Integrated General Biology · Some Perpendicular Lines: y = 𝟏 𝟓 x + 1 y = 𝟏 𝟓 x + 2 y =...

Integrated General Biology A Contextualized Approach

Math In Science

Instructor’s Version FIRST EDITION

Jason E. Banks

Julianna L. Johns

Diane K. Vorbroker, PhD

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Publisher’s Cataloging-in-Publication data

Jason E. Banks, Julianna L. Johns, and Diane K. Vorbroker

Integrated Biology & Skills for Success in Science, A Contextualized Approach: Integrated

Biology / Jason E. Banks, Julianna L. Johns, and Diane K. Vorbroker. In R. Olsen (Ed.)

p. cm.

Domestic 14 13 12 11 10 / 10 9 8 7 6 5 4 3 2 1

ISBN 978-1-942573-28-9

1. The main category of the book —Biology – General Biology – Freshman Biology—

Integrated Science– Introductory Science. 2. Prerequisite to Anatomy and Physiology – Pre-

A&P Biology – Introduction to Anatomy and Physiology Biology— 3. Contextualized Biology

—4. Skills for Success in Science – Science Skills – Prepare for Anatomy and Physiology

Skills.

Case # 1-1661570877 2014

First Edition

Integrated General Biology

Math in Science (G) Geometry 3

Math in Science (G) Geometry

Share Your Thoughts

Directions: Take a moment to write down your thoughts concerning the information below. Don’t look

in a book or search the internet for any answers, just write what you think at this moment in time. It

doesn’t matter if you are right or wrong—the important thing is that you are sharing what is already in

your head. If you are right about everything, great! If there are some misconceptions, this activity will

give you the opportunity to identify and correct them.

1. What is the difference between a line, ray and line segment?

2. Name this ray.

3. Under what circumstances would you be able to find an acute angle represented by the human body?

a. A right angle?



b. An obtuse angle?

c. A reflex angle?

4. Find the equation of the line below.

a. Find a line that is parallel to this line.

b. Find another line that is perpendicular.

5. Determine if these figures have symmetry. If a figure does have symmetry, draw all of the lines of symmetry.



G.1 Lines and Angles in the Body

What’s the Big Idea?

Lines, line segments and rays are named for points on them. When two lines meet, and angle is formed.

The human body is capable of a wide range of angle formation.

[G.1.1] Geometry—Seeing the Answer in a Picture

Geometry is a way to look at the problem in the form of a picture. It includes working with shapes,

areas, volumes, etc., but it also could include any graphical representation of a problem. Geometry had

its beginnings in several different cultures, but it was Euclid who really got things moving in the 3rd

Century BC. In more recent times, geometry has been used to support many different theories,

including string theory. Here, we’ll use geometry to help us make sense of the structure of organisms.

[G.1.2] Lines, Rays, and Line Segments

In our everyday lives, if something is straight, we usually call it a line—but it is probably more accurately

described as a line segment, or maybe a ray. A line, in mathematics, has some particular aspects that

must be adhered to. To understand a line, let’s first see what a point is.

A point represents a unique location in space, and is usually represented by coordinates (x and y for

two dimensions—z would be used for the 3rd dimension). A point has no length, width, area or volume.

However, a group of points can link together to form any imaginable figure.

A line is an infinite series of points (infinite in both directions) that all lie in a single plane that make

an infinitely small curve—a line in mathematics may be referred to as a curve, even though lines are

straight. It can be named using any two points on the line

Figure G–1: Line AB, or



A ray is also an infinite series of points, but it has one endpoint and is only infinite in one direction. A

ray is named with its endpoint first, followed by any other point on the ray.

Figure G–2: Ray BA, or

A line segment is a finite series of points that has two endpoints, for which it is named. A line and a ray

have an infinite length, but a line segment does not.

Figure G–3: Line segment AB, or

[G.1.3] Looking at it from different angles

An angle can be described as the amount of turn or the number of degrees between two intersecting

lines, line segments or rays. This number can vary in value. Here are some different names for some

angles.

Figure G–4: Various types of angles, all of which can be found in the human body

If someone is said to have acute pain that means that the pain is sharp or sudden. And if the

condition is not long lasting, that may also be referred to as acute. With these definitions in mind, we



can see that an angle that is less than 90 degrees has a shape that fits these usages—sharp, like the edge

of a knife, and short lived.

A right angle is exactly 90 degrees, and an obtuse angle is greater than 90 degrees, but less than 180

degrees. If a person is obtuse, this usually means that he is not the sharpest tool in the shed. The

straight angle is exactly 180 degrees. These angles are part of the normal range of motion in our arms

and legs. A reflex angle, greater than 180 degrees, would represent a hyperextension of the arm or

leg—something that would not be very comfortable. There are hyperextension exercises for the back,

however, that can be very helpful in strengthening the lower back muscles.

Section Review

1) Give the mathematical definition and a scientific definition for acute and reflex. G.1.3

2) What parts of the body can be hyperextended without injury? G.1.3

a) What parts would be injured if hyperextended? G.1.3

b) What is the minimum number of degrees produced by a hyperextension? G.1.3



G.2 Parallel and Perpendicular Lines


Parallel lines have the same slope, while perpendicular lines have the opposite inverse slope.

[G.2.1] Going My Way?

Parallel and perpendicular lines are found in the same plane, like on a coordinate plane. Parallel lines

are lines that have the same slope, and will never meet (some mathematicians may say that they will

meet at infinity). Perpendicular lines meet at a right angle.

Parallel Lines

Perpendicular Lines

Figure G–5: Parallel and Perpendicular Lines

Some interesting things happen to the slopes of parallel and perpendicular lines.

Figure G–6: Perpendicular and parallel lines

Try to determine the equations for each of the lines (AB, CD and EF) on your own before you move on.



Lines AB and CD are parallel. Line EF is perpendicular to both lines AB and CD. When we view the

equations for each of the lines, an interesting pattern seems to emerge. What is the pattern?

Line AB y = 2x + 1 Line CD y = 2x – 2

Line EF y = −1

2 x + 3 OR y = − (

1

2) x + 3 OR y =

−𝑥

2 + 3

Parallel lines have the same slope. One could find an infinite number of parallel lines—you would only

need to change the y-intercept.

Given Line: y = -5x + 1

Some Parallel Lines: y = -5x + 2

y = -5x + 3

y = -5x – 1

Perpendicular lines have opposite inverse slopes—in other words, take the inverse of the slope (flip the

fraction form of the slope) and take its opposite. You could also find an infinite number of lines that are

perpendicular to a given line—you would only need to change the slope to the opposite of the inverse,

and then choose any y-intercept.

Given Line: y = -5x + 1

Some Perpendicular Lines: y = 𝟏

𝟓x + 1

y = 𝟏

𝟓x + 2

y = 𝟏

𝟓x + 3

y = 𝟏

𝟓x – 1

Section Review

For each pair of equations, tell whether the lines formed are parallel, perpendicular or neither. You may need to rearrange the formula into slope-intercept form by solving for “y”.

G.2.1

1) 𝑦 = 0.8(𝑥 − 4) + 3 𝑦 = −1.25(𝑥 − 3) + 9

2) 𝑦 = 3 − 2𝑥 𝑦 = −2𝑥 + 5

3) 6𝑥 − 4𝑦 = 7 −9𝑥 + 6𝑦 = −2

4) 3𝑥 + 2𝑦 = 6 10𝑥 − 15𝑦 = 14



G.3 Symmetry and Sections


Lines of symmetry divide the subject into mirror images. Organisms sometimes have body plans that demonstrate asymmetry, radial symmetry or bilateral symmetry.

In mathematics, the term “cross section” is used to describe a two-dimensional form of a three-dimensional figure. However, in anatomy, there are three different planes of dissection, each with a different name describing the plane of dissection.

[G.3.1] A Sense of Balance

There are several types of symmetry; however, we will just deal with one in this text—reflection

symmetry. This type of symmetry, as the name suggests, involves reflections. The line of symmetry, in

this case, would be the line that splits the figure in two halves that are mirror images of each other—in

other words, if the two halves were folded over this line, the two halves would be identical.

Figure G–7: A Rectangle and Triangle, and their Lines of Symmetry

Lines of reflective symmetry can often be drawn along more than one line, like in both of the geometric

figures above.

A sponge has no line of symmetry—this is called asymmetry (the prefix “a” means without). An

anemone has radial symmetry, meaning is has an infinite number of lines of symmetry or many lines of

symmetry as you move around it. A human standing in anatomical position would only have one line of

symmetry, called bilateral symmetry.



Asymmetrical Radial Symmetry Bilateral Symmetry

Figure G–8: A Sponge, Anemones, and a Human, and their Type of Symmetry

[G.3.2] From Three Dimensions to Two

Bodies are three dimensional. However, a cross section of a three dimensional figure will be in two

dimensions. Cross sections can be viewed as slices of the original object.

Figure G–9: A Three-Dimensional Rectangular Prism and its Two-Dimensional Cross Section

Figure G–10: A Three-Dimensional Cylinder and its Two-Dimensional Cross Section

http://upload.wikimedia.org/wikipedia/commons/c/c9/Venus_Flower_Basket.jpg

http://upload.wikimedia.org/wikipedia/commons/a/a9/Haeckel_Actiniae.jpg

http://upload.wikimedia.org/wikipedia/commons/1/11/Uomo_Vitruviano.jpg



A geometric understanding of figures and their corresponding two-dimensional cross section is very

important to a proper visualization of the human body. In mathematics, the term cross section can be

used to describe most two-dimensional forms of the three-dimensional figure. However, in anatomy

there are three different planes of dissection, each with a different name describing the angle of the

plane of dissection. These anatomical planes intersect at 90 degree angles, and will be discussed in the

next chapter.

Section Review

1) Draw a figure with at least one line of symmetry. G.3.1

2) Draw a figure with no lines of symmetry. G.3.1



G.4 Review by Section

CONCEPTS VOCABULARY

15.1 Lines, line segments and rays are named for points on them. When two lines meet, and angle is formed.

The human body is capable of a wide range of angle formation.

Point

Line

Ray

Line Segment

Acute Angle

Right Angle

Obtuse Angle

Straight Angle

Reflex Angle

Hyperextension

15.2 Parallel lines have the same slope, while perpendicular lines have the opposite inverse slope

Parallel

Perpendicular

15.3 Lines of symmetry divide the subject into mirror images. Organisms sometimes have body plans that demonstrate asymmetry, radial symmetry or bilateral symmetry.

In mathematics, the term “cross section” is used to describe a two-dimensional form of a three-dimensional figure. However, in anatomy, there are three different planes of dissection, each with a different name describing the plane of dissection.

Reflection Symmetry

Asymmetry

Radial Symmetry

Bilateral Symmetry

Cross Section



G.5 Practice

1) Name an organ in the human body that does not have bi-lateral symmetry.

2) Name an organ in the human body that does have bi-lateral symmetry.

3) Draw an organ that does have bi-lateral symmetry and its line of symmetry.

4) Use graph paper to plot the two points given, then connect the two points with lines. Determine if the two lines formed by the points are parallel, perpendicular or neither.

a) Line 1: (-5, 0), (1, 4) and Line 2: (6, 3), (-3, -3)

b) Line 1: (-3, -2), (3, 2) and Line 2: (5, -3), (-1, 6)

c) Line 1: (-4, -1), (-2, 7) and Line 2: (2, 6), (3, 3)



Closing Remarks

Before leaving this chapter, go back and review your answers to the “Share Your Thoughts” section at

the beginning of this chapter.

1) Do you still agree with everything you originally stated?

2) What changed your thinking?

3) What surprised you the most?

4) How might you apply this new thinking in the future?

“Since the beginning of physics, symmetry considerations have

provided us with an extremely powerful and useful tool in our

effort to understand nature. Gradually they have become the

backbone of our theoretical formulation of physical laws.”

– Tsung-Dao Lee

Integrated General Biology · Some Perpendicular Lines: y = 𝟏 𝟓 x + 1 y = 𝟏 𝟓 x + 2 y =...

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