Parts of the Circle

Checking and securing understanding of the parts of a circle

Sets of Numbers: Natural, Whole, Integers, Rational Numbers, Irrational Numbers, Real Numbers, Complex Numbers Interval Notation Set Builder Notation Number Lines, open circles, closed circles Sets and Subsets - Union and Intersection Parts of the Cartesian Plane: origin, quadrants, signs on numbers in each quadrant, x-axis, y-axis Points being symmetric to each other about the x-axis, y-axis and origin Distance Formula-calculating distance between 2 given points Midpoint-calculating midpoint between 2 given points

How Glooskap Found Summer Long ago, it grew very cold. Ice and snow covered the land. Fires could not keep people warm, and corn could not grow. Glooskap, the leader of the people, had to do something. Glooskap traveled far to the north. Everywhere he looked was cold and white with snow. He came to a house made of solid ice where a giant named Winter lived. Winter greeted Glooskap and invited him inside his house. Winter began to tell stories of the time when he ruled the Earth. Soon Glooskap fell asleep under Winter's spell. But Glooskap's messenger, Tatler, woke him. "Wake up, Glooskap!" said the bird. "In the south, you will find a woman who can defeat Winter," said Tatler. Glooskap traveled far to the south. He came to a land where it was warm and sunny. Grass grew and flowers bloomed in the beautiful land. Glooskap saw spirits dancing in a circle. At the center of the circle was Summer. She wore a crown of flowers in her long brown hair. Glooskap asked Summer to come north with him. She followed him to Winter's house of ice. Winter invited them in and asked them to sit down. He began to tell stories again. But Winter's spells could not capture Summer. She began to chant her own spell, and sweat ran down Winter's face. "I am stronger than you," said Summer. "You must leave this land and thaw your icy breath," she said. 10 Winter wept, and his tears became rivers of melted snow and ice. The corn grew, and flowers bloomed again. Summer told Winter, "You will have your own land in the north. It will always be winter there. You may come and visit other lands for part of the year. But in the spring, I will drive you out." Since that day, Winter has ruled for part of the year. But every spring, Summer drives him away. Sometimes it seems as if winter will never end. But Summer is stronger than Winter. Spring will always come.

The Ship of Shapes One day, Elder Decagon saw that the shapes on Shape Island had become lazy. They sat in their huts, fanning themselves until it was time to eat. The different shapes didn't like to spend time with each other. The Rectangles stayed with the Rectangles, the Circles with the Circles, and so on. Elder Decagon came up with a plan. "Oh, oh, oh!" she exclaimed. Worried, the shapes gathered around her. "Big Scary Fire Mountain just spoke," she said. "It will erupt soon, and all our pants will be on fire." "We must leave the island!" The shapes were confused and scared. "Didn't you hear me?" "Pants will be on fire!" Elder Decagon yelled. "What should we do?" the shapes asked. "You should build a ship," she said very slowly. The shapes cheered for the great idea and hurried off to begin. The next day, Elder Decagon was surprised to see many ships on the beach. Each ship was meant for only one kind of shape. "None of these ships are shipshape, she said. "The Triangles' boat will tip in the water." "The Ovals' ship will float, but it won't move." "The Squares' ship will move, but too slowly." "What should we do?" the shapes asked. "You should build one big ship," Elder Decagon said very slowly. This time, the shapes didn't cheer. They weren't sure how to work together. Elder Decagon picked up a stick and started to draw. She showed them how the Squares sails would move the Ovals' boat. The Triangles' bottom would keep it from tipping. The Stars' propeller and the Hearts oars would help the ship move faster. In the end, all of the shapes went into the ship. The shapes stared at the drawing, but no one moved. "Pants will be on fire!" Elder Decagon yelled. The shapes went to work. When it was finished, all the shapes climbed onto the ship. They waited for Big Scary Fire Mountain to erupt, but it never did. The shapes asked Elder Decagon why it didn't. She just said, "Look at this wonderful, shipshape ship." "It shows that if you work hard together, you can go anywhere and do anything.' After some thought, the shapes agreed. They decided to work together to make Shape Island a better place. They also decided to explore the seas in their shipshape ship.

1. What is the meaning of the word "Izhaar"? A) To hide the sound B) To make it clear C) To change the sound D) To merge two letters 2. Which part of the body is used to pronounce Izhaar Halqi letters? A) The lips B) The tongue only C) The throat D) The nose 3. How many letters are there for Izhaar Halqi? A) 4 letters B) 6 letters C) 15 letters D) 2 letters 4. When do we apply the rule of Izhaar Halqi? A) When any letter comes after Meem Sakinah B) When an Izhaar letter comes after Noon Sakinah or Tanween C) When we see a Shaddah D) Only at the end of a Surah 5. Which of the following is NOT an Izhaar Halqi letter? A) Hamzah (أ) B) Haa (هـ) C) Baa (ب) D) 'Ayn (ع) 6. Which pair of letters comes from the deepest part of the throat (closest to the chest)? A) ع and ح B) غ and خ C) ء and هـ D) ق and ك 7. When you do Izhaar, do you make a long Ghunnah (nasal sound)? A) Yes, a very long one B) No, we pronounce the Noon clearly without extra Ghunnah C) Only if we want to D) Yes, for 2 counts 8. Which letter comes from the top part of the throat (closest to the mouth)? A) Khaad (خ) B) Haa (ح) C) Hamzah (أ) D) Meem (م) 9. What are the middle throat letters? A) ء and هـ B) ع and ح C) غ and خ D) ت and د 10. In the phrase "مَنْ عَمِلَ" (Man 'Amila), which rule is applied? A) Idghaam B) Ikhfaa C) Izhaar Halqi D) Iqlaab 11. Why do we do Izhaar in "مَنْ عَمِلَ"? A) Because the letter 'Ayn (ع) comes after Noon Sakinah B) Because it is easy to say C) Because Meem has a Fathah D) Because the Noon has a Shaddah 12. What does "Noon Sakinah" mean? A) A Noon with a Fathah B) A Noon with a Kasrah C) A Noon with no vowel (has a Sukoon) D) A double Noon 13. What is Tanween? A) A double vowel sign (Fathatain, Kasratain, Dammatain) at the end of a word D) A small Meem on top of a letter C) A stretching sign D) A stop sign 14. Can Izhaar Halqi happen within a single word? A) No, never B) Yes, it can happen in one word or between two words C) Only in short words D) Only in Surah Al-Fatihah 15. Look at the word "وَانْحَرْ" (Wanhar). What is the Izhaar letter here? A) Waw (و) B) Noon (ن) C) Haa (ح) D) Raa (ر) 16. In the Quran, what sign is usually placed on the Noon Sakinah to show it is Izhaar? A) A small circle or head of a Khaa (Sukoon sign) B) A Shaddah C) Nothing at all D) A little Meem 17. What happens to the Tanween vowels when there is Izhaar? A) They are written far apart from each other B) They are aligned perfectly parallel above/below each other C) One vowel is deleted D) They change color 18. Which of the following words contains an Izhaar Halqi rule? A) أَنْعَمْتَ B) مَنْ يَقُولُ C) مِنْ بَعْدِ D) كُنْتُمْ 19. Choose the group that contains ONLY Izhaar Halqi letters: A) ي ، ر ، م ، ل B) ء ، هـ ، ع ، ح ، غ ، خ C) ك ، ق ، ج ، د D) ب ، ت ، ث 20. In the phrase "عَذَابٌ أَلِيمٌ" ( 'Adhaabun Aleem), why is there Izhaar? A) Because Tanween is followed by Hamzah (أ) B) Because it ends with Meem C) Because the word is long D) Because of the letter Laam 21. What is the correct way to read "مِنْ حَكِيمٍ"? A) Mi---hakeem (hide the Noon) B) Min Hakeem (read Noon clearly and quickly) C) Mih-hakeem (mix them together) D) Mim-hakeem (change Noon to Meem) 22. "Ghain" (غ) and "Khaa" (خ) come from which part of the throat? A) Deep throat B) Middle throat C) Top throat D) The lips 23. If a Noon Sakinah is followed by the letter "هـ" (Haa), how do we pronounce it? A) Clear Noon B) Hidden Noon C) Double Noon D) Silent Noon 24. Which of these is a middle throat letter? A) ء B) خ C) ح D) هـ 25. Complete the sentence: Izhaar Halqi means to pronounce the Noon Sakinah or Tanween cleanly from its articulation point without any ________. A) Breathing B) Vowel (Harakah) C) Extra Ghunnah (nasalization) D) Stopping

Understanding Quantum Theory of Electrons in Atoms The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is a precursor to chemical bonding. As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels. The energy levels are labeled with an n value, where n = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of n. This number, n, is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure 9.4.1). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has. This quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom. The following equation summarizes these relationships and is based on the hydrogen atom: The values nf and ni are the final and initial energy states of the electron. The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital, which is distinct from an orbit, is a general region in an atom within which an electron is most probable to reside. The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in a multi-electron atoms and ions are located. Another quantum number is l, the angular momentum quantum number. It is an integer that defines the shape of the orbital, and takes on the values, l = 0, 1, 2, …, n – 1. This means that an orbital with n = 1 can have only one value of l, l = 0, whereas n = 2 permits l = 0 and l = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The l value specifies the shape of the orbital. Orbitals with the same value of l form a subshell. In addition, the greater the angular momentum quantum number, the greater is the angular momentum of an electron at this orbital. Orbitals with l = 0 are called s orbitals (or the s subshells). The value l = 1 corresponds to the p orbitals. For a given n, p orbitals constitute a p subshell (e.g., 3p if n = 3). The orbitals with l = 2 are called the d orbitals, followed by the f-, g-, and h-orbitals for l = 3, 4, 5, and there are higher values we will not consider. There are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. In other words, the value of the wavefunction ψ is zero at this distance for this orbital. Such a value of radius r is called a radial node. The number of radial nodes in an orbital is n – l – 1. Consider the examples in Figure 9.4.2. The orbitals depicted are of the s type, thus l = 0 for all of them. It can be seen from the graphs of the probability densities that there are 1 – 0 – 1 = 0 places where the density is zero (nodes) for 1s (n = 1), 2 – 0 – 1 = 1 node for 2s, and 3 – 0 – 1 = 2 nodes for the 3s orbitals. The s subshell electron density distribution is spherical and the p subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found. Principal quantum number (n) & Orbital angular momentum (l): The Orbital Subshell: https://youtu.be/ms7WR149fAY If an electron has an angular momentum (l ≠ 0), then this vector can point in different directions. In addition, the z component of the angular momentum can have more than one value. This means that if a magnetic field is applied in the z direction, orbitals with different values of the z component of the angular momentum will have different energies resulting from interacting with the field. The magnetic quantum number, called ml, specifies the z component of the angular momentum for a particular orbital. For example, for an s orbital, l = 0, and the only value of ml is zero. For p orbitals, l = 1, and ml can be equal to –1, 0, or +1. Generally speaking, ml can be equal to –l, –(l – 1), …, –1, 0, +1, …, (l – 1), l. The total number of possible orbitals with the same value of l (a subshell) is 2l + 1. Thus, there is one s-orbital for ml = 0, there are three p-orbitals for ml = 1, five d-orbitals for ml = 2, seven f-orbitals for ml = 3, and so forth. The principal quantum number defines the general value of the electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies orientation of the orbital in space, as can be seen in Figure 9.4.3. Figure 9.4.4 illustrates the energy levels for various orbitals. The number before the orbital name (such as 2s, 3p, and so forth) stands for the principal quantum number, n. The letter in the orbital name defines the subshell with a specific angular momentum quantum number l = 0 for s orbitals, 1 for p orbitals, 2 for d orbitals. Finally, there are more than one possible orbitals for l ≥ 1, each corresponding to a specific value of ml. In the case of a hydrogen atom or a one-electron ion (such as He+, Li2+, and so on), energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n, are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and orbitals that belong to different subshells have different energies. Orbitals within the same subshell (for example ns, np, nd, nf, such as 2p, 3s) are still degenerate and have the same energy. While the three quantum numbers discussed in the previous paragraphs work well for describing electron orbitals, some experiments showed that they were not sufficient to explain all observed results. It was demonstrated in the 1920s that when hydrogen-line spectra are examined at extremely high resolution, some lines are actually not single peaks but, rather, pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number, or ms. The other three quantum numbers, n, l, and ml, are properties of specific atomic orbitals that also define in what part of the space an electron is most likely to be located. Orbitals are a result of solving the Schrödinger equation for electrons in atoms. The electron spin is a different kind of property. It is a completely quantum phenomenon with no analogues in the classical realm. In addition, it cannot be derived from solving the Schrödinger equation and is not related to the normal spatial coordinates (such as the Cartesian x, y, and z). Electron spin describes an intrinsic electron “rotation” or “spinning.” Each electron acts as a tiny magnet or a tiny rotating object with an angular momentum, even though this rotation cannot be observed in terms of the spatial coordinates. The magnitude of the overall electron spin can only have one value, and an electron can only “spin” in one of two quantized states. One is termed the α state, with the z component of the spin being in the positive direction of the z axis. This corresponds to the spin quantum number ms=12. The other is called the β state, with the z component of the spin being negative and ms=−12. Any electron, regardless of the atomic orbital it is located in, can only have one of those two values of the spin quantum number. The energies of electrons having ms=−12 and ms=12 are different if an external magnetic field is applied. Figure 9.4.5 illustrates this phenomenon. An electron acts like a tiny magnet. Its moment is directed up (in the positive direction of the z axis) for the 12 spin quantum number and down (in the negative z direction) for the spin quantum number of −12. A magnet has a lower energy if its magnetic moment is aligned with the external magnetic field (the left electron) and a higher energy for the magnetic moment being opposite to the applied field. This is why an electron with ms=12 has a slightly lower energy in an external field in the positive z direction, and an electron with ms=−12 has a slightly higher energy in the same field. This is true even for an electron occupying the same orbital in an atom. A spectral line corresponding to a transition for electrons from the same orbital but with different spin quantum numbers has two possible values of energy; thus, the line in the spectrum will show a fine structure splitting. The Pauli Exclusion Principle An electron in an atom is completely described by four quantum numbers: n, l, ml, and ms. The first three quantum numbers define the orbital and the fourth quantum number describes the intrinsic electron property called spin. An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same set of all the four quantum numbers. What this means is that electrons can share the same orbital (the same set of the quantum numbers n, l, and ml), but only if their spin quantum numbers ms have different values. Since the spin quantum number can only have two values (±12), no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons. The properties and meaning of the quantum numbers of electrons in atoms are briefly

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