4.2.3: Using Diagrams to Find the Number of Groups

4.2.3: Using Diagrams to Find the Number of Groups

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Let's draw tape diagrams to think about division with fractions.

Exercise (PageIndex{1}): How Many of These in That?

  1. We can think of the division expression (10div 2frac{1}{2}) as the question: “How many groups of (2frac{1}{2}) are in 10?” Complete the tape diagram to represent this question. Then find the answer.
  1. Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then find the answer.

Exercise (PageIndex{2}): Representing Groups of Fractions with Tape Diagrams

To make sense of the question “How many (frac{2}{3})s are in 1?,” Andre wrote equations and drew a tape diagram.



  1. In an earlier task, we used pattern blocks to help us solve the equation (1divfrac{2}{3}=?). Explain how Andre’s tape diagram can also help us solve the equation.
  2. Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.
  1. How many (frac{3}{4})s are in (1)?
  1. How many (frac{2}{3})s are in (3)?
  1. How many (frac{3}{2})s are in (5)?

Exercise (PageIndex{3}): Finding Number of Groups

  1. Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning.
    1. How many (frac{3}{8})-inch thick books make a stack that is 6 inches tall?
    2. How many groups of (frac{1}{2}) pound are in (2frac{3}{4}) pounds?
  2. Write a question that can be represented by the division equation (5div 1frac{1}{2}=?). Then, find the answer and explain or show your reasoning.


A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for (frac{2}{5}) kilogram of flour per batch. How many batches did she make?

We can think of the question as: “How many groups of (frac{2}{5}) kilogram make 2 kilograms?” and represent that question with the equations:



To help us make sense of the question, we can draw a tape diagram. This diagram shows 2 whole kilograms, with each kilogram partitioned into fifths.

We can see there are 5 groups of (frac{2}{5}) in 2. Multiplying 5 and (frac{2}{5}) allows us to check this answer: (5cdotfrac{2}{5}=frac{10}{5}) and (frac{10}{5}=2), so the answer is correct.

Notice the number of groups that result from (2divfrac{2}{5}) is a whole number. Sometimes the number of groups we find from dividing may not be a whole number. Here is an example:

Suppose one serving of rice is (frac{3}{4}) cup. How many servings are there in (3frac{1}{2}) cups?



Looking at the diagram, we can see there are 4 full groups of (frac{3}{4}), plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make (frac{2}{3}) of a group. So the number of servings (the “?” in each equation) is (4frac{2}{3}). We can check this by multiplying (4frac{2}{3}) and (frac{3}{4}).

(4frac{2}{3}cdotfrac{3}{4}=frac{14}{3}cdotfrac{3}{4}), and (frac{14}{3}cdotfrac{3}{4}=frac{14}{4}), which is indeed equivalent to (3frac{1}{2}).


Exercise (PageIndex{4})

We can think of (3divfrac{1}{4}) as the question “How many groups of (frac{1}{4}) are in (3)?” Draw a tape diagram to represent this question. Then find the answer.

Exercise (PageIndex{5})

Describe how to draw a tape diagram to represent and answer (3divfrac{3}{5}=?) for a friend who was absent.

Exercise (PageIndex{6})

How many groups of (frac{1}{2}) day are in 1 week?

  1. Write a multiplication equation or a division equation to represent the question.
  2. Draw a tape diagram to show the relationship between the quantities and to answer the question. Use graph paper, if needed.

Exercise (PageIndex{7})

Diego said that the answer to the question “How many groups of (frac{5}{6}) are in (1)?” is (frac{6}{5}) or (1frac{1}{5}). Do you agree with him? Explain or show your reasoning.

Exercise (PageIndex{8})

Select all the equations that can represent the question: “How many groups of (frac{4}{5}) are in (1)?”

  1. (?cdot 1=frac{4}{5})
  2. (1cdotfrac{4}{5}=?)
  3. (frac{4}{5}div 1=?)
  4. (?cdotfrac{4}{5}=1)
  5. (1divfrac{4}{5}=?)

(From Unit 4.2.2)

Exercise (PageIndex{9})

Calculate each percentage mentally.

  1. What is (10)% of (70)?
  2. What is (10)% of (110)?
  3. What is (25)% of (160)?
  4. What is (25)% of (48)?
  5. What is (50)% of (90)?
  6. What is (50)% of (350)?
  7. What is (75)% of (300)?
  8. What is (75)% of (48)?

(From Unit 3.4.5)

The purpose of introducing quantum numbers has been to show that similarities in the electron arrangement or electron configuration lead to the similarities and differences in the properties of elements. But writing the quantum numbers of electrons of an element in set notation like <2,1,-1,1&frasl2>is time consuming and difficult to compare so an abbreviated form was developed. An electron configuration lists only the first two quantum numbers, n and (ell), and then shows how many electrons exist in each orbital. For example, write the electron configuration of scandium, Sc: 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 1 . So for scandium the 1 st and 2 nd electron must be in 1s orbital, the 3 rd and 4 th in the 2s, the 5 th through 10 th in the 2p orbitals, etc.

This is a memory device to remember the order of orbitals for the first two quantum numbers. Follow the arrow starting in the upper right, when the arrow ends go to the next arrow and start again.

In Scandium, the 4s has lower energy and appears before 3d (the complexity of the d-orbitals leads to its higher energy), so it is written before adding 3d to the electron configuration. But it is common to to keep all the principle quantum numbers together so you may see the electron configuration written as Sc: 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 1 . Writing electron configurations like this can cause difficulties in determining the element that matches an electron configuration. But if you just count the number of electrons it will equal the number of protons which equals the atomic number which is unique for each element. For example: &ldquoWhich element has the electron configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4s 2 4p 6 4d 8 5s 2 ?&rdquo Counting the electrons gives 46, which is the atomic number of palladium.

Here&rsquos a diagram of the first several electron configurations. David&rsquos Whizzy Periodic Table is a visual way of looking at the changing electron configuration of elements.

Note the 3d orbital follows the 4s in the lowest row, but starting with Ga (#31) it is next to the 3p orbital. It is most commonly listed with the other 3 orbitals, but sometimes it follows the 4s orbital to indicate that the 3d orbital is lower in energy than the 4s while it is being filled.

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UML Diagram Types Guide: Learn About All Types of UML Diagrams with Examples

UML stands for Unified Modeling Language. It’s a rich language to model software solutions, application structures, system behavior and business processes. There are 14 UML diagram types to help you model these behaviors.

You can draw UML diagrams online using our software, or check out some UML diagram examples at our diagramming community.

List of UML Diagram Types

So what are the different UML diagram types? There are two main categories structure diagrams and behavioral diagrams. Click on the links to learn more about a specific diagram type.

Structure diagrams show the things in the modeled system. In a more technical term, they show different objects in a system. Behavioral diagrams show what should happen in a system. They describe how the objects interact with each other to create a functioning system.

Class Diagram

Click on the image to edit the above class diagram (opens in new window)

Get More UML Class Diagram Examples >>

Component Diagram

A component diagram displays the structural relationship of components of a software system. These are mostly used when working with complex systems with many components. Components communicate with each other using interfaces. The interfaces are linked using connectors. The image below shows a component diagram.

You can use this component diagram template by clicking on the image

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Deployment Diagram

A deployment diagram shows the hardware of your system and the software in that hardware. Deployment diagrams are useful when your software solution is deployed across multiple machines with each having a unique configuration. Below is an example deployment diagram.

Click on the image to use this deployment diagram as a template

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Object Diagram

Object Diagrams, sometimes referred to as Instance diagrams are very similar to class diagrams. Like class diagrams, they also show the relationship between objects but they use real-world examples.

They show how a system will look like at a given time. Because there is data available in the objects, they are used to explain complex relationships between objects.

Click on the image to use the object diagram as a template

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Package Diagram

As the name suggests, a package diagram shows the dependencies between different packages in a system. Check out this wiki article to learn more about the dependencies and elements found in package diagrams.

Profile Diagram

Profile diagram is a new diagram type introduced in UML 2. This is a diagram type that is very rarely used in any specification. For more profile diagram templates, visit our diagram community.

Composite Structure Diagram

Composite structure diagrams are used to show the internal structure of a class. Some of the common composite structure diagrams.

Use Case Diagram

As the most known diagram type of the behavioral UML types, Use case diagrams give a graphic overview of the actors involved in a system, different functions needed by those actors and how these different functions interact.

It’s a great starting point for any project discussion because you can easily identify the main actors involved and the main processes of the system. You can create use case diagrams using our tool and/or get started instantly using our use case templates.

Click on the image to edit this template

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Activity Diagram

Activity diagrams represent workflows in a graphical way. They can be used to describe the business workflow or the operational workflow of any component in a system. Sometimes activity diagrams are used as an alternative to State machine diagrams. Check out this wiki article to learn about symbols and usage of activity diagrams. You can also refer this easy guide to activity diagrams.

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State Machine Diagram

State machine diagrams are similar to activity diagrams, although notations and usage change a bit. They are sometimes known as state diagrams or state chart diagrams as well. These are very useful to describe the behavior of objects that act differently according to the state they are in at the moment. The State machine diagram below shows the basic states and actions.

State Machine diagram in UML, sometimes referred to as State or State chart diagram

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Sequence Diagram

Sequence diagrams in UML show how objects interact with each other and the order those interactions occur. It’s important to note that they show the interactions for a particular scenario. The processes are represented vertically and interactions are shown as arrows. This article explains the purpose and the basics of Sequence diagrams. Also, check out this complete Sequence Diagram Tutorial to learn more about sequence diagrams.

You can also instantly start drawing using our sequence diagram templates.

Sequence diagram drawn using Creately

Communication Diagram

In UML 1 they were called collaboration diagrams. Communication diagrams are similar to sequence diagrams, but the focus is on messages passed between objects. The same information can be represented using a sequence diagram and different objects. Click here to understand the differences using an example.

Interaction Overview Diagram

Interaction overview diagrams are very similar to activity diagrams. While activity diagrams show a sequence of processes, Interaction overview diagrams show a sequence of interaction diagrams.

They are a collection of interaction diagrams and the order they happen. As mentioned before, there are seven types of interaction diagrams, so any one of them can be a node in an interaction overview diagram.

Timing Diagram

Timing diagrams are very similar to sequence diagrams. They represent the behavior of objects in a given time frame. If it’s only one object, the diagram is straightforward. But, if there is more than one object is involved, a Timing diagram is used to show interactions between objects during that time frame.

Click here to create your timing diagram.

Mentioned above are all the UML diagram types. UML offers many diagram types, and sometimes two diagrams can explain the same thing using different notations.

Check out this blog post to learn which UML diagram best suits you. If you have any questions or suggestions, feel free to leave a comment.

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For light atoms, the spin–orbit interaction (or coupling) is small so that the total orbital angular momentum L and total spin S are good quantum numbers. The interaction between L and S is known as LS coupling, Russell–Saunders coupling (named after Henry Norris Russell and Frederick Albert Saunders, who described this in 1925. [2] ) or spin-orbit coupling. Atomic states are then well described by term symbols of the form

S is the total spin quantum number. 2S + 1 is the spin multiplicity, which represents the number of possible states of J for a given L and S, provided that LS. (If L < S, the maximum number of possible J is 2L + 1). [3] This is easily proven by using Jmax = L + S and Jmin = |LS|, so that the number of possible J with given L and S is simply JmaxJmin + 1 as J varies in unit steps. J is the total angular momentum quantum number. L is the total orbital quantum number in spectroscopic notation. The first 17 symbols of L are:

L = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .
S P D F G H I K L M N O Q R T U V (continued alphabetically) [note 1]

The nomenclature (S, P, D, F) is derived from the characteristics of the spectroscopic lines corresponding to (s, p, d, f) orbitals: sharp, principal, diffuse, and fundamental the rest being named in alphabetical order from G onwards, except that J is omitted. When used to describe electron states in an atom, the term symbol usually follows the electron configuration. For example, one low-lying energy level of the carbon atom state is written as 1s 2 2s 2 2p 2 3 P2. The superscript 3 indicates that the spin state is a triplet, and therefore S = 1 (2S + 1 = 3), the P is spectroscopic notation for L = 1, and the subscript 2 is the value of J. Using the same notation, the ground state of carbon is 1s 2 2s 2 2p 2 3 P0. [1]

Small letters refer to individual orbitals or one-electron quantum numbers, whereas capital letters refer to many-electron states or their quantum numbers.

The term symbol is also used to describe compound systems such as mesons or atomic nuclei, or molecules (see molecular term symbol). For molecules, Greek letters are used to designate the component of orbital angular momenta along the molecular axis.

For a given electron configuration

  • The combination of an S value and an L value is called a term, and has a statistical weight (i.e., number of possible microstates) equal to (2S+1)(2L+1)
  • A combination of S, L and J is called a level. A given level has a statistical weight of (2J+1), which is the number of possible microstates associated with this level in the corresponding term
  • A combination of S, L, J and MJ determines a single state.

The parity of a term symbol is calculated as

When it is odd, the parity of the term symbol is indicated by a superscript letter "o", otherwise it is omitted:

2 P o
½ has odd parity, but 3 P0 has even parity.

Alternatively, parity may be indicated with a subscript letter "g" or "u", standing for gerade (German for "even") or ungerade ("odd"):

2 P½,u for odd parity, and 3 P0,g for even.

It is relatively easy to calculate the term symbol for the ground state of an atom using Hund's rules. It corresponds with a state with maximum S and L.

  1. Start with the most stable electron configuration. Full shells and subshells do not contribute to the overall angular momentum, so they are discarded.
    • If all shells and subshells are full then the term symbol is 1 S0.
  2. Distribute the electrons in the available orbitals, following the Pauli exclusion principle. First, fill the orbitals with highest m ℓ > value with one electron each, and assign a maximal ms to them (i.e. +½). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning ms = −½ to them.
  3. The overall S is calculated by adding the ms values for each electron. According to Hund's first rule, the ground state has all unpaired electron spins parallel with the same value of ms, conventionally chosen as +½. The overall S is then ½ times the number of unpaired electrons. The overall L is calculated by adding the m ℓ > values for each electron (so if there are two electrons in the same orbital, add twice that orbital's m ℓ > ).
  4. Calculate J as
    • if less than half of the subshell is occupied, take the minimum value J = |LS|
    • if more than half-filled, take the maximum value J = L + S
    • if the subshell is half-filled, then L will be 0, so J = S .

As an example, in the case of fluorine, the electronic configuration is 1s 2 2s 2 2p 5 .

  1. Discard the full subshells and keep the 2p 5 part. So there are five electrons to place in subshell p ( ℓ = 1 ).
  2. There are three orbitals ( m ℓ = 1 , 0 , − 1 =1,0,-1> ) that can hold up to 2 ( 2 ℓ + 1 ) = 6 electrons . The first three electrons can take ms = ½ (↑) but the Pauli exclusion principle forces the next two to have ms = −½ (↓) because they go to already occupied orbitals.
    m ℓ >
    m s >↑↓↑↓
  3. S = ½ + ½ + ½ − ½ − ½ = ½ and L = 1 + 0 − 1 + 1 + 0 = 1 , which is "P" in spectroscopic notation.
  4. As fluorine 2p subshell is more than half filled, J = L + S = 3 ⁄ 2 . Its ground state term symbol is then 2S+1 LJ = 2 P
  5. 3 ⁄ 2 .

Atomic term symbols of the chemical elements Edit

Term symbols for the ground states of most chemical elements [4] are given in the collapsed table below (with citations for the heaviest elements here). In the d-block and f-block, the term symbols are not always the same for elements in the same column of the periodic table, because open shells of several d or f electrons have several closely spaced terms whose energy ordering is often perturbed by the addition of an extra complete shell to form the next element in the column.

For example, the table shows that the first pair of vertically adjacent atoms with different ground-state term symbols are V and Nb. The 6 D1/2 ground state of Nb corresponds to an excited state of V 2112 cm −1 above the 4 F3/2 ground state of V, which in turn corresponds to an excited state of Nb 1143 cm −1 above the Nb ground state. [1] These energy differences are small compared to the 15158 cm −1 difference between the ground and first excited state of Ca, [1] which is the last element before V with no d electrons.

Background color shows category:

The process to calculate all possible term symbols for a given electron configuration is somewhat longer.

Case of three equivalent electrons Edit

Alternative method using group theory Edit

For configurations with at most two electrons (or holes) per subshell, an alternative and much quicker method of arriving at the same result can be obtained from group theory. The configuration 2p 2 has the symmetry of the following direct product in the full rotation group:

which, using the familiar labels Γ (0) = S , Γ (1) = P and Γ (2) = D , can be written as

The square brackets enclose the anti-symmetric square. Hence the 2p 2 configuration has components with the following symmetries:

S + D (from the symmetric square and hence having symmetric spatial wavefunctions) P (from the anti-symmetric square and hence having an anti-symmetric spatial wavefunction).

The Pauli principle and the requirement for electrons to be described by anti-symmetric wavefunctions imply that only the following combinations of spatial and spin symmetry are allowed:

1 S + 1 D (spatially symmetric, spin anti-symmetric) 3 P (spatially anti-symmetric, spin symmetric).

Then one can move to step five in the procedure above, applying Hund's rules.

The group theory method can be carried out for other such configurations, like 3d 2 , using the general formula

Γ (j) × Γ (j) = Γ (2j) + Γ (2j−2) + ⋯ + Γ (0) + [Γ (2j−1) + ⋯ + Γ (1) ].

The symmetric square will give rise to singlets (such as 1 S, 1 D, & 1 G), while the anti-symmetric square gives rise to triplets (such as 3 P & 3 F).

More generally, one can use

Γ (j) × Γ (k) = Γ (j+k) + Γ (j+k−1) + ⋯ + Γ (|jk|)

where, since the product is not a square, it is not split into symmetric and anti-symmetric parts. Where two electrons come from inequivalent orbitals, both a singlet and a triplet are allowed in each case. [6]

Basic concepts for all coupling schemes:

  • l → >> : individual orbital angular momentum vector for an electron, s → >> : individual spin vector for an electron, j → >> : individual total angular momentum vector for an electron, j → = l → + s → >=>+>> .
  • L → >> : Total orbital angular momentum vector for all electrons in an atom ( L → = ∑ i l i → >=sum _>>> ).
  • S → >> : total spin vector for all electrons ( S → = ∑ i s i → >=sum _>>> ).
  • J → >> : total angular momentum vector for all electrons. The way the angular momenta are combined to form J → >> depends on the coupling scheme: J → = L → + S → >=>+>> for LS coupling, J → = ∑ i j i → >=sum _>>> for jj coupling, etc.
  • A quantum number corresponding to the magnitude of a vector is a letter without an arrow (ex: l is the orbital angular momentum quantum number for l → >> and l ^ 2 | l , m , … ⟩ = ℏ 2 l ( l + 1 ) | l , m , … ⟩ >^<2>>left|l,m,ldots ight angle =<^<2>>lleft(l+1 ight)left|l,m,ldots ight angle > )
  • The parameter called multiplicity represents the number of possible values of the total angular momentum quantum number J for certain conditions.
  • For a single electron, the term symbol is not written as S is always 1/2, and L is obvious from the orbital type.
  • For two electron groups A and B with their own terms, each term may represent S, L and J which are quantum numbers corresponding to the S → >> , L → >> and J → >> vectors for each group. "Coupling" of terms A and B to form a new term C means finding quantum numbers for new vectors S → = S A → + S B → >=>+>>> , L → = L A → + L B → >=>+>>> and J → = L → + S → >=>+>> . This example is for LS coupling and which vectors are summed in a coupling is depending on which scheme of coupling is taken. Of course, the angular momentum addition rule is that X = X A + X B , X A + X B − 1 , . . . , | X A − X B | ,X_+X_-1. |X_-X_|> where X can be s, l, j, S, L, J or any other angular momentum-magnitude-related quantum number.

LS coupling (Russell–Saunders coupling) Edit

  • Coupling scheme: L → >> and S → >> are calculated first then J → = L → + S → >=>+>> is obtained. From a practical point of view, it means L, S and J are obtained by using an addition rule of the angular momenta of given electron groups that are to be coupled.
  • Electronic configuration + Term symbol: n ℓ N ( ( 2 S + 1 ) L J ) ^><<(>^<(2S+1)>><_>)> . ( ( 2 S + 1 ) L J ) ^<(2S+1)>><_>)> is a Term which is from coupling of electrons in n ℓ N ^>> group. n , ℓ are principle quantum number, orbital quantum number and n ℓ N ^>> means there are N (equivalent) electrons in n ℓ subshell. For L > S , ( 2 S + 1 ) is equal to multiplicity, a number of possible values in J (final total angular momentum quantum number) from given S and L. For S > L , multiplicity is ( 2 L + 1 ) but ( 2 S + 1 ) is still written in the Term symbol. Strictly speaking, ( ( 2 S + 1 ) L J ) ^<(2S+1)>><_>)> is called Level and ( 2 S + 1 ) L >> is called Term. Sometimes superscript o is attached to the Term, means the parity P = ( − 1 ) ∑ i ℓ i ^<>>,<_>>>> of group is odd ( P = − 1 ).
  • Example:
    1. 3d 7 4 F7/2: 4 F7/2 is Level of 3d 7 group in which are equivalent 7 electrons are in 3d subshell.
    2. 3d 7 ( 4 F)4s4p( 3 P 0 ) 6 F 0
      9/2 : [7] Terms are assigned for each group (with different principal quantum number n) and rightmost Level 6 F o
      9/2 is from coupling of Terms of these groups so 6 F o
      9/2 represents final total spin quantum number S, total orbital angular momentum quantum number L and total angular momentum quantum number J in this atomic energy level. The symbols 4 F and 3 P o refer to seven and two electrons respectively so capital letters are used.
    3. 4f 7 ( 8 S 0 )5d ( 7 D o )6p 8 F13/2: There is a space between 5d and ( 7 D o ). It means ( 8 S 0 ) and 5d are coupled to get ( 7 D o ). Final level 8 F o
      13/2 is from coupling of ( 7 D o ) and 6p.
    4. 4f( 2 F 0 ) 5d 2 ( 1 G) 6s( 2 G) 1 P 0
      1 : There is only one Term 2 F o which is isolated in the left of the leftmost space. It means ( 2 F o ) is coupled lastly ( 1 G) and 6s are coupled to get ( 2 G) then ( 2 G) and ( 2 F o ) are coupled to get final Term 1 P o
      1 .

Jj Coupling Edit

J1L2 coupling Edit

LS1 coupling Edit

    3d 7 ( 4 P)4s4p( 3 P o ) D o 3 [5/2] o
    7/2 : L 1 = 1 , L 2 = 1 , S 1 = 3 2 , S 2 = 1 _<1>>=1,

Most famous coupling schemes are introduced here but these schemes can be mixed to express the energy state of an atom. This summary is based on [1].

These are notations for describing states of singly excited atoms, especially noble gas atoms. Racah notation is basically a combination of LS or Russell–Saunders coupling and J1L2 coupling. LS coupling is for a parent ion and J1L2 coupling is for a coupling of the parent ion and the excited electron. The parent ion is an unexcited part of the atom. For example, in Ar atom excited from a ground state . 3p 6 to an excited state . 3p 5 4p in electronic configuration, 3p 5 is for the parent ion while 4p is for the excited electron. [8]

Paschen notation is a somewhat odd notation it is an old notation made to attempt to fit an emission spectrum of neon to a hydrogen-like theory. It has a rather simple structure to indicate energy levels of an excited atom. The energy levels are denoted as n′l#. l is just an orbital quantum number of the excited electron. n′l is written in a way that 1s for (n = N + 1, l = 0), 2p for (n = N + 1, l = 1), 2s for (n = N + 2, l = 0), 3p for (n = N + 2, l = 1), 3s for (n = N + 3, l = 0), etc. Rules of writing n′l from the lowest electronic configuration of the excited electron are: (1) l is written first, (2) n′ is consecutively written from 1 and the relation of l = n′ − 1, n′ − 2, . , 0 (like a relation between n and l) is kept. n′l is an attempt to describe electronic configuration of the excited electron in a way of describing electronic configuration of hydrogen atom. # is an additional number denoted to each energy level of given n′l (there can be multiple energy levels of given electronic configuration, denoted by the term symbol). # denotes each level in order, for example, # = 10 is for a lower energy level than # = 9 level and # = 1 is for the highest level in a given n′l. An example of Paschen notation is below.


Create a Box Plot

Create a box plot of the miles per gallon ( MPG ) measurements. Add a title and label the axes.

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Create a box plot of the miles per gallon ( MPG ) measurements from the sample data, grouped by the vehicles' country of origin ( Origin ). Add a title and label the axes.

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Generate two sets of sample data. The first sample, x1 , contains random numbers generated from a normal distribution with mu = 5 and sigma = 1 . The second sample, x2 , contains random numbers generated from a normal distribution with mu = 6 and sigma = 1 .

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Create a 100-by-25 matrix of random numbers generated from a standard normal distribution to use as sample data.

Create two box plots for the data in x on the same figure. Use the default formatting for the top plot, and compact formatting for the bottom plot.

Each plot presents the same data, but the compact formatting may improve readability for plots with many boxes.

Box Plots for Vectors of Varying Length

Create box plots for data vectors of varying length by using a grouping variable.

Randomly generate three column vectors of varying length: one of length 5 , one of length 10 , and one of length 15 . Combine the data into a single column vector of length 30 .

Create a grouping variable that assigns the same value to rows that correspond to the same vector in x . For example, the first five rows of g have the same value, First , because the first five rows of x all come from the same vector, x1 .

Deploy application to Azure Container Instances

Next, change to the ACI context. Subsequent Docker commands run in this context.

Run docker compose up to start the application in Azure Container Instances. The azure-vote-front image is pulled from your container registry and the container group is created in Azure Container Instances.

Docker Compose commands currently available in an ACI context are docker compose up and docker compose down . There is no hyphen between docker and compose in these commands.

In a short time, the container group is deployed. Sample output:

Run docker ps to see the running containers and the IP address assigned to the container group.

To see the running application in the cloud, enter the displayed IP address in a local web browser. In this example, enter . The sample application loads, as shown in the following example:

To see the logs of the front-end container, run the docker logs command. For example:

You can also use the Azure portal or other Azure tools to see the properties and status of the container group you deployed.

When you finish trying the application, stop the application and containers with docker compose down :

This command deletes the container group in Azure Container Instances.

2.0 Indirect and total effects

One of the appealing aspects of path models is the ability to assess indirect, as well as total effects (i.e., relationships among variables). Note that the total effect is the combination of the direct effect and indirect effects. In this example we will request the estimated indirect effect of hs on grad (through gre). Below is the diagram corresponding to this model with the desired indirect effect shown in blue. We can obtain the estimate of the indirect effect by adding the model indirect: command to our input file, and specifying grad ind hs.

Here is the entire program. Notice that the model indirect has been added.

The output for this model is shown below. The output is the same as the output from the previous example because we have estimated the same model adding the indirect effects requests additional output from Mplus, but that does not change the model itself. The breakdown of the total, indirect, and direct effects appears below the MODEL RESULTS and STANDARDIZED MODEL RESULTS in a section labeled TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS. Because standardized coefficients were requested, the standardized total, indirect, and direct effects appear below the unstandardized effects.

Under Specific indirect, the effect labeled GRAD GRE HS (note that each appears on its own line and the final outcome is listed first), gives the estimated coefficient for the indirect effect of hs on grad, through GRE . The coefficient labeled Direct is the direct effect of hs on grad. We can say that part of the total effect of hs on grad is mediated by gre scores, but the significant direct path from hs to grad suggests only partial mediation.

Division Word Problems (1-step word problems)

Here are some examples of division word problems that can be solved in one step. We will illustrate how block diagrams or tape diagrams can be used to help you to visualize the division word problems in terms of the information given and the data that needs to be found.

We use division or multiplication when the problem involves equal parts of a whole. The following diagram shows how to use division to find unknown size of parts or groups or to find unknown number of parts or groups. Scroll down the page for examples and solutions.

There are 160 grade 3 students in a school. The students are to be equally divided into 5 classes. How many students do we have in each class?

We have 32 students in each class.

Melissa made 326 cupcakes. She packed 4 cupcakes into each box. How many boxes of cupcakes did she pack? How many cupcakes were left unpacked?

She packed 81 boxes of cupcakes.

2 cupcakes were left unpacked.

Practice solving the following multiplication and division word problems.
1. Dan went to the market on Friday. He bought two tomatoes. On Sunday, he bought six times as many. How many tomatoes did he buy on Sunday?
2. In July, a construction company built 360 miles of road. In February, the company laid down 60 miles of road. How many times more road did the company complete in July?
3. Linh ran 21 miles. Linh ran three times as far as Sophie. How far did Sophie run?
4. Molly's bedroom is 220 square feet. Molly's dining room is five times the size of her bedroom. How large is her dining room?

Using Division Tape Diagrams to find Unknown Number of Groups
After playing Belmont, the 24 Islander players traveled to South Boston. This time they went by car and 3 players rode in each care. How many cars did they need?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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