topic badge

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.

1
2
3
4
x
2
4
6
8
10
12
14
16
18
20
y
  1. What do you think the units labeled on the axes of the graph could be?
  2. 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.

1
2
3
4
5
6
7
x
20
40
60
80
100
120
y
a

Explain why the units on the axes cannot be meters and seconds.

Worked Solution
Create a strategy

We know from the problem that Qin is tracking his time and distance and that time is independent of distance.

Apply the idea

If the time and distance were in meters and seconds, we would see that at 13.8 seconds, the distance would be 100 meters. This is not what we see on the graph.

1
2
3
4
5
6
7
\text{Time, } x
20
40
60
80
100
120
\text{Distance, } y
b

Convert Qin's speed to miles per hour.

Worked Solution
Create a strategy

Starting with the rate at which Qin runs, we can choose appropriate conversion factors to multiply by. Then, dividing out common units in the conversion will simplify the rate.

Apply the idea
A solution process. 100 m is divided by 13.8 sec times 1 km over 1000 m times 0.625 mi over 1 km times 60 sec over 1 min times 60 min over 1 hr is equal to 225000 mi over 13800 hr which reduces to 16.30 mi over 1 hr. The units m, km, sec, and min were eliminated. The conversion factors are 1 km is 1000 m, 1.06 mi is 1 km, 60 sec is 1 min and 60 min is 1 hr.

Qin runs 16.30 miles per hour.

c

Determine if the axes on the treadmill's graph can be labeled in miles and hours.

Worked Solution
Create a strategy

The unit rate we converted to in part (b) was miles per hour. We can confirm that the graph matches the unit rate by matching points.

Apply the idea

The graph begins at zero, and after 1 hour, we see the distance is marked around 16 miles. This matches the units converted to in the unit rate from part (b).

1
2
3
4
5
6
7
\text{Time in hours, } x
20
40
60
80
100
120
\text{Distance in miles, } y
Reflect and check

It is unlikely that Qin can maintain his sprint pace for hours, so the graph provided by the treadmill is not realistic.

Example 2

A stand at a farmer's market offers three bag sizes of peaches: small, medium, and large.

a

Describe the quantities that should be considered if we want to maximize the amount of peaches we can purchase.

Worked Solution
Create a strategy

Consider what you know about how the cost of fruit, like peaches, is usually determined.

Apply the idea

The quantities we want to measure are the peaches and their cost. The units we could use to measure them may be the number of peaches, their weight in pounds, ounces, kilograms, or grams, and the cost will likely be in dollars but could be measured in other units of value.

b

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.

Worked Solution
Create a strategy

A comparison in cost per pound of peaches is useful in helping us decide how to combine bags.

Apply the idea

For each of the bags, we can compare the cost per pound of peaches:

Small bag: \dfrac{\$5.40}{2 \text{ lb}}= \dfrac{\$2.70}{\text{lb}}

Medium bag: \dfrac{\$21.35}{10 \text{ lb}}= \dfrac{\$2.14}{\text{lb}}

Large bag: \dfrac{\$39.14}{20 \text{ lb}}= \dfrac{\$1.96}{\text{lb}}

The largest bag is the most expensive, but by comparing the rate for each, the largest bag offers the lowest price per pound of peaches. This means that we will get the most weight by purchasing more large bags than small ones.

The following combinations would cost \$100 or less:

  • 36 \text{ lb} with 18 small bags
  • 44 \text{ lb} with 4 medium bags and 2 small bags
  • 48 \text{ lb} with 2 large bags and 4 small bags
  • 50 \text{ lb} with 2 large bags and 1 medium bag

Since we know that the large bags of peaches are the best deal and the medium bags are the next best deal, we should buy the most large bags possible with the most medium bags to get the most peaches for \$100. The combination of 50 \text{ lb} with 2 large bags and 1 medium bag will be the most peaches.

Reflect and check

When calculating the cost per pound for each bag, we rounded our rates for the medium bag from 2.135 to 2.14 and 1.957 to 1.96. Since these amounts are in dollars, rounding to the hundredths place is reasonable.

Example 3

The owner of a cafe wants to determine how busy at lunch the shop is on the weekends versus weekdays.

a

Describe appropriate independent and dependent variables of the relationship.

Worked Solution
Create a strategy

The owner is attempting to determine how busy the cafe could be during certain times. The times could be measured in minutes or hours. How busy the cafe is could be measured by the amount of money the cafe brings in, the number of meals they sell, or the number of customers in the cafe.

Apply the idea

The independent variable could be time in hours. The dependent variable could be the number of customers in the cafe.

b

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.

x
y
Worked Solution
Create a strategy

Consider the independent and dependent variables from part (a) and determine if they are reasonable with the given graph.

Apply the idea

The independent variable is time in hours. The x-axis scale could be by 1, showing a total of 4 hours for a cafe's lunch crowd.

The dependent variable of customers in the cafe could be reasonable with this graph. If the scale on the y-axis is also 1, it looks like there are about 8 guests each hour in the cafe for the flatter graph and more for the steeper graph.

Reflect and check

The dependent variable could also be measured in the number of meals or the amount of money the cafe is making each hour. The scale for the amount of money could be by 5 or 10 or another dollar amount that makes sense aligned to the cost of items in the cafe.

Idea summary

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.

Significant figures

The digits considered to be significant in reporting a measurement, regardless of the location of the decimal point.

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.

The first image is an elephant that is being measured using a meter ruler with labelled markings from 1 to 4. The feet of the elephant lines up with the 0 mark and the top of the head lines up with the 4th mark after 2. The meter ruler has the smallest unit of 0.2 m, absolute error of 0.1 m, closest mark of 2.8 m and measurement of 2.8 plus or minus 0.1 m. The second image is the head of an elephant that is being measured using a centimeter ruler with labelled markings from 150 to 300. Half of the trunk lines up with 100 cm mark and the top of the head lines up with 3rd mark after 250 cm. The centimeter ruler has the smallest unit of 10 cm, absolute error of 5 cm, closest mark of 280 cm and measurement of 280 plus or minus 5 cm.
The head of an elephant is measured using a centimeter ruler with labelled markings from 275 to 295. Half of the head of the elephant lines up with the 275 cm mark and the top of the head lines up in the middle of the 4th and 5th mark after 280 cm. The centimeter ruler has the smallest unit of 1 cm, absolute error of 0.5 cm, closest mark of 284 cm and measurement of 284 plus or minus 0.5 cm.

Examples

Example 4

Round the following numbers to the indicated significant figures.

a

Round 461\,585 to three significant figures.

Worked Solution
Create a strategy

When rounding to n significant figures, we count (n+1) digits starting with the first nonzero digit from the left and round the nth digit up or down according to the (n+1) \text{th} digit. Then replace all other digits with zeros.

Apply the idea

462\,000

b

Round 0.006\,377\,36 to two significant figures.

Worked Solution
Apply the idea

0.0064

Reflect and check

We started counting two digits from the first nonzero digit, 6, and then determined what to round 3 to based on the 7 to its right. We rounded up the digit from 3 to 4, leaving us with the rounded number 0.0064 at two significant figures.

Example 5

A measuring tape has markings every 20 \text{ cm.}

a

What is the absolute error of the measuring tape?

Worked Solution
Create a strategy

The absolute error of a measuring tool is equal to \dfrac{1}{2} the smallest measurement of that tool. We are given that the measuring tape has markings every 20 \text{ cm,} so the smallest measurement of the measuring tape will be 20 \text{ cm.}The absolute error will be equal to half this amount.

Apply the idea

The absolute error of the measuring tape is 10 \text{ cm}.

b

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?

Worked Solution
Create a strategy

We can express the range of values as \text{Range of values} = \text{Marking value } \pm \text{ Absolute error}

Apply the idea

The second measurement should lie within 110 \text{ cm} and 130 \text{ cm.}

Idea summary

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.

Outcomes

N.Q.A.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N.Q.A.2

Define appropriate quantities for the purpose of descriptive modeling.

N.Q.A.3

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

What is Mathspace

About Mathspace