = 2 times ____
2a. 14 = 2 times ____
3a. 16 = 2 times ____
4a. 18 = 2 times ____
1b. 12 = 2  ____
2b. 14 = 2  ____
3b. 16 = 2  ____
4b. 18 = 2  ____
5a. 24 = 2 times ____
6a. 30 = 2 times ____
7a. 64 = 2 times ____
8a. 96 = 2 times ____
5b. 24 = 2  ____
6b. 30 = 2  ____
7b. 64 = 2  ____
8b. 96 = 2  _____
1a. 12 = 2 times 6
2a. 14 = 2 times 7
3a. 16 = 2 times 8
4a. 18 = 2 times 9
1b. 12 = 2  6
2b. 14 = 2  7
3b. 16 = 2  8
4b. 18 = 2  9
5a. 24 = 2 times 12
6a. 30 = 2 times 15
7a. 64 = 2 times 32
8a. 96 = 2 times 48
5b. 24 = 2  12
6b. 30 = 2  15
7b. 64 = 2  32
8b. 96 = 2  48
Answers
And, tra la, tra la, you can also write any even number as a natural number times 2.
2 = 1 times 2
4 = 2 times 2
6 = 3 times 2
8 = 4 times 2
10 = 5 times 2
2=21
4=22
6=32
8=42
10 = 2  2
Your Turn Write each even number as a) a natural number times 2 and b) a natural number  2.
1a. 12 = ____ times 2
2a. 14 = ____ times 2
3a. 16 = ____ times 2
4a. 18 = ____ times 2
1b. 12 = ____  2
2b. 14 = ____  2
3b. 16 = ____  2
4b. 18 = ____  2
1a. 12 = 6 times 2
2a. 14 = 7 times 2
3a. 16 = 8 times 2
4a. 18 = 9 times 2
1b. 12 = 6  2
2b. 14 = 7  2
3b. 16 = 8  2
4b. 18 = 9  2
Answers
7
Algebra Backpack
Algebra-Numbers-01
You can use little squares to model an even number by arranging an even number of little
squares into two rows with the same number of little squares in each row.

Arrange an even number of little squares into two rows with the same number of little
squares in each row.
Even
Even number of little squares
number
6
■■■■■■
10
■■■■■■■■■■
20
■■■■■■■■■■■■■■■■■■■■
Arranged in two rows
■■■
■■■
■■■■■
■■■■■
■■■■■■■■■■
■■■■■■■■■■
2  natural number
23
25
2  10
Your Turn Fill in the missing stuff in this table. Drawing lots of little squares is labor intensive,
so it is OK if you use dots instead of carefully drawn little squares.
Even
Even number of little squares
number
Arranged in two rows
2  natural number
4
■■■■
22
8
■■■■■■■■
24
12
■■■■■■■■■■■■
26
Answers
4
■■■■
8
■■■■■■■■
12
■■■■■■■■■■■■
■■
■■
■■■■
■■■■
■■■■■■
■■■■■■
22
24
26
Imagine
In your mind's eye, imagine an even number of little squares, and then
imagine them moving into an arrangement of two rows with the same
number of little squares in each row.
Algebra is a great tool for modeling real world phenomena.
Little squares are great tools for modeling algebra.
8
Algebra Backpack
Algebra-Numbers-01
Alas, alack, and oh heck, an odd number cannot be split into two equal natural numbers. Hang
on for the good news. Any odd number except 1 can be split into two equal natural numbers and
remainder 1.
Numbers
Little squares
Split 3 into 1 and 1, and remainder 1. Split ■■■ into ■ and ■, and remainder ■
Split 5 into 2 and 2, and remainder 1. Split ■■■■■ into ■■ and ■■, and remainder ■
Split 7 into 3 and 3, and remainder 1. Split ■■■■■■■ into ■■■ and ■■■, and remainder ■
Split 9 into 4 and 4, and remainder 1. Split ■■■■■■■■■ into ■■■■ and ■■■■,
and remainder ■
Your Turn Split each odd number into two equal natural numbers and remainder 1.
1. Split ■■■■■■■■■■■ into _____________ and _____________, remainder _____
2. Split ■■■■■■■■■■■■■ into ____________ and ____________ , remainder _____
3. Split 11 into ____ and ____, remainder ___.
4. Split 13 into ____ and ____, remainder ___.
5. Split 15 into ____ and ____, remainder ___.
6. Split 17 into ____ and ____, remainder ___.
7. Split 19 into ____ and ____, remainder ___.
8. Split 21 into ____ and ____, remainder ___.
Answers
1. Split ■■■■■■■■■■■ into ■■■■■ and ■■■■■, remainder ■
2. Split ■■■■■■■■■■■■■ into ■■■■■■ and ■■■■■■, remainder ■
3. Split 11 into 5 and 5, remainder 1.
4. Split 13 into 6 and 6, remainder 1.
5. Split 15 into 7 and 7, remainder 1.
6. Split 17 into 8 and 8, remainder 1.
7. Split 19 into 9 and 9, remainder 1.
8. Split 21 into 10 and 10, remainder 1.
Remember: You can split any odd number except 1 into
two equal natural numbers, and remainder 1.
9
Algebra Backpack
Algebra-Numbers-01
You can use little squares to model an odd number by arranging an odd number of little squares
into two rows with the same number of little squares in each row and a remainder of 1 little
square.

Arrange an odd number of little squares into two rows with the same number of little
squares in each row and a remainder of 1 little square.
Even
Odd number of little squares
number
7
■■■■■■■
11
■■■■■■■■■■■
19
■■■■■■■■■■■■■■■■■■■
Arranged in two rows
and a remainder
■■■
■■■ ■
■■■■■
■■■■■ ■
■■■■■■■■■
■■■■■■■■■ ■
2  natural number + 1
23+1
25+1
29+1
Your Turn Fill in the missing stuff in this table. Drawing lots of little black squares is labor
intensive, so it is A-OK to use dots instead of carefully drawn little black squares.
Even
Even number of little black
number squares
Arranged in two rows
and a remainder
2  natural number + 1
5
■■■■■
22+1
9
■■■■■■■■■
24+1
13
■■■■■■■■■■■■■
26+1
Answers
5
■■■■■
9
■■■■■■■■■
13
■■■■■■■■■■■■■
■■
■■ ■
■■■■
■■■■ ■
■■■■■■
■■■■■■ ■
22+1
24+1
26+1
Remember:
You can split any even number into two equal natural numbers.
You can split any odd number except 1 into two equal natural numbers,
and remainder 1.
10
Algebra Backpack
Algebra-Numbers-01
If you divide an odd number by 2, you get a natural number quotient and remainder 1.
_1 ← quotient
2)3
2
1 ← remainder
_2 ← quotient
2)5
4
1 ← remainder
_3 ← quotient
2)7
6
1 ← remainder
_4 ← quotient
2)9
8
1 ← remainder
Your Turn Divide each odd number by 2 and get a natural number quotient and remainder 1.
___ ← quotient
2)11
__
← remainder
___ ← quotient
2)13
__
← remainder
___ ← quotient
2)19
__
← remainder
___ ← quotient
2)31
__
← remainder
__6 ← quotient
2)13
12
1 ← remainder
__9 ← quotient
2)19
18
1 ← remainder
_15 ← quotient
2)31
30
1 ← remainder
Answers
__5 ← quotient
2)11
10
1 ← remainder
Any odd number except 1 can be written as 2 times a natural number plus 1.

Odd numbers: 3 = 2  1 + 1
5=22+1
7=23+1
9=24+1
Your Turn Write each odd number as 2 times a natural number plus 1.
1. 11 = 2  ____ + ____
2. 13 = 2  ____ + ____
3. 15 = 2  ____ + ____
4. 17 = 2  ____ + ____
5. 19 = 2  ____ + ____
6. 21 = 2  ____ + ____
1. 11 = 2  5 + 1
2. 13 = 2  6 + 1
3. 15 = 2  7 + 1
4. 17 = 2  8 + 1
5. 19 = 2  9 + 1
6. 21 = 2  10 + 1
Answers
Remember:
You can write any even number as 2 times a natural number.
You can write any odd number except 1 as
2 times a natural number plus 1.
11
Algebra Backpack
Algebra-Numbers-01
You can write any odd number as 2 times a natural number minus 1:

Odd numbers: 1 = 2  1 – 1
3=22–1
5=23–1
7=24–1
9=25–1
Your Turn Write each odd number as 2 times a natural number minus 1.
1. 11 = 2  ____  ____
2. 13 = 2  ____  ____
3. 15 = 2  ____  ____
4. 17 = 2  ____  ____
5. 19 = 2  ____  ____
6. 21 = 2  ____  ____
1. 1 = 2  6  1
2. 13 = 2  7  1
3. 15 = 2  8  1
4. 17 = 2  9  1
5. 19 = 2  10  1
6. 21 = 2  11  1
Answers
Remember these odd number and even number alakazams:
Examples
The sum of two odd numbers is an even number.
3+5=8
9 + 7 = 16
The sum of two even numbers is an even number.
2+4=6
8 + 6 = 14
The sum of an odd number and an even number is an odd number.
3+4=7
9 + 6 = 15
The sum of an even number and an odd number is an odd number
4+3=7
6 + 9 = 15
The product of two odd numbers is an odd number.
3  5 = 15
9  7 = 63
The product of two even numbers is an even number.
2  6 = 12
8  4 = 32
The product of an odd number and an even number is an even
number.
3  8 = 24
15  4 = 60
8  3 = 24
4  15 = 60
The product of an even number and an odd number is an even
number.
Sum or product
odd number
odd number + odd number
even number
x
odd number + even number
x
even number + odd number
x
even number + even number
x
odd number  odd number
x
odd number  even number
x
even number  odd number
x
even number  even number
x
12
Algebra Backpack
Algebra-Numbers-01
Your Turn Put an x under each heading (odd number or even number) that describes the sum
or product in column 1.
Sum or product
odd number
even number
1. odd number + even number
___
___
2. odd number + odd number
___
___
3. even number  even number
___
___
4. even number + odd number
___
___
5. odd number  even number
___
___
6. even number + even number
___
___
7. even number  odd number
___
___
8. odd number  odd number
___
___
odd number
even number
Answers
Sum or product
1. odd number + even number
x
2. odd number + odd number
x
3. even number  even number
x
x
4. even number + odd number
5. odd number  even number
x
6. even number + even number
x
7. even number  odd number
x
8. odd number  odd number
x
odd + odd = even
odd + even = odd
even + odd = odd
even + even = even
odd  odd = odd
odd  even = odd
even  odd = even
even  even = even
13
Algebra Backpack
Algebra-Numbers-01
Whole Numbers | TOC
You can use natural numbers to count objects: 1 object, 2 objects, 3 objects, and so on. But how
do you count no objects? Easy – use the number zero (0) to mean no objects. If we tack on zero
(0) to the set (bunch; collection) of natural numbers, we get a new set of numbers called the
whole numbers.
 The whole numbers are 0 and the natural numbers. 

Natural numbers:

Whole numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
Let's use whole numbers to count little squares (■) from no square to 5 squares.
0
1
■
2
■■
3
■■■
4
5
and so on
■■■■ ■■■■■
Here are natural numbers and whole numbers displayed on number lines:

┼──┼──┼──┼──┼──┼──┼──┼──┼───►
Natural numbers:
1

Whole numbers:
2
3
4
5
6
7
8
9 ...
┼──┼──┼──┼──┼──┼──┼──┼──┼──┼───►
0
1
2
3
4
5
6
7
8
9 ...
Some math sources say that zero is a natural number. In this unit, we
assume that zero is a whole number, but not a natural number.
Is zero an odd number or an even number? There are differences of
opinion, but we will go with zero as an even number, so remember:
Zero is an even number.
14
Algebra Backpack
Algebra-Numbers-01
In the table below, we put an x under each heading that describes the number in the first column.
Number
Natural number
0
Whole number
Odd number
Even number
x
1
x
x
2
x
x
3
x
x
4
x
x
5
x
x
x
x
x
x
x
x
Your Turn Put an x under each heading that describes the number in the first column.
Number
Natural number
Whole number
Odd number
Even number
0
_____
_____
_____
_____
6
_____
_____
_____
_____
9
_____
_____
_____
_____
13
_____
_____
_____
_____
24
____
_____
_____
_____
37
____
_____
_____
_____
Natural number
Whole number
Odd number
Even number
Answers
Number
0
x
x
x
6
x
x
9
x
x
x
13
x
x
x
24
x
x
37
x
x
15
x
x
Algebra Backpack
Algebra-Numbers-01
Here is a handy summary of natural numbers, whole numbers, odd numbers, and even numbers.

Natural numbers:

Whole numbers:

Odd numbers:

Even numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
1,
0,
3,
2,
5,
4,
7,
6,
9, ...
8,
...
Below we show number lines with natural numbers and whole numbers, and then number lines
with odd numbers and even numbers graphed by dots (●).

┼──┼──┼──┼──┼──┼──┼──┼──┼───►
Natural numbers:
1

Whole numbers:
1
Even numbers:
4
5
6
7
8
9 ...
2
3
4
5
6
7
8
9 ...
●──┼──●──┼──●──┼──●──┼──●───►
Odd numbers:
1

3
┼──┼──┼──┼──┼──┼──┼──┼──┼──┼───►
0

2
2
3
4
5
6
7
8
9 ...
●──┼──●──┼──●──┼──●──┼──●──┼───►
0
1
2
3
16
4
5
6
7
8
9 ...
Algebra Backpack
Algebra-Numbers-01
Zero (0) and One (1) | TOC
The whole numbers zero (0) and one (1) are very interesting. In this section, we'll tell you some
interesting properties of zero and one.
0 (zero). If you add 0 to any whole number, then the sum is that whole number.
If n is any whole number, then n + 0 = n and 0 + n = n.
0+0=0
1+0=1
0+1=1
2+0=2
0+2=2
3+0=3
0+3=3
4+0=4
0+4=4
5+0=5
0+5=5
et cetera
and so on
If you subtract any whole number from itself, then the result is zero (0).
If n is a whole number, then n – n = 0.
00=0
11=0
22=0
33=0
44=0
5  5= 0
et cetera
If you multiply any whole number by zero, then the result is zero (0).
If n is a whole number, then n  0 = 0 and 0  n = 0.
00=0
10=0
01=0
20=0
02=0
30=0
03=0
40=0
04=0
50=0
05=0
et cetera
and so on
Division by 0 is a no no, not allowed. Division by 0 is undefined.
Your Turn Show off your knowhow about the mathemagical number zero (0).
1. 7 + 0 = ____
2. 0 + 7 = ____
3. 8 + 0 = ____
4. 0 + 8 = ____
5. 7  0 = ____
6. 0  7 = ____
7. 8  0 = ____
8. 0  8 = ____
9. 7  7 = ____
10. 8  8 = ____
11. 0 – 0 = ____
12. 10 – 10 = 0
13. 5  0 is _________________________
15. 0  0 is ___________________________
Answers
1. 7 + 0 = 7
2. 0 + 7 = 7
3. 8 + 0 = 8
4. 0 + 8 = 8
5. 7  0 = 0
6. 0  7 = 0
7. 8  0 = 0
8. 0  8 = 0
9. 7  7 = 0
10. 8  8 = 0
11. 0 – 0 = 0
12. 10 – 10 = 0
13. 5  0 is a no no, not allowed.
15. 0  0 is a no no, not allowed.
17
Algebra Backpack
Algebra-Numbers-01
1 (one ). If you multiply any whole number by 1, then the product is that whole number, the
number that you multiplied by 1.
If n is a whole number, then n  1 = n and 1  n = n.
01=0
11=1
21=2
31=3
41=4
51=5
et cetera
10=0
11=1
12=2
13=3
14=4
15=5
and so on
Your Turn Show off your knowhow about the mathemagical number one (1).
1. 7  1 = ____
2. 8  1 = ____
3. 10  1 = ____
4. 20  1 = ____
5. 1  7 = ____
6. 1  8 = ____
7. 1  10 = ____
8. 1  20 = ____
9. 12  ____ = 12
10. ____  13 = 13
11. 16  ____ = 16
12. ____  24 = 24
10. 1  ____ = 9
11. ____  1 = 9
12. 1  ____ = 37
13. ____  1 = 37
13. If w is any whole number, then w  1 = ____ and 1  w = ____.
Answers
1. 7  1 = 7
2. 8  1 = 8
3. 10  1 = 10
4. 20  1 = 20
5. 1  7 = 7
6. 1  8 = 8
7. 1  10 = 10
8. 1  20 = 20
9. 12  1 = 12
10. 1  13 = 13
11. 16  1 = 16
12. 1  24 = 24
10. 1  9 = 9
11. 9  1 = 9
12. 1  37 = 37
13. 37  1 = 37
13. If w is any whole number, then w  1 = w and 1  w = w.
If n is a whole number, then n + 0 = n and 0 + n = n.
If n is a whole number, then n – n = 0.
If n is a whole number, then n  0 = 0 and 0  n = 0.
If n is a whole number, then n  1 = n and 1  n = n.
Division by 0 is not allowed. It is undefined. Don’t do it!
n  0 is undefined and n / 0 is undefined.
If you divide by 0 on a calculator, you will get an error message.
18
Algebra Backpack
Algebra-Numbers-01
Square Numbers | TOC
If you multiply a natural number by itself, the product is the square of the number. The set
(bunch, collection, whole shebang) of these numbers is the set of square numbers.
The first six square numbers are 1, 4, 9, 16, 25, and 36.
11=1
22=4
33=9
4  4 = 16
5  5 = 25
6  6 = 36
You can model a square number by a square arrangement of rows and columns of little squares
with the same number of little squares in each row and column.
■
■ ■
■ ■
■ ■ ■
■ ■ ■
■ ■ ■
1=11
4=22
9=33
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
16 = 4  4
■ ■ ■ ■ ■
■ ■ ■ ■ ■
■ ■ ■ ■ ■
■ ■ ■ ■ ■
■ ■ ■ ■ ■
25 = 5  5
Your Turn To test what you have learned about square numbers, romp through these exercises.
1. Calculate (a) the square of 7 ____ and (b) the square of 8. ____
2. The square number 100 is the square of what number? ____
3. The square number 36 can be modeled by a square arrangement of rows and columns of little
squares. How many rows? ____. How many columns? ____.
5. The number 121 is a square number. 121 = ____  ____.
Answers
1. (a) The square of 7 is 7  7 = 49. (b) The square of 8 is 8  8 = 64.
2. The square number 100 is the square of 10. 10  10 = 100.
3. The square number 36 can be modeled by a square arrangement of 6 rows and 6 columns of
little squares.
5. The number 121 is a square number. 121 = 11  11.
121 is a palindrome, a number that is the same from left to right and right to left.
19
Algebra Backpack
Algebra-Numbers-01
Alakazam! Every square number is the sum of consecutive odd numbers beginning with 1.
Consecutive odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and so on.

Square number 1 is equal to 1.
The one-and-only first odd number.

Square number 4 is equal to 1 + 3 = 4.
Sum of the first two odd numbers.

Square number 9 is equal to 1 + 3 + 5 = 9
Sum of the first three odd numbers.
Aha! A pattern. Whenever we see a pattern, we like to make a table like the one below.
Square number
Sum of consecutive odd numbers beginning with 1
11=1
1
22=4
1+3=4
33=9
4  4 = 16
5  5 = 25
1+ 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
Your Turn. Continue the above table (pattern) by filling in the missing stuff.
Square number
6  6 = 36
___  ___ = 49
Sum of consecutive odd numbers beginning with 1
1 + 3 + 5 + 7 + ___ + ___ = 36
1 + 3 + 5 + 7 + ___ + ___ + ___ = 49
8  8 = ___
___ + ___ + ___ + ___ + 9 + 11 + 13 + 15 = 64
___  ___ = 81
1 + 3 + 5 + 7 + ___ + ___ + ___ + 15 + 17 = 81
10  10 = _____
____________________________________________________
Answers
6  6 = 36
1 + 3 + 5 + 7 + 9 + 11 = 36
7  7 = 49
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
8  8 = 64
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64
9  9 = 81
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
10  10 = 100
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
20
Algebra Backpack
Algebra-Numbers-01
Triangular Numbers | TOC
Triangular numbers are among our favorite things. The first triangular number is 1. The second
triangular number is 1 + 2 = 3. The third triangular number is 1 + 2 + 3 = 6. Do you see the
pattern? Each and every triangular number is the sum of consecutive natural numbers.

Triangular numbers keep on going and going and going – never ending.
You can model a triangular number by a triangular arrangement of little squares. Here we go
modeling triangular numbers 1, 3, 6, 10, and 15 with little squares.
■
■
■ ■
■
■ ■
■ ■ ■
■
■ ■
■ ■ ■
■ ■ ■ ■
1
3=1+2
6=1+2+3
10 = 1 + 2 + 3 + 4
■
■ ■
■ ■ ■
■ ■ ■ ■
■ ■ ■ ■ ■
15 = 1 + 2 + 3 + 4 + 5
Your Turn. Complete the following sentences by filling in the missing stuff.
1. The fourth triangular number is 1 + 2 + 3 + 4 = ____
2. The fifth triangular number is 1 + 2 + 3 + 4 + ____ = 15
3. The sixth triangular number is 1 + 2 + 3 + 4 + 5 + ____ = ____
4. The seventh triangular number is ______________________________________________
Answers
1. The fourth triangular number is 1 + 2 + 3 + 4 = _10_
2. The fifth triangular number is 1 + 2 + 3 + 4 + 5 = 15
3. The sixth triangular number is 1 + 2 + 3 + 4 + 5 + 6 = _21_
4. The seventh triangular number is 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
Pick a secret natural number. If we call your secret natural number n, then the nth triangular
number is the sum of the natural numbers from 1 up to your secret natural number (n). 1 plus 2
plus 3 and so on up to n. Mathematicians like to write the sum of natural numbers from 1 to n as
1+2+3+…+n
Learn more about triangular numbers at http://en.wikipedia.org/wiki/Triangular_number.
21
Algebra Backpack
Algebra-Numbers-01
Mathemagical Black Hole 123 | TOC
Start with a number from a designated set of numbers. For example, start with a natural
number. Apply a process (a sequence of mathematical operations, a step-by-step procedure) to
the starting number and get a new number. If the new number is the same as the starting number,
then the starting number is a mathemagical black hole using that process.
The first mathemagical black hole that we learned about was Dr. Michael Ecker's Mathemagical
Black Hole 123. We recommend it to amaze and amuse friends and students from first grade up.
It requires only counting and catenating (putting together). Catenate 1, 2, and 3 and you get
123. Catenate 12, 34, and 56 and you get 123456. Catenate g, a, m, and e and you get game.
Catenate. Put together, put side by side.
Catenate f, u, and n to get fun.
Catenate 12, 34, and 56 to get 123456.
(catenate is also called concatenate)
Start with 123.
Count the number of even digits. The number 123 has 1 even digit.
Count the number of odd digits. The number 123 has 2 odd digits.
Count the total number of digits. The number 123 has 3 total digits.
Catenate (put together) the three numbers in the order: even odd total. You get 123. Voila!
Black hole 123.
Below is a handy table showing the count & catenate process applied to the number 123.
Start with 123
number
# even
# odd
total #
123
1
2
3
123
Black hole 123
The number 123 is a mathemagical black hole using the count & catenate process. Start with any
natural number, apply this process repeatedly, and you will end up in black hole 123.
Even digits are 0, 2, 4, 6, and 8.
Odd digits are 1, 3, 5, 7, and 9.
22
Algebra Backpack
Algebra-Numbers-01
Hey! To play this black hole game, all you need to know is how to count and catenate (put
together) three counting numbers. That’s why we call it the count & catenate process.
Apply the count & catenate process to the numbers 213 and 321.
Start with 213.
Count the number of even digits. The number 213 has 1 even digit.
Count the number of odd digits. The number 213 has 2 odd digits.
Count the total number of digits. The number 213 has 3 digits.
Catenate the three numbers in the order: even odd total. You get 123. The number 213 collapses
into black hole 123 in one application of the count & catenate process.
Start with 213
number
# even
# odd
total #
213
1
2
3
123
Black hole 123
Start with 321.
Count the number of even digits. The number 321 has 1 even digit.
Count the number of odd digits. The number 321 has 2 odd digits.
Count the total number of digits. The number 321 has 3 digits.
Catenate the three numbers in the order: even odd total. You get 123. The number 321 moseys
into black hole 123 in one application of the count & catenate process.
Start with 321
number
# even
# odd
total #
321
1
2
3
123
Black hole 123
The numbers 123, 213, and 321 have the same digits arranged in different order. They are
permutations. The digits 1, 2, and 3 can be arranged in six permutations: 123, 132, 213, 231,
312, and 321.
23
Algebra Backpack
Algebra-Numbers-01
Your Turn Apply the count & catenate process to the number 231. Complete the numbered
steps and the table.
1. Count the number of even digits. The number 231 has ___ even digit(s).
2. Count the number of odd digits. The number 231 has ___ odd digit(s).
3. Count the total number of digits. The number 231 has ___ digits.
4. Catenate your three answers above. ______
Did 231 plop into black hole 123? ______
Start with 231
number
# even
# odd
total #
231
___
___
___
______
Black hole 123? ______
Answers
1. Count the number of even digits. The number 231 has 1 even digit.
2. Count the number of odd digits. The number 231 has 2 odd digits.
3. Count the total number of digits. The number 231 has 3 digits.
4. Catenate your three answers above. 123
Did 231 plop into black hole 123? Yes
Start with 231
number
# even
# odd
total #
231
1
2
3
123
Black hole 123? Yes
On the next page we will apply the count and catenate process to the numbers 357 and 246.


357 has no even digits. The number of even digits is 0.
246 has no odd digits. The number of odd digits is 0.
Think about how it might go, and then amble on down to the next page.
24
Algebra Backpack
Algebra-Numbers-01
Start with 357.
Count the number of even digits. The number 357 has 0 even digits.
Count the number of odd digits. The number 357 has 3 odd digits.
Count the total number of digits. The number 357 has 3 digits.
Catenate 0, 3, and 3 and get 033. Be sure to include the leading 0 when you catenate.
Count the number of even digits. The number 033 has 1 even digit.
Count the number of odd digits. The number 033 has 2 odd digits.
Count the total number of digits. The number 033 has 3 digits.
Catenate 1, 2, and 3 and get 123. The number 357 is sucked into black hole 123 in two
applications of the count & catenate process.
Start with 357
number
# even
# odd
total #
357
0
3
3
033
1
2
3
123
Black hole 123
Question. Permutations: 357, 375, 537, 573, 735, and 753. If you apply the count & catenate
process to one of these permutations, will it be gobbled up by black hole 123?
Start with 246.
Count the number of even digits. The number 246 has 3 even digits.
Count the number of odd digits. The number 246 has 0 odd digits.
Count the total number of digits. The number 246 has 3 digits.
Catenate 3, 0, and 3 and get 303. Apply the process to 303.
Count the number of even digits. The number 303 has 1 even digit.
Count the number of odd digits. The number 303 has 2 odd digits.
Count the total number of digits. The number 303 has 3 digits.
Catenate 1, 2, and 3 and get 123. The number 246 is zoomed into black hole 123 in two
applications of the count & catenate process.
Start with 246
number
# even
# odd
total #
246
3
0
3
303
1
2
3
123
Black hole 123
25
Algebra Backpack
Algebra-Numbers-01
Your Turn Apply the count & catenate process to the number 537. Complete the table.
Start with 537
number
# even
# odd
total #
537
___
___
___
______
___
___
___
______
___
___
___
______
Black hole 123? ______
Answers
Start with 1234
number
# even
# odd
total #
537
0
3
3
033
1
2
3
123
Black hole 123? Yes
Together we have applied the count & catenate process to 3-digit numbers. They all got pulled
into black hole 123. Does the process work for 1-digit numbers and 2-digit numbers? Let's find
out.
Start with 0
number
# even
# odd
total #
0
1
0
1
101
1
2
3
123
Black hole 123
Two applications of the count & catenate process nudge 0 into black hole 123. Recall that 0 is a
whole number, but not a natural number according to the definitions in this unit.
26
Algebra Backpack
Algebra-Numbers-01
Apply the process to 1. [Recall: 1 is a whole number and 1 is a natural number.]
Start with 1
number
# even
# odd
total #
1
0
1
1
011*
1
2
3
123
Black hole 123
*Remember to include 0 when you catenate.
Your Turn Apply the count & catenate process to the 1-digit number 7.
Start with 11
number
# even
# odd
total #
7
___
___
___
_____
___
___
___
_____
Black hole 123? _____
Answers
Start with 7
number
# even
# odd
total #
7
0
1
1
011
1
2
3
123
Black hole 123? Yes
Let's boldly move on to much larger numbers.
27
Algebra Backpack
Algebra-Numbers-01
Start with 1223334444.
Count the number of even digits. The number 1223334444 has 6 even digits.
Count the number of odd digits. The number 1223334444 has 4 odd digits.
Count the total number of digits. The number 1223334444 has 10 digits.
Catenate 6, 4, and 10 to get 6410. Apply the process to 6410.
Count the number of even digits. The number 6410 has 3 even digits.
Count the number of odd digits. The number 6410 has 1 odd digit.
Count the total number of digits. The number 6410 has 4 digits.
Catenate 3, 1, and 4 to get 314. Apply the process to 314.
Count the number of even digits. The number 314 has 1 even digit.
Count the number of odd digits. The number 314 has 2 odd digits.
Count the total number of digits. The number 314 has 3 digits.
Catenate 1, 2, and 3 to get 123. Black hole.
The number 122333444 was gently eased into black hole 123 in three applications of the count &
catenate process. Here are the steps in handy table form.
Start with 1223334444
number
# even
# odd
total #
1223334444
6
4
10
6410
3
1
4
314
1
2
3
123
Black hole.
Serendipity! The table shows the action in a compact way compared to the verbose description
above the table. It is easy to see every step in the count & catenate process. We like it. What do
you think?
28
Algebra Backpack
Algebra-Numbers-01
Your Turn Apply the count & catenate process to the number 123456789.
Start with 123456789
number
# even
# odd
total #
123456789
___
___
___
_____
___
___
___
123
Black hole.
Answers
Start with 123456789
number
# even
# odd
total #
123456789
4
5
9
459
1
2
3
123
Black hole.
Investigate
1. Start with 1234567890. Arrive at black hole 123 in three applications of the count & catenate
process. Show the steps in a table.
2. Start with 122333444455555666666777777788888888999999999. Tumble into black hole
123 in four applications of the count & catenate process.
3. Find a number that requires five applications of the count & catenate process to zap it into
black hole 123.
4. Find a number that requires n applications of the count & catenate process to whirl it into
black hole 123. You choose the value of n.
29
Algebra Backpack
Algebra-Numbers-01
Questions
1. There are six ways to arrange the digits 1, 2, and 3 into a 3-digit number. These ways are
called permutations.
The permutations are 123, 132, 213, 231, 312, and 321.
You know that 123 is a black hole. How many applications of the count & catenate process will
it take to stuff 132, 213, 231, 312, or 321 into black hole 123?
2. The permutations of 235 are 235, 253, 325, 352, 523, and 532. The number 235 homed in on
black hole 123 in one application of the count & catenate process. How many applications of the
process will it take to plunge 253, 325, 352, 523, or 532 into black hole 123?
3. The permutations of 357 are 357, 375, 537, 573, 735, and 753. The number 357 arrived at
black hole 123 in two applications of the count & catenate process. How many applications of
the count & catenate process will it take to drag 375, 537, 573, 735, or 753 into black hole 123?
4. Examine these numbers: 101, 121, 141, 161, and 181. For each of these numbers, how many
applications of the count & catenate process will it take to slurp it into black hole 123?
5. Examine these numbers: 131, 151, 171, and 191. For each of these numbers, how many
applications of the count & catenate process will it take to suck it into black hole 123?
6. Examine these numbers: 212, 232, 252, 272, and 292. For each of these numbers, how many
applications of the count & catenate process will it take to suck it into black hole 123?
Answers
1. One application. Each of these numbers has 1 even digit, 2 odd digits, and 3 total digits.
Catenate 1, 2, and 3 to get 123. Black hole.
2. One application. Each of these numbers has 1 even digit, 2 odd digits, and 3 total digits.
Catenate 1, 2, and 3 to get 123. Black hole.
3. Two applications. Each permutation of 357 starts out with 0 even digits, 3 odd digits, and 3
total digits. Catenate 0, 3, and 3 to get 033. The number 033 has 1 even digit, 2 odd digits, and 3
total digits. Catenate 1, 2, and 3 to get 123. Black hole.
4. One application. Each of these numbers has 1 even digit, 2 odd digits, and 3 total digits.
Catenate 1, 2, and 3 to get 123. Black hole.
5. Two applications. Each of these numbers has 0 even digits, 3 odd digits, and 3 total digits.
Catenate 0, 3, and 3 to get 0333. The number 033 has 1 even digit, 2 odd digits, and 3 total
digits. Catenate 1, 2, and 3 to get 123. Black hole.
6. Two applications. Each of these numbers has 2 even digits, 1 odd digit, and 3 total digits.
Catenate 2, 1, and 3 to get 213. The number 213 has 1 even digit, 2 odd digits, and 3 total digits.
Catenate 1, 2, and 3 to get 123. Black hole.
30
Algebra Backpack
Algebra-Numbers-01
Self-Test Exercises (followed by answers) | TOC
1. Answer T for True or F for False to each statement.
a. The first natural number is 1. ____
b. The first whole number is 1. ____
c. The least natural number is 0. ____
c. The least whole number is 0. ____
c. Every natural number has a successor that is a natural number. ____
d. Every natural number has a predecessor that is a natural number. ____
e. The last natural number is the number 999,999,999,999,999,999. ____
f. The greatest natural number is the number 999,999,999,999,999,999. ____
2. Graph the whole numbers 0, 4, and 7 on the number line below:
┼──┼──┼──┼──┼──┼──┼──┼──┼──┼───►
0
1
2
3
4
5
6
7
8
9 ...
3. Graph odd numbers and even numbers on these number lines.
┼──┼──┼──┼──┼──┼──┼──┼──┼───►
Odd numbers:
1
Even numbers:
2
3
4
5
6
7
8
9 ...
┼──┼──┼──┼──┼──┼──┼──┼──┼──┼───►
0
1
2
3
4
5
6
7
8
9 ...
4. Complete each sentence by writing the successor or predecessor of each natural number.
a. The successor of 1 is _____.
b. The predecessor of 1 is _____.
c. The successor of 9 is _____.
d. The predecessor of 9 is _____.
e. The successor of 100 is _____.
f. The predecessor of 100 is _____.
31
Algebra Backpack
Algebra-Numbers-01
5. Split each even number into two equal natural numbers.
1. Split 4 into _____ and _____.
2. Split ■■■■ into ______ and ______.
3. Split ■■■■■■ into ________ and ________.
4. Split 6 into _____ and _____.
5. Split 16 into _____ and _____.
6. Split 24 into ____ and ____.
6. Write each even number as a) 2  a natural number and b) a natural number  2.
1a. 10 = 2  ____
1b. 10 = ____  2
2a. 16 = 2  ____
2b. 16 = ____  2
3a. 24 = 2  ____
3b. 24 = ____  2
4a. 60 = 2  ___
4b. 60 = ____  2
7. Split each odd number into two equal natural numbers and remainder 1.
a. Split ■■■■■■■ into _____________ and _____________, and remainder _____
b. Split 9 into ____ and ____, remainder ___.
c. Split 13 into ____ and ____, remainder ___.
d. Split 19 into ____ and ____, remainder ___.
e. Split 21 into ____ and ____, remainder ___.
8. Write each odd number as 2 times a natural number plus 1.
a. 9 = 2  ____ + ____
b. 13 = 2  ____ + ____
c. 15 = 2  ____ + ____
d. 17 = 2  ____ + ____
e. 19 = 2  ____ + ____
f. 21 = 2  ____ + ____
Reminder: You cannot write 1 as 2 times a natural number + 1.
9. Write each odd number as 2 times a natural number minus 1.
a. 1 = 2  ____  ____
b. 13 = 2  ____  ____
c. 15 = 2  ____  ____
d. 17 = 2  ____  ____
e. 19 = 2  ____  ____
f. 21 = 2  ____  ____
32
Algebra Backpack
Algebra-Numbers-01
10. Put an x under each heading that describes the sum or product in column 1.
Sum or product
odd number
even number
a. odd number + odd number
___
___
b. odd number + even number
___
___
c. even number + even number
___
___
d. odd number  odd number
___
___
e. odd number  even number
___
___
f. even number  even number
___
___
11. Answer T for True or F for False to each statement about properties of 1 and 0.
a. 7  1 = 7. ___
b. 7 + 0 = 7. ___
c. If a is a whole number, then a  1 = a. ___
d. If a is a whole number, then a + 0 = a. ___
e. If a is a whole number, then 1  a = a. ___
f. If a is a whole number, then 0 + a = a. ___
g. If a ≠ 0, then a / a = 1. ___
h. a – a = 0. ___
i. 1 / 0 = the largest whole number. ___
j. 0 / 0 = 1. ___
k. If a ≠ 0, then 0 / a = 0. ___
l. 0 / 0 = 0. ___
12. In the list of square numbers from 1 to 100 below, there are a few omissions. Complete the
list by filling in the blanks.
1
4
____
16
____
36
____
64
13. Write each square number as the sum of consecutive odd numbers.
Square number
Sum of consecutive odd numbers beginning at 1
4  4 = 16
_______________________________________________
6  6 = 36
_______________________________________________
33
____
100
Algebra Backpack
Algebra-Numbers-01
14. In the list of triangular numbers from 1 to 55 below, there are a few omissions. Complete the
list by filling in the blanks.
1
3
____
10
____
21
____
36
____
55
15. Write each triangular number as the sum of consecutive natural numbers beginning with 1.
Triangular number
Sum of consecutive natural numbers beginning with 1
10
_______________________________________________
21
_______________________________________________
36
_______________________________________________
16. Put an x under each heading that describes the number in column 1.
Number
Natural
number
Whole
number
Odd
number
Even
number
Square
number
Triangular
number
0
1
9
10
36
17. Show that the count & catenate process swoops 1,000,000,000 into black hole 123 in three
applications of the process.
Start with 1,000,000,000
number
# even
# odd
1,000,000,000
123
Black hole 123
34
total #
Algebra Backpack
Algebra-Numbers-01
Self-Test Answers | TOC
1. Answer T for True or F for False to each statement.
a. The first natural number is 1. T
b. The first whole number is 1. F
c. The least natural number is 0. F [It is 1]
c. The least whole number is 0. T
[It is 0]
c. Every natural number has a successor that is a natural number. T
d. Every natural number has a predecessor that is a natural number. F
[1 has no predecessor]
e. The last natural number is the number 999,999,999,999,999,999. F
[No last number]
f. The greatest natural number is the number 999,999,999,999,999,999. F
[No greatest number]
2. Graph the whole numbers 0, 4, and 7 on the number line below:
●──┼──┼──┼──●──┼──┼──●──┼──┼───►
0 1 2 3 4 5 6 7 8 9 ...
3. Graph odd numbers and even numbers on these number lines.
●──┼──●──┼──●──┼──●──┼──●───►
Odd numbers:
1
Even numbers:
2
3
4
5
6
7
8
9 ...
●──┼──●──┼──●──┼──●──┼──●──┼───►
0
1
2
3
4
5
6
7
8
9 ...
4. Complete each sentence by writing the successor or predecessor of each natural number.
a. The successor of 1 is _____.
b. The predecessor of 1 is _____.
c. The successor of 9 is _____.
d. The predecessor of 9 is _____.
e. The successor of 100 is _____.
f. The predecessor of 100 is _____.
35
Algebra Backpack
Algebra-Numbers-01
5. Split each even number into two equal natural numbers.
1. Split 4 into 2 and 2.
2. Split ■■■■ into ■■ and ■■.
3. Split ■■■■■■ into ■■■ and ■■■.
4. Split 6 into 3 and 3.
5. Split 16 into 8 and 8.
6. Split 24 into 12 and 12.
6. Write each even number as a) 2  a natural number and b) a natural number  2.
1a. 10 = 2  5
1b. 10 = 5  2
2a. 16 = 2  8
2b. 16 = 8  2
3a. 24 = 2  12
3b. 24 = 12  2
4a. 60 = 2  30
4b. 60 = 30  2
7. Split each odd number into two equal natural numbers and a remainder of 1.
a. Split ■■■■■■■ into ■■■ and ■■■, and remainder ■
b. Split 9 into 4 and 4, remainder 1.
c. Split 13 into 6 and 6, remainder 1.
d. Split 19 into 9 and 9, remainder 1.
e. Split 21 into 10 and 10, remainder 1.
8. Write each odd number as 2 times a natural number plus 1.
a. 9 = 2  4 + 1
b. 13 = 2  6 + 1
c. 15 = 2  7 + 1
d. 17 = 2  8 + 1
e. 19 = 2  9 + 1
f. 21 = 2  10 + 1
Reminder: You cannot write 1 as 2 times a natural number plus 1.
9. Write each odd number as 2 times a natural number minus 1.
a. 1 = 2  1  1
b. 13 = 2  7  1
c. 15 = 2  8  1
d. 17 = 2  9  1
e. 19 = 2  10  1
f. 21 = 2  11  1
36
Algebra Backpack
Algebra-Numbers-01
10. Put an x under each heading that describes the sum or product in column 1.
Sum or product
odd number
even number
a. odd number + odd number
1 + 3 = 4, 7 + 5 = 12, …
x
b. odd number + even number
1 + 2 = 3, 5 + 4 = 9, …
x
c. even number + even number
2 + 4 = 6, 8 + 6 = 14, …
x
d. odd number  odd number
examples
1  3 = 3, 7  5 = 35, …
x
e. odd number  even number
x
3  4 = 6, 5  2 = 10, …
f. even number  even number
x
2  4 = 8, 10  6 = 60, …
11. Answer T for True or F for False to each statement about properties of 1 and 0.
a. 7  1 = 7. T
b. 7 + 0 = 7. T
c. If a is a whole number, then a  1 = a. T
d. If a is a whole number, then a + 0 = a. T
e. If a is a whole number, then 1  a = a. T
f. If a is a whole number, then 0 + a = a. T
g. If a ≠ 0, then a / a = 1. T
h. a – a = 0. T
i. 1 / 0 = the largest whole number. F
j. 0 / 0 = 1. F
k. If a ≠ 0, then 0 / a = 0. T
l. 0 / 0 = 0. F
12. In the list of square numbers from 1 to 100 below, there are a few omissions. Complete the
list by filling in the blanks.
1
4
9
16
25
36
49
13. Write each square number as the sum of consecutive odd numbers.
Square number
Sum of consecutive odd numbers beginning at 1
4  4 = 16
16 = 1 + 3 + 5 + 7
6  6 = 36
36 = 1 + 3 + 5 + 7 + 9 + 11
37
64
81
100
Algebra Backpack
Algebra-Numbers-01
14. In the list of triangular numbers from 1 to 55 below, there are a few omissions. Complete the
list by filling in the blanks.
1
3
6
10
15
21
28
36
45
55
15. Write each triangular number as the sum of consecutive natural numbers beginning with 1.
Triangular number
Sum of consecutive natural numbers beginning with 1
10
10 = 1 + 2 + 3 + 4
21
21 = 1 + 2 + 3 + 4 + 5 + 6
36
36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8
16. Put an x under each heading that describes the number in column 1.
Number
Natural
number
Whole
number
0
Odd
number
Even
number
x
Square
number
Triangular
number
x
x
1
x
x
x
x
9
x
x
x
x
10
x
x
x
36
x
x
x
x
x
x
17. Show that the count & catenate process swoops 1,000,000,000 into black hole 123 in three
applications of the process.
Start with 1,000,000,000
number
# even
# odd
total #
1,000,000,000
9
1
10
9110
1
3
4
134
1
2
3
123
Black hole 123
The end – of the beginning.
38