Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Define and write ratios in different ways.
• Simplify ratios to their simplest form.
• Understand and calculate rates like speed (km/h).
• Relate percentages to fractions and decimals.
• Calculate simple percentages of whole numbers.

Lesson summary

This week, we begin exploring the fascinating world of ratios, rates, and percentages. These mathematical tools are essential for understanding proportions, comparisons, and changes in our everyday lives. From understanding the nutritional information on our favourite snacks to calculating discounts at the local spaza shop, ratios, rates, and percentages are used constantly around us. They also underpin important concepts in economics, finance, and science. This week serves as an introduction, laying the foundation for more advanced applications later on. Understanding these concepts will empower you to make informed decisions and become more critical consumers.

Lesson notes

Ratios A ratio is a way to compare two or more quantities. It shows the relative sizes of these quantities.

Ratios can be written in several ways: Using a colon: a : b (e.g., 2:3)

As a fraction: a/b (e.g., 2/3) Using the word "to": a to b (e.g., 2 to 3)

Example 1: In a class, there are 15 boys and 10 girls. What is the ratio of boys to girls?

Boys : Girls = 15 : 10 This ratio can be simplified by dividing both sides by their greatest common factor, which is 5. 15 : 10 = (15/5) : (10/5) = 3 : 2 So, the ratio of boys to girls is 3:

2. This means for every 3 boys, there are 2 girls.

Example 2: Thabo has 20 apples and Maria has 30 oranges. What is the ratio of apples to oranges?

Apples : Oranges = 20 : 30 Simplifying the ratio by dividing by 10: 20 : 30 = (20/10) : (30/10) = 2 : 3 The ratio of apples to oranges is 2:

3. Simplifying Ratios: To simplify a ratio, divide all parts of the ratio by their greatest common factor (GCF). This makes the numbers in the ratio as small as possible while keeping the proportion the same. Rates A rate is a comparison of two quantities with different units. This is what distinguishes it from a ratio. Common examples of rates include speed (distance per time) and price per unit.

Example 1: Speed A car travels 200 kilometers in 4 hours. What is its average speed? Speed = Distance / Time Speed = 200 km / 4 hours Speed = 50 km/h (kilometers per hour) The rate is 50 km/h.

Example 2: Price A bag of potatoes weighing 5 kg costs R

4

0. What is the price per kilogram? Price per kg = Total Price / Weight Price per kg = R40 / 5 kg Price per kg = R8/kg (Rands per kilogram) The rate is R8 per kilogram. Important

Note: When working with rates, make sure to include the units in your answer. The units tell you what is being compared. Percentages A percentage is a way of expressing a number as a fraction of

1

0

0. The word "percent" means "out of one hundred". The symbol for percent is %.

Understanding Percentages: 100% means the whole thing (100 out of 100). 50% means half (50 out of 100), which is equivalent to 1/2 or 0.5. 25% means a quarter (25 out of 100), which is equivalent to 1/4 or 0.25. 10% means one-tenth (10 out of 100), which is equivalent to 1/10 or 0.

1. Converting Percentages to Fractions and Decimals: To convert a percentage to a fraction, write the percentage as a fraction with a denominator of 100 and simplify if possible. For example, 60% = 60/100 = 3/

5. To convert a percentage to a decimal, divide the percentage by

1

0

0. For example, 60% = 60/100 = 0.

6. Calculating Percentages of Whole Numbers: To find a percentage of a whole number, convert the percentage to a decimal or fraction and multiply it by the number.

Example 1: What is 20% of 80?

Method 1 (Decimal): 20% = 0.20. 0.20 x 80 = 16 Method 2 (Fraction): 20% = 20/100 = 1/5. (1/5) x 80 = 80/5 = 16 So, 20% of 80 is

1

6. Example 2: What is 50% of 150?

Method 1 (Decimal): 50% = 0.50. 0.50 x 150 = 75 Method 2 (Fraction): 50% = 50/100 = 1/2. (1/2) x 150 = 150/2 = 75 So, 50% of 150 is

7

5. Guided Practice (With Solutions)

Question 1: A fruit basket contains 8 apples and 12 bananas. What is the ratio of apples to bananas in its simplest form?

Solution: Apples : Bananas = 8 : 12 Find the GCF of 8 and 12, which is

4. Divide both sides by 4: (8/4) : (12/4) = 2 : 3 The simplified ratio of apples to bananas is 2:

3. Commentary: This question reinforces the basic concept of ratios and simplifying them. The GCF is crucial for simplification.

Question 2: A cyclist travels 60 km in 3 hours. Calculate the cyclist's average speed.

Solution: Speed = Distance / Time Speed = 60 km / 3 hours Speed = 20 km/h

Commentary: This question introduces the concept of rate, specifically speed. Remember to include the correct units (km/h).

Question 3: Calculate 25% of

2

0

0. Solution: Method 1 (Decimal): 25% = 0.25. 0.25 x 200 = 50 Method 2 (Fraction): 25% = 25/100 = 1/4. (1/4) x 200 = 200/4 = 50

Commentary: This question focuses on calculating percentages of whole numbers using both decimal and fraction methods. Both methods lead to the same answer.

Question 4: In a group of learners, the ratio of girls to boys is 4:

5. If there are 20 boys, how many girls are there?

Solution: Ratio of Girls : Boys = 4 : 5 We know there are 20 boys, so the '5' in the ratio represents 20 boys. To find what one part of the ratio represents, divide 20 by 5: 20/5 = 4 So, each part of the ratio represents 4 learners. Since the ratio of girls is 4, multiply 4 by 4: 4 x 4 = 16 Therefore, there are 16 girls.

Commentary: This question combines ratios with a problem-solving element. It requires understanding the proportional relationship within the ratio.

Question 5: A shop is offering a 10% discount on a pair of shoes that originally cost R

3

0

0. How much is the discount?

Solution: Discount = 10% of R300 Method 1 (Decimal): 10% = 0.10. 0.10 x R300 = R30 Method 2 (Fraction): 10% = 10/100 = 1/10. (1/10) x R300 = R300/10 = R30 The discount is R30.