Lesson Notes By Weeks and Term v4 - SHS 2
PROPORTIONAL REASONING
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PROPORTIONAL REASONING
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Subject: Mathematics
Class: SHS 2
Term: 1st Term
Week: 7
Grade code: 2.1.2.LI.2
Strand code: 1
Sub-strand code: 2
Content standard code: 2.1.2.CS.1
Indicator code: 2.1.2.LI.2
Theme: NUMBERS FOR EVERYDAY LIFE
Subtheme: PROPORTIONAL REASONING
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Proportional reasoning is a foundational mathematical skill that we use every day, often without even realising it. When we cook Jollof rice for a large family gathering based on a recipe for four people, we use proportions. When a tailor takes measurements to sew a perfectly fitting dress, they are using ratios. When we discuss the speed of a tro-tro in kilometres per hour, we are talking about rates. Understanding the relationship between ratios, rates, and proportions allows us to make fair comparisons, scale quantities up or down, and solve a wide range of practical problems in science, business, and daily life in Ghana.
Part 1: Defining the Core Ideas
a) Ratio A ratio is a comparison of two or more quantities of the *same kind* and in the *same unit*. It shows the relative size of the quantities. Notation: A ratio can be written in three ways: Using a colon, e.g., `a : b` As a fraction, e.g., `a/b` In words, e.g., "a to b" Key Points: The order in a ratio is very important. The ratio of boys to girls is different from the ratio of girls to boys. Ratios should be simplified to their lowest terms, just like fractions. For example, a ratio of `10 : 15` is simplified to `2 : 3` by dividing both parts by 5. Ghanaian Context Example: In a class, there are 20 girls and 15 boys. The ratio of girls to boys is `20 : 15`. Simplified, this is `4 : 3`. The ratio of boys to the total number of students (20 + 15 = 35) is `15 : 35`. Simplified, this is `3 : 7`.
b) Rate A rate is a special type of ratio that compares two quantities with *different units*. Key Points: The units are always stated. A unit rate is a rate where the second quantity is 1. This is very useful for comparisons. Ghanaian Context Examples: Speed: A bus travels 240 kilometres in 3 hours. The rate is `240 km / 3 hours`. The unit rate (speed) is `80 km/h` (80 kilometres per hour). Cost: A 5kg bag of Gari costs GHS 30. The rate is `GHS 30 / 5 kg`. The unit rate (unit price) is `GHS 6 per kg`. Work: A student can solve 20 math problems in 40 minutes. The rate is `20 problems / 40 minutes`. The unit rate is `0.5 problems per minute`.
c) Proportion A proportion is an equation that states that two ratios (or rates) are equal. Notation: A proportion is written as: `a : b = c : d` or `a/b = c/d` Reading a Proportion: "a is to b as c is to d". The Cross-Product Property: This is the most powerful tool for solving proportions. For any proportion `a/b = c/d`, the cross-products are equal. `a × d = b × c` Part 2: Solving Proportions