OBJECTIVES - EDUCARM



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:

• 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

▪ 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. Write three multiples of: a) 7; b) 11; c) 15; d) 24.

2. Write three factors (or divisors) of: a) 15; b) 50; c) 60; d) 77.

3. Mark the multiples of 3: 111, 270, 210, 816, 325

4. Mark the multiples of 4: 544, 3221, 1136, 7732, 7745

5. Mark the multiples of 5: 881, 345, 657, 650, 776

6. Mark the multiples of 11: 495, 913, 2794, 3915, 4191

7. Among the following numbers find:

i. 275, 333, 495, 540, 1202, 8580, 1155, 2873, 3330, 6655

a. Multiples of 3.

b. Multiples of 4.

c. Multiples of 5.

d. 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)  to express each of the numbers as a product of powers of its prime factors;

2)  to write out all common factors in these factorisations; 

3)  to take the least 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

                       GCF(168, 180, 3024) = 22  · 31  = 12

Problem: Your gym teacher is setting up teams for a soccer 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 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:

Find the LCM and the GCF:

a) 24, 15 and 27. f) 1000 and 2100.

b) 12, 144 and 36. g) 2500 and 1750.

c) 100, 120 and 160. h) 11, 33, 55 and 121.

d) 210, 220 and 250. i) 13, 26, 39 and 169.

e) 150, 200 and 250. 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:

[pic]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

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