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Algebra 1 View details
7. Scaling and Descriptive Modeling
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Chapter 1
7. 

Scaling and Descriptive Modeling

Scaling and descriptive modeling are essential tools in Algebra 1 for understanding relationships between quantities. The lesson covers various real-life situations, such as calculating population density, determining the cost of paint needed for a room, and understanding unit rates in different contexts like phone plans and shopping deals. These mathematical tools help in making informed decisions, whether it is choosing the most cost-effective paint or selecting a phone plan based on call duration and cost. The importance of using appropriate units for different quantities is also emphasized, making the concepts more relatable and easier to apply in everyday life.
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Lesson Settings & Tools
10 Theory slides
8 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Scaling and Descriptive Modeling
Slide of 10
In real-world situations, quantities are accompanied by appropriate units so that the situations can be easily understood and interpreted. Before analyzing situations involving too large or too small quantities, their models are created by keeping the ratio between corresponding lengths constant. In this lesson, this ratio will be named and used to analyze real-life situations.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Scaling the Solar System

In the model of the Solar System below, the initial labeled distance corresponds to miles. Zoom in or out to see how the labeled distance is affected.
Solar System with a moving scale
Think about the following questions:
  • Is it possible to make a model of the Solar System using the actual sizes of planets and distances between planets?
  • Is it possible to draw a model of the Solar System on a sheet of paper? If so, how many miles should be represented by inch?
Discussion

Different Presentations in Modeling

When modeling a real-life situation, it is sometimes impossible to represent an object or scenario using the original dimensions. In these situations, it is more convenient to work with manageable units while still being able to maintain the same properties as the original. This can be done by making a scale drawing.

Concept

Scale Drawing

A scale drawing is a two-dimensional drawing that is similar to an actual object or place. In a scale drawing, the ratio of any length on the drawing to the actual length is always the same.
Possible examples of scale drawing are floor plans, blueprints, and maps. Plan.jpg

In the case of the original real-life situation involving a three-dimensional object, making a model is more useful than a drawing. The idea behind a model is the same as a scale drawing, but the model has three dimensions instead of two.

Concept

Scale Model

A scale model is a three-dimensional model that is similar to a three-dimensional object. The ratio of a linear measurement of a model to the corresponding linear measurement of the actual object is always the same.
Here is an example scale model of a building. Building-Model.jpg
 In any case, it is important to define appropriate quantities to represent the distances of the original object or situation. This relationship between the new quantities used and the actual distances is known as the scale.
Discussion

Scale

The scale of a model or drawing is the ratio between any length on the model or drawing and its corresponding length on the actual object or place.

Suppose a drawing has a scale of This means that inch on the drawing represents feet on the actual object. Apart from the colon notation, a scale can be expressed using an equals sign or as a fraction, as it is a ratio.

Denoting a Scale
Ratio
Equals Sign
Fraction

When a scale is written without specifying the units, it is understood that both numbers have the same units of measure. For example, a scale of means that the actual object is twice the size of the model. A scale of means that the actual object is half the size of the model.

Big fire truck, scalable fire truck, and scale between the two trucks
External credits: Derek Quinn
 Knowing the scale is helpful in different scenarios, like when reading a map. The scale allows the reader to compute actual distances.
Example

Using the Scale to Find Actual Dimensions

The Sweden Solar System is the world's largest model of the planetary system. In this model, the Globe Arena in Stockholm represents the Sun, while the planets align to the north. Click on the dots below, representing the planets, to show the dimensions, location, and distance to the Globe Arena.
Swedish Solar System
The scale used to make this model is million.
a What is the actual diameter of the Earth? Give the answer in kilometers.
b What is the actual distance from Pluto to the Sun?

Hint
a Note that the scale does not specify the units used. This means that both numbers share the same units of length.
b To find the actual dimension, multiply both numbers in the scale by the matching dimension on the model.

Solution
a Since the scale does not specify the units, it means that both numbers have the same units. Then, to find the actual diameter of the Earth, multiply both numbers by the diameter of the Earth in the model, centimeters.
This shows that centimeters on the model corresponds to centimeters in real life. This means that the actual diameter of the Earth is centimeters. A more appropriate unit to represent this is kilometers, so next, use a conversion factor to convert the centimeters to kilometers.
MultII

Multiply by

▼
Simplify
CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

MultFrac

Multiply fractions

CalcQuot

Calculate quotient

b Similarly, to find the actual distance from Pluto to the Sun, multiply both numbers in the scale by
In conclusion, the distance from Pluto to the Sun is billion kilometers.
When the scale is not given, it can be found using a dimension on the model and the corresponding dimension on the actual object.
Example

Finding the Scale of a 3D Model

Tom made a model of the Empire State Building for a school project. The model has a height of centimeters.
3D model of the Empire State Building
The actual height of the Empire State Building is meters, including its antenna.
a According to the scale Tom used, how many meters correspond to centimeter on the model?
b What is the actual width of the Empire State Building in meters?

Hint
a The scale compares a dimension on the model to the corresponding dimension on the original object.
b Multiply each side of the scale by the width of the model.

Solution
a By definition, the scale of a model compares a dimension on the model to the corresponding dimension on the actual object. In this case, compare the height of the model and the actual height of the Empire State Building in order to find the scale.
According to this scale, centimeters correspond to meters. The scale can be simplified by dividing both numbers by
In the scale used by Tom, centimeter corresponds to meters.
b Although the scale was found using the height, the same scale is used for the width as well. Therefore, the actual width can be calculated by multiplying each side of the scale by the width of the model, which is centimeters. This time the scale in the form of an equation will be used.
Therefore, the Empire State Building is meters wide.
A scale can also be seen as a rate. For example, consider inch : miles. This rate can then be written as a ratio.
In a rate, however, the numerator and denominator do not have to represent units of distance. For example, when the numerator represents a population and the denominator represents an area, this unit of measurement is called population density.
Example

Calculating Population Density

The United States of America has an area of square kilometers. In mid the total population was people.

People on the map of USA
External credits: @rocketpixel
Find the population density in people per square mile. Round the answer to the nearest integer. Then, based on the units, write a few sentences to describe what population density measures.

Answer

In the United States of America, the population density is people per square mile. The population density measures the number of people per unit of area.

Hint

People per square mile can be written as This implies that the population density is expressed as a ratio. Remember that square kilometer is equal to square mile.

Solution
According to the problem, the population density needs to be given using the unit of people per square mile. This means that the population density is found by dividing the number of people by the area of a certain region.
Before calculating the quotient, note that the area given is in square kilometers. Therefore, a unit conversion from square kilometers to square miles should be performed. To do this, use the fact that square kilometer is equal to about square miles.
MultII

Multiply by

▼
Simplify
CrossCommonFac

Cross out common factors

MultFrac

Multiply fractions

The next step is to divide the population by the area, which gives the population density. Here, the result is rounded to the nearest integer.
Now that the population density is known, write the answer in a sentence form.

The population density of the United States of America is approximately per

This statement assumes that the total population is spread out evenly across the United States.

94 people in a rectangle of the area of 1 square mile
The population density measures the number of people per unit of area.
Example

Practicing Converting a Scale

Mark wants to paint all of the walls of his bedroom except for the wall that contains the door. Each rectangular wall is meters high. The paint he will use is sold in liter cans — the price per liter is

Dimensions of the bedroom

After painting for a few minutes, Mark noticed that square meters can be covered with one liter of paint.

a What is the total area, in square meters, that Mark has to paint?
b What is the minimum number of liters of paint Mark needs?
c How many cans should Mark buy?
d How many dollars could Mark save if the paint was sold in liter cans?

Hint
a Use the formula for the area of a rectangle to calculate the area of each wall.
b How many liters of paint are needed to cover one square meter of wall?
c Compare the number of liters needed to cover all three walls to the amount of paint in one can.
d Calculate the difference between the amount of money that Mark paid and the cost of liters of paint.

Solution
a Start by writing the dimensions of the walls to be painted. Note that two walls have the same dimensions. This means that they also have the same area.
Dimensions
Wall
Wall
Wall
The area of a rectangular wall is found by multiplying the length by the height.
SubstituteII

,

Multiply

Multiply

Remember that Walls and are the same size and have the same area. The area of Wall can be calculated similarly.
Now, to find the total area Mark has to paint, find the sum of the areas of all three walls.
b Saying that square meters can be covered with a liter of paint is equivalent to saying that of a liter of paint can cover square meter.
Multiplying by the total area gives the number of liters that Mark needs to paint the three walls.
c Mark needs liters of paint, but each can of paint contains liters. This means that Mark will actually have to buy cans to have enough paint to cover all three walls.
d The cost of the cans can be calculated by multiplying the number of liters in both cans by the price per liter. Each can contains liters, so in total there are liters of paint.
If the paint was sold in liter cans, Mark would need to buy cans. In this case, he would pay The amount of money that Mark could have saved can be found by calculating the difference between the price of liters of paint and liters of paint.
Note that using the appropriate units is critical when making decisions based on comparing two or more quantities.
Pop Quiz

Practicing Finding or Using the Scale

On the applet, the model and the actual object are shown. Using the given information, find the scale or the size of either the model or the actual object.

  • When entering the scale, write it in the form of a fraction with the numerator of
  • When entering the size of a model or an actual objects, round to the closest integer.
Applet that randomly generates the images of actual objects and corresponding models with the given scale or sizes of the objects.
Closure

Calculating the Scale to Choose Better Phone Plan

Tadeo wants to buy a cheap phone plan for calling his friends and family. He asked Ali and Ramsha how much they pay per call.

  • Ali said he paid for his last call.
  • Ramsha said she paid for her last call.

Initially, Tadeo decided to take Ramsha's plan since she paid less. Later, he realized that this information is not helpful since he does not know the duration of each of the calls made by Ali and Ramsha.

Price per call is not an appropriate unit.

Therefore, Tadeo decided to ask Ali and Ramsha how long each phone call lasted.

  • Ali said his call lasted minutes.
  • Ramsha said her call lasted minutes.
Since the duration of the calls is different, Tadeo became confused and made the following diagram to think about the situation.
After this, Tadeo realized that dividing each call's cost by its duration will give him the price per minute, which is an appropriate unit to compare the plans.
Person Scale of the Plan
Ali per minute
Ramsha per minute
Consequently, Tadeo decided to take Ali's plan because the price per minute is cheaper. What are other possible situations where calculating the scale can help to make best decision?



Level 1
Level 2
Level 3
Scaling and Descriptive Modeling
Exercises

Zain is painting a fence with slats. On average, they paint about slat every minutes.

A fence with 120 slats
How many hours will it take Zain to paint the entire fence?

We know that Zain paints 1 slat in 5 minutes. We can use this piece of information to find his speed or unit rate. Speed → 1 slat/5 minutes It is also given that Zain needs to paint 120 slats. Note that distance can be measured by any unit we want, as long as it has a length. This is why we can use the Speed Formula. Speed=Distance/Time Let's substitute the known values of the speed and the distance to calculate the time.

It takes Zain 600 minutes to paint the fence. The answer should be given in hours, so this value must be converted from minutes to hours. Note that 1 hour equals 60 minutes, which gives the following conversion factor. 1 hour/60 minutes Let's convert the time by multiplying 600 minutes by the conversion factor.

Ignacio bought a package of disposable razors for If a single razor lasts for a week, count in cents the average cost per day for the razors. Round the value to the nearest cent, if necessary.

Since there are 20 razors in each pack, and every razor lasts for 1 week, the entire pack lasts for 20* 1=20 weeks. This means that Ignacio pays $10 in the span of 20 weeks. Divide $10 by 20 weeks to determine the cost per week, or unit rate. $10/20 weeks = $0.5/1 week Weeks can be converted into days to determine how much a pack of razors costs Ignacio per day. Since there are 7 days in one week, the following conversion factor can be used. 1 week/7 days We can find the cost per day by multiplying the cost per week by this conversion factor from weeks to days.

Ignacio paid about $0.07 per day for the razors. Since $1 equals 100 cents, we get a cost of 7 cents per day.

Which offer is the better deal, bagels for or bagels for

A six-bagel batch with a price of $3.29 and an eight-bagel batch with a price of $4.15

We can find the cheapest price per bagel by comparing the unit rate of each offer. Begin by finding the unit rate for the six bagel offer. Rate &→ Price/Number of bagels & ⇓ Rate &→ $ 3.29/6 bagels Next, to find the corresponding unit rate, we need to determine the price of 1 bagel.

Using the same method, calculate the unit rate for the eight bagel offer.

Offer Price/Number of bagels Price/1bagel
6 bagels for $ 3.29 $3.29/6bagels ≈ $ 0.55/1bagel
8 bagels for $4.15 $4.15/8bagels \dfrac{\approx \$ 0.52} {1 \text{ bagel}}

The second offer is cheaper than the first.

Consider the following units of length: millimeters (mm), centimeters (cm), kilometers (km), micrometer (μm), and lightyears (ly). Assign them to the appropriate situation below.

1)Distance to star; 2)Height of a human; 3)Length of a ladybug; 4)Length of a bacteria; 5)Distance to cities

Let's go situation by situation and assign appropriate units of length to each object.

Distance Between Stars

Distances between stars are gargantuous. They are so large that measuring them in normal everyday units, such as kilometers or meters, becomes relatively meaningless.

Think of it as measuring the distance from California to New York in millimeters. It can be done, but it does not convey meaningful data compared to measuring that distance in kilometers. For a similar reason, we use lightyears — where 1 lightyear is the distance that light travels in one year — when measuring the distance to stars.

Height of a Human

Here, we have two possible units. We can measure a person in centimeters or in meters. However, it is more common to talk about someone's height in centimeters. This is because most people fall within 1 and 2 meters of height.

By expressing the height of a human in centimeters, we avoid working with pesky decimals yet we can still have precise and accurate measurements.

Length of a Ladybug

Let's think of a time when we saw a ladybug. Was it bigger than a fingernail? Likely not, right? But it was probably not that much smaller either.

Since a fingernail has a length of about 1 centimeter, we should use millimeters to measure the length of a ladybug that is slightly smaller than 1 centimeter.

Length of a Bacteria

Bacteria are so small that people can not even see them.

Because of that, they require the smallest unit of the given list. This is μm, which stands for micrometers.

Distance Between Cities

Think of two major cities like New York and Boston. If someone asked us the distance between them, we would probably answer in miles or kilometers.

In this case, miles is not one of the given units, so we will assign kilometers to this situation.

In one household, the electricity bill cost for a day period.

Electricity bill
What does the household pay per hour? Round to the nearest cent.

The rate per hour can be written as the following ratio. Rate → Price/1 hour To find this ratio for the presented situation, we first have to convert 30 days into hours. Since there are 24 hours in a day, we get the following conversion factor. 24 hours/1 day By multiplying the days in the month by this conversion factor, we can convert 30 days into hours.

As we can see, the family pays $119.22 for 720 hours. Let's substitute these values into the expression for rate. Rate → $119.22/720 hours Next, we can simplify this ratio to obtain a unit rate.

The family pays about $0.17 per hour. Since $1 equals 100 ¢, we get a cost of 17 ¢ per hour.

Scaling and Descriptive Modeling

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