7. Scaling and Descriptive Modeling
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Chapter 1
7.
Scaling and Descriptive Modeling
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Why
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.
| | 10 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
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.
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Explore
Scaling the Solar System
In the model of the Solar System below, the initial labeled distance corresponds to 93 000 000 miles. Zoom in or out to see how the labeled distance is affected.
- 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 1 inch?
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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. drawing→/actual→ l_1/L_1 = l_2/L_2 ←drawing/←actual Possible examples of scale drawing are floor plans, blueprints, and maps.
In the case of the original real-life situation involving a three-dimensional object, making a 3D model is more useful than a drawing. The idea behind a 3D 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. model→/actual→ l_1/L_1 = l_2/L_2 ←model/←actual Here is an example scale model of a building.
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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.
lLength on the drawing : lCorresponding length on the actual object
Suppose a drawing has a scale of 1 in:100 ft. This means that 1 inch on the drawing represents 100 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 | 1 in : 100 ft |
| Equals Sign | 1 in = 100 ft |
| Fraction | 1 in/100 ft |
When a scale is written without specifying the units, it is understood that both numbers have the same unit of measure. For example, a scale of 1:2 means that the actual object is twice the size of the model. A scale of 1:0.5 means that the actual object is half the size of the model — whether it be in meters, inches, yards, and so on.
External credits: Derek Quinn
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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.
a
What is the actual diameter of the Earth? Give the answer in kilometers.
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b
What is the actual distance from Pluto to the Sun?
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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, 65 centimeters.
65cm* 1= 20 000 000^(Original scale) * 65cm ⇕ 65 cm=1 300 000 000 cm This shows that 65 centimeters on the model corresponds to 1 300 000 000 centimeters in real life. This means that the actual diameter of the Earth is 1 300 000 000 centimeters. A more appropriate unit to represent this is kilometers, so next, use a conversion factor to convert the centimeters to kilometers.
1 300 000 000 cm
MultII
Multiply 1 300 000 000 cm by 1 m/100 cm*1 km/1000 m
1 300 000 000 cm * 1 m/100 cm*1 km/1000 m
▼
Simplify
CrossCommonFac
Cross out common factors
1 300 000 000 cm * 1 m/100 cm*1 km/1000 m
CancelCommonFac
Cancel out common factors
1 300 000 000 * 1/100* 1 km/1000
MultFrac
Multiply fractions
1 300 000 000/100 000 km
CalcQuot
Calculate quotient
13 000 km
b
Similarly, to find the actual distance from Pluto to the Sun, multiply both numbers in the scale by 300 km.
300km* 1= 20 000 000^(Original scale) * 300km ⇕ 300 km=6 000 000 000 km In conclusion, the distance from Pluto to the Sun is 6 billion kilometers.
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Example
Finding the Scale of a 3D Model
Tom made a 3D model of the Empire State Building for a school project. The model has a height of 55.375 centimeters.
a
According to the scale Tom used, how many meters correspond to 1 centimeter on the model?
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b
What is the actual width of the Empire State Building in meters?
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Hint
a
The scale compares a dimension on the 3D model to the corresponding dimension on the original object.
b
Multiply each side of the scale by the width of the 3D model.
Solution
a
By definition, the scale of a 3D 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.
ccc Model Height& &Actual Height 55.375 cm &:& 443 m According to this scale, 55.375 centimeters correspond to 443 meters. The scale can be simplified by dividing both numbers by 55.375. 55.375/55.375 cm : 443/55.375 m ⇕ 1 cm : 8 m In the scale used by Tom, 1 centimeter corresponds to 8 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 3D model, which is 7.125 centimeters. This time the scale in the form of an equation will be used.
7.125* 1 cm= 8 m * 7.125 ⇕ 7.125 cm= 57 m Therefore, the Empire State Building is 57 meters wide.
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Example
Calculating Population Density
The United States of America has an area of 9 147 420 square kilometers. In mid-2020, the total population was 331 923 317 people.
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 94 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 peoplemi^2. This implies that the population density is expressed as a ratio. Remember that 1 square kilometer is equal to 0.386102 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.
Population:& 331 923 317 people Area:& 9 147 420 km^2
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 1 square kilometer is equal to about 0.386102 square miles.
9 147 420 km^2
MultII
Multiply 9 147 420 km^2 by 0.386102 mi^2/1 km^2
9 147 420 km^2 * 0.386102 mi^2/1 km^2
▼
Simplify
CrossCommonFac
Cross out common factors
9 147 420 km^2 * 0.386102 mi^2/1 km^2
MultFrac
Multiply fractions
3 531 837.15684 mi^2
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. Pop. density &= 331 923 317 people/3 531 837.15684 mi^2 [0.25cm] Pop. density &≈ 94 people/mi^2 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 94 people per square mile.
This statement assumes that the total population is spread out evenly across the United States.
The population density measures the number of people per unit of area.
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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 2.8 meters high. The paint he will use is sold in 5-liter cans — the price per liter is $ 25.
After painting for a few minutes, Mark noticed that 5 square meters can be covered with one liter of paint.
a
What is the total area, in square meters, that Mark has to paint?
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b
What is the minimum number of liters of paint Mark needs?
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c
How many cans should Mark buy?
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d
How many dollars could Mark save if the paint was sold in 1-liter cans?
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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 7 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 1 | l_1 &= 4.75 m h &= 2.8 m |
| Wall 2 | l_2 &= 4.75 m h &= 2.8 m |
| Wall 3 | l_3 &= 3 m h &= 2.8 m |
The area of a rectangular wall is found by multiplying the length by the height.
A_1 = l_1 * h
SubstituteII
l_1= 4.75 m, h= 2.8 m
A_1 = ( 4.75 m)( 2.8 m)
Multiply
Multiply
A_1 = 13.3 m^2
Remember that Walls 1 and 2 are the same size and have the same area. The area of Wall 3 can be calculated similarly. A_3 = (3 m)(2.8 m) = 8.4 m^2 Now, to find the total area Mark has to paint, find the sum of the areas of all three walls. A_(total) &= 13.3 + 13.3 + 8.4 &= 35 m^2
b
Saying that 5 square meters can be covered with a liter of paint is equivalent to saying that 15 of a liter of paint can cover 1 square meter.
5 m^2/liter ⇔ 1 liter/5 m^2 Multiplying 15 liters/m^2 by the total area gives the number of liters that Mark needs to paint the three walls. 35 m^2 * 1 liter/5 m^2 = 7 liters
c
Mark needs 7 liters of paint, but each can of paint contains 5 liters. This means that Mark will actually have to buy 2 cans to have enough paint to cover all three walls.
d
The cost of the 2 cans can be calculated by multiplying the number of liters in both cans by the price per liter. Each can contains 5 liters, so in total there are 10 liters of paint.
Price of2cans &= 10 liters* 25 dollars/liter [0.8em] &= 250 dollars If the paint was sold in 1-liter cans, Mark would need to buy 7 cans. In this case, he would pay 7* 25=$175. The amount of money that Mark could have saved can be found by calculating the difference between the price of 10 liters of paint and 7 liters of paint. 250-175=$ 75
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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 1.
- When entering the size of a model or an actual objects, round to the closest integer.
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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 $10 for his last call.
- Ramsha said she paid $8 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 40 minutes.
- Ramsha said her call lasted 30 minutes.
Since the duration of the calls is different, Tadeo became confused and made the following diagram to think about the situation. Ali: & $10 → 40 minutes Ramsha: & $8 → 30 minutes 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 | $10/40 min=$0.25 per minute |
| Ramsha | $8/30 min≈ $0.27 per minute |
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Scaling and Descriptive Modeling
Exercise 2.1
Complete the statement. 75¢/h= $/day
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First, note that the left-hand side of the given equation can be written as a ratio. 75cents/hour We are asked to convert cents per hour into dollars per day. Since this will involve multiple conversion factors, we will begin by organizing them in a table.
| Measure | Conversion Factor |
|---|---|
| 100 cents equals 1 dollar | 1dollar/100 cents |
| 24 hours equals 1 day | 24 hours/1 day |
The first conversion factor allows to convert dollars into cents. The second conversion factor allows to convert hours into days. To perform the whole conversion, we need to multiply the given value by both conversion factors in the table.
The statement can now be completed. 75¢/h=18$/day
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Hint & Answer
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Solution
How many grams of a medicine must be added to 2000 gallons of distilled water for it to have a concentration of 3 ppm? 1 ppm ≈ 0.00379 grams/gallon
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We want the concentration to be 3 ppm. This means we need to multiply the equation showing the conversion factor between ppm and g/gal by 3.
Next, we will label the amount of medicine needed in grams as x. If we divide x by 2000 gallons, this ratio should equal 0.01137 for it to have a concentration of 3 ppm. With this information, we can write the following proportion. x grams/2000 gallons=0.01137 grams/gallon Let's solve this for x.
To get a concentration of 3 ppm, we need to add 22.74 grams of medicine to 2000 gallons of distilled water.
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Solution
Scaling and Descriptive Modeling
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