Download Unit 1:Integers and divisibility

Unit 1:Integers
and divisibility
In this lesson you will learn about:

Integers and hierarchy about how to
operate with them

Powers, roots and their properties

Absolute value
Vocabulary you need in this lesson:
You are going to work with some vocabulary related to this
lesson and the problems you will work with. The same words
appear in the crosswords, in the word scramble and in the
word search.
Some theory
Integers are the set of whole numbers and their opposites. The number line can be used to represent the set of integers. Look carefully at the
number line below and the definitions that follow:
Definitions
 The number line goes on forever in both directions. This is indicated by the
arrows.
 Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.
 Whole numbers less than zero are called negative integers. These numbers
are to the left of zero on the number line.
 The integer zero is neutral. It is neither positive nor negative.
 The sign of an integer is either positive (+) or negative (-), except zero,
which has no sign.
Two integers are opposites if they are each the same distance away from
zero, but on opposite sides of the number line. One will have a positive sign,
the other a negative sign. In the number line above, +3 and -3 are labelled
as opposites.
Rules of operations
 Addition 1) To add two numbers with the same sign, add their absolute value
and write the common sign. 2) To add two numbers with different signs, subtract their absolute values and use the sign of the number with the larger absolute value.
 Subtraction 1) Change the subtraction symbol to addition. 2) Write the additive inverse of the number being subtracted. 3) Use the rules for addition.
Multiplication/Division 1) If both numbers have the same sign, the product
(quotient) is positive. 2) If the numbers have different signs, the product
(quotient) is negative.
Powers and their properties
A power whose base is an integer and whose exponent is a natural number is
the product of a number by itself as many times as the exponent indicates.
Properties
a
0
a
1

a



a
a
a
a

1

a
 b




a
b
m
n

n
 a
n
n
m

n


a
b

a

m
a
n
a
n ´ 
 b
n
n
m
 n
n 
a
m
m
Square root
If b is a non-negative integer:
b/ameans:
 a 2  b

   a  2  b
n
Now it´s time to practice
1.
Exercises
1. Put these temperatures in order on the thermometer: 16°C, 37°C, -33°C, -6°C
25°C, 2°C, 25°C, 1°C, 14°C
2. Tell me an integer to describe each situation:
5 degrees below zero
a loss of 7 pounds
a gain of 10 yards
3 feet below sea level
e. 2 degrees above zero
a.
b.
c.
d.
3. Tell me the opposite of each integer: 3, -5, -151, 1420, -1
4. Write 4 consecutive, even integers beginning with each of the
following integers: -26, 20
5. What number are you on if you:
a. Start at 2 and take away 4
b. Start at -7 and add on 3
c. Start at 5, take away 8 and add on 1
d. Start at -3m take away 3 and add on 4
6. .- Write the correct symbol ( < or > ) between each pair of numbers:
a) 3
5
b) 4
1
c) 6
10
d) 7
1
e) 1
6
f) 7
15
7. Work out:
a. 1  ( 7  2  10 )  (3  8)
b. 13  8  ( 6  3)  4  3  (  7 )
c. 2  7  5  2  3  (10  4 ) 
d . 1  3  4  15  12  4  2  7   5 
e.     2     7 
f.

3

2
4  3  27  5  36
-
2. Problems
Intergalactic Temperature Quest
1. Your Space Commander sends you on a quest around the universe to collect temperatures for
research purposes.
You have the use of a spaceship:
…and a thermometer:
First find the difference between the hottest and coldest temperatures you collected for
each planet. Copy the table into your book. The first one is done for you.
Planet
Hottest recorded
temperature
Coldest recorded
temperature
Difference
Kark
10°C
-5°C
15°C
Hikl
126°C
37°C
Topa
54°C
12°C
Uni-7
-13°C
-179°C
Jolarg
54°C
-68°C
Vortan
104°C
-97°C
Intop
-95°C
-214°C
Yurg
-27°C
-163°C
2. The record high temperature in Nowhere, USA is 98 degrees and the record low is -23
degrees. What is the temperature variation between the record high and the record
low?
3. Jenny has $2. She earns $5, spends $10, earns $4, then spends $3. How many dollars
does she have or owe?
4. The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea
level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is
the distance from the top of Mt. McKinley to the bottom of Death Valley?
5. The temperature in Anchorage, Alaska was 8°F in the morning and dropped to -5°F in the
evening. What is the difference between these temperatures?
6. A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its
new position? Use integer numbers to solve this problem.
7. The Roman Civilization began in 509 B.C. and ended in 476 A.D. How long did the Roman
Civilization last?
8. The Punic Wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last?
9. A proton has a charge of positive one. An electron has a charge of negative one. Find the
total charge of an ion with 14 protons and 19 electrons.
Some interesting web sites!
In the next page you will find a presentation of how to represent
integers on the number line:
http://www.nebo.edu/misc/learning_resources/ppt/6-12/
integers_numberline.ppt#256,1,Integers and the Number Line
To take a self-test and check if you have learnt about integers and
absolute values correctly, go to:
http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?
isbn=0-07-829633-1&chapter=3&lesson=1&headerFile=4&state=na
To read a bit more and do some more practice, take a look at this
page:
http://www.glencoe.com/sec/math/msmath/mac04/course2/
study_guide/pdfs/mac2_pssg03.pdf
To play a game with integers, click here:
http://www.glencoe.com/sec/math/msmath/mac04/course2/
group_activities/lesson.php/na/3l
To take a quiz about integers, where you can choose the length and
the difficulty, go to:
http://www.thatquiz.org/
If you want to read about integers across history and how they appear
and are used by mathematicians, click here:
http://www.bookrags.com/research/integers-wom/