DIDACTIC UNIT: INTEGERS (2ND ESO MATHS)
By Carmen Cano Sarabia.
1. INTRODUCTION AND DESCRIPTION OF THE UNIT
This is the first unit programmed for the 2nd year ESO.
This is the starting unit of the course. It is basically a review of the concepts already
studied last year. Concepts such as multiple, divisor, LCM, GCF, integers, operations
with integers, etc. All this knowledge will be necessary for the later study of other units,
particularly for Fractions or Algebra.
This unit should be developed in 14 sessions.
2. OBJECTIVES
The student will be able to:
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Find divisors and multiples of a given number.
Apply the criteria of divisibility by 2, 3, 5 and 10.
Identify the prime numbers less than 50.
Perform the prime factor decomposition of a number.
Obtain the LCM and the GCF of two or more numbers.
Solve real life situations involving the LCM or the GCF.
Define the set of integers, positive numbers, negative numbers, opposites and
signs.
Identify an integer to represent a given real-life situation.
Identify the opposite of an integer.
Define absolute value.
Determine the absolute value of an integer using the proper notation.
Describe the relationship between distance and absolute value.
Differentiate between the inequality symbols < and >.
Compare two integers, using the proper inequality symbol.
Order a set of integers.
Perform addition of two negative integers and of integers with unlike signs.
Perform subtraction of integers using the arithmetic procedure or using the
number line.
Perform multiplication of two integers with like signs or with unlike signs.
Know and apply the right order in combined operations with integer (BIDMAS).
Perform indices with an integer as base and natural exponent.
Properties of the indices.
Analyze each word problem to identify the given information and develop
problem-solving skills.
Connect integers to the real world.
Examine the solution for each exercise presented in this unit.
Identify and evaluate incorrect answers.
3. CONTENTS
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Multiples and divisors.
Criteria of divisibility by 2, 3, 5 and 10.
Prime numbers and composite numbres.
Greatest common factor and Lowest common multiple.
Integers on the number line.
The absolute value of an integer.
Addition and subtraction of integers.
Multiplication and division of integers.
Combined operations with integers.
Indices of integers.
Rules of indices.
Roots of integers.
4. SESSIONS AND LEARNING ACTIVITIES
The unit will be developed in 14 sessions.
1st Session : Divisibility
Let n be a natural number.
A factor or divisor of n is a natural number that divides n evenly (without a
remainder).
A multiple of n is a natural number that can be divided evenly by n.
A prime number is a natural number (greater than 1) that has no positive divisors other
than 1 and itself. Numbers that have also other factors are called composite numbers.
Prime numbers up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Divisibility Criteria
Divisibility by 2. A number is divisible by 2 if its last digit is 0 or is divisible by 2.
Numbers, which are divisible by 2 are called even numbers. Otherwise, numbers are
called odd numbers.
Divisibility by 3.. A number is divisible by 3, if the sum of its digits is divisible by 3.
Divisibility by 5. A number is divisible by 5, if its last digit is 0 or 5.
Divisibility by 10. A number is divisible by 10, if its last digit is 0.
Activities:
1.
2.
3.
4.
5.
6.
7.
Write three multiples of: a) 7; b) 11; c) 15; d) 24.
Write three factors (or divisors) of: a) 15; b) 50; c) 60; d) 77.
Mark the multiples of 3: 111, 270, 210, 816, 325
Mark the multiples of 4: 544, 3221, 1136, 7732, 7745
Mark the multiples of 5: 881, 345, 657, 650, 776
Mark the multiples of 11: 495, 913, 2794, 3915, 4191
Among the following numbers find:
i. 275, 333, 495, 540, 1202, 8580, 1155, 2873, 3330, 6655
b. Multiples of 3.
c. Multiples of 4.
d. Multiples of 5.
e. Multiples of 11.
2nd Session : GCF and LCM
Greatest common factor (GCF) is the greatest number that divides two given numbers.
Procedure to obtain the GCF
1)
2)
3)
4)
to express each of the numbers as a product of powers of its prime factors;
to write out all common factors in these factorisations;
to take the least power of each of them;
to multiply these powers.
Example:
168 = 2 · 2 · 2 · 3 · 7 = 23 · 31 · 71
180 = 2 · 2 · 3 · 3 · 5 = 22 · 32 · 51
3024 = 2 · 2 · 2 · 2 · 3 · 3 · 3 · 7 = 24 · 33 · 71
GCF(168, 180, 3024) = 22 · 31 = 12
Problem: Your gym teacher is setting up teams for a soccer game. There are 24 fifthgrade students and 28 fourth-grade students on the field. He wants each team to have as
many players as possible. What is the greatest number of teams he can create?
Problem: A scientist is setting up some study tanks. She has collected 12 identical fish
and 18 identical plants. She wants all tanks to be alike and contain as many fish and
plants as possible. What is the greatest number of tanks she can set up?
Least common multiple (LCM) is the smallest (positive) number that is a multiple of
two given numbers.
Procedure to obtain the GCF:
1) to express each of the numbers as a product of powers of its prime factors
(factorise each of the numbers);
2) to write out all factors in these factorisations;
3) to take the greatest power of each of them;
4) to multiply these powers.
Example: 168 = 2 · 2 · 2 · 3 · 7 = 23 · 31 · 71 ,
180 = 2 · 2 · 3 · 3 · 5 = 22 · 32 · 51 ,
3024 = 2 · 2 · 2 · 2 · 3 · 3 · 3 · 7 = 24 · 33 · 71
LCM[168, 180, 3024] = 24 · 33 · 5 · 7 = 15120
Problem: During the summer months, one ice cream truck visits Jeannette's
neighbourhood every 14 days and another ice cream truck visits her neighbourhood
every 15 days. If both trucks visited today, when is the next time both trucks will visit
on the same day?
Problem: Mrs. Hernandez waters one of her plants every 10 days and another plant
every 14 days. If she waters both plants today, when is the next time both plants will be
watered on the same day?
Activities:
a)
b)
c)
d)
e)
Find the LCM and the GCF:
24, 15 and 27.
12, 144 and 36.
100, 120 and 160.
210, 220 and 250.
150, 200 and 250.
f) 1000 and 2100.
g) 2500 and 1750.
h) 11, 33, 55 and 121.
i) 13, 26, 39 and 169.
j) 490 and 363.
3rd Session : Integers
Positive and Negative Integers
We can use integers to represent the following situations:
20320 feet above sea level: +20320
282 feet below sea level: -282
10 degrees (above zero): +10
12 degrees below zero: -12
509 B.C: -509
476 A.D: +476
a loss of 16 dollars: -16
a gain of 5 points: +5
8 steps backward: -8
7 steps forward: +7
Positive integers are all greater than zero: 1, 2, 3, 4, 5, ...
Negative integers are all less than zero: -1, -2, -3, -4, -5, …
We do not consider zero to be a positive or negative number.
The Number Line
The number line is a line labelled with the integers in increasing order from left to right,
that extends in both directions:
For any two different places on the number line, the integer on the right is greater than
the integer on the left.
Examples: 9 > 4, 6 > -9, -2 > -8, and 0 > -5
-2 < 1, 8 < 11, -7 < -5, and -10 < 0
Absolute Value of an Integer
The absolute value of a number is always a positive number (or zero). We specify the
absolute value of a number n by writing n in between two vertical bars: |n|.
Examples: |6| = 6; |-12| = 12; |0| = 0; |1234| = 1234; |-1234| = 1234
Exercise: Complete:
a
Opposite
3
-5
0
-4
Absolute value
Adding Integers
1) When adding integers of the same sign, we add their absolute values, and give the
result the same sign.
Examples: 2 + 5 = 7
(-7) + (-2) = -(7 + 2) = -9
2) When adding integers of the opposite signs, we take their absolute values, subtract
the smaller from the larger, and give the result the sign of the integer with the largest
absolute value.
Examples: 8 + (-3) = 5
8 + (-17) = -15
-22 + 11 = -11
53 + (-53) = 0
Subtracting Integers
Subtracting an integer is the same as adding its opposite.
In the following examples, we convert the subtracted integer to its opposite, and add the
two integers.
Examples: 7 - 4 = 7 + (-4) = 3
12 - (-5) = 12 + (5) = 17
-8 - 7 = -8 + (-7) = -15
-22 - (-40) = -22 + (40) = 18
Exercise. Calculate:
a) 3  5  7  3  7  5 1 
b) 2  5   3  6  5   2 
c) 12  3   7  5 
d)  3  5   3  5 
e)  9  5  1  6 
f)  5  7  8   6  1  12 
Example: 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? Solution: 20320 - (-282) = 20320 + 282 =20602
Example: In Buffalo, New York, the temperature was -14°F in the morning. If the
temperature dropped 7°F, what is the temperature now? Solution: -14 –7= -21
Example: Roman Civilization began in 509 B.C. and ended in 476 A.D. How long did
Roman Civilization last? Solution: 476 –(-509) = 476 + 509 =985
4th Session : Operations with integers
Multiplying Integers
To multiply a pair of integers if both numbers have the same sign, their product is the
product of their absolute values (their product is positive). If the numbers have opposite
signs, their product is the opposite of the product of their absolute values (their product
is negative). If one or both of the integers is 0, the product is 0.
Examples: 4 × 3 = 12
(-4) × (-5) = |-4| × |-5| = 4 × 5 = 20
(-7) × 6 = -42.
12 × (-2) = -24
4 × (-2) × 3 × (-11) × (-5) = -1320.
Dividing Integers
To divide a pair of integers if both integers have the same sign, divide the absolute
value of the first integer by the absolute value of the second integer.
To divide a pair of integers if both integers have different signs, divide the absolute
values and give this result a negative sign.
Examples: 4 ÷ 2 = 2
(-24) ÷ (-3) = 8
(-100) ÷ 25 = -4
98 ÷ (-7) = -14
5th, 6th and 7th Sessions : Combined Operations with integers
THE ORDER OF OPERATIONS (BIDMAS)
1) Brackets
2) Indices or Powers
3) Divisions and Multiplications
4) Additions and Subtractions
Examples:
a) 3 + 28 : 4 + 23 x 6 = 3 + 28 : 4 + 8 x 6 = 3 + 7 + 48 = 58
b) 28 ÷ (4 + 3) + 23 x 6 = 28 ÷ 7 + 23 x 6 = 28 ÷ 7 + 8 x 6 = 4 + 48 = 52
c) (28: 4 + 3) x 23 + 36 : 32 = (7 + 3) x 8 + 36 : 9= 80 + 4 =84
Activities:
1. Work out:
a) 6  2   3  2  5 
b)  3  5   4  2  
c) 3   6  3  3 : 2 
d) 5   2  4  6  2  3  4 
2. Complete:
a b c a · b a - b a + b a - c a·b·c (a-b)·(a+b) a - b·c
-1 3 2
2
-1
3
4
5
-3
-5
-3
3
-2
1
-2
2  3  2  5  1  2  3  6 
b) 5  2  8  1 : 2  3  2  5 
c) 2  4  3  7  2 3 
d)  3  2  5  7  3  2 
e) 2  33  5  2 : 2 
f)  5  24  3  7  6 : 2  5  3 
3. Combined
operations:
a)
g) 4  6   25  3 
2
h) 2  47  2  3  6  
2
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i)  3 5  2  35  3  3  7   4  5 
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3
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j) 2  5  4  7   2  9  5 : 2 
k) 3  26  8  3  5  3  32 
2
8th and 9th Sessions : Indices
Powers or Indices
The expression xn is called a power, where n is the exponent or index and x is the
base:
“x to the nth power” or “x raised to the nth power”
32: 3 squared, 33: 3 cubed, 34: 3 to the forth, etc
Example: 210 = 2·2·2·2·2·2·2·2·2·2 = 1024, 1024 is a power of 2 (it is 2 multiplied by
itself 10 times).
“Anything” to the zero power is 1: x0 = 1
“Anything” to the first power is itself: x1 = x
Negative integers raised to an odd exponent will always result in a negative integer,
while if the exponent is even the result will be a positive integer.
Example: (-2)5 = -32; (-2)4 =16;
But, -(-2)5 = -(-32)=32; -(-2)4 =-16;
RULES OF INDICES
1.
2.
3.
4.
5.
Product of two powers with like bases: xa · xb = xa+b
Quotient of two powers with like bases: xa / xb = xa-b
Power of a power: (xa)b = xab
Power of a product: (xy)a = xa ya
Power of a quotient: (x/y)a = xa / ya
Activities
1. Complete:
Base Exponent
Power
Result
4
3
4
3 =3·3·3·3=81
81
-3
4
3
0
-3
0
7
3
-7
3
-7
0
-7
2
2. Express the result of these calculations as a single power:
a) 39 : 35
b)
e) 39 : 3
f)
8 
5 
2 7
3 7
c ) 29  7 9
d ) (15)6 : 36
g ) 25  75
h) ( 27) 6 : 36
10th Session : Roots
ROOTS
A square root of a given number n is a number that must be multiplied by itself 2 times
to equal n.
A cube root of a given number n is a number that must be multiplied by itself 3 times to
equal n.
Examples:
36  6 2  6;
3
8  3 2 3  2;
4
16  4 2 4  2;
100  10 2  10;
3
125  3 5 3  5;
4
625  4 5 4  5;
11th, 12th and 13th Sessions : Review
Activities
1. Work out:
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a) 55 : 5  2  (1  2) 5  3
b) 2  7  (3  5)  4  22 : 11
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c) 18  2  5  (3)  2 4 : 2
d)  2   3 2   2   33  (2) 0
3
4
e) 1  3  1  3   1  3: 2
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f) 55 : 11  3  (2  1) 5  4
g) 2  2  (3  5)  (5)  22 : 11
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h) 18  2  5  (3)  2 3 : 4
i) 1  2  1  2   1  2: 13
j) 53  5 2  35 : 33  (5) 3  5 2  (2) 2
WORD PROBLEMS
1. The temperature in Alaska was 8°F in the morning and dropped to -5°F in the
evening. What is the difference between these temperatures?
2. Mt. Everest, the highest elevation in Asia, is 29 028 feet above sea level.
The Dead Sea, the lowest elevation, is 1 312 feet below sea level. What is
the difference between these two elevations?
3. A submarine was situated 800 feet below sea level. If it ascends 250 feet, what
is its new position?
4. Maggie owes the candy store $35 and she has $7. Five friends will help
her pay off her debt. How much will each friend pay?
5. Lilly bought 4 pairs of blue jeans at $32 each. How much money did she give to
the clerk if she got $22 back as the change?
6. A submarine was situated 450 feet below sea level. If it descends 300 feet,
what is its new position?
7. In the Sahara Desert one day it was 136°F. In the Gobi Desert a temperature of
- 50°F was recorded. What is the difference between these two temperatures?
8. The Punic Wars began in 264 B.C. and ended in 146 B.C. How long did the
Punic Wars last?
9. Metal mercury at room temperature is a liquid. Its melting point is - 39°C. The
freezing point of alcohol is -114°C. What is the difference between these two
temperatures?
10. Each week I receive 5 euros and I spend 3 euros, how much money will I
have saved in 5 weeks? How many weeks do I have to wait before having
enough money to buy a toy that costs 29 euros?
11. John and Tony have together 77 euros and Tony has 9 euros more than John,
how much money does each of them have?
12. Mary wants to buy a coat and a pullover and she has 100 euros. The
pullover costs 35 euros and the price of the coat is the double, does she
have enough money for buying both things?
13. Edward and Oscar have 45 picture cards and Edward has 13 picture cards
more than Oscar, how many cards does each of them have?
14. Two buses leave the terminal at 8 am. Bus A takes 60 minutes to complete
its route; Bus B takes 75 minutes. When is the next time the two buses
will arrive together at the terminal?
15. Samantha has two pieces of cloth. One piece is 72 inches wide and the other
piece is 90 inches wide. She wants to cut both pieces into trips of equal width
as wide as possible. How wide should she cut the strips?
16. By selling cookies at 24 cents each. Jose made enough money to buy
several cans of pop costing 45 cents per can and he had no money left.
What is the least number of cookies he could have sold? In that case, how
many cans could he buy?
17. Rose will make a game board that is 16 inches by 24 inches for a game she
has invented. She wants to use square tiles. What is the largest tile Rosa can
use?
18. Your gym teacher is setting up teams for a game. There are 24 fifth-grade
students and 28 fourth-grade students on the field. He wants each team to
have as many players as possible. What is the number of students in each
team? How many teams can he create?
19. Mrs. Hernandez waters one of her plants every 10 days and another plant every
14 days. If she waters both plants today, when is the next time both plants will
be watered on the same day?
14th Session : EXAM
5. DIDACTIC RESOURCES
We use the same book as the other students of 2nd ESO: Anaya 2º ESO-Matemáticas.
Although it is written in Spanish it is a good reference for the students so that they can
have a look to the theory explained in English in the class but this time in Spanish.
The teacher also provides the students with some photocopies with exercises written in
English, specially those real life problems with long statements.
Finally, several webs can be used to revise the unit. Specially interesting are those with
interactive activities. For example:
http://www.mathgoodies.com/lessons/toc_vol5.html
http://recursostic.educacion.es/descartes/web/ingles/index.html
http://www.mathsisfun.com/whole-numbers.html
6. ASSESSMENT
Exam: At the end of the unit an exam will be set for checking the theory (definitions
and vocabulary) and practical exercises. The exam will be marked 70%.
Notebook : Since we are still in the compulsory education, notebooks will be
required and marked. Very important the cleanliness and the way of expressing ideas in
English. (10%)
Homework: Nearly everyday the teacher will see whether the students have done
the homework or not. This is an important point because it is the way of practising the
procedures learnt and check if they are clear enough for the student.(10%)
Attitude: Students must behave correctly in class, try to listen the explanations and
ask questions politely. They should show interest for the matter explained and make an
effort to express their questions, answers, opinions, etc in English during the lesson.(10%)
Carmen Cano Sarabia