The Number System
Dividing Fractions and Rational Numbers
Grade 6
- ✓By the end of this lesson students will be able to divide fractions by fractions, including mixed numbers.
- ✓By the end of this lesson students will be able to understand and use negative numbers to represent quantities in real-world contexts.
- ✓By the end of this lesson students will be able to locate and order rational numbers on a number line.
- ✓By the end of this lesson students will be able to understand and find the absolute value of rational numbers.
- ✓By the end of this lesson students will be able to solve word problems involving division of fractions and rational numbers.
Key concepts
To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is found by flipping the numerator and the denominator. For example, the reciprocal of 2/3 is 3/2. This method is often remembered as 'Keep, Change, Flip': Keep the first fraction, Change the division sign to multiplication, and Flip the second fraction (find its reciprocal). If you have mixed numbers, always convert them to improper fractions before dividing.
Negative numbers are numbers less than zero. They are used to represent quantities that have an opposite direction or value to positive numbers. For example, temperatures below zero, debt, or elevations below sea level. On a number line, negative numbers are to the left of zero, and positive numbers are to the right. The further a negative number is from zero to the left, the smaller its value.
A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. This includes all integers (since any integer 'n' can be written as n/1), fractions, mixed numbers, and terminating or repeating decimals. For example, -3, 0, 1/2, 4.75, and -2/3 are all rational numbers.
A number line is a visual representation of numbers. Zero is at the center. Positive numbers extend to the right, and negative numbers extend to the left. To plot a rational number like a fraction or decimal, you find its position relative to the integers. When comparing rational numbers, the number further to the right on the number line is always greater. For example, -2 is greater than -5 because -2 is to the right of -5 on the number line.
The absolute value of a number is its distance from zero on the number line. Since distance is always a positive quantity (or zero), the absolute value of any non-zero number is always positive. The absolute value of zero is zero. The symbol for absolute value is two vertical bars around the number, for example, |x|. So, |-5| = 5 because -5 is 5 units away from zero, and |5| = 5 because 5 is also 5 units away from zero.
Key facts to remember
- 1To divide by a fraction, multiply by its reciprocal.
- 2The reciprocal of a fraction a/b is b/a.
- 3Negative numbers are numbers less than zero and are located to the left of zero on a number line.
- 4A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0.
- 5The absolute value of a number is its distance from zero on the number line.
- 6The absolute value of any number is always positive or zero.
- 7On a number line, numbers increase in value from left to right.
Worked examples
Example 1
Divide: 3/4 ÷ 1/8
Answer
6
Remember 'Keep, Change, Flip' (KCF) to help you remember the steps.
Example 2
A recipe calls for 2 and 1/2 cups of flour. If you only want to make 1/4 of the recipe, how much flour do you need?
Answer
10 servings
Always convert mixed numbers to improper fractions before performing multiplication or division.
Example 3
Find the absolute value of -8 and 3.5. Then, compare |-8| and |3.5| using <, >, or =.
Answer
|-8| = 8, |3.5| = 3.5, and |-8| > |3.5|
The absolute value of a number is always non-negative.
Common mistakes
- ✗Forgetting to 'flip' (find the reciprocal of) the *second* fraction when dividing.
- ✗Confusing absolute value with negative numbers (e.g., thinking |-7| = -7).
- ✗Incorrectly ordering negative numbers (e.g., thinking -10 is greater than -5 because 10 is greater than 5).
- ✗Not converting mixed numbers to improper fractions before dividing or multiplying.
Exam tips
- ★Always convert mixed numbers to improper fractions before performing division or multiplication with fractions.
- ★Remember the 'Keep, Change, Flip' (KCF) rule for dividing fractions: Keep the first fraction, Change the division sign to multiplication, Flip the second fraction.
- ★Use a number line to visualize and compare rational numbers, especially negative numbers.
- ★Remember that absolute value represents distance from zero, so it is always a positive value or zero.
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