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Integrated General
Biology
A Contextualized Approach
Active Learning Activities
FIRST EDITION
Jason E. Banks
Julianna L. Johns
Diane K. Vorbroker, PhD
Math - The Language of the Cosmos
Chapter 1
Active Learning Activities
Math - The Language of the Cosmos
Section 1.1 Discovering the Atom
Directions for
the Student:
This lesson is designed for you to complete, on your own or in your study group. Use
your notes and follow along in the text, as you find necessary.
Objectives:
Describe the three subatomic particles, their places in the atom, and their charges.
Describe the process of science and how it involves many people over a long period of
time.
Even though the Greek word atomos means unable to be cut, atoms are made up of smaller particles.
There are three basic subatomic particles. The first one discovered was the electron. In 1897, J.J.
Thomson shot electrons out of a cathode ray tube (just like an old TV or computer monitor with a large
back) through a magnet.
Electrons moving in a cathode ray
tube (left to right) with a magnet in
the middle
Notice how the electrons are moving in a straight line until they reach the magnetic field and then are
deflected.
1. Remembering that opposites attract and likes Electrons are negatively charged
repel, what is the charge on the electrons?
No matter what metal Thomson used, the same type of particle (electrons) came out of the cathode ray
tube, and the overall mass of the metal used had no noticeable change in its mass. This means that . . .
2. All atoms may or may not have electrons?
All atoms have electrons
3. Electrons have large or very small mass?
Electrons have very small mass
After the discovery of the electrons, came the discovery of the nucleus. In 1909, we did not have the
technology to see the inside of an atom, so we really had very little idea about what else may be inside
of it. Around that time, Ernest Marsden and Hans Geiger (working under the direction of Ernest
Rutherford) begin a series of experiments to find out what the center (or nucleus) of the atom is like by
shooting alpha particles (which have a positive charge) into gold foil. Their experiments were very
similar to playing games of battleship, where a player shoots missiles at a target that he/she cannot see.
Discovering the Atom
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Active Learning Activities
You can simulate their experiments by playing your own game of battleship. On the grid provided,
slightly darken one square anywhere you choose—this will be your ship if you play against someone
else, or it will be your target if you play by yourself. You can play this game of battleship with a partner
(only mark your hits and misses, not your opponent’s), or you can close your eyes and drop your pencil
over the grid and see how many shots it takes you to hit the square that is the target. Either way you
play, notice how difficult it is to get a “hit” and do not spend too long playing.
A
B
C
D
E
F
G
H
I
J
1
2
3
4
5
6
7
8
9
10
4. What were the odds of you hitting the target
on your first shot? Give your answer in
fraction and then decimal form
1
100
or
0.01
The results of the actual experiment were probably somewhat similar to the results from your game,
except that the odds of hitting the nucleus with one of their missiles (the alpha particles) were
extremely low. The vast majority of particles went straight through the gold foil; they expected all of the
particles to go through the foil. But a few were deflected or bounced back—the particles were repelled
by the nucleus of the atom.
The bouncing-back of particles was not expected by many scientists of that time in history, but the fact
the experiment was observable and repeatable helped advance the understanding of atoms. In a very
general sense, this is the scientific method in action. The scientific method includes making
observations, or gathering evidence. The interpretation of this evidence becomes a claim. The following
table offers a way to practice the scientific method using experimental data just discussed.
Discovering the Atom
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Active Learning Activities
5. Complete the table:
Evidence
Claim
(Observations)
(What it means)
Most particles pass right through the atom. This means that most of the atom is . . .
Empty space
Very few of the particles hit the nucleus.
This means that the nucleus is . .
Very small
A few of the positive particles bounced
back, being repelled by the nucleus.
This means that the nucleus has a . . .
Positive charge
J.J. Thomson suggested a model for the atom that was called the “plum pudding” model because he
thought the atom was a sphere of positive charge with the negative electrons spread throughout. This
model, of course, proved to be incorrect. Marsden and Geiger showed that the nucleus was very small
(but not nearly as small as the electrons) and that the nucleus had a positive charge.
6. Place a large “X” through the incorrect model of the atom, and complete the sentences.
Marsden and Geiger expected the alpha particles to . . .
Pass directly through the atom
Instead, most of the particles passed through and a few . . .
Bounced back or were deflected after hitting the nucleus
After these experiments, we knew several things about the atom:



Electrons are very small and have a negative charge.
Most of the atom is empty space.
The nucleus is much larger than electrons, but still not very much of the overall space in the
atom.
 The nucleus is much more massive than electrons.
But what was in the nucleus? What are the nucleons (particles in the nucleus)?
Next, Rutherford shot alpha particles into nitrogen gas, and found that a proton comes out of the
nitrogen atom. This meant that atoms have protons in their nuclei. But the nucleus was very massive—
Discovering the Atom
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Math - The Language of the Cosmos
Active Learning Activities
there had to be another particle in the nucleus of most atoms. The neutron was discovered by James
Chadwick in 1932.
7. Complete the table:
Evidence
Claim
(Observations)
(What it means)
Chadwick found that the neutron was not
affected by an electrical field (with + and -)
This means that the neutron…
The neutron is slightly more massive than
the proton
This means that most of the mass of an
atom is from the…(name the two particles)
Does not have a charge
1) Neutrons
2) Protons
Now we know all three subatomic particles, where they are found and what their charges are. Let’s
review.
8. Name the three subatomic particles
Electrons
Protons
Neutrons
9. Where is each particle found?
Protons and Neutrons are found in the nucleus;
Electrons are found in sub-orbital shells
10. What is the charge on each particle?
Neutrons = None/Neutral
Protons = positive
Electrons = negative
11. What particles determine the overall charge
on the atom?
Electrons
12. What particles make up most of the mass of
an atom?
Protons and Neutrons
13. What is most of the space in an atom made
up of?
Empty space
14. If two atoms were to bump into each other,
what particles would touch?
Electrons
Discovering the Atom
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Math - The Language of the Cosmos
15. In your own words, describe how different
researchers over long period of time
contributed to discover the three subatomic
particles.
Active Learning Activities
In 1897, J.J. Thomson discovered electrons. Next
Marsden and Geiger showed that the nucleus was
very small and that the nucleus had a positive
charge. After that, Rutherford found that a proton
comes out of the nitrogen atom. This meant that
atoms have protons in their nuclei. The neutron
was discovered by James Chadwick in 1932.
Study Hint:
Go back through this activity and through this section of the text, and look for important words. Include
a definition or any other important information about the vocabulary word. You could also provide an
illustration, and you could use the word in a sentence. Place the word on one side of an index card and
all of its information on the other side, but be careful not to put too much on the back. It is best to only
have a few bullet points that highlight the most important parts of the term. Show yourself one side,
and try to guess the other.
Example:
FRONT OF CARD
Electron
BACK OF CARD
Very small subatomic particle
Negative charge
Discovered by J. J. Thomson using a cathode ray tube
Very little mass
Deflected away from the negative part of the magnet
Using the Word in a Sentence:
With the discovery of the electron, Thomson showed
the atom was made of smaller parts.
Discovering the Atom
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Active Learning Activities
Section 1.2 Metals, Nonmetals and Metalloids, and the Periodic Table
Directions for
the Student:
Objectives:
Materials
Required
This lesson is designed for you to complete, on your own or in your study group. Use
your notes and follow along in the text, as you find necessary.
1. Determine the atomic number, the number of protons and the number of
electrons in an atom.
2. Differentiate between an atom and an ion.
3. Determine an atom’s number of valence electrons.
4. Explain how metals and non-metals hold their valence electrons.
5. Determine the most common isotope of an element, and explain how it differs
from other isotopes of that element.
6. Predict chemical behavior based on the number of valence electrons.
 Blank Periodic Table worksheet
 Blank Periodic Table instructions
 Periodic Table
To understand much of the world around us, we must understand the building blocks of the matter that
makes up this world and most everything in it. To help us understand these building blocks, we have
invented a “User’s Manual” with most of the information that we will need. At first glance, the periodic
table can be overwhelming. It holds a large amount of information in a small amount of space and it has
an unusual shape. However, we can start to make sense of this very important tool by taking out these
confusing parts and just focusing on the shape of the table and the patterns found within it.
1. Complete the blank periodic table using the instructions.
There are three particles that make up the atom: protons, neutrons and electrons. But it is the number
of protons that determines which element the atom is. Find the number of protons, and you know what
element that atom is.
The boxes on the periodic table are numbered from left to right, and top to bottom. The number of the
square is also the number of protons that element has. It is the number of protons that tells us the
“atom” number, or atomic number.
Use both your “blank” periodic table and the periodic table to answer these questions.
2. Hydrogen is in square number one. How many protons does hydrogen have?
1
3. Carbon is in square number six. How many protons does carbon have?
6
4. If an atom has eight protons, which element is it?
Oxygen
5. Find chlorine on the periodic table. How many protons does it have?
17
6. How many protons does calcium have?
20
7. What is the atomic number of boron?
5
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
The number of protons defines the element, but atoms sometimes may be the same element with the
same number of protons, but be different in some other way.
If you were to go outside and start collecting carbon atoms, they would all have the same number of
protons (carbon has an atomic number of six). But not all of the carbons would have the same number
of neutrons—these variations of the same element are called isotopes (they have the same number of
protons so they are all the same element, but different numbers of neutrons). Remembering that
protons and neutrons are the nucleons (particles in the nucleus), let’s just focus on what the nucleus of
these varieties would look like, with no electrons around the nucleus.
8. Why are all three of these isotopes
considered to be carbon?
They have the same number of protons, but
different number of neutrons
9. Explain what the numbers 12, 13, and 14
written after the word "carbon" refer to.
The total number of protons
10. Lithium has two naturally occurring isotopes,
lithium-6 and lithium-7. How many protons
does lithium have?
3 protons
11. Draw the nuclei of the two isotopes of lithium. Use the same designs as the ones used for carbon.
Lithium-6
Lithium-7
# Protons = 3
# Protons = 3
# Neutrons = 3
# Neutrons = 4
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
12. Oxygen has three naturally occurring isotopes, 8 protons
oxygen-16, oxygen-17 and oxygen-18. How
many protons does oxygen have?
13. Draw the nuclei of the three isotopes of oxygen. Use the same designs as the ones used for carbon.
Oxygen-16
Oxygen-17
# Protons = 8
# Protons = 8
# Neutrons = 8
# Neutrons = 9
Oxygen-18
# Protons = 8
# Neutrons = 10
But how can you tell which one is more common? Are there more lithium-6 or lithium-7 isotopes?
Imagine you could go out and catch a few million carbon atoms (1, 2, 3, 4, 5, . . .) and then you added
them all up and divided by how many carbon atoms you caught. This would tell you the average of mass
of the carbon atoms, and the average mass will likely be very close to the most common isotope! (And
the average atomic mass is 12.011 amu, which is close to 12.)
Generally speaking, to determine the most common isotope you should find the average atomic mass
on the periodic table and round it to the nearest whole number. (Remember: 7.5 would round to 8, and
7.4 would round to 7.) This will give you the most common mass number for most of the element.
14. What is the most common isotope for
carbon?
Carbon-12
(Carbon’s average atomic mass is 12.011 atomic
mass units. So, 12.0 rounds to “12”.)
15. What is the most common isotope for
lithium?
Lithium-7
16. What is the most common isotope for
oxygen?
Oxygen-16
17. What is the most common isotope for
chlorine?
Chlorine-35
(Focus on the “35.4” part of the average atomic
mass, and don’t worry about the rest of it.)
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
While the nucleus is important (the number of protons tells us what element an atom is and the number
of protons and neutrons tells us how much mass it has), the nucleus really does not predict how the
atom will behave, chemically speaking. If you want to know how an atom will behave, you must look at
its electrons!
18. What subatomic particle determines the
chemical behavior of an atom?
Electrons
Atoms are considered to be electrically neutral, meaning they have the same number of positive
particles (protons) as they do negative particles (electrons). But being electrically neutral does not mean
it is stable—it is the number of electrons that determines the atom’s stability. So the atom will gain
electrons or lose electrons until it is stable. When this happens, the atom will no longer have the same
number of protons and electrons. This means that it will not have a balance between the positive
protons and negative electrons anymore, and it will be a charged atom or an ion.
19. When an atom is electrically neutral, that
means it has the same number of
Protons and electrons
20. When an atom has gained or lost an electron, Ion
it is now known as an
However, there is a column of elements (also known as a group) that already has very stable elements.
These elements do not need to gain or lose any electrons. This is column 8A (or column 18), and this
group is known as the Noble Gases. All the other atoms try to get an electron arrangement like this
group. But what is the secret of these Noble Gases? We will look at our “blank” periodic table for the
answers.
21. Helium is the first Noble Gas. How many
electrons does helium have?
2
(Hint: If it is in its atomic state, the protons and
the electrons are equal in number.)
22. How many electrons does neon have?
10
23. How many electrons does argon have?
18
But not all electrons are equal—some electrons are going to interact more than others. Imagine two
atoms bumping into each other. Which electrons will interact? The electrons in the outermost (or
valence) shell will do the interacting.
Electrons are held in orbitals around the nucleus. These are also referred to as electron clouds, shells or
energy levels. The periodic table tells us how many electrons are found in each shell.
Metals, Nonmetals and Metalloids, and the Periodic Table
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Math - The Language of the Cosmos
Active Learning Activities
24. Use your periodic table to find the element, then use your “blank” periodic table to answer the
questions. Remember that helium is in column 8A on the periodic table and in column 2A on the
“blank” periodic table.
How many electrons does helium have, total?
2
How many electrons does helium have in the first energy level?
2
(Hint: Count the number of squares in the first row.)
What is helium’s outermost (valence) shell (highest energy level)?
2
(Hint: What row is helium in?)
How many electrons does helium have in its valence shell?
2
What column (on the “blank” periodic table) is helium in?
8A
How many electrons does neon have, total?
10
How many electrons does neon have in the first energy level?
2
(Hint: This will be the same as it was for helium.)
How many electrons does neon have in the second energy level?
8
(Hint: Count the number of squares in the second row until you get to neon.)
What is neon’s outermost shell (highest energy level)?
8
How many electrons does neon have in its valence (outermost) shell?
8
What column is neon in?
8A
How many electrons does argon have, total?
18
How many electrons does argon have in the first energy level?
2
How many electrons does argon have the second energy level?
8
How many electrons does argon have in the third energy level?
8
What is argon’s highest energy level?
8
How many valence electrons does argon have?
8
(Hint: It means the same as electrons in the outermost shell.)
What column is argon in?
8A
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
Here are some diagrams of helium, neon and argon with the nucleus represented by the symbol:
25. Explain how looking at the periodic table can
tell you how many electrons can fit into an
energy level.
In the periodic table, the atoms are arranged by
either groups or periods. The groups are arranged
by vertical columns; each column signifies the
number of valence shell electrons in an element's
atom. The periods are organized by horizontal
rows; each row signifies the total number of
electron shells in an element's atom.
We looked at the Noble Gases first because they are stable (with a stable electron arrangement), and all
of the other elements are trying to be stable too. But how can an atom become stable? What can an
atom change? How will an atom get to be like a Noble Gas?
Reactions involving the nucleus (or nuclear reactions) are rare compared to reactions involving electrons
(chemical reactions). Therefore, if an atom is going to change, it is likely going to add or give away
electrons until it has the same electron arrangement as a Noble Gas—a complete outermost shell.
26. Find sodium on the periodic table and answer the following questions.
How many protons does sodium have?
11
How many electrons does sodium have, total?
11
How many electrons will fit into the first orbital?
2
How many electrons will fit into the second orbital?
8
How many electrons are remaining and will go into the third orbital?
1
27. Draw a sodium atom with the symbol “Na” representing the nucleus and dots on circles around the
nucleus (similar to how the Noble Gases were drawn previously). Be sure to put two electrons into
the first shell, eight into the second shell, and only one into the third shell.
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
28. If sodium was going to become similar to a Noble Gas, which Noble Gas would it
be? (Which Noble Gas is sodium closest to on the periodic table? You may have to
go backwards!)
Neon
If sodium is going to have the same electrons arrangement as neon, it will have to lose one electron. Go
back up to the sodium atom you drew in #23 and cover up sodium’s valence electron with your finger.
29. If sodium loses its valence electron, what will be its new valence number of
electrons?
8
30. Is this new arrangement a stable one, like a Noble Gas?
Yes
When sodium loses its outermost electron, it becomes an ion or charged atom. Its new electron
arrangement is similar to that of a Noble Gas, but its nucleus has not changed, so it still has 11 protons.
31. If sodium now has 11 positive protons and only 10 negative electrons, what is the
charge on a sodium ion?
+1
(Hint +11 – 10 =)
Here is a model of aluminum. Notice how it has two
electrons in the first orbital, eight in the second, and three
in the third.
32. What is the closest Noble Gas to aluminum?
Neon
(Hint: again, go backwards!)
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
33. Cover up aluminum’s outermost electrons.
What does aluminum have to do to become
more like a Noble Gas?
Lose 3 electrons
(Hint: Lose 3…)
34. Now that aluminum has lost 3 electrons and it 13+ - 10- = 3+
has formed an ion, what is its ionic charge?
(Remember, protons – electrons.)
(Hint: _____-______=_____)
Generally speaking, atoms that lose electrons are called metals. They usually have one, two or three
electrons in their valence shell. Since they lose electrons when they form an ion, they become positively
charged. Non-metals, however, tend to gain electrons when they form an ion.
35. How many protons does fluorine have?
9
36. How many electrons does fluorine have (total)?
9
37. How many electrons will fit into the first orbital?
2
38. How many electrons does that leave to fit into the second orbital?
7
Let’s now apply what we have learned to another atom.
39. Draw a fluorine atom with the symbol “F”
representing the nucleus (similar to #21).
Be sure to put two electrons into the first shell and
seven into the second shell.
40. Which Noble Gas is fluorine closest to?
Neon
(No need to go backwards this time!)
41. What is the charge on this fluorine ion?
Negative
(9 protons – 10 electrons)
Metals usually have a valence shell with one, two or three electrons. Metals will lose these outermost
electrons to become more stable. Non-metals have five, six or seven electrons in their outermost shell.
Non-metals will gain electrons until they have a complete valence shell. Metals hold their electrons
loosely, while non-metals hold their electrons tightly with a strong pull. This is called electronegativity,
and non-metals tend to have a much higher electronegativity than metals.
Most atoms are not stable and they will react until they achieve a stable arrangement. Knowing their
number of valence electrons is the key to predicting how they will react.
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
Here’s a review of the major concepts covered in this section.
42. Complete the table:
Element Atomic
Mass
Valence
“A”
(Symbol) Number Protons Neutrons number Electrons Electrons Column
Li
Na
3
11
Ga
Fl
S
35
12
7
3
23
39
11
31
31
9
9
10
19
9
16
16
32
16
18
Br
4
11
16
Ar
3
18
70
22
35
80
18
# of
Orbitals
2
2
1
1
31
40
45
1
Row
1
3
3A
7
7A
3
3
4
4
2
2
6
6A
3
3
8
8A
3
3
35
7
7
4
4
Review the table from the previous task and notice the patterns that emerge.
43. The element is determined by the number of
protons, which is the same as the . . .
Atomic Number
44. The mass number is determined by the two
nucleons.
Protons
plus
Neutrons
45. As long as the atom has not formed an ion (gained or Electrons
lost electrons), the number of protons is equal to the
number of . . .
46. The number of valence electrons is equal to what
column?
A column
47. The number of orbitals or electrons shells is equal to Row #
the . . .
Metals, Nonmetals and Metalloids, and the Periodic Table
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Active Learning Activities
48. Explain the difference between how metals and non- Metals are the elements on the left side of
metals react to achieve stable arrangements of
the Periodic Table. Metals tend to lose
electrons.
electrons to attain Noble Gas electron
configuration. Non-metals are limited to
the elements in the upper right hand corner
of the Periodic Table. Non-metals tend to
gain electrons to attain Noble Gas
configurations. Metals tend to lose
electrons and non-metals tend to gain
electrons, so in reactions involving these
two groups, there is electron transfer from
the metal to the non-metal.
49. Give specific examples of how metals and non-metals react, with drawings.
Metals, Nonmetals and Metalloids, and the Periodic Table
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Section 1.3 What's the Matter?
Directions for
the Student:
This lesson is designed for you to complete, on your own or in your study group. Use
your notes and follow along in the text, as you find necessary.
Objectives:
1. Describe and differentiate between the four states of matter commonly found on
Earth.
 Glass of water, and a small plate
 Food coloring (any color)
 2-liter bottle (or some other plastic bottle with a lid)
Materials
Required:
Take a look around you. Think about the atoms that make up everything you see. Are those atoms
moving around, or do they simply stay in their places without moving at all? Find an example of a solid, a
liquid and a gas, and answer the question below.
1. Write a few sentences that explain what the atoms are doing in each of these states of matter.
State of Matter
Behavior of Atoms
Solid
Atoms in a solid are tightly packed, usually in a regular pattern. These solid
atoms usually vibrate but generally do not move from place to place.
Liquid
Atoms in a liquid are close together with no regular arrangement. These
liquid atoms: usually vibrate, move about, and slide past each other.
Gas
Atoms in a gas are well separated with no regular arrangement. These
gaseous atoms vibrate and move freely at high speeds.
2. To find out some of the properties of these three states of matter, fill a glass about half full of water
and put a small plate on top of the glass, and answer the following questions.
Question
Answer/Observation
What shape is the solid (the glass)?
retains a fixed volume and shape
What shape is the liquid (the water)?
assumes the shape of the part of the container
which it occupies
What shape is the gas (the air in the glass)?
assumes the shape and volume of its container
What's the Matter?
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Active Learning Activities
3. Which of the following states of matter take the shape of their container?
State of Matter
Take Shape of Container [YES / NO]
Solid
NO
Liquid
YES
Gas
YES
We can see that solids do not change their shape very easily. Solids have a fixed shape. Liquids and
gases, however, seem to take the shape of whatever container they are in.
But what about their volume? Can you easily change how much space they take up? (Don't confuse
volume with mass. Remember that volume is how much space something takes up and mass is how
much matter or stuff is in something.)
Put a lid on an empty plastic bottle. We say that it is empty, but really, it is full of air or gas. Give the
bottle a good squeeze and see how far you can compress the bottle. Repeat this "squeeze test" with a
bottle that is half full of water and with a bottle that is completely full of water.
4. Record your observations below and make a CLAIM that is based on your observations.
Observation
Claim
Bottle with no water, all air
Compressible (lots of free space between
particles)
Bottle with half water, half air
Semi-compressible (half liquid & half gas)
Bottle with all water, no air
not easily compressible
(little free space between particles)
Since the air was able to be squeezed and its volume was changed, we say that gas is compressible.
Liquids and solids are not readily compressible—you could not easily squeeze the bottle full of water,
but the bottle full of air was easily squeezed. Solids are not readily compressible either.
Now let's focus on the atoms, or groups of atoms (called molecules), that make up these states of
matter. Can these atoms move around, and change their order? Or are these atoms fixed in place and
cannot change their order?
Take a drop of food coloring and put it into the glass of water.
5. Record your observations and make a claim about how the atoms in a liquid behave.
What's the Matter?
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Math - The Language of the Cosmos
Active Learning Activities
Observation
The only water that is colored initially is where the
food coloring makes contact in the container. Over
time the food coloring diffuses throughout the
container causing all water molecules to turn the
color of the food dye.
Claim
Atoms in a liquid flow easily as the particles can
move/slide past one another with little
intramolecular friction.
Notice that the food coloring drop moves from the area of HIGH concentration of food coloring to the
area of LOW concentration of food coloring. This could not happen if the H2O molecules were fixed in an
order—they must be free to move around. A similar experiment could be performed with perfume in
the air. Since perfume spreads out, the air molecules must also be free to move around. This is why both
liquid and gas are considered to be fluids. Fluids flow, and solids are not fluids because their atoms are
fixed in order and do not change their positions.
The last property of solids, liquids and gases that we will discover is that of density. We can test the
relative density of a solid by whether it sinks or floats in a liquid. If it sinks, it is denser. If it floats, it is
less dense. Of course, you must keep the substance the same—you cannot choose solid copper and put
it in liquid iron. Let's just use copper.
6. Will solid copper float or sink in liquid copper?
sinks
Solid copper sinks in liquid copper, and this is true of almost any substance, solids are denser than their
liquid counterparts. The very rare and unusual exception to this is water. What happens when you put
ice (solid water) in to your glass of (liquid) water? It floats! This means that liquid water is denser than
solid water—in this way, water is a very unusual substance!
Plasma
A fourth state of matter found on Earth is plasma. Do not mistake the state of matter called plasma for
blood plasma. Blood plasma is the yellow liquid part of the blood that holds the blood cells in suspension
(like water holds mud particles in suspension in muddy water), and makes up about 55% of the blood.
The state of matter called plasma is something very different.
Plasma is a state of matter where the atoms have seemingly fallen apart. Normally, the positivelycharged nucleus holds the negatively-charged electrons. In plasma, all or most of these particles are
loose, and free to move around. This type of matter requires an energy source, and is not found in the
human body. It is found in lightning, fluorescent lights, plasma TVs, and stars. Since stars are made of
plasma and are the largest objects around, plasma is the most common state of matter in the universe.
What's the Matter?
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Math - The Language of the Cosmos
Active Learning Activities
Review
7. Try to complete the table as a summary of the four fundamental states of matter.
Solid
Molecules:
Liquid
Gas
Plasma
In fixed order
Free to move
Free to move
Free to move
Spacing:
Very close
Close
Spread out
Varies
Volume:
Fixed
Fixed
Varies
Varies
Shape:
Fixed
Fixed
Varies
Varies
One final note about the four states of matter: all of them have particles that are constantly moving or
at least vibrating back and forth. Even atoms in a solid are vibrating—they don't change position, but
they shake back and forth. If the atoms were to stop shaking, they would be the coldest temperature
possible, which is absolute zero (-273.15 degrees Celsius or –459.67 degrees Fahrenheit). We can never
reach this temperature where all atomic motion stops, but we have gotten really close.
What's the Matter?
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Math - The Language of the Cosmos
Active Learning Activities
Section 1.4 How to Read the Numbers
Directions for
the Student:
Objectives:
This lesson is designed for you to complete, on your own or in your study group. Use
your notes and follow along in the text, as you find necessary.
1. Describe the importance of units in solving problems.
2. Properly track units throughout problems.
3. Identify and properly combine like terms.
4. Properly use the order of operations, inverse operations, the commutative and
associative rules of addition, and the distributive property when solving problems.
Take a look around wherever you may be, and make some quantitative observations. Avoid the
qualitative observations that focus on the qualities of an object, like the chair is yellow. Instead, focus on
how many there are, or the quantity of an object.
1. Make at least 5 quantitative observations.
Observation #
1
Quantitative Observation
There are ___ people in this room
2
3
4
5
In each of your observations, the number (or quantity) described the object. There are really not
many numbers in our lives that are true numbers. Most of the numbers that we deal with are attached
to a unit, even if we often forget to write that unit. If we pay close attention to that unit, however, many
problems that arise can usually be avoided.
Review the examples you choose. Which ones can we combine?
Let's find a rule for addition that will help us track out units and combine terms.
2. Complete the table with several more examples and make a generalized rule about addition.
Evidence
Claim
(4 + 5) + 6 = 15
and
4 + (5 + 6) = 15
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Active Learning Activities
Now we have established a pattern and can be confident in making a CLAIM about our ability to regroup the terms to our liking when performing addition. (Please remember that all operations must be
addition for this rule to be true.)
The Associative Rule of Addition states that a + (b + c) = (a + b) + c, which means that we can put these
terms in any order we want to (as long as we are only performing addition).
Re-group the following problems so that only the terms that are alike are in parentheses. We will group
the like terms together to combine them.
3. Complete the table with 5 more examples of your own.
Problem
Regrouped
Answer
7 oranges + 3 peaches + 1 orange
(7 oranges + 1 orange) + 3
peaches
8 oranges + 3 peaches
3x2 + 2x + 5x2 + 8x + 1
(3x2 + 5x2) + (2x + 8x) + 1
8x2 + 10x + 1
The Associative Rule of Addition is a powerful tool, but it only works for addition problems. What can
you do if subtraction and addition problems are mixed together?
4. Find a pattern in the problems below as you complete the table, and make a claim based on that
pattern that will help you deal with changing subtraction in to addition. Be sure to draw use the
model for each.
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Math - The Language of the Cosmos
Problem
Active Learning Activities
Model
5–6
[+]
(-)
[+]
(-)
[+]
(-)
[+]
(-)
[+]
(-)
Equivalent Problem
Claim
5 + -6
(-)
-7 – 9
7–4
(-) (-)
(-) (-)
(-) (-)
(-) (-)
(-)
(-) (-)
(-)
(-) (-)
(-)
(-)
[+] [+] (-) (-)
-7 + -9
7 + -4
[+] [+] (-) (-)
[+]
[+]
[+]
6 – -3
[+] [+] [+]
[+]
[+]
[+]
[+]
6+3
[+]
[+]
-9 – -7
(-) (-)
[+] [+]
(-) (-)
[+] [+]
(-) (-)
[+] [+]
(-) (-)
[+]
-9 + 7
(-)
Now we can see that subtraction is the same as adding the opposite. Whenever you see a subtraction
problem, it is always a good idea to change the subtraction to adding the opposite. Be careful not to
change anything about the first term, however.
Our activity would not be complete without an introduction to the Distributive Property. This very
important concept can best be seen by using a geometric model. Geometry is simply math that you can
see, using shapes and diagrams. The rectangle will teach us all about the Distributive Property.
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Math - The Language of the Cosmos
Active Learning Activities
Remembering that area is length x width, what is the area of this
rectangle?
a * c = ac
Now, we will add another part to side “a” to make the sides “a + b”
and “c”. What is the area of this rectangle? (Be sure that you don’t
combine terms that are unlike. “ac” and “bc” are not like terms, so
you must keep your answer as simply “ac + bc”.)
(a*c) + (b*c) = ac + bc
or
(a+b) * c
Finally, we add another part to side “c” to make the sides “a + b” and (a*c) + (a*d) + (b*c) + (b*d) =
“c + d”. What is the area of this rectangle?
ac + ad + bc + bd
From these problems, the Distributive Property and its extension become obvious.
Distributive Property
(a + b) times c = (a + b) c = c (a + b) = ac + bc
Extended Distributive Property
(a + b) (c + d) = c (a + b) + d (a + b) = ac + bc + ad + bd
OR
(a + b) (c + d) = a (c + d) + b (c + d) = ac + ad + bc + bd
With this new idea in mind, we will be able to more accurately solve many complicated problems. Try
these problems out, while remembering the order of operations—Parentheses, Exponents,
Multiplication and Division at the same time from left to right, and Addition and Subtraction at the same
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Math - The Language of the Cosmos
Active Learning Activities
time from left to right. Many people like to use the phrase “Please Excuse My Dear Aunt Sally” to help
them remember the order of operations.
Example:
4x2 – (2x + 7x2) = 4x2 + - (2x + 7x2) =
4x2 + -2x + -7x2 =
(4x2 + -7x2) + -2x
= -3x2 + -2x
Notice that changing the subtraction sign to adding the opposite shows us that we must distribute the
negative sign to each of the terms inside the parentheses. Also, it is best practice to write the term with
the highest exponent first.
-5x3 + -2x4 – 2 + (7 – 3x3)
-5x3 + -2x4 + -3x3 + 5
8cd + 2c(d + 5w)
10cd + 10cw
If we are given the value of the variables, we can evaluate the expression by placing the value for the
variable in place of the variable.
Example:
Evaluate for x = 2
6x2 – 5x + 3 = 6(2)2 – 5(2) + 3 = 6(4) – 5(2) + 3 = 24 – 10 + 3 = 14 + 3 = 17
Notice that whenever you substitute a number for a variable, you should always use
parentheses.
Evaluate for a = 5 and b = 3
-5(52) + 3(5) – (5)(3) + 8 = -125 + 15 – 15 + 8 = 133
-5a2 + 3a – ab + 8
Evaluate for x = -3, y = 6 and z = 2
4xy + 3x – 7yz + xyz – 9
4(-3)(6) + 3(-3) – 7(6)(2) + (-3)(6)(2) – 9 =
-72 + -9 – 84 + -36 – 9 = -210
Review
Take some time to review this ALA. Go through each section and make your own examples of the
problems within. Make flashcards for the Associative Property of Addition, the Distributive Property and
the Order of Operations.
How to Read the Numbers
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