1. Chapter 18: Growth and Form Lesson Plan
For All Practical Purposes
Size and Growth
Geometric Similarity
How Much Is That In ?
Scaling a Mountain
Sorry, No King Kongs
Dimension Tension
How to Grow
Mathematical Literacy in Today’s World, 7th ed. 1
© 2006, W.H. Freeman and Company

2. Chapter 18: Growth and Form Size and Growth
Understanding the underlying principles of scaling will help to appreciate why objects and living creatures in the world have the particular shapes and sizes that fit their function and survival.
Problem of Scale
How every living species survives by adapting at the different sizes from the beginning of life to the final size of a mature adult. 2

3. Chapter 18: Growth and Form Geometric Similarity
The mathematical idea that two objects are geometrically similar if the corresponding linear dimensions have the same factor of proportionality.
Example: To enlarge a photo, it is enlarged by the same factor in
Geometrically Similar – When two objects have the same shape, regardless of the materials of which they are made.
both the horizontal and vertical directions—in fact, in any direction (such as diagonal).
Linear Scaling Factor
The number by which each linear dimension of an object is multiplied when it is scaled up or down.
The linear scaling factor of two geometrically similar objects is the ratio of the length of any part of the second to the corresponding part of the first. See figure 18.2 on p. 665. 3

4. Chapter 18: Growth and Form Geometric Similarity
How Area Scales
The area of a scaled-up object goes up with the square of the linear scaling factor.
We symbolize the relationship between the area A and the linear scaling factor L by:
A  L2
where the symbol  is read as “is proportional to” or “scales as.”
How Volume Scales
The volume of a scaled-up object goes up with the cube
of the linear scaling factor.
Denoting the volume by V, we can write:
V  L3 4

5. Chapter 18: Growth and Form Geometric Similarity
Example: the length of each side of cube A is 2 inches and the length of each side of cube B is 12 inches.
What is the scaling factor of the enlargement from cube A to cube B?
What is the ratio of the diagonal of cube B to the diagonal of cube A?
What is the ratio of the surface area of the large cube B to that of the small cube A?
If the small cube A weighs 3 ounces, how much does the large cube B weigh?

6. Chapter 18: Growth and Form Geometric Similarity
The Language of Growth, Enlargement, Decrease
Here are the correct ways to discuss meanings of percentages:
“x% of A” or “x% as large as” means × A.
“x% more than A” means A plus x% of A, in other words, ( ) × A.
Saying that A has “increased by x%” means the same thing.
“x% less than A” means A minus x% of A, in other words, (1 − ) × A.
Saying that A has “decreased by x%” means the same thing.
Note the following meanings:
The terms of, times, and as much as refer to multiplication of the original amount.
The terms more, larger, and greater refer to adding to the original amount.
Do not use times with more or with less.
X 100
X 100
X 100 6

7. Chapter 18: Growth and Form How Much Is That In…?
U.S. Customary System – United State’s measurement system
Distance
1 mile (mi) = 1760 yards (yd) = 5280 feet (ft) = 63,360 inches (in)
1 yard (yd) = 3 feet (ft) = 36 inches (in)
1 foot (ft) = 12 inches (in)
Area
1 square mile = 1 mi × 1 mi = 5280 ft × 5280 ft
1 square mile = 640 acres
1 acre = 43,560 ft2
Volume
1 cubic mile = 1 mi × 1 mi × 1 mi = (5280 ft )3
1 U.S. gallon (gal) = 4 U.S. quarts (qt) = 231 in3, exactly
Mass
1 ton (t) = 2000 pounds (lb) 7

8. Chapter 18: Growth and Form How Much Is That In…?
Metric System – The rest of the world uses the metric system in science, industry, and commerce (meter for length; kilogram for mass).
Distance
1 meter (m) = 100 centimeters (cm)
1 kilometer (km) = 1000 meters (m) = 100,000 cm = 1 × 105 cm
Area
1 square meter (m2) = 1 m × 1 m
= 100 cm × 100cm = 10,000 cm2 = 1 × 104 cm2
1 hectare (ha) = 10,000m2
Volume
1 liter (L) = 1000 cm3 = m3
1 cubic meter (m3) = 1 m × 1 m × 1 m
= (100cm)3 = 1,000,000 cm3 = 1 × 106 cm3 (or cc)
Mass
1 kilogram (kg) = 1000 grams (g) 8

9. Chapter 18: Growth and Form How Much Is That In…?
Conversions Between U.S. Customary and Metric Systems
Distance
1 in = 2.54 cm, exactly
1 ft = 12 in = 12 × 2.54 cm = cm = m, exactly
1 yd = m, exactly
1 mi = 5280 ft = 5280 × cm = 160,934.4 cm ≈ 1.61 km
Area
1 hectare (ha) ≈ 2.47 acres
Volume
1 cubic meter (m3) = 1000 liters ≈ U.S. gallons ≈ ft3
1 liter (L) = 1000 cm3 ≈ U.S. quarts (qt) ≈ U.S. gallons
Mass
1 lb = kg, exactly
1 kg ≈ lb
Example: How much is 160 cm tall in feet? cm × × ≈ ft
Units cancel out, leaving feet in the answer.
1 in cm
1 ft in 9

10. Chapter 18: Growth and Form How Much Is That In…?
Examples:
A weight of 5 lbs is approximately the same as _____ kg.
A distance of 4 yards is the same as _____ feet.
A volume of 8 quarts is the same as _____ gallons.
An area of 2000 in2 is approximately the same as _____ m2.

11. Chapter 18: Growth and Form Scaling a Mountain
Weight – Force under gravity.
Pressure – Force per unit area, P = W /A.
Density – Mass per unit volume density = mass/V.
Scaling Up Objects
Matter is determined by its mass, which is its volume multiplied by its density.
The force of gravity is what gives an object its weight.
For all matter there is an upper limit to the amount of pressure or weight that it can support without distortion. This limit is known as the crushing strength that is inherent in all materials.
As a result, there is a limit to which objects can be scaled up because the pressure increases proportionally to the scaling factor.
If the size increases enough, scaled up beyond what the material can hold, the object will buckle under its own weight.
To solve the problem of scale, it would be necessary to change the material composition (to one that has higher crushing strength), or change the shape of the object, or both. 11

12. Chapter 18: Growth and Form Scaling a Mountain
Scaling Up Objects
The height of a mountain or building is limited to gravity, its composition (material strength), and its shape.
Crushing strength of steel = 7.5 million lb/ft2, and crushing strength of granite = 4 million lb/ft2.
A Mile-High Building?
In 1956 Frank Lloyd Wright (1867–1959) proposed a mile-high tower for the Chicago lakefront.
There are many limitations to how high to make a building:
Bending of the building in wind.
More room for elevators, fire escapes, heat, and air conditioning for the taller the building.
There could be limitations on human physiology, such as the difference in air pressure from top to bottom. 12

13. Chapter 18: Growth and Form Sorry, No King Kongs
The resistance of bone to crushing is not nearly as great as that of steel or granite. This explains why there cannot be any King Kongs (unless they were made of steel or granite!).
How Tall Can a Tree Be?
Galileo suggested that no tree could grow taller than 300 ft, but he did not know of the sequoias on the West Coast of the United States.
The tallest tree today is 369 ft, and the tallest that was reliably measured was 413 ft.
Limitations of tree height:
Root system must anchor the tree properly.
The wood at the bottom could crush if there is too much weight above.
Limit to how far the tree can lift water and minerals from the roots to the leaves. 13

14. Chapter 18: Growth and Form Dimension Tension
Large change in scale forces a change in either materials (change to a stronger material) or form (may need to redesign the object).
The problem resulting from scaling up is the tension between weight and the need to support it.
Area-Volume Tension – The result of the fact that, as an object is scaled up, the volume increases faster than the surface area and faster than areas of cross sections.
Example: The diagram on the left shows the weight being distributed better by having two rows of cubes down on the floor level and cutting another row of cubes in half and distributing them across the top. 14

15. Chapter 18: Growth and Form How to Grow
Within narrow limits (up to a factor of 2), most creatures can grow according to geometric similarity, or grow proportionally.
However, most living things grow by a factor greater than 2.
Growth laws are much more complicated than for proportional growth.
Computerized morphing techniques take a person’s picture and change it smoothly, with different scalings for different parts of the face.
National Center for Missing and Exploited Children (NCMEC) uses computer programs to show age progression.
Image stretching is applied to a photograph to progress the age to obtain a rough idea of what a child may look like 10 years later.
Age 7
Age 17 15

16. Chapter 18: Growth and Form How to Grow
Human Growth
Basically, a human adult is not simply a scaled-up baby.
Allometric Growth – Growth of the length of one feature at a rate proportional to a power of the length of another.
Before 9 months, the arm length increases relatively faster than height.
Proportional Growth – After 9 months, a change in pattern occurs. Growth occurs more proportionally, also known as isometric growth.
Different parts of the body scale geometrically, each with a different scale factor.
Babies’ eyes grow to perhaps twice their original size, while the arms grow to about four times their original size.
The proportions of the human body change with age. Notice that the baby’s head is larger in proportion to the rest of its body. 16