integrated general biology · some perpendicular lines: y = 𝟏 𝟓 x + 1 y = 𝟏 𝟓 x + 2 y =...
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Integrated General Biology A Contextualized Approach
Math In Science
Instructor’s Version FIRST EDITION
Jason E. Banks
Julianna L. Johns
Diane K. Vorbroker, PhD
Copyright © 2016 by SOCRATIS, LLC
All rights reserved. No part of this publication may be reproduced, distributed, or
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www.integratedbiology.com
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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
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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?
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Math in Science (G) Geometry 4
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.
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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
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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
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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
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G.2 Parallel and Perpendicular Lines
What’s the Big Idea?
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.
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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
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G.3 Symmetry and Sections
What’s the Big Idea?
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.
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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
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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
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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
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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)
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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