TOPIC 1: NUMBERS

 

A) WHOLE NUMBERS

OBJECTIVES: know and use the concept of factor, multiple, common factor, lowest common multiple, prime and composite numbers

FACTORS

A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. All numbers have a factor of one since one multiplied by any number equals that number. All numbers can be divided by themselves to produce the number one. Therefore, we normally ignore one and the number itself as useful factors.

The number fifteen can be divided into two factors which are three and five.

The number twelve could be divided into two factors which are 6 and 2. Six could be divided into two further factors of 2 and 3. Therefore the factors of twelve are 2, 2, and 3.

If twelve was first divided into the factors 3 and 4, the four could be divided into factors of 2 and 2. Therefore the factors of twelve are still 2, 2, and 3.

There are several clues to help determine factors.

Any even number has a factor of two

Any number ending in 5 has a factor of five

Any number above 0 that ends with 0 (such as 10, 30, 1200) has factors of two and five.

To determine factors see if one of the above rules apply (ends in 5, 0 or an even number). If none of the rules apply, there still may be factors of 3 or 7 or some other number.

LESSONS: 

1. Factors and Primes

 

Divisibility Rules

Easily test if one number can be evenly divided by another

Divisible By:

"Divisible By" means "when you divide one number by another the result is a whole number"

Examples:

14 is divisible by 7, because 14÷7 = 2 exactly

But 15 is not divisible by 7, because 15÷7 = 2 1/7 (i.e., the result is not a whole number)

"Divisible by" and "can be evenly divided by" mean the same thing

 

LESSONS:

1. Apply the rules of divisibility of 2, 3 and 5

 Multiples of numbers smaller than 10

Multiply and Divide Whole Numbers

Distributive Property

The Distributive Property states that when you multiply the sum of two or more addends by a factor, the product is the same as if you multiplied each addend by the factor and then added the partial products. The Distributive Property is illustrated below graphically, arithmetically, and algebraically. At this time, students do not need to know the algebraic explanation of the Distributive Property.

LESSONS:

1. Multiply and Divide Whole Numbers

 

 Exponential Notation

This lesson is foundation for the topic of properties of integer exponents. For the first time in this lesson, students are seeing the use of exponents with negative valued bases. It is important that students explore and understand the importance of parentheses in such cases, just as with rational base values. It may also be the first time that students are seeing the notation (dots and braces) used in this lesson. If students have already mastered the skills in this lesson, it is optional to move forward and begin with Lesson 2 or provide opportunities for students to explore how to rewrite expressions in a different base, 4 2 as 2 4 , for example. 

LESSONS: 

1. Exponential Notation

Generate common whole number sequences, including odd and even numbers, prime numbers, multiples, square numbers and cube numbers.

LESSONS: 

1.Multiples, factors and powers

 

 

(b) Powers and roots

OBJECTIVE: understand and use the notation and terminology for squares, square roots, cubes, and cube.

 

Roots

The exponent of a number says how many times to use the number in a multiplication.

In 82 the "2" says to use 8 twice in a multiplication,

so 82 = 8 × 8 = 64

In words: 82 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

Exponents are also called Powers or Indices.

Where a function equals zero:

In this example, −2 and 2 are the roots of the function x2 − 4

But sometimes "root" is used as a quick way of saying "square root", for example "root 2" means √2

Lessons

1. Power and roots

2. Power and roots

 

(C) Common and decimal fractions

LEARNING OBJECTIVES:

Understand the relationship between common and decimal fractions realize that a common fraction is either a terminating or a recurring decimal

Understand the concept of multiplying and dividing by fractional quantities

 

Lessons

1. Ordering fraction

 

(D) Percentages

 
OBJECTIVE: understand and use percentages

 

What Is a Percent?

Picture a big pizza with your favorite toppings on it. Now, I like pepperoni, so mine has lots of pepperoni slices. Now picture the pizza being sliced into fourths, then eights, then sixteenths. Yum! My stomach's rumbling - is yours? Picture holding a slice of pizza in your hand and you're holding a percentage of the pizza. So a percentage, or percent, is a part of a whole. The word has its roots in the Latin language, from the phrase 'per centum,' which literally means 'per hundred.'

What Does It Mean?

We can think of a hundred as a whole or all of something. A hundred percent of a pizza is the whole pizza. Half a pizza would be 50 percent, or half of a hundred percent. The percentage tells you how much of the whole you have.

If we divided our pizza into 100 little slices, then each slice would represent 1 percent of the pizza. This is another way to visualize percentages. You can take a whole of something and divide it into 100 little pieces and then figure out how many little pieces are in the portion you are interested in. If I wanted a quarter of the pizza, I would see that it would require 25 little slices out of 100, or 25 percent of the pizza.

It is definitely possible to have a percentage that represents more than a whole or more than a hundred of something. What if you had a friend with a very good appetite and he happened to eat 2 whole pizzas? Let's visualize this. If we divided each pizza into our little 100 slices, how many slices did he eat? Yes, he would have eaten 200 little slices. That sounds like a lot, but your friend has a big appetite and he's won several eating contests because of it. Anyways, if he ate 200 little slices, then he would have eaten 200 percent of one pizza.

We know that a hundred percent is a whole of something. Percent’s less than a hundred mean that they are less than a whole. Percent’s more than a hundred mean that they include a whole of something plus more of the something.

 

Writing Percentages

There are three ways to write our percentages. We can use the mathematical percentage symbol or we can write it in either decimal or fraction form. Let's take a look at how we can write 50 percent. Using the percentage symbol, it will look like 50%.

The percentage symbol looks like two little zeroes with a slash separating them. In decimal form, 50% is 0.50. In decimal form, 1 is the same as a 100%. In fraction form, 50% is 1/2. 1 in this form also means a 100%. If you divide 1 by 2, you will see that you get the decimal form of the fraction, 0.50.

Let's try writing 25 percent in the three different forms. Using the symbol, it's 25%. In decimal form, it is 0.25 and in fraction form, it is 1/4. Can you guess what 1 divided by 4 equals?

Lessons

1.       writing percentages

 

 

(E) Calculator skills

Objectives: Use more advanced functions of the scientific calculator.

Lessons

1.calculator skills

 

 

 

(F) Estimation

OBJECTIVE: Learners will: Know how to use approximation to check that their answers are reasonable.

 

What is estimate - Definition and Meaning?

Estimate:

Calculation of the approximate value of the result is called the estimation.

Example:

53 + 69. Here we can calculate as 50 + 70 = 120. But the actual value is 53 + 69 = 12

 

Estimate:

Calculation of the approximate value of the result is called the estimation.

Example:

53 + 69. Here we can calculate as 50 + 70 = 120. But the actual value is 53 + 69 = 122.

 

Lessons

1. Estimation