Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Number System → Fractions ↓


Comparing and Ordering Fractions


Understanding fractions

Fractions are parts of a whole. When we talk about fractions, we are referring to a way of expressing a number that is not a whole. A fraction is made up of two parts, the numerator and the denominator. The numerator is the top number, and the denominator is the bottom number.

1 - 2

In this example, 1 is the numerator and 2 is the denominator. The numerator tells us that we are talking about a portion of something that is divided into two equal parts.

Comparing fractions with the same denominators

When fractions have the same denominator, it's easy to compare them. You just compare the numerators because the parts are the same size. For example:

3 5 - vs - 7 7

The denominators are the same: 7 So, you can compare the numerators: 3 and 5 Obviously, 5 is greater than 3, so 5/7 is greater than 3/7.

Comparing fractions with different denominators

When fractions have different denominators, it's not as easy to compare them directly. We need to find a common denominator to accurately compare their sizes. Here's how you can do that:

Finding a common denominator

Let's look at an example:

2 3 - vs - 5 4

The LCM (least common multiple) of the denominators 5 and 4 is 20 We convert each fraction into an equivalent fraction with a denominator of 20.

For 2/5, multiply the numerator and denominator by 4 because 20 ÷ 5 = 4:

2 × 4 8 ------ = --- 5 × 4 20

For 3/4 multiply the numerator and denominator by 5 because 20 ÷ 4 = 5:

3 × 5 15 ------ = --- 4 × 5 20

now we have:

8 15 - vs -- 20 20

Comparing 8 and 15, 15 is greater than 8, so 3/4 is greater than 2/5.

Visualization of comparison of fractions

Visual aids can be useful for better understanding fraction comparison. Consider plotting the fractions on a number line or using strips of equal length divided into parts to show relative sizes.

Fraction Bar: 1/4 |xxxx|----|----|----| 1/3 |xxxxxxx|xxxxxxx|---| Number Line: 0 1/3 1/2 2/3 1 -----o-----×-----×-----×-----o-----o---

Here, each different bar and number line representation makes the comparison clear.

Ordering fractions

Ordering fractions means arranging them in ascending or descending order. Just like comparing, this can be done for fractions with the same denominator or different denominators.

Sequences with the same denominators

Similar to comparing, when ordering fractions with the same denominators, you only need to compare the numerators. For example, here is a list of fractions:

3 1 4 - , - , - 8 8 8

To order them from smallest to largest:

1 3 4 - , - , - 8 8 8

In increasing order, fractions are arranged according to their fraction size.

Sequences with different denominators

Just like comparing, you need to find a common denominator to order fractions with different denominators. Consider the following fractions:

2 5 3 - , - , - 5 6 4

The LCM of 5, 6 and 4 is 60 Convert each fraction using 60 as the common denominator:

2/5 = (2×12)/(5×12) = 24/60 5/6 = (5×10)/(6×10) = 50/60 3/4 = (3×15)/(4×15) = 45/60

now we have:

24 45 50 -- , -- , -- 60 60 60

These can be ordered from smallest to largest:

24 45 50 -- , -- , -- 60 60 60

This means that 2/5, 3/4, 5/6 is an ordered sequence.

Solve problems involving comparison and ordering of fractions

To understand how to implement comparison and ordering, consider practical examples:

For example, suppose you have three pieces of cake that represent 1/3, 1/2, and 1/6 of a cake. To find out who gets the most cake, you compare the fractions.

Find a common denominator for 3, 2 and 6, which is 6 Convert each fraction:

1/3 = 2/6 1/2 = 3/6 1/6 = 1/6

From smallest to largest, the fractions are 1/6, 2/6, 3/6. So, 1/2 is the largest and represents the largest piece of cake.

Practice problems

Let's practice comparing and ordering fractions:

  1. Compare fractions: 4/9 and 7/8. 
    Solution: - Find LCM of 9 and 8: 72 - Convert 4/9 to 32/72 - Convert 7/8 to 63/72 - Therefore, 7/8 > 4/9
  2. Order the fractions: 3/10, 2/5, 7/15. 
    Solution: - Find LCM of 10, 5, and 15: 30 - Convert 3/10 to 9/30 - Convert 2/5 to 12/30 - Convert 7/15 to 14/30 - In order: 3/10, 2/5, 7/15

Closing thoughts on fractions

Understanding how to compare and order fractions is an essential part of mathematical literacy. It allows you to make educated decisions based on numerical data and find patterns in different scenarios. Regular practice with fractions will increase your confidence and improve your arithmetic skills.


Grade 6 → 1.3.6


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Comparing and Ordering Fractions

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