Click on the two quarters under the highlighted one half. As the two quarters are the same size as the one half, one half and two quarters are known as equivalent fractions. Click on the four eighths under the highlighted two quarters. Again, this is the same size or amount as one half. We can say that. These three fractions are equivalent fractions; they all have the same value.
The next bar has been divided into sixteen equal parts, or sixteenths. Click on the sixteenth until the shaded area is equivalent to one half. You will see that the list of equivalent fractions in the right hand column now includes eight sixteenths.
Understanding 2 Fractions are Numbers All numbers can be located on a number line. Think of the fractions wall as a number line below.
Understanding 3 Fractions on a Number Line The number line below represents eighths, that is, fractions with a denominator of eight. We can see that the fractions on the same point on the number line are equivalent fractions. Learning Activity 2 Battleships Number Line You can use the Battleship Number Line Game to practice and build your estimation and visualisation skills when placing fraction and decimal numbers on a number line.
For example: If placing on the number line it may be easier to visualise where the equivalent fraction would lie. If placing on the number line, it is a little further to the right than or. If placing 0. Battleship Number Line Game Number lines can assist with addition, subtraction, multiplication and division of fractions also see Fractions as operators further down on this page.
Understanding 4 Fractions Greater than One The term improper fractions is used to describe fractions greater than one, such as ten quarters. We can clearly see that or is equivalent to The decimal notation for this is 2. Understanding 5 Fractions as measures Fractions are commonly used as measures.
Fractions are often used to measure time, for example: The term half an hour is more commonly used than 30 minutes. Understanding 6 Fractions as Operators A fraction acts as an operator when it is applied to a number, set or quantity to find a certain proportion of that number, set or quantity.
Example 1 Find three quarters of twenty four. Understanding 7 Fractions as Division Fractions can be used as a representation of division. Numbers that can be expressed in this way As are known as rational numbers and can be derived from dividing the numerator by the denominator. Example 2 Three people have to share two choc bars.
How much does each person get? Here is one way to visually represent this problem:. Understanding 8 Fractions as Ratios Fractions can be interpreted as ratios. Example 1 In the pictures below there are 5 puppies, 3 females and 2 males. Image from Math is Fun. Practice Task 1 1. Starting at write a sequence of eight numbers counting by: a One half b One quarter c One tenth d One twelfth 2. Find equivalent fractions for: a b. Practice Task 2 Solve the following problems where fractions are used as operators.
Genni was renovating her laundry. The total number of equal sized tiles on the floor was How many tiles had to be replaced? Which is the larger, of 45 or of 40? We cannot assume that the second one is the largest just because is a larger fraction than , as they relate to different wholes.
Practice Task 3 Draw a visual model to show how six pizzas can be equally shared among eight people if: The pizzas are cut into eighths The pizzas are cut into quarters Identify whether the following problems are Part-to-part or Part-to-whole representations and represent the answers using a ratio: Timber Town has a population of people, 40 of these being children.
Check your Understanding of Big Idea 1 The purpose of this module was to identify how: Fractions occur in a wide range of contexts. Fractions have many different interpretations. To demonstrate the following understandings: Fractions can be used to describe parts of a whole. To order, compare, add or subtract fractions, they must relate to the same unit or whole. This would not be the case if we were talking about of a small pizza and of a large pizza. Fractions can be used as numbers as placed on the number line.
The system of decimal numbers is an extension of the whole-number number system. Decimal numbers are one way of representing fractions, ratios and percents. Decimals are widely used today, especially in areas such as finances, commerce and science. They are also used when measurements require a given accuracy. Although decimal notation is often used to represent fractions, sometimes fraction notation is more appropriate.
For example, it is much simpler to find one third of twenty four x 24 than to use the decimal equivalent of one third 0. The threes in this decimal go on indefinitely. Learning Activity 1 Where do fractions and decimals fit into the base-ten number system? Watch the following video about the classification of numbers The video shows how fractions and decimals fit into our number system. Example The fraction three quarters can be written as a decimal. The following link shows Pi to one million decimal places!
Common Misconceptions about Decimals 1. Longer is larger A common misunderstanding when comparing numbers originates from the separation of decimal numbers into two whole numbers; that is, the sets of numbers on each side of the decimal point are treated as whole numbers.
The following examples show how this misunderstanding can go unrecognised because sometimes it results in a 'fluke' correct answer: Which is smaller? Incorrect reasoning Correct reasoning Correct answer 5. The seven hundredths are not relevant as they are smaller than tenths 5. They are both have the same value because 5 tenths is the same as 50 hundredths Both the same 2. Shorter is larger This is another common misunderstanding when comparing numbers.
The following examples show how this misunderstanding can go unrecognised, as sometimes it results in a 'fluke' correct answer: Which is smaller? Incorrect reasoning Correct reasoning Correct answer 7. Write each of the following numbers as fractions and decimals.
Sample answers have been given for the first number. Number Fraction Expressed as a Decimal One quarter 0. Practice Task 2 Write each of the following numbers in expanded notation, renaming in terms of the place value parts: a Check your Understanding of Big Idea 2 The purpose of this module was to identify how: fractions and decimals fit into our number system.
To demonstrate the following understandings: Each digit in a number has a place value depending on its position or place. Numbers can be partitioned and named renamed in terms of the position of each digit.
The decimal point separates the whole number places from the decimal number places. Does this make sense to you now? Fractions, decimals and percentages are related and can be used to express the same number, or proportion in different ways.
By the end of this module you should be able to fill in a chart similar to this one. Number Fraction Decimal Percent five 5. Learning Activity 1 Relating Decimals, Fractions and Percent Please go to the link below and complete the activities suggested below.
Math Is Fun Virtual Manipulative Activities to demonstrate the relationship between fractions, decimals and percents, and reinforce and extend your understandings of percents being another way to represent fractions: 1.
Understanding 2 Representing Decimals to Thousandths A one thousand grid can be used to represent one whole 1 , and to demonstrate decimals up to thousandths. The entire grid represents one 1 , or one whole. The following statements can be made: The red area is one tenth or zero point one 0. The yellow area is one hundredth or zero point zero one 0.
The blue area is one thousandth or zero point zero zero one 0. The shaded area of the grid is one hundred and eleven thousands of the grid, which can also be expressed as the decimal fraction zero point one one one 0. Learning Activity 2 One Thousand Grid: A visual model for decimal fractions The following video uses a thousandths grid in a similar way, to demonstrate writing decimal fractions: The second example in the video focuses on the shaded area being one thousandths of a whole comprising one thousandths.
This sets the numerator range at the bottom of the screen as 0 — , and the denominator range at 1 — The fractions will therefore be improper, or greater than 1, because the numerator will be greater than the denominator.
Use the plus and minus tabs either side of the numerator and denominator settings to select a numerator of 5 and a denominator of 3. You will see five thirds represented on the area model on the screen. Above this you will see how this number is expressed as a fraction or improper fraction , a mixed number , a decimal 1. Note that the decimal and percent have been rounded up; otherwise they would go on forever. Look at the different models length, area, region, set. Try other numbers greater than one, looking at the different visual representations.
Note how they are expressed in improper fractions, mixed numbers, decimals and percents. Understanding 3 Relating Decimals, Fractions and Percent using a Number Line The number line below is marked in increments of one hundredths from zero to 0.
Notice where the following decimal numbers, all containing similar digits but in different places, are placed on the number line: 0. Common Misconceptions for Ordering Fractions 1. The larger the denominator, the bigger the fraction This is true for unit fractions fractions with a numerator of one.
Practice Task 1 1 Complete the table so that the numbers in each row represented by fractions, decimals and percents are equivalent: Fraction Decimal Percent 1. Practice Task 2 1 Relating decimals, fractions and percent using a number line Click on the link below and complete the activity by placing all of the fractions, decimals and percents on the number lines from ICT games.
Check your understanding of Big Idea 3 The purpose of this big idea was to demonstrate the following understandings; A number can be represented as a fraction or a decimal.
A percent is a fraction out of one hundred and are a very commonly used in everyday life. Percents can also be understood as hundredths Does this make sense to you now? An understanding of percent relationships helps us to compare and represent increasing and decreasing proportions. In numerical notation this can be expressed as One way to convert a fraction into a percent is to remember that any fraction can be interpreted as a quotient division see FDRP BI2 Fractions as division.
Understanding 2 Percent Decrease and Increase Example 1 Mathematical language and data presented in media is not always clear. Example 2. Image provided by: Mathematics Assessment Resource Service. Note that 0. Look at the same paragraph as the previous section: Alarming figures show a Understanding 4 Using Percentages in Simple Interest Calculations When money is borrowed or invested for a fixed time at a fixed interest rate and the interest rate is calculated only on the initial investment , simple interest sometimes known as flat interest is calculated.
Practice task 1 You want to buy two T-shirts. Which is the best buy, Shop A or Shop B? Shop A. Shop B. Practice Task 3 The staff of a company decreased from 55 to What was the percent decrease?
Place value materials can be used to support students to develop conceptual links between fractions and decimals. By applying place value it is simple to convert from a decimal to fraction. Understanding that the place to the right of the decimal point is the tenths place makes it immediately clear that, for example,.
It is possible to convert fractions to decimals by i converting the denominator to a power of ten, or ii by using equivalent fractions or iii by using division. It is often more useful to work with numbers in the form of decimals than fractions. The term percent is another name for hundredths. A fraction expressed as a hundredth can simply be renamed as a percent. An understanding of the role of the decimal point in naming decimals can help in understanding the link between hundredths and percents.
The decimal point identifies the units. You know that all percents are out of , so you can skip making the percent into a fraction. You have to divide the percent by to get a decimal, but there's a quick way to do that. Just move the decimal point two spaces to the left! This way, you can get the same answer with just one easy step. We'll reverse what we did in the last section.
This time, we'll move the decimal point two places to the right. We've finished converting our decimal into a percent. Let's try another example. This time our decimal has three numbers to the right of the decimal point.
We still have a number to the right of the decimal point. The number isn't a 0 , so we can't drop it. Instead, we'll keep the decimal point and add a percent sign at the end of the number.
Knowing how to write percents as fractions and vice versa can help you in your everyday life. When your teacher grades the test, she may do the opposite. When you're converting a percent into a fraction, it helps to remember that percents are always out of You can practice with percents in our Introduction to Percentages lesson. In Introduction to Percents , you learned that all percents are out of In fact, that's what the word percent means.
So any percent is equal to itself over So we can divide both parts of the fraction by We'll divide the numerator of the fraction first. Write these percentages as fractions.
Make sure to reduce each fraction to its simplest form. Converting a fraction uses two of the skills you just learned: writing a fraction as a decimal , and writing a decimal as a percent. Let's see how we can use these skills to convert a fraction into a percent. Just like when we converted a fraction into a decimal, we'll divide the numerator by the denominator.
Now we'll turn the decimal into a percent by moving the decimal point two spaces to the right. We'll also change the decimal point into a percent sign. Percents: Converting Percentages, Decimals, and Fractions.
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