Types of numbers (Talmängder)

Ämne / Topic

Natural, integer, rational, irrational, and real numbers.

Order of operations.

Addition, subtraction, multiplication, and division of fractions.

Inlärningsmål / Learning Goals

After this lesson you should know / understand....

How to classify numbers as natural, integer, rational, irrational, or real.

How to follow the Order of Operations (PEDMAS) to solve problems.

How to add, subtract, multiply, and divide fractions.

Google Slides & Videos

Lektionssteg / Steps in the lesson: 

1. Lektionsuppvärmning / Lesson warm up / Hook / Review from previous lesson

• Lesson 1:Students will complete warm-up questions involving matching relative mathematical terms with the corresponding definitions.

• Lesson 2:Students will complete a lateral-thinking question, based on the topic to be discussed in class (Order of Operations)

• Lesson 3:Students will complete a lateral-thinking question, based on the topic to be discussed in class (Adding and subtracting fractions)

• Lesson 4:Students will complete a lateral-thinking question, based on the topic to be discussed in class (Multiplying and dividing fractions)

2. Introduktion till lektionen / Introduction to the lesson content 

• Lesson 1: Types of numbers (see file at foot of page)

• Lesson 2: Order of Operations (see file at foot of page)

• Lesson 3: Adding and Subtracting Fractions (see file at foot of page)

• Lesson 4: Multiplying and dividing fractions (see file at foot of page)

3. Aktiviteter / Key Activities 

Lesson 1:

• Identify the different types of numbers i.e Natural numbers, Whole numbers etc.

• Provide definitions for the different types of numbers.

• Create a types of numbers poster, with examples to refer to as reinforcement of the concept.

• Using the poster, answer additional questions on classifying numbers.

Types of Numbers

Understanding the different types of numbers is key to other areas of maths. Types of numbers is all about terminology and knowing what each number actually is.

Type 1: Integers vs Non-Integers

The word integer is just another way of saying whole number. This is a number with no decimals or fractions.

Examples of integers:  7, 23, −11, 3, 0, 583

Non-integer numbers is just a way of referring to all numbers that are not whole numbers.

Examples of non-integers:  0.25, −5.5, π , 13, 2

Type 2: Special Integers 

There are some notable integers that you should be able to recognise, these include…

• Square Numbers

A square number is the result of multiplying any integer by itself.

Examples of square numbers:  1, 4, 9, 16, 25, 36, 49, 64, 81, 100

• Cube Numbers

A cube number is the result of multiplying any integer by itself twice

Examples of cube numbers: 1, 8, 27, 64, 125

• Prime Numbers

A prime number is only divisible by 1 and itself. Every whole number is made up of prime numbers.

Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23

Type 3: Rational Numbers

A rational number is any number that we can write as a fraction. Specifically, a fraction that has an integer on the top and the bottom.

Numbers that are rational include:

• Integers – all integers are rational numbers as they can be written as a fraction over 1 e.g. 6=1/6

• Decimals – decimals that are recurring e.g. decimals that are recurring e.g. 0.166˙ or terminate e.g. 0.375 are rational.

• Fractions – all fractions that are in the form a/b where a and b are integers e.g. 2/3

Remember: Just because a number isn’t written as a fraction doesn’t mean it can’t be.

Type 4: Irrational Numbers

An irrational number is any number that we can’t write as a fraction. In other words, it is the opposite of rational. Another way to see irrational numbers is decimals that go on forever and never repeat.

• Square roots – if the square root of a positive whole number is not an integer then it is irrational, i.e. √9 = 3 is an integer whereas √3 = 1.732050808...  is a non-terminating and non-repeating decimal so it is irrational. Such numbers containing irrational roots are called surds.

Examples of irrational numbers: π, √2, √7

Type 5: Multiples and Factors 

Factors

A factor is a number that goes into another number. For example, we say that “2 is a factor of 8” because 2 goes into 8 with no remainder: 

8÷2=4

Most integers have multiple factors.

All the factors of 12 are: 1, 2, 3, 4, 6, 12

Highest Common Factor – HCF

The Highest Common Factor, or HCF, of two numbers is the biggest number that goes into both of them.

Example: Consider the numbers 12 and 20

The factors of 12 are: 1, 2, 3, 4, 6, and 12

The factors of 20 are: 1, 2, 4, 5, 10, and 20

They have a few factors in common, but the biggest factor they have in common is 4, therefore 4 is the HCF of 12 and 20.

Product of prime factors

Example: Determine the prime factorisation of 60.

Step 1: To construct a factor tree, think of 2 numbers which multiply together to make 60 – here, we’ve gone with 10 and 6

Step 2: Draw two branches coming down from 60, and at the end of the branches write the two factors that you chose.

Step 3: If a factor is prime, then  circle it. If a factor is not prime, then repeat the process as shown in the factor tree below.

Step 4: The prime factorisation of 60 is therefore 60=2×2×3×5

Step 5: We write this prime factorisation in index form, where if there is more than one of the same factor, we write it as a power instead, where the power is the number of times it occurs. 

So 60 = 2²×3×5

Example of the diagram will be shown in class

Multiples

A multiple of a number is any value that appears in the times tables for that number. For example, we say that “30 is a multiple of 6” because

6×5=30

Every number has an infinite number of multiples.

Some multiples of 8 are: 8, 24, 64, 112, 888, 2008

Lowest Common Multiple – LCM

The lowest common multiple, or LCM, of two numbers is the smallest number that is a multiple of both of them.

Example: Consider the numbers 5 and 7

Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, …Multiples of 7 are: 7, 14, 21, 28, 35, 42, … and so on.

So, we can see that the first occurrence of a number which is a multiple of both of these numbers is 35, therefore 35 is the LCM of 5 and 7. 

Lesson 2:

• Understand the concept of PEDMAS and what each letter stands for and how to use it.

• Introducing students to a step by step process on how to solve work using PEDMAS (students will be able to refer to this when completing work).

• Work through example problems as a class (order of operations)

• Students will work in pairs to solve problems, thus encouraging cooperative learning.

• Students will complete a an Order of Operations worksheet.

Using PEMDAS

When performing calculations, always follow the PEMDAS order of operations.

Example: Work out the value of 3 × (3² + 4) − 8

Step 1: The first letter of PEMDAS is P, meaning the first thing we should do is look to what’s inside the parentheses. (If there are no parentheses, move onto exponents, then multiply and so on..)

Here, we have two operations happening: a power/exponent, and an addition. The letter E comes before the letter A in PEMDAS which means we first work out the result of 

3² and then add 4 to it.

(3² + 4) = (9 + 4) = 13

Step 2: We are left with a multiplication and a subtraction, so because M comes before S, we do the multiplication first and the subtraction second,

3 × 13 − 8 = 39 − 8 = 31

PEMDAS and Fractions

For fractions, we work out what the values of the top (numerator) and bottom (denominator) are separately (using the rules of PEMDAS), and then lastly, we look at the fraction we have and see if it can be simplified.

Example: Simplify the fraction: 3 x 4 - 5

                                                     11 + (9 / 3)  

Step 1: First, considering the numerator. There’s a multiplication and a subtraction, so we do the multiplication first and the subtraction second.

3 × 4 − 5 = 12 − 5 = 7

Step 2: Now, the denominator. That contains a division inside brackets, so that will be the first bit of the calculation, and then the addition will be second.

11 + (9 ÷ 3) = 11 + (3) = 14

Step 3: Therefore, our fraction is 7/14. Both top and bottom have a common factor of 7, so the simplified answer is:

7/14 = 1/2                           

Lesson 3:

• What are like and unlike fractions and key language with regard to fractions (examples provided).

• As a class, work through addition and subtraction of fractions using an easy to follow step by step process, that can be referred to at any time to solve fraction questions.

Example of work that will be done:

Adding and Subtracting Fractions

In order to add and subtract fractions, you need to find a common denominator – some value that can become the denominator of both fractions. There are two main methods for choosing a common denominator:

• Use the lowest common multiple (LCM) of the two denominators.

• Use the product of the two denominators.

Adding and Subtracting Fractions with Different Denominators

1

Find the lowest common denominator. This means the lowest number both denominators have in common.[4] Let's take the fractions 2/3 and 3/4. What are the denominators? 3 and 4. To find the lowest common denominator of the two, you can do this one of three ways:

• Write out the multiples. The multiples of 3 are 3, 6, 9, 12, 15, 18...and so on. The multiples of 4? 4, 8, 12, 16, 20, etc. 

• What's the lowest number seen in both of the sets? 

• 12! That's your lowest common denominator, or LCD.

• Prime factorisation. If you know what factors are, you can do prime factorisation. That's finding out what numbers can make your denominators. For 3, the factors are 3 and 1. For 4, the factors are 2 and 2. Then, you multiply them together. 3 x 2 x 2 = 12. Your LCD!

  • Multiply the numbers together for small numbers. In some cases, like this one, you could just multiply the numbers together – 3 x 4 = 12. However, if your denominators are big, don't do this! You don't want to multiply 56 x 44 and have to work with 2,464 as your answer!

2

Multiply the denominator by the number needed to get the LCD.[5] In other words, you want each of your denominators to be the same number – the LCD. 

In our example, we want our denominator to be 12. 

To turn 3 into 12, you need 3 x 4. 

To turn 4 into twelve, you need 4 x 3. 

The resulting like denominator will be the denominator for your final answer.

• So our 2/3 turns into 2/3 x 4 and 3/4 turns into 3/4 x 3. That means we now have 2/12 and 3/12. But we're not done yet!

  • You'll notice that the denominators, in this instance, are multiplied by each other. This works in this situation, but not all situations. Sometimes, instead of multiplying the two denominators together, you can multiply both denominators by different numbers to get one small number.

  • And then in other cases, sometimes you only have to multiply one denominator to make it equal to the denominator of the other fraction in the equation.

3

Multiple the numerator by that number, too. When you multiply the denominator by a certain number, you also have to multiply the numerator by the same number. What we did in the last step was just half of the multiplication necessary.

• We had 2/3x4 and 3/4x3 as our first step – to add the second step, it's really 2 x 4/3 x 4 and 3 x 3/4 x 3. 

• That means our new numbers are 8/12 and 9/12. Perfect!

4

Add (or subtract) the numerators to get your answer. To add 8/12 + 9/12, all you have to do is add the numerators. Remember: you leave the denominator alone now. The number you got with the LCD is your final denominator.

• For this example, (8+9)/12 = 17/12. 

• To turn this into a mixed fraction we need to do the following:

  •  subtract the denominator from the numerator and see what's left over. In this case, 17/12 = 1 5/12

Adding and Subtracting Mixed and Improper Fractions

1

Convert your mixed fractions into improper fractions. 

A mixed fraction is when you have a whole number and a fraction, like in the above example (1 5/12). 

Meanwhile, an improper fraction is one where the numerator (the top number) is bigger than the denominator (the bottom number). 

That's also seen in the above step, with 17/12.

• For the example for this section, let's work with 13/12 and 17/8.

2

Find the common denominator. 

Remember the three ways you can find the LCD? 

By either writing out the multiples, using prime factorisation, or by multiplying the denominators.

• Let's figure out the multiples of our example, 12 and 8. What's the smallest number these two go into? 24. 8, 16, 24 and 12, 24 – bingo!

3

Multiply your numerators and denominators to get your like fraction. 

Both denominators now need to be turned into 24. 

How do you get 12 to 24? 

Multiply it by 2. 8 to 24? 

Multiply it by three. 

But don't forget – you need to multiply the numerators, too!

• So 13 x 2/12 x 2 = 26/24. And 17 x 3/8 x 3 = 51/24. We're well on our way to solving the problem!

4

Add or subtract your fractions. Now that you have the same denominator, you can add these two numbers together with ease. Remember, leave the denominator alone!

• 26/24 + 51/24 = 77/24. There's your one fraction! That top number is mighty big, though....

5

Convert your answer back into a mixed fraction. Having such a large number on top is a little weird – you can't quite tell the size of your fraction. All you have to do is put the denominator into the numerator until in can't be repeated again and then see what you have leftover.

• For this example, 24 goes into 77 three times. 

  • That is, 24 x 3 = 72. 

  • But there's 5 leftover! 

  • So what's your final answer? 3 5/24. That's it!

• Differentiated worksheets on addition and subtraction of fractions, challenging learners of all learning abilities.

• Problem solving (word problems) questions based on adding and subtracting fractions (relating what was done in class to the real world).

Lesson 4:

• As a class, work through multiplication and division fraction problems using an easy to follow step by step process, which can be referred to at any time when solving fractions.

Example of work that will be done:

Multiplying Fractions

1

Multiply the numerators of the fractions.

 The numerator is the number on the top of a fraction and the denominator is the number on the bottom.[1] 

The first step to multiplying fractions is to line them up so that their numerators and denominators are next to each other. 

If you're multiplying the fraction 1/2 by 12/48, then the first thing you want to do is multiply the numerators, 1 and 12. 1 x 12 = 12. 

Write the product, 12, in the numerator of the answer.

2

Multiply the denominators of the fractions. 

Now, you'll just have to do the same thing with the denominators.[2] 

Multiply 2 and 48 to find the new denominator. 2 x 48 = 96. 

Write this answer in the denominator of the new fraction. 

Therefore, the new fraction is 12/96.

3

Simplify the fraction. The last step is to simplify the result if it can be done. 

To simplify a fraction, you have to find the greatest common factor (GCF) of both the numerator and the denominator. 

The GCF is the largest number that can be evenly divided into both numbers.

In the case of 12 and 96, 12 happens to evenly divide into 96. 

So, divide 12 by 12 to get 1, and divide 96 by 12 to get 8. Therefore, 12/96 ÷ 12/12 = 1/8.

• If both numbers are even, you can also start by dividing them both by 2 and keep going. 12/96 ÷ 2/2 = 6/48 ÷ 2/2 = 3/24. Then you may notice that 3 goes evenly into 24, so you can divide both the numerator and denominator by 3 to get 1/8. 3/24 ÷ 3/3 = 1/8.

Dividing Fractions

1

Reverse the numerator and denominator of the second fraction and change the division sign to a multiplication sign.[4] Let's say you're dividing the fraction 1/2 by 18/20. 

Now, find the reciprocal of 18/20 which is 20/18 and change the division sign to a multiplication sign. 

So, 1/2 ÷ 18/20 = 1/2 x 20/18.

2

Multiply the numerators and denominators of the fractions and simplify your answer. 

Now, do the same thing you would do to multiply.[5] 

If you multiply the numerators, 1 and 20, you get the result of 20 in the numerator. 

If you multiply the denominators, 2 and 18, you get 36 in the numerator. 

The product of the fractions is 20/36. 

4 is the largest number that is evenly divisible by both the numerator and the denominator of this fraction, so divide each by 4 to get the simplified answer. 20/36 ÷ 4/4 = 5/9.

• Differentiated worksheets on multiplication and division of fractions, challenging learners of all learning abilities.

• Problem solving (word problems) questions involving multiplying and dividing fractions, with relation to the real world.

4. Kolla vad eleverna kan / Check for understanding / Exit ticket

• Students will be asked to work backwards, coming up with a question of their own based on the work done in class. The students will then swap questions with the learner closest to them, and try to solve their peers question.

5.  Avslutning / Closure / Summary of lesson

• Students will be selected at random (each lesson a different learner will be selected) to report back on what their understanding of the concept done in class for that day.

Advanced Order of Operations.pdf