Welcome back to class! This week, we are tackling two fundamental areas of quantitative literacy: understanding the sheer scale of numbers and analyzing how quantities change over time. These concepts form the bedrock of critical thinking in mathematics, helping us verify whether the statistics we see in the news or advertisements actually make sense.

Section 1-5: Critical Thinking and Number Sense

In this section, we focus on magnitudes and the power of estimation. We often encounter numbers that are incredibly large (like the national debt) or incredibly small (like the width of a human hair). To handle these, we use powers of 10.

It is important to remember your exponent rules. For example, a negative exponent indicates a reciprocal:

$$a^{-n} = \frac{1}{a^n}$$

So, $10^{-3}$ is equal to $\frac{1}{1000}$ or $0.001$.

Common Pitfall: Area and Volume Conversions
A major focus of the lecture notes is unit conversion, specifically regarding area and volume. A common consumer error is assuming that because 1 yard = 3 feet, a square yard is 3 square feet. This is incorrect.

  • Area: Since area is length $\times$ width, 1 square yard is $3 \text{ ft} \times 3 \text{ ft} = 9 \text{ sq ft}$.
  • Volume: Similarly, 1 cubic yard is $3 \text{ ft} \times 3 \text{ ft} \times 3 \text{ ft} = 27 \text{ cubic ft}$.

Keep this in mind when estimating costs for things like carpeting or concrete!

Section 2-1: Measurements of Growth

In the second half of this week's lesson, we look at Analysis of Growth. We begin by defining a function, which describes how a dependent variable (output) relies on an independent variable (input). In many real-world examples, the independent variable is time.

To measure how fast something is changing, we use two specific formulas derived from the class notes:

1. Percentage Change
This measures the relative change over a period:

$$\text{Percentage Change} = \frac{\text{Change in function}}{\text{Previous function value}} \times 100\%$$

2. Average Growth Rate
This tells us the rate of change over a specific interval:

$$\text{Average Growth Rate} = \frac{\text{Change in function}}{\text{Change in independent variable}}$$

Interpolation vs. Extrapolation

Finally, we discussed using these rates to predict data.

  • Interpolation is estimating an unknown value between two known data points.
  • Extrapolation is estimating a value beyond the known data points.

Warning: Be very careful with extrapolation! As shown in our lecture slides, if you extrapolate the age of first-time mothers out to the year 3000, you might predict a mother being 95 years old. Always consider if your mathematical prediction makes sense in the real world.

Please review the attached PDFs for the detailed examples on SARS cases and computer memory, and don't forget to take the quizzes for both sections!