1.01 Descriptive modeling
Introduction
Choosing appropriate units, defining the quantities we are working with, and the level of accuracy with which we work in mathematics are all important skills for problem-solving. We will focus on these in real-world contexts to strengthen our understanding and ability to reason with these skills.
Units and quantities
Understanding units in context is a foundational skill when solving problems in math. Choosing and interpreting units are ways to build on our graphing skills when defining quantities in problems and determining relationships between quantities defined as rates.
A rate is the ratio of one quantity to a corresponding quantity. A rate is often indicated by the word "per," and when a rate is reduced to a ratio with a denominator of 1, we call it the unit rate.
Exploration
Pura is attempting to do as many pull-ups as she can. Consider the graph that models this context.
- What do you think the units labeled on the axes of the graph could be?
- Explain what the ordered pair \left(3, 15\right) could mean in context.
Understanding the context helps us make decisions about defining quantities.
In the exploration, since Pura is attempting to track her pull-ups, we can define the dependent variable, y, as the number of pull-ups Pura completes. The independent variable, x, will depend on how Pura is measuring her pull-ups. It's likely that she is tracking the number of pull-ups she completes over time, but different units give different meanings: If Pura is tracking her number of pull-ups after x seconds, then the point \left(3,15\right) means she completes 15 pull-ups in 3 seconds. We can also determine that her rate is 5 pull-ups per second. We should consider what units of time make the most sense in this context, be it seconds, minutes, days, or weeks of practice.
Examples
Example 1
Qin is training for the 100-meter sprint on a treadmill where he can set a constant running speed. His best time during training was 13.8 seconds.
The graph below comes from the display on Qin's treadmill.
Explain why the units on the axes cannot be meters and seconds.
Convert Qin's speed to miles per hour.
Determine if the axes on the treadmill's graph can be labeled in miles and hours.
Example 2
A stand at a farmer's market offers three bag sizes of peaches: small, medium, and large.
Describe the quantities that should be considered if we want to maximize the amount of peaches we can purchase.
Suppose that the charges for the three bag sizes of peaches are as follows: a small bag at 2 \text{ lb} for \$5.40; a medium bag at 10 \text{ lb} for \$21.35; a large bag at 20 \text{ lb} for \$39.14. Use this information to determine the combination of bags that would weigh the most for \$100.
Example 3
The owner of a cafe wants to determine how busy at lunch the shop is on the weekends versus weekdays.
Describe appropriate independent and dependent variables of the relationship.
The graph below shows the comparison between lunch on the weekends versus weekdays. Choose appropriate scales and unit labels for the axes. Justify your reasoning.
A unit rate is indicated by words such as "per," "each," and "every," when comparing quantities. The independent and dependent variables are usually stated in this order for a rate:\text{Dependent variable per Independent variable} = \dfrac{\text{Dependent variable}}{\text{Independent variable}}
Accuracy and measurement
There are often situations, typically involving measurement, where it is necessary and practical to round the values that have been obtained.
Although we are familiar with rounding values to a certain number of decimal places, rounding using significant figures can be applied to all numbers, whether or not they have a decimal point.
For example, if 95 \,446 people attend a football game, the media may report this figure as 95 \,000. It's an easier number for the public to remember, and no real meaning is lost in using it.
If a chef wants to divide 500 \text{ g} of pasta among 6 people, each portion should be 83.333 \ldots \text{ g}. It is not possible to measure to this degree of accuracy, so the chef may choose instead to weigh each portion as 80 \text{ g}, depending on the precision of the measuring device.
The aim of a measurement is to obtain the "true" value of a quantity: the height of a tree, the temperature of a room, the mass of a rock, or whatever we want to know.
But is a tree ever exactly 50 \text{ ft} tall? Is a room ever exactly 73 \degree? Is a rock ever exactly 2 \text{ kg} in weight?
We can carefully design a measurement procedure to make more and more precise measurements, which makes the number of significant figures in our measurement increase. But we can only ever report the closest marking, and at some point, the object we are measuring will fall between the markings.
For any measurement tool, we say its absolute error is equal to half the distance of its smallest unit. Any measurement we make with that tool must be given as plus or minus the absolute error.
Any subsequent measurement that is more precise will fall within this range, but we can't know exactly where until we try with a better tool.
Examples
Example 4
Round the following numbers to the indicated significant figures.
Round 461\,585 to three significant figures.
Round 0.006\,377\,36 to two significant figures.
Example 5
A measuring tape has markings every 20 \text{ cm.}
What is the absolute error of the measuring tape?
The length of an object is measured as 120 \text{ cm} by the measuring tape. A second measurement is then taken, measuring its length to the nearest centimeter. What is the range we should expect this second measurement to lie within?
Rounding to significant figures is a skill when reporting information in a practical manner. Understanding that absolute error is equal to half the distance of a measurement tool's smallest unit also helps us report appropriate measurements.