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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 2 Math - The Language of the Cosmos 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 3 Math - The Language of the Cosmos 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 4 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 5 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 6 Math - The Language of the Cosmos 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 7 Math - The Language of the Cosmos 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 8 Math - The Language of the Cosmos 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 9 Math - The Language of the Cosmos 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 10 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 11 Math - The Language of the Cosmos 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 12 Math - The Language of the Cosmos 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 13 Math - The Language of the Cosmos 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 14 Math - The Language of the Cosmos 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 15 Math - The Language of the Cosmos 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 16 Math - The Language of the Cosmos Active Learning Activities 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? 17 Math - The Language of the Cosmos 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? 18 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? 19 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? 20 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 How to Read the Numbers 21 Math - The Language of the Cosmos 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. How to Read the Numbers 22 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. How to Read the Numbers 23 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 How to Read the Numbers 24 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 25