Lesson Notes By Weeks and Term v5 - Grade 11
Numbers and calculations with numbers (revision and extension) – Week 5 focus
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Numbers and calculations with numbers (revision and extension) – Week 5 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematical Literacy
Class: Grade 11
Term: 1st Term
Week: 5
Theme: General lesson support
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This week, we're revisiting and expanding on our understanding of numbers and calculations. This is absolutely crucial in Mathematical Literacy because it forms the foundation for understanding budgets, salaries, loan repayments, interpreting statistics, and making informed financial decisions – all things you'll encounter daily, both now and as adults in South Africa. Whether you're calculating taxi fares, understanding data bundles, or planning your future finances, a solid grasp of numbers is essential. We'll be focusing on accuracy and efficiency in performing calculations, as well as interpreting the results in context.
2.1 Fractions, Decimals, Percentages, and Ratios: Fractions: Represent parts of a whole. Important for sharing resources, calculating proportions, and understanding measurements (e.g., fabric needed for clothing). A fraction has a numerator (top number) and a denominator (bottom number).
Example: 1/4 means one part out of four.
Decimals: Another way to represent parts of a whole, using a base-10 system. Easily used with calculators and for precise measurements.
Example: 0.75 is equivalent to 3/
4. Percentages: A way of expressing a number as a fraction of
1
0
0. Crucial for understanding discounts, interest rates, and statistics. "%" symbol means "out of one hundred."
Example: 25% is equivalent to 25/100 or 0.
2
5. To convert a decimal to a percentage, multiply by
1
0
0. To convert a percentage to a decimal, divide by
1
0
0. Ratios: Compare two or more quantities. Useful for recipes, sharing costs, and understanding scale models.
Written as a:b (e.g., 1:2 means one part of the first quantity for every two parts of the second quantity). Example 1 (Fractions, Decimals, and Percentages): A soccer team won 7 out of 10 matches.
As a fraction: 7/10 As a decimal: 7 ÷ 10 = 0.7 As a percentage: 0.7 × 100 = 70% Example 2 (Ratios): To make a traditional 'vetkoek' recipe, the ratio of flour to water is 3:
2. If you want to use 6 cups of flour, how much water do you need? The ratio is 3 parts flour to 2 parts water. 6 cups of flour represent 3 parts. Each part is therefore 6 ÷ 3 = 2 cups.
Water needed: 2 parts × 2 cups/part = 4 cups of water. 2.2 Direct and Inverse Proportion: Direct Proportion: When one quantity increases, the other quantity increases proportionally. The relationship can be represented as y = kx, where k is a constant.
Example: The more hours you work, the more money you earn (assuming a fixed hourly rate).
Inverse Proportion: When one quantity increases, the other quantity decreases proportionally. The relationship can be represented as y = k/x, where k is a constant.
Example: The more people who share a pizza, the smaller each person's slice will be.
Example 3 (Direct Proportion): If 3 oranges cost R12, how much will 7 oranges cost?
Cost per orange: R12 ÷ 3 = R4 Cost of 7 oranges: R4 × 7 = R28 Example 4 (Inverse Proportion): It takes 4 workers 6 hours to complete a task. How long will it take 8 workers to complete the same task, assuming they work at the same rate?
Total work required: 4 workers × 6 hours = 24 worker-hours Time for 8 workers: 24 worker-hours ÷ 8 workers = 3 hours 2.3 Rounding: Rounding simplifies numbers and makes them easier to work with.
However, it introduces a degree of inaccuracy.
Rules: If the digit to the right of the rounding place is 5 or greater, round up. If the digit to the right of the rounding place is less than 5, round down. Understanding the context is crucial. Rounding to the nearest Rand is appropriate for costs, while rounding to several decimal places might be necessary for precise measurements.
Example 5 (Rounding): A loaf of bread costs R12.
7
8. Round the price to the nearest Rand: R
1
3. A scientific calculation results in 3.
1
4
1
5
9. Round to two decimal places: 3.14. 2.4 Calculator Use: Calculators are essential tools, but it's crucial to understand how to use them effectively and interpret the results. Familiarize yourself with the basic functions (+, -, ×, ÷, %, √, x², memory functions). Be aware of the order of operations (BODMAS/PEMDAS). Use the memory functions (M+, M-, MR, MC) to store intermediate results in complex calculations. Practice using the calculator to solve various problems, including those involving fractions, decimals, and percentages. Guided Practice (With Solutions)
Question 1: A shop offers a 15% discount on a shirt that originally costs R
2
5
0. What is the discounted price?
Solution: Calculate the discount amount: 15% of R250 = (15/100) × R250 = R37.50 Subtract the discount from the original price: R250 - R37.50 = R212.50 Therefore, the discounted price is R212.
5
0. Question 2: You want to share R500 between two siblings, Ayanda and Bongani, in the ratio 2:
3. How much does each sibling receive?
Solution: Find the total number of parts in the ratio: 2 + 3 = 5 parts Calculate the value of one part: R500 ÷ 5 parts = R100 per part Calculate Ayanda's share: 2 parts × R100/part = R200 Calculate Bongani's share: 3 parts × R100/part = R300 Therefore, Ayanda receives R200 and Bongani receives R
3
0
0. Question 3: If it takes 6 hours to paint a room with 2 painters, how long will it take with 3 painters, assuming they work at the same rate?
Solution: This is an inverse proportion problem. Calculate the total work required in painter-hours: 6 hours * 2 painters = 12 painter-hours Divide the total work by the new number of painters to find the new time: 12 painter-hours / 3 painters = 4 hours Therefore, it will take 4 hours.
Question 4: A farmer has 250 sheep. He estimates that 12% will have twins this lambing season. Approximately how many lambs will be born?