Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Analyze complex real-life scenarios.
• Identify the correct mathematical concepts to use.
• Perform accurate calculations involving finance and measurement.
• Interpret data from tables and graphs.
• Critically evaluate your solutions.

Lesson summary

This week, we delve into integrated exam preparation using mixed real-life tasks. This is a crucial skill for Grade 12 Mathematical Literacy students, as the final exam often presents scenarios that require you to draw upon multiple mathematical concepts and apply them in practical contexts.

Think about it: Managing a budget involves understanding percentages, income tax, interest, and data interpretation. Planning a road trip requires calculating distances, fuel consumption, time, and interpreting maps. These are the kinds of integrated tasks you will face, not only in exams but also in your daily lives.

Lesson notes

This week is all about bringing together what you've learned throughout the year and applying it to complex situations. Here's a breakdown of key areas we'll be focusing on: Financial Mathematics: This is often the cornerstone of integrated tasks.

Understanding Income Tax: South Africa operates on a progressive tax system. This means the higher your income, the higher the tax bracket you fall into and the higher the percentage of tax you pay. You need to be able to calculate taxable income (gross income minus deductions like pension contributions), and then calculate tax based on the tax brackets provided in the annual budget.

Example: Sipho earns a gross monthly salary of R15,

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0. He contributes 7.5% of his salary to a pension fund. His medical aid contribution is R1,500 per month. Assuming the annual tax rebates for medical aid are applicable, calculate his monthly taxable income and estimated monthly tax, given the following simplified annual tax brackets (for illustrative purposes): R0 - R80,000: 18% R80,001 - R150,000: 26% Solution: Pension Contribution: R15,000 * 0.075 = R1,125 Monthly Taxable Income: R15,000 - R1,125 = R13,875 Annual Taxable Income: R13,875 * 12 = R166,500 Annual Tax: (R80,000 0.18) + ((R166,500 - R80,000) 0.26) = R14,400 + R22,490 = R36,890 Monthly Tax: R36,890 / 12 = R3,074.17 (approximately)

Important Notes: This example ignores medical aid rebates to keep things simple. In a real exam, they would be provided. It also uses simplified tax brackets.

Compound and Simple Interest: Understanding how interest works is crucial for investments and loans.

Simple Interest: Interest calculated only on the principal amount.

Formula: A = P(1 + rt), where A = final amount, P = principal, r = interest rate, t = time.

Compound Interest: Interest calculated on the principal and accumulated interest.

Formula: A = P(1 + r/n)^(nt), where n = number of times interest is compounded per year.

Example: Maria invests R5,000 in a fixed deposit account that offers 8% interest per annum compounded quarterly. How much will she have after 3 years?

Solution: P = R5,000 r = 0.08 n = 4 (quarterly) t = 3 A = 5000(1 + 0.08/4)^(43) = 5000(1.02)^12 = R6,341.66 (approximately)

Loans and Repayments: You need to be able to calculate loan repayments using provided amortization schedules or formulas.

Measurement and Conversions: Units of Measurement: Be comfortable converting between different units (e.g., meters to kilometers, liters to milliliters, kilograms to grams).

Scale Drawings and Maps: Understand how to interpret scale and calculate actual distances from map measurements.

Example: A map has a scale of 1:50,

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0. Two towns are 7 cm apart on the map. What is the actual distance between the towns in kilometers?

Solution: Actual distance in cm: 7 cm * 50,000 = 350,000 cm Convert cm to meters: 350,000 cm / 100 = 3,500 m Convert meters to kilometers: 3,500 m / 1000 = 3.5 km Data Handling: Data Collection and Organisation: Understanding how data is collected and organized into tables.

Data Representation: Interpreting and creating different types of graphs (bar graphs, pie charts, line graphs, histograms) and understanding their strengths and weaknesses.

Data Analysis: Calculating measures of central tendency (mean, median, mode) and measures of spread (range).

Probability: Calculating basic probabilities in real-life scenarios.

Time Management and Planning: Timetables and Schedules: Interpreting and creating timetables, calculating travel times, and scheduling activities.

Project Planning: Using timelines and Gantt charts to plan and manage projects. Guided Practice (With Solutions)

Question 1: A small business owner, Thando, is planning a market day stall to sell her crafts.

She estimates the following costs: Stall rental: R250 Materials: R10 per item Transport: R100 She plans to sell each item for R

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5. She estimates she can sell 50 items. a) Calculate her total costs. b) Calculate her total revenue. c) Calculate her profit or loss. d) What is her profit/loss per item sold?

Solution: a)

Total Costs: Materials cost: 50 items R10/item = R500 Total cost: R250 (stall) + R500 (materials) + R100 (transport) = R850 b)

Total Revenue: Revenue: 50 items R35/item = R1750 c)

Profit/Loss: Profit: R1750 (revenue) - R850 (cost) = R900 d)

Profit/Loss per item: Profit per item: R900 / 50 items = R18/item

Commentary: This question integrates cost calculations, revenue calculations, and profit/loss determination – all crucial for basic business understanding.

Question 2: A family of four (2 adults, 2 children) is planning a road trip from Johannesburg to Durban. The distance is approximately 560 km. Their car consumes 8 liters of petrol per 100 km. The current petrol price is R22.50 per liter. They plan to stop for lunch which costs R350. a) Calculate the total amount of petrol needed for the trip (one way). b) Calculate the total cost of petrol for the trip (one way).