3.9: Inventory Costing Calculations- A Closer Look at the COGS and Ending Inventory Computations
First, it must be recognized that the sales amount is independent of any costing method. For every sale, there is an invoice, and the dollar amount is not at all subject to any choice of accounting inventory costing method.
Under FIFO, the first- in were assumed to be the first- out. So, the cost of goods sold (COGS) figure was the sum of the first four (since four units were sold) costs: $100 + 150 + 200 + 250 = $700. The income statement will show a $700 COGS number as a deduction from sales to arrive at the gross profit of $2,000 – 700 = $1,300. This left $300 in the ending inventory on the balance sheet. Notice that $700 + 300 = $1,000 or the entire inventory cost.
Under LIFO, we must take the last-in , first-out order to compute the sum of the units sold and to arrive at the COGS number. Therefore, we add in reverse chronological order: $300 + 250 + 200 + 150 = $900. The ending inventory will be the first unit acquired, costing $100. Again, the two numbers add up to the sum of the inventory cost: $100 + 900 = $1,000. Gross profits will be lower: $2,000 – 900 = $1,100. Taxes are based on profits. In an inflationary environment, taxes payable will therefore be lower.
Assuming inventory costs rise as often happens when there is inflation , FIFO will always produce a higher gross profit. When corporations expect high inflation (as occurred in the 1970s and in the early 2020s), many will switch to LIFO in order to reduce taxes. Companies pay higher taxes when profits are higher, as might be the case using FIFO , assuming all else equal . To switch to LIFO, the firm must complete IRS Form 970 . The corporation cannot go back and forth from one method to the other as the wind blows. You will find the form on the Internet.
The company will prefer LIFO to reduce taxes. It will prefer FIFO to report higher inventory levels and thus show “more” assets to lenders and shareholders. Well, which of the two is preferable in the end? It is permissible for the company to maintain two sets of books, one for filing taxes to the Internal Revenue Service (the IRS) and the other for “reporting.” Moreover, it can use varying methods for different inventory items – FIFO for this and LIFO for that. How do you like that?!
Once again, a company may use one inventory costing method for one inventory item, and a different method for another item. In the end, the various items will be lumped together on one inventory line on the Balance Sheet and reflected as such in COGS . There might be a footnote indicating that inventory was valued using both FIFO and LIFO (and perhaps still another method), but further details may be absent. Talk about confusing!


1.2: Mathematics Milestones
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- Janet Stramel
- Fort Hays State University
Children grow faster in the first two years of life; more than at any other period of development. During the first two years, the brain grows from 25% of its adult weight to 75% of its final adult weight. Additionally, gross motor skills develop; they use large muscle groups to crawl, sit, stand, and lift things. Fine motor skills develop more slowly: stacking blocks, stringing beads, zipping zippers, and buttoning buttons.
Children are learning about the world around them through experiencing it through sight, smell, sound, touch, and taste. They are also learning by thinking about things, forming hypotheses, testing them, and problem solving. This is Piaget’s sensorimotor stage of development. This stage is from birth to approximately 2 years of age, and is a period of rapid cognitive growth and development. Object permanence is the main development during the sensorimotor stage and is vital to mathematics because it is the understanding that an object continues to exist even if it is not seen. According to Geist, “object permanence is the first step toward representation. When a child knows something exists even when he can’t see it, it means he has a mental concept of the object in the absence of its presence. This is one of the bases upon which more advanced mathematics will be built. Numbers are representative of objects and quantities even if we do not see them” (Geist, 2009).
At a very early age, babies use mathematics concepts to make sense of their world. For example, they can signal “more” when they want more food. In addition, they know the difference between familiar and unfamiliar adults which is “sorting and classifying.” They have “spatial reasoning” as they play with boxes of different sizes, and they use “patterns” as they say words and phrases.
As early as 6 months old , children begin to develop an understanding of “more.” They ask for “more milk” or “more food.” Babies from 0-12 months begin to predict the sequence of events, such as running water means bath time, they begin to classify things into certain categories, such as toys that make noise and toys that don’t, and they begin to understand relative size, such as parents are big and babies are small (Morin, 2014).
At 12 to 18 months old , children start to recognize patterns and understand shapes as they play with objects. They begin to sort familiar objects by characteristics, and they enjoy filling and emptying containers. Many children at this age can complete simple puzzles when the puzzle pieces show the whole object.
Although each child progresses at different levels, generally children of 18 months to 24 months old can show “one” and “two” on their fingers. They know some number words. Although they do not understand quantity, they do begin to understand “take one” or “give me one.” Children can match same-sized shapes. Puzzles and toys of squares, triangles, and circles help children develop spatial reasoning.
At 24 to 30 months old , children are learning important mathematical concepts and skills through their play. They recognize patterns and begin to use logical reasoning to solve everyday problems. They can sort shapes and stack blocks by size, and be able to distinguish between “big and small” and “fast and slow.” Children will be able to say some numbers, although they may skip some numbers in the sequence. For example, they may say, “one, two, three, five, ten…” By the age of 3, children should be able to count one to ten. Children at this stage of development generally understand addition and subtraction with the numbers one and two.
At approximately three years old , a child may attend a preschool. But what does mathematics teaching and learning look like in a preschool classroom? All children have the potential to learn mathematics that is challenging and at a high-level (Sarama & Clements, 2009). There is a wide range of development between ages three and five. There will be children who are “almost potty trained” as well as children who have begun learning to read. “Sit-and-listen” approach is only effective if it is highly engaging and exciting, for example, a story being read to the class. Therefore, teaching should consist almost entirely of “active learning.” Teaching should emphasize student-directed/teacher-facilitated activities. Worksheets are inappropriate at this level.
At 3-4 years old , children can count up to 30; and be able to count backwards from ten. Additionally, they can use ordinal numbers “first,” “second,” and “third.” At this age, children will be able to use superlatives such as “big, bigger, biggest.” At age 4, children will understand the concept of addition as putting together and subtraction as taking from when objects are in front of them. They also recognize shapes, begin to sort things by color, shape, size, or purpose, understand that numerals stand for numbers, and have the spatial awareness to complete puzzles (Morin, 2014).
At age four , children begin to have one-to-one correspondence in which numbers correspond to specific quantities. For example, a child would point to the number of blocks on the table and say, “1, 2, 3, 4, 5. I have five blocks.”
At age five , children can identify the larger of two numbers and recognize numerals up to 20. They can copy or draw shapes, and understand basic time concepts such as days of the week. By age six , children can add and subtract within ten mentally and solve simple word problems.
Read more about Mathematical Milestones: Children’s Development of Mathematical Concepts: Ages 0-4 (Infants, Toddlers, & Preschoolers).
Infants and Toddlers
Mathematics is all around us! And we use mathematical language all the time, even without realizing it. For example, when we sort laundry into colors and whites, we are sorting and classifying. Keeping score at a game is number and operations. Spatial relationships occur when we give someone directions, and when we make comparisons such as big and little, that is measurement. Everyday activities include patterns and order. Teachers must make mathematics visible to children through math talks (Greenberg, 2012).
Infants and toddlers understand number and operations as we talk with them about numbers, quantity, and one-to-one correspondence. According to Greenberg (2012), we can say the following to young children:
- You have two eyes, and so does your bear. Let’s count: 1-2.
- I have more crackers than you do. See, I have 1, 2, 3, and you have 1, 2.  Now I have the same as you!
- That’s the third time you have said mama. You’ve said mama three times!
Infants and toddlers understand geometry by recognizing and naming shapes. According to Greenberg (2012), we can talk with our children about understanding shapes and spatial relationships:
- Look, Samantha went under the stool and Hartley is on top.

- Some of the cookies are square and some are round .
Algebraic thinking begins as children see patterns, relationships, and change as they recognize relationships that make up a pattern or create repetitions. For example, you can say to infants and toddlers:
- Grandpa has stripes on his shirt – red, white, blue, red, white, blue.
- Let’s clap to the beat of this song.
- I put the blocks in the bucket; you dump them out .
Data analysis begins as infants and toddlers gather, sort, classify, and analyze information to make sense of their world. Teachers can talk with children about data by saying the following:
- Let’s put the big lid on the big bowl.
- You always smile when mommy sings.
Adapted from Greenberg, 2012
Math talks begin as soon as a child is born. During bath time, diaper time, meal time, or on walks or shopping trips is the perfect time to talk about shapes and sizes, talk about patterns, and describe things that are different. Mathematics comprehension is more than just numbers. Math is also about sorting things into categories such as big, bigger, and biggest; solving problems using patterns; solving spatial problems; and using logic.
Read more about Math Talks with infants and toddlers in this article, More, All Gone, Empty, Full: Math Talk Every Day in Every Way .
Kindergarten and First Grade
As children enter kindergarten and first grade, there are many changes in the physical, cognitive, and social-emotional development. Children in kindergarten can sit and attend to a formal lesson for approximately 10 minutes. At this level, the focus of mathematics instruction should be through experiences. By first grade, children can sit for a 15-20 minute formal lesson and although they are developing selective attention (the ability to decide what to attend to and what to ignore), the focus should still be on mathematics teaching through experiences.
Children in kindergarten and first grade are developing their gross and fine motor skills. According to Thelen and Smith, the development of coordination comes from physical play such as running and jumping and skipping (Thelen & Smith, 1996). Fine motor skills are developed as children write numbers and letters and use a pencil.
Kindergarteners can add by counting on their fingers, and start with six on the other hand; they can copy or draw symmetrical shapes, and can follow multi-step directions using first and next. In first grade, students can predict what comes next in a pattern, know the difference between two- and three-dimensional shapes, count to 100 by ones, twos, fives, and tens, and can write and recognize numbers, and do basic addition and subtraction up to 20 (Morin, 2014).
The Value of Play
Play does not guarantee mathematical development, but it offers rich possibilities. As the teacher, it is up to you to observe and develop a classroom when children have many opportunities to discover mathematical concepts through play. Douglas Clements says, “Anyone who is pushing arithmetic onto preschoolers is wrong. Do not hurry children.” (Clements, 2001, pg. 270). Mathematics should be about joy and challenges, not pressure and compulsion.
For example, children begin to recognize mathematical patterns by listening to music. They naturally make relationships between different-size and different-shaped blocks by sorting. According to Clements (2001), “Good early mathematics is broader and deeper than mere practice in counting and adding. It includes debating which child is bigger and drawing maps to ‘treasure’ buried outside. Quality mathematics instruction includes providing loads of unit blocks, along with loads of time to use them; asking children to get just enough pencils for everyone in the group; and challenging children to estimate and check how many steps are required to walk to the playground.”
Classrooms that offer children opportunities to engage in play-based learning activities help children grow not only academically. “Play is a particular attitude or approach to materials, behaviors, and ideas and not the materials or activities or ideas themselves; play is a special mode of thinking and doing” (McLane, 2003, p. 11). As discussed in an earlier chapter, there is a strong connection between early mathematics learning and later success. One determining factor of later achievement is quality early mathematical experiences.
Play is critical to a young child’s learning. Mathematics activities must be developmentally appropriate by “integrating logical thinking, decision making comparison, and the making of relationships throughout the child’s day into everything they do” (Geist, 2009). This is the goal of emergent mathematics – to immerse children in mathematics, number, and number concepts. Mathematics at this level must be taught through experiences. Engaging young children in play-based learning, teachers can include the following:
- Rhythm and music allows children to recognize mathematical patterns,
- Blocks and shapes gives children opportunities to make relationships between them, such as same and different,
- Everyday activities such as counting help children develop number concepts,
- Manipulatives such as stringing beads, multicolored balls, blocks, etc. encourage children in imaginative play,
- Everyday routines and common activities such as snack time or distributing plates and napkins, and
- Math games encourage children to make mathematical relationships.
Adapted from Geist, 2009
Read the article, “Mathematics in the Preschool” by Douglas H. Clements for more information about quality early mathematical experiences.
Watch the video, “ The Decline of Play ,” by Peter Gray.
Play is how children learn! Mathematics games, fantasy play, puzzles, manipulatives, and games allow children to learn and grow.


14.3: Burnside's Counting Theorem
Suppose that we wish to color the vertices of a square with two different colors, say black and white. We might suspect that there would be \(2^4=16\) different colorings. However, some of these colorings are equivalent. If we color the first vertex black and the remaining vertices white, it is the same as coloring the second vertex black and the remaining ones white since we could obtain the second coloring simply by rotating the square \(90^\circ\) (Figure \(14.17\)).
\(Figure \text { } 14.17.\) Equivalent colorings of square.
Burnside's Counting Theorem offers a method of computing the number of distinguishable ways in which something can be done. In addition to its geometric applications, the theorem has interesting applications to areas in switching theory and chemistry. The proof of Burnside's Counting Theorem depends on the following lemma.
Lemma \(14.18\)
Let \(X\) be a \(G\) -set and suppose that \(x \sim y\text{.}\) Then \(G_x\) is isomorphic to \(G_y\text{.}\) In particular, \(|G_x| = |G_y|\text{.}\)
- Proof
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Let \(G\) act on \(X\) by \((g,x) \mapsto g \cdot x\text{.}\) Since \(x \sim y\text{,}\) there exists a \(g \in G\) such that \(g \cdot x=y\text{.}\) Let \(a \in G_x\text{.}\) Since
\[ gag^{-1} \cdot y = ga \cdot g^{-1}y = ga \cdot x = g \cdot x = y\text{,} \nonumber \]we can define a map \(\phi: G_x \rightarrow G_y\) by \(\phi(a) = gag^{-1}\text{.}\) The map \(\phi\) is a homomorphism since
\[ \phi(ab) = gabg^{-1} = gag^{-1} gbg^{-1} = \phi(a) \phi(b)\text{.} \nonumber \]Suppose that \(\phi(a) = \phi(b)\text{.}\) Then \(gag^{-1}= gbg^{-1}\) or \(a=b\text{;}\) hence, the map is injective. To show that \(\phi\) is onto, let \(b\) be in \(G_y\text{;}\) then \(g^{-1}bg\) is in \(G_x\) since
\[ g^{-1}bg \cdot x = g^{-1}b \cdot gx = g^{-1}b \cdot y = g^{-1} \cdot y = x; \nonumber \]and \(\phi(g^{-1}bg ) = b\text{.}\)
Theorem \(14.19\). Burnside
Let \(G\) be a finite group acting on a set \(X\) and let \(k\) denote the number of orbits of \(X\text{.}\) Then
- Proof
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We look at all the fixed points \(x\) of all the elements in \(g \in G\text{;}\) that is, we look at all \(g\)'s and all \(x\)'s such that \(gx =x\text{.}\) If viewed in terms of fixed point sets, the number of all \(g\)'s fixing \(x\)'s is
\[ \sum_{g \in G} |X_g|\text{.} \nonumber \]However, if viewed in terms of the stabilizer subgroups, this number is
\[ \sum_{x \in X} |G_x|; \nonumber \]hence, \(\sum_{g \in G} |X_g| = \sum_{x \in X} |G_x|\text{.}\) By Lemma \(14.18\),
\[ \sum_{y \in {\mathcal O}_x} |G_y| = | {\mathcal O}_x| \cdot |G_x|\text{.} \nonumber \]By Theorem \(14.11\) and Lagrange's Theorem, this expression is equal to \(|G|\text{.}\) Summing over all of the \(k\) distinct orbits, we conclude that
\[ \sum_{g \in G} |X_g| = \sum_{x \in X} |G_x| = k \cdot |G|\text{.} \nonumber \]
Example \(14.20\)
Let \(X = \{1, 2, 3, 4, 5 \}\) and suppose that \(G\) is the permutation group \(G= \{(1), (1 \, 3), (1 \, 3)(2 \, 5), (2 \, 5) \}\text{.}\)
Solution
The orbits of \(X\) are \(\{1, 3\}\text{,}\) \(\{2, 5\}\text{,}\) and \(\{4\}\text{.}\) The fixed point sets are
Burnside's Theorem says that
A Geometric Example
Before we apply Burnside's Theorem to switching-theory problems, let us examine the number of ways in which the vertices of a square can be colored black or white. Notice that we can sometimes obtain equivalent colorings by simply applying a rigid motion to the square. For instance, as we have pointed out, if we color one of the vertices black and the remaining three white, it does not matter which vertex was colored black since a rotation will give an equivalent coloring.
The symmetry group of a square, \(D_4\text{,}\) is given by the following permutations:
The group \(G\) acts on the set of vertices \(\{ 1, 2, 3, 4\}\) in the usual manner. We can describe the different colorings by mappings from \(X\) into \(Y = \{ B, W \}\) where \(B\) and \(W\) represent the colors black and white, respectively. Each map \(f : X \rightarrow Y\) describes a way to color the corners of the square. Every \(\sigma \in D_4\) induces a permutation \(\widetilde{ \sigma }\) of the possible colorings given by \(\widetilde{\sigma}(f) = f \circ \sigma\) for \(f : X \rightarrow Y\text{.}\) For example, suppose that \(f\) is defined by
and \(\sigma = (1 2)(3 4)\text{.}\) Then \(\widetilde{\sigma}(f) = f \circ \sigma\) sends vertex \(2\) to \(B\) and the remaining vertices to \(W\text{.}\) The set of all such \(\widetilde{\sigma}\) is a permutation group \(\widetilde{G}\) on the set of possible colorings. Let \(\widetilde{X}\) denote the set of all possible colorings; that is, \(\widetilde{X}\) is the set of all possible maps from \(X\) to \(Y\text{.}\) Now we must compute the number of \(\widetilde{G}\)-equivalence classes.
- \(\widetilde{X}_{(1)} = \widetilde{X}\) since the identity fixes every possible coloring. \(|\widetilde{X}| = 2^4 =~16\text{.}\)
- \(\widetilde{X}_{(1 \, 2 \, 3 \, 4)}\) consists of all \(f \in \widetilde{X}\) such that \(f\) is unchanged by the permutation \((1 \, 2 \, 3 \, 4)\text{.}\) In this case \(f(1) = f(2) = f(3) = f(4)\text{,}\) so that all values of \(f\) must be the same; that is, either \(f(x)= B\) or \(f(x)= W\) for every vertex \(x\) of the square. So \(|\widetilde{X}_{(1 \, 2 \, 3 \, 4)}| = 2\text{.}\)
- \(|\widetilde{X}_{(1 \, 4 \, 3 \, 2)}| = 2\text{.}\)
- For \(\widetilde{X}_{(1 \, 3)(2 \, 4)}\text{,}\) \(f(1) = f(3)\) and \(f(2) = f(4)\text{.}\) Thus, \(|\widetilde{X}_{(1 \, 3)(2 \, 4)}| = 2^2 = 4\text{.}\)
- \(|\widetilde{X}_{(1 \, 2)(3 \, 4)}| = 4\text{.}\)
- \(|\widetilde{X}_{(1 \, 4)(2 \, 3)}| = 4\text{.}\)
- For \(\widetilde{X}_{(1 \, 3 )}\text{,}\) \(f(1) = f(3)\) and the other corners can be of any color; hence, \(|\widetilde{X}_{(1 \, 3)}| = 2^3 = 8\text{.}\)
- \(|\widetilde{X}_{(2 \, 4)}| = 8\text{.}\)
By Burnside's Theorem, we can conclude that there are exactly
ways to color the vertices of the square.
Proposition \(14.21\)
Let \(G\) be a permutation group of \(X\) and \(\widetilde{X}\) the set of functions from \(X\) to \(Y\text{.}\) Then there exists a permutation group \(\widetilde{G}\) acting on \(\widetilde{X}\text{,}\) where \(\widetilde{\sigma} \in \widetilde{G}\) is defined by \(\widetilde{\sigma}(f) = f \circ \sigma\) for \(\sigma \in G\) and \(f \in \widetilde{X}\text{.}\) Furthermore, if \(n\) is the number of cycles in the cycle decomposition of \(\sigma\text{,}\) then \(|\widetilde{X}_{\sigma}| = |Y|^n\text{.}\)
- Proof
-
Let \(\sigma \in G\) and \(f \in \widetilde{X}\text{.}\) Clearly, \(f \circ \sigma\) is also in \(\widetilde{X}\text{.}\) Suppose that \(g\) is another function from \(X\) to \(Y\) such that \(\widetilde{\sigma}(f) = \widetilde{\sigma}(g)\text{.}\) Then for each \(x \in X\text{,}\)
\[ f( \sigma(x )) = \widetilde{\sigma}(f)(x) = \widetilde{\sigma}(g)(x) = g( \sigma(x ))\text{.} \nonumber \]Since \(\sigma\) is a permutation of \(X\text{,}\) every element \(x'\) in \(X\) is the image of some \(x\) in \(X\) under \(\sigma\text{;}\) hence, \(f\) and \(g\) agree on all elements of \(X\text{.}\) Therefore, \(f=g\) and \(\widetilde{\sigma}\) is injective. The map \(\sigma \mapsto \widetilde{\sigma}\) is onto, since the two sets are the same size.
Suppose that \(\sigma\) is a permutation of \(X\) with cycle decomposition \(\sigma = \sigma_1 \sigma_2 \cdots \sigma_n\text{.}\) Any \(f\) in \({\widetilde{X}}_{\sigma}\) must have the same value on each cycle of \(\sigma\text{.}\) Since there are \(n\) cycles and \(|Y|\) possible values for each cycle, \(|{\widetilde{X}}_{\sigma}| = |Y|^n\text{.}\)
Example \(14.22\)
Let \(X = \{1, 2, \ldots, 7\}\) and suppose that \(Y = \{ A, B, C \}\text{.}\) If \(g\) is the permutation of \(X\) given by \((1 \, 3)(2 \, 4 \, 5) = (1 \, 3)(2 \, 4 \, 5)(6)(7)\text{,}\)
Solution
then \(n = 4\text{.}\) Any \(f \in \widetilde{X}_g\) must have the same value on each cycle in \(g\text{.}\) There are \(|Y|=3\) such choices for any value, so \(|\widetilde{X}_g| = 3^4 = 81\text{.}\)
Example \(14.23\)
Suppose that we wish to color the vertices of a square using four different colors. By Proposition \(14.21\), we can immediately decide that there are
Solution
possible ways.
Switching Functions
In switching theory we are concerned with the design of electronic circuits with binary inputs and outputs. The simplest of these circuits is a switching function that has \(n\) inputs and a single output (Figure \(14.24\)). Large electronic circuits can often be constructed by combining smaller modules of this kind. The inherent problem here is that even for a simple circuit a large number of different switching functions can be constructed. With only four inputs and a single output, we can construct 65,536 different switching functions. However, we can often replace one switching function with another merely by permuting the input leads to the circuit (Figure \(14.25\)).
\(Figure \text { } 14.24.\) A switching function of \(n\) variables
We define a switching or Boolean function of \(n\) variables to be a function from \({\mathbb Z}_2^n\) to \({\mathbb Z}_2\text{.}\) Since any switching function can have two possible values for each binary \(n\)-tuple and there are \(2^n\) binary \(n\)-tuples, \(2^{2^n}\) switching functions are possible for \(n\) variables. In general, allowing permutations of the inputs greatly reduces the number of different kinds of modules that are needed to build a large circuit.
\(Figure \text { } 14.25. A switching function of two variables
The possible switching functions with two input variables \(a\) and \(b\) are listed in Table \(14.26\). Two switching functions \(f\) and \(g\) are equivalent if \(g\) can be obtained from \(f\) by a permutation of the input variables. For example, \(g(a, b, c) = f(b, c, a)\text{.}\) In this case \(g \sim f\) via the permutation \((a,c,b)\text{.}\) In the case of switching functions of two variables, the permutation \((a,b)\) reduces 16 possible switching functions to 12 equivalent functions since
\( Table \text { } 14.26\). Switching functions in two variables
| Inputs | Outputs | ||||||||
| \(f_0\) | \(f_1\) | \(f_2\) | \(f_3\) | \(f_4\) | \(f_5\) | \(f_6\) | \(f_7\) | ||
| \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) |
| \(0\) | \(1\) | \(0\) | \(0\) | \(0\) | \(0\) | \(1\) | \(1\) | \(1\) | \(1\) |
| \(1\) | \(0\) | \(0\) | \(0\) | \(1\) | \(1\) | \(0\) | \(0\) | \(1\) | \(1\) |
| \(1\) | \(1\) | \(0\) | \(1\) | \(0\) | \(1\) | \(0\) | \(1\) | \(0\) | \(1\) |
| Inputs | Outputs | ||||||||
| \(f_8\) | \(f_9\) | \(f_{10}\) | \(f_{11}\) | \(f_{12}\) | \(f_{13}\) | \(f_{14}\) | \(f_{15}\) | ||
| \(0\) | \(0\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
| \(0\) | \(1\) | \(0\) | \(0\) | \(0\) | \(0\) | \(1\) | \(1\) | \(1\) | \(1\) |
| \(1\) | \(0\) | \(0\) | \(0\) | \(1\) | \(1\) | \(0\) | \(0\) | \(1\) | \(1\) |
| \(1\) | \(1\) | \(0\) | \(1\) | \(0\) | \(1\) | \(0\) | \(1\) | \(0\) | \(1\) |
For three input variables there are \(2^{2^3} = 256\) possible switching functions; in the case of four variables there are \(2^{2^4} =65{,}536\text{.}\) The number of equivalence classes is too large to reasonably calculate directly. It is necessary to employ Burnside's Theorem.
Consider a switching function with three possible inputs, \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\) As we have mentioned, two switching functions \(f\) and \(g\) are equivalent if a permutation of the input variables of \(f\) gives \(g\text{.}\) It is important to notice that a permutation of the switching functions is not simply a permutation of the input values \(\{a, b, c\}\text{.}\) A switching function is a set of output values for the inputs \(a\text{,}\) \(b\text{,}\) and \(c\text{,}\) so when we consider equivalent switching functions, we are permuting \(2^3\) possible outputs, not just three input values. For example, each binary triple \((a, b, c)\) has a specific output associated with it. The permutation \((acb)\) changes outputs as follows:
Let \(X\) be the set of output values for a switching function in \(n\) variables. Then \(|X|=2^n\text{.}\) We can enumerate these values as follows:
Now let us consider a circuit with four input variables and a single output. Suppose that we can permute the leads of any circuit according to the following permutation group:
The permutations of the four possible input variables induce the permutations of the output values in Table \(14.27\).
Hence, there are
possible switching functions of four variables under this group of permutations. This number will be even smaller if we consider the full symmetric group on four letters.
\(Table \text { } 14.27\). Permutations of switching functions in four variables
| Group | Number | |
| Permutation | Switching Function Permutation | of Cycles |
| \((a)\) | \((0)\) | 16 |
| \((a, c)\) | \((2,8)(3,9)(6,12)(7,13)\) | 12 |
| \((b, d)\) | \((1,4)(3,6)(9,12)(11,14)\) | 12 |
| \((a, d, c, b)\) | \((1,2,4,8)(3,6.12,9)(5,10)(7,14,13,11)\) | 6 |
| \((a, b, c, d)\) | \((1,8,4,2)(3,9,12,6)(5,10)(7,11,13,14)\) | 6 |
| \((a, b)(c, d)\) | \((1,2)(4,8)(5,10)(6,9)(7,11)(13,14)\) | 10 |
| \((a, d)(b, c)\) | \((1,8)(2,4)(3,12)(5,10)(7,14)(11,13)\) | 10 |
| \((a, c)(b, d)\) | \((1,4)(2,8)(3,12)(6,9)(7,13)(11,14)\) | 10 |
Historical Note
William Burnside was born in London in 1852. He attended Cambridge University from 1871 to 1875 and won the Smith's Prize in his last year. After his graduation he lectured at Cambridge. He was made a member of the Royal Society in 1893. Burnside wrote approximately 150 papers on topics in applied mathematics, differential geometry, and probability, but his most famous contributions were in group theory. Several of Burnside's conjectures have stimulated research to this day. One such conjecture was that every group of odd order is solvable; that is, for a group \(G\) of odd order, there exists a sequence of subgroups
such that \(H_i\) is normal in \(H_{i+1}\) and \(H_{i+1} / H_i\) is abelian. This conjecture was finally proven by W. Feit and J. Thompson in 1963. Burnside's The Theory of Groups of Finite Order , published in 1897, was one of the first books to treat groups in a modern context as opposed to permutation groups. The second edition, published in 1911, is still a classic.


10.2: The Correlation Coefficient r
As we begin this section we note that the type of data we will be working with has changed. Perhaps unnoticed, all the data we have been using is for a single variable. It may be from two samples, but it is still a single variable ( univariate ) . The type of data described in the examples above and for any model of cause and effect is bivariate data ("bi" for two variables). In reality, statisticians use multivariate data, meaning they use many variables in their analyses.
For our work we can classify data into three broad categories, time series data, cross-section data, and panel data. We met the first two very early on. Time series data measures a single unit of observation; say a person, or a company or a country, as time passes. What are measured will be at least two characteristics, say the person’s income, the quantity of a particular good they buy and the price they paid. This would be three pieces of information in one time period, say 1985. If we followed that person across time we would have those same pieces of information for 1985, 1986, 1987, etc. This would constitute a times series data set. If we did this for 10 years we would have 30 pieces of information concerning this person’s consumption habits of this good for the past decade and we would know their income and the price they paid.
A second type of data set is for cross-section data. Here the variation is not across time for a single unit of observation, but across units of observation during one point in time. For a particular period of time we would gather the price paid, amount purchased, and income of many individual people.
A third type of data set is panel data. Here a panel of units of observation is followed across time. If we take our example from above we might follow 500 people, the unit of observation, through time, ten years, and observe their income, price paid and quantity of the good purchased. If we had 500 people and data for ten years for price, income and quantity purchased we would have 15,000 pieces of information. These types of data sets are very expensive to construct and maintain. They do, however, provide a tremendous amount of information that can be used to answer very important questions. As an example, what is the effect on the labor force participation rate of women as their family of origin, mother and father, age? Or are there differential effects on health outcomes depending upon the age at which a person started smoking? Only panel data can give answers to these and related questions because we must follow multiple people across time. The work we do here however will not be fully appropriate for data sets such as these.
Beginning with a set of data with two independent variables we ask the question: are these related? One way to visually answer this question is to create a scatter plot of the data. We could not do that before when we were doing descriptive statistics because those data were univariate. Now we have bivariate data so we can plot in two dimensions.
To provide mathematical precision to the measurement of what we see we use the correlation coefficient. The correlation tells us something about the co-movement of two variables, but nothing about why this movement occurred. Formally, correlation analysis assumes that both variables being analyzed are independent variables. This means that neither one causes the movement in the other. Further, it means that neither variable is dependent on the other, or for that matter, on any other variable. Even with these limitations, correlation analysis can yield some interesting results.
The correlation coefficient, \(\rho\) (pronounced rho), is the mathematical statistic for a population that provides us with a measurement of the strength of a linear relationship between the two variables. For a sample of data, the statistic, \(r\), developed by Karl Pearson in the early 1900s, is an estimate of the population correlation and is defined mathematically as:
\[r_{X Y}=\frac{\sum (X_{i}-\overline{X})*(Y_{i}-\overline{Y})}{\sqrt{\sum (X_{i}-\overline{X})^{2}*\sum (Y_{i}-\overline{Y})^{2}}}\nonumber\]
\[\text{or}\nonumber\]
\[r_{X Y}=\frac{\sum X_{i} Y_{i}-\frac{\left(\sum X_{i}\right)\left(\sum Y_{i}\right)}{n}}{\sqrt{\left[\sum X_{i}^{2}-\frac{\left(\sum X_{i}\right)^{2}}{n}\right]*\left[\sum Y_{i}^{2}-\frac{\left(\sum Y_{i}\right)^{2}}{n}\right]}}\nonumber\]
where \(\overline{X}\) and \(\overline{Y}\) are the sample means of the two independent variables \(X\) and \(Y\), \(X_{i}\) and \(Y_{i}\) are the individual observations of \(X\) and \(Y\), and \(n\) is our sample size. The correlation coefficient \(r\) ranges in value from -1 to 1. The second equivalent formula is often used because it may be computationally easier. As scary as these formulas look, they are really just the ratio of the covariance between the two variables and the product of their two standard deviations. That is to say, it is a measure of relative variances.
To visualize any linear relationship that may exist review the plot of a scatter diagrams of the standardized data. Figure \(\PageIndex{1}\) presents several scatter diagrams and the calculated value of \(r\). In panels (a) and (b) notice that the data generally trend together, (a) upward and (b) downward. Panel (a) is an example of a positive correlation and panel (b) is an example of a negative correlation, or relationship. The sign of the correlation coefficient tells us if the relationship is a positive or negative (inverse) one. If all the values of two variables \(X_1\) and \(X_2\) are on a straight line the correlation coefficient will be either \(1\) or \(-1\) depending on whether the line has a positive or negative slope and the closer to one or negative one the stronger the relationship between the two variables.
Remember, all the correlation coefficient tells us is whether or not the data are linearly related. In panel (d) the variables obviously have some type of very specific relationship to each other, but the correlation coefficient is zero, indicating no linear relationship exists. In panel (c) the variables are not related and the correlation coefficient is understandably equal to zero.
If you suspect a linear relationship between two variables \(X\) and \(Y\), then \(r\) can measure how strong the linear relationship is.
What the VALUE of \(r\) tells us:
- The value of \(r\) is always between \(–1\) and \(+1\): \(–1 \leq r \leq 1\).
- The size of the correlation \(r\) indicates the strength of the linear relationship between variables \(X\) and \(Y\). Values of \(r\) close to \(–1\) or to \(+1\) indicate a stronger linear relationship between the two variables.
- If \(r = 0\) there is absolutely no linear relationship between variables \(X\) and \(Y\) (no linear correlation) .
- If \(r = 1\), there is perfect positive correlation. If \(r = –1\), there is perfect negative correlation. In both these cases, all of the original data points lie on a straight line. Of course, in the real world, this will not generally happen.
What the SIGN of \(r\) tells us
- A positive value of \(r\) means that when \(X\) increases, \(Y\) tends to increase and when \(X\) decreases, \(Y\) tends to decrease (positive correlation) .
- A negative value of \(r\) means that when \(X\) increases, \(Y\) tends to decrease and when \(X\) decreases, \(Y\) tends to increase (negative correlation) .
Note
Strong correlation does not suggest that \(X\) causes \(Y\) or that \(Y\) causes \(X\). We say "correlation does not imply causation."


2.10: Entropy and Equilibrium in an Isolated System
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In an isolated system, the probability of population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\) is \(W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }\), where \({\rho }_{MS,N,\left\langle E\right\rangle }\) is a constant. It follows that \(W=W\left(N_i,g_i\right)\) is proportional to the probability that the system is in one of the microstates associated with the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). Likewise, \(W^{\#}=W\left(N^{\#}_i,g_i\right)\) is proportional to the probability that the system is in one of the microstates associated with the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\). Suppose that we observe the isolated system for a long time. Let \(F\) be the fraction of the time that the system is in microstates of population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\) and \(F^{\#}\) be the fraction of the time that the system is in microstates of the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\). The principle of equal a priori probabilities implies that we would find
\[\frac{F^{\#}}{F}=\frac{W^{\#}}{W} \nonumber \]
Suppose that \(W^{\#}\) is much larger than \(W\). This means there are many more microstates for \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\) than there are for \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). The fraction of the time that the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\) characterizes the system is much greater than the fraction of the time \(\{N_1,\ N_2,\dots ,N_i,\dots \}\) characterizes it. Alternatively, if we examine the system at an arbitrary instant, we are much more likely to find the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\) than the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). The larger \(W\left(N_1,g_1,\ N_2,g_2,\dots ,N_i,g_i,\dots \right)\), the more likely it is that the system will be in one of the microstates associated with the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). In short, \(W\) predicts the state of the system; it is a measure of the probability that the macroscopic properties of the system are those of the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\).
If an isolated system can undergo change, and we re-examine it at after a few molecules have moved to different energy levels, we expect to find it in one of the microstates of a more-probable population set; that is, in one of the microstates of a population set for which \(W\) is larger. At still later times, we expect to see a more-or-less smooth progression: the system is in microstates of population sets for which the values of \(W\) are increasingly larger. This can continue only until the system occupies one of the microstates of the population set for which \(W\) is a maximum or a microstate of one of the population sets whose macroscopic properties are essentially the same as those of the constant-\(N\)-\(V\)-\(E\) population set for which \(W\) is a maximum.
Once this occurs, later inspection may find the system in other microstates, but it is overwhelmingly probable that the new microstate will still be one of those belonging to the largest-\(W\) population set or one of those that are macroscopically indistinguishable from it. Any of these microstates will belong to a population set for which \(W\) is very well approximated by \(W\left(\ N^{\textrm{⦁}}_1,g_1,\ N^{\textrm{⦁}}_2,g_2,\dots ,N^{\textrm{⦁}}_i,g_i,\dots \right)\). Evidently, the largest-\(W\) population set characterizes the equilibrium state of the either the constant-\(N\)-\(V\)-\(T\) system or the constant–\(N\)-\(V\)-\(E\) system. Either system can undergo change until \(W\) reaches a maximum. Thereafter, it is at equilibrium and can undergo no further macroscopically observable change.
Boltzmann recognized this relationship between \(W\), the thermodynamic probability, and equilibrium. He noted that the unidirectional behavior of \(W\) in an isolated system undergoing spontaneous change is like the behavior we found for the entropy function. Boltzmann proposed that, for an isolated (constant energy) system, \(S\) and \(W\) are related by the equation \(S=k{ \ln W\ }\), where \(k\) is Boltzmann’s constant. This relationship associates an entropy value with every population set. For an isolated macroscopic system, equilibrium corresponds to a state of maximum entropy. In our microscopic model, equilibrium corresponds to the population set for which \(W\) is a maximum. By the argument we make in § 6, this population set must be well approximated by the most probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,.,,,\}\). That is, the entropy of the equilibrium state of the macroscopic system is
\[ \begin{align*} S &= k ~ { \ln W_{max}\ } \\[4pt] &=k ~ { \ln \frac{N!}{N^{\textrm{⦁}}_i!N^{\textrm{⦁}}_i!\dots N^{\textrm{⦁}}_i!\dots }\ }+k ~ \sum^{\infty }_{i=1}{N^{\textrm{⦁}}_i{ \ln g_i\ }} \end{align*} \nonumber \]
This equation can be taken as the definition of entropy. Clearly, this definition is different from the thermochemical definition, \(S={q^{rev}}/{T}\). We can characterize—imperfectly—the situation by saying that the two definitions provide alternative scales for measuring the same physical property. As we see below, our statistical theory enables us to define entropy in still more ways, all of which prove to be functionally equivalent. Gibbs characterized these alternatives as “entropy analogues;” that is, functions whose properties parallel those of the thermochemically defined entropy.
We infer that the most probable population set characterizes the equilibrium state of either the constant-temperature or the constant-energy system. Since our procedure for isolating the constant-temperature system affects only the thermal interaction between the system and its surroundings, the entropy of the constant-temperature system must be the same as that of the constant-energy system. Using \(N^{\textrm{⦁}}_i=NP_i=Ng_i\rho \left({\epsilon }_i\right)\) and assuming that the approximation \({ \ln N^{\textrm{⦁}}_i!\ }=N^{\textrm{⦁}}_i{ \ln N^{\textrm{⦁}}_i\ }-N^{\textrm{⦁}}_i\) is adequate for all of the energy levels that make a significant contribution to \(S\), substitution shows that the entropy of either system depends only on probabilities:
\[ \begin{align*} S &= kN ~ { \ln N - kN - k\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{\left[NP_i{ \ln \left(NP_i\right)\ } - NP_i\right]}\ } + k\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{NP_i{ \ln g_i\ }} \\[4pt]&= kN ~ { \ln N\ }\mathrm{-kN} -kN\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{\left[P_i{ \ln \left(N\right)\ } + P_i{ \ln P_i\ } - P_i - P_i{ \ln g_i\ }\right]} \\[4pt] &= k ~ \left(N{ \ln N\ } - N\right) - k\left(N{ \ln N\ } - N\right)\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{P_i} - kN\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{P_i}\left[{ \ln P_i - { \ln g_i\ }\ }\right]\mathrm{=-}kN\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{P_i{ \ln \rho \left({\epsilon }_i\right)\ }} \end{align*} \nonumber \]
The entropy per molecule, \({S}/{N}\), is proportional to the expected value of \({ \ln \rho \left({\epsilon }_i\right)\ }\); Boltzmann’s constant is the proportionality constant. At constant temperature, \(\rho \left({\epsilon }_i\right)\) depends only on \({\epsilon }_i\). The entropy per molecule depends only on the quantum state properties, \(g_i\) and \({\epsilon }_i\).


37.1: Rubric
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- Darcy Ernst, May Chen, Katie Foltz, and Bridget Greuel
- Evergreeen Valley College via Open Educational Resource Initiative at Evergreen Valley College
The following rubric will be used to grade the lab reports:
|
Excellent
5 points |
Satisfactory
2.5 points |
Unsatisfactory
0 points |
|
|---|---|---|---|
| Title Page | Contains title, student name, instructor name and section | Missing either instructor name or section | No title page |
| Formatting: typed, spacing | Typed and single spaced | Typed, but not single spaced | Not typed |
| Grammar and Spelling | No errors; contains complete sentences and no misspellings | A few minor errors in grammar and spelling | Several major errors in grammar and spelling |
| Formatting: headings | Each section has a heading as described in template | Some sections lack headings | No headings |
| Hypothesis | Predictions are clearly stated and written as a testable statement | Predictions/expected outcomes are not clearly stated | Not written as a testable statement |
| Materials | All equipment and materials described; identify variables, controls and constants | Materials incompletely described | No identification of variables, controls and constants |
| Procedure | Clear step-by step description | Description missing details making it difficult for another scientist to repeat experiment | Description missing so much detail it would be impossible to repeat |
| Results | Clearly written description of results comparing controls and variables | Results are presented but no comparison between controls and variables are made | No written description of results |
| Data tables, graphs or charts | Easy to interpret, clear labels, all data, including calculated averages, included | Disorganized (not easy to understand, missing labels) but all data included | Disorganized and or data clearly missing |
| Conclusion | Clearly explains acceptance or rejection of hypothesis using data to support conclusion; identifies sources of error | Accepts or rejects hypothesis but does not use data to explain why; or does not identify sources of error | Does not explain conclusion and does not identify sources of error |
| Total ______out of 50 points |
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, ORIGINAL
- Lab Report Grading Rubric. Authored by : Lynette Hauser. Provided by : Tidewater Community College. Located at : http://www.tcc.edu/ . License : CC BY: Attribution


12.5: The Kidneys and Urinary System
The kidneys in mammals are bean-shaped organs that lie in the abdominal cavity attached to the dorsal wall on either side of the spine (see diagram 12.2). An artery from the dorsal aorta called the renal artery supplies blood to them and the renal vein drains them.
Diagram 12.3 - The urinary system
To the naked eye kidneys seem simple enough organs. They are covered by a fibrous coat or capsule and if cut in half lengthways (longitudinally) two distinct regions can be seen - an inner region or medulla and the outer cortex . A cavity within the kidney called the pelvis collects the urine and carries it to the ureter , which connects with the bladder where the urine is stored temporarily. Rings of muscle ( sphincters ) control the release of urine from the bladder and the urine leaves the body through the urethra (see diagrams 12.3 and 12.4).
Diagram 12.4 - The dissected kidney


14.2: Consent
We’ve touched on consent throughout this book and the ways it gets complicated by power and privilege. Being communicative about what is ok and not ok throughout an intimate encounter is necessary for consent to authentically occur. Often, this is not adequately addressed, and then the encounter becomes coercive in some form. This chapter will be discussing some ways in which this coercion plays out. It is important to understand and distinguish between consent and non-consent, in any relationship between individuals who wish to participate in any sexual act. An introduction to understanding consent can be seen in this video previously mentioned in Chapter 4, Tea Consent . Consent can be defined as “an agreement between participants to engage in sexual activity… [and] should be clearly and freely communicated” (What Consent Looks Like, n.d.).
Concent in Relationships
In relationships, it is mandatory to ask for consent, regardless of the duration or status of the relationship.  People have the right to change their minds at any time, even during a sexual act, and even if they agreed to something once before, does not mean they will feel the same in the future; consent to one sexual activity does not guarantee consent to every sexual activity [thereafter] (Secretary of State for the Home Department, n.d.).
In healthy relationships, partners are able to communicate comfortably, continuously, and honestly, and inform one another of their desired levels of physical activity, from hand holding to touching, and/or having intercourse (What is Sexual Coercion, n.d). When getting physical with a crush, romantic partner, significant other, or a hookup, you absolutely have a voice and control over how far you wish to take things, if at all (What is Sexual Coercion, n.d).
The women in my life who are mothers to young girls often talk about what a scary time it is to have daughters. As a woman, I understand why. But as a mother to a young boy, I feel equally as fearful of the potential future he faces being raised in a society that normalizes and dismisses sexual violence against women. The first time I became aware that I would need to start teaching my son about consent occurred much earlier than I anticipated. When he was about a year and a half old, I was still breast feeding him, and he had gotten into the habit of walking up to me and pulling my shirt down when he wanted to nurse. He was a full blown toddler with a hearty appetite for solid food, meaning breastmilk was no longer his primary source of nutrition, and I no longer needed to nurse on demand. I decided it was time to create some boundaries around when and where I would allow him to nurse. He did not like being told no, and although it was heartbreaking for me to hold these boundaries at first, the more I did it, the more I felt that it was the perfect first opportunity to lovingly teach him about consent and bodily autonomy. As he grows, I make sure to offer plenty of opportunities for him to assert his own bodily autonomy. He knows that if we’re playing and he says “no” or “stop”, that his words will be honored immediately. He has also never been forced to give kisses or hugs to any family member, including me. I can’t know if what I’m doing is “enough” to counterbalance everything he will inevitably be exposed to out in the world. What I do know is that introducing these concepts from a very early age is becoming more and more mainstream, and this gives me great hope for a generation of children who will have a healthy and clear understanding of consent from the start.
-Juliana Garcia, mom of Wylder
Consent in Law
There are legal contexts to be aware of when giving and/or receiving consent. According to the free public resource site, AgeofConsent.net, the legal age of consent in the U.S. varies from 16-18 years old between states, and any sexual relations with someone under the age of consent is illegal and considered statutory rape . In California, the legal age of consent is 18 years old and there is no “close in age exemption,” a law that decriminalizes consensual sex between those who are underage (AgeofConsent.net, n.d.). Other factors to be taken into consideration with consent include developmental disabilities, intoxication [by alcohol and/or illegal/prescribed drugs], physical disabilities, unconsciousness, vulnerable adults, and the relationship of the victim and perpetrator (Legal Role of Consent, n.d.). Based on all of the above factors, it is understandable that there is confusion with the term consent; therefore, the need for a concrete legal definition is necessary for our judicial system.
There is no single legal definition of consent. RAINN or Rape, Abuse, & Incest National Network, the largest anti-sexual violence organization breaks down the complexities of consent. Each state sets its own definition, either in law or through court cases. In general, there are three main ways that states analyze consent in relation to sexual acts:
- Affirmative consent: Did the person express overt actions or words indicating agreement for sexual acts?
- Freely given consent: Was the consent offered of the person’s own free will, without being induced by fraud, coercion, violence, or threat of violence?
- Capacity to consent: Did the individual have the capacity, or legal ability, to consent?
Source: Legal role of consent . RAINN. (n.d.). Retrieved February 28, 2022, from https://www.rainn.org/articles/legal-role-consent
In the legal field, one prominent issue is that the definition of consent also varies between states or is altogether lacking, which can be detrimental in prosecuting sexual assault cases. Joyce Short, CEO of Consent Awareness Network (CAN), proposes a legal definition of consent as, “freely given, knowledgeable, and informed agreement” [ #FGKIA ] and stresses that consent is more complex than yes or no (McCrystal, 2021).
Planned Parenthood also offers an acronym to help define consent, stating “consent is as easy as FRIES”:
- Freely given. Consenting is a choice you make without pressure, manipulation, or under the influence of drugs or alcohol.
- Reversible. Anyone can change their mind about what they feel like doing, anytime. Even if you’ve done it before, and even if you’re both naked in bed.
- Informed. You can only consent to something if you have the full story. For example, if someone says they’ll use a condom and then they don’t, there isn’t full consent.
- Enthusiastic. When it comes to sex, you should only do stuff you WANT to do, not things that you feel you’re expected to do.
- Specific. Saying yes to one thing (like going to the bedroom to make out) doesn’t mean you’ve said yes to others (like having sex) (Sexual Consent, n.d.).
Communication is key. Yes means yes.
In an effort to combat sexual assault in college and how they are handled, the affirmative consent campaign, or “Yes Means Yes” (SB-967) was proposed to the senate. Our society has a deep history of protecting perpetrators, and silencing and blaming survivors. In 2014, California was the first state to sign the bill SB-967 into law. The law requires all colleges receiving state funding to have policies that “adopt an affirmative consent standard to be used when investigating sexual assault complaints and proceeding through the disciplinary process” (Emba, 2015). Affirmative consent is defined as:
a knowing, voluntary, and mutual decision among all participants to engage in sexual activity. Consent can be given by words or actions, as long as those words or actions create clear permission regarding willingness to engage in the sexual activity. Silence or a lack of resistance does not demonstrate consent. The definition of consent does not vary based upon a participant's sex, sexual orientation, gender identity, or gender expression (SUNY, n.d.).
This further breakdown of consent is a necessary step to protect all survivors of assault, and teach sexually active individuals what consent looks and sounds like, compared to when they are not. While there is progress among state and local governments in adopting policies such as these to end sexual violence, it is imperative that our society as a whole understands the boundaries of consent and when it has been breached, therefore, ending victim-blaming and establishing a new standard for solutions.
 The phrase "Me Too" was initially used in this context on social media in 2006, on Myspace, by sexual assault survivor and activist Tarana Burke.   As with all other aspects of social stratification, those who carry greater privilege have always received better treatment under the eyes of the law. Women of color experience lesser support and belief by others that it took place when reporting sexual assault or harassment than white women. Structural racism and misogyny are often to blame for these differing responses. Because of white privilege and an expectation of the law being there to protect them, studies have found that white women are more likely to report incidents than women of color. These discrepancies disadvantage women of color, and make getting support harder (Berdahl and Moore, 2006). The #metoo movement recognizes that these cultural hierarchies stand in the way of true progress for all of us who have experienced sexual assault. While this movement was started to support those often unheard voices of the marginalized, it is embedded within larger cultural scripts around who is a legitimate victim and who is an abuser, which is inherently problematic. Shifting the focus from specific individuals who were hurt or who hurt someone to an examination of the social structure that maintains rape culture can help move in a direction where sexual violence is less prevalent. In that regard, #METOO has helped to push legislation forward in terms of workplace harassment policies, and has opened the door for more people to share their stories.


4.3: Metric Notation
The metric system, besides being a collection of measurement units for all sorts of physical quantities, is structured around the concept of scientific notation. The primary difference is that the powers-of-ten are represented with alphabetical prefixes instead of by literal powers-of-ten. The following number line shows some of the more common prefixes and their respective powers-of-ten:
Looking at this scale, we can see that 2.5 Gigabytes would mean 2.5 x 10 9 bytes, or 2.5 billion bytes. Likewise, 3.21 picoamps would mean 3.21 x 10 -12 amps, or 3.21 1/trillionths of an amp.
Other metric prefixes exist to symbolize powers of ten for extremely small and extremely large multipliers. On the extremely small end of the spectrum, femto (f) = 10 -15 , atto (a) = 10 -18 , zepto (z) = 10 -21 , and yocto (y) = 10 -24 . On the extremely large end of the spectrum, Peta (P) = 10 15 , Exa (E) = 10 18 , Zetta (Z) = 10 21 , and Yotta (Y) = 10 24 .
Because the major prefixes in the metric system refer to powers of 10 that are multiples of 3 (from “kilo” on up, and from “milli” on down), metric notation differs from regular scientific notation in that the mantissa can be anywhere between 1 and 999, depending on which prefix is chosen. For example, if a laboratory sample weighs 0.000267 grams, scientific notation and metric notation would express it differently:
2.67 x 10
-4
grams (scientific notation)
267 µgrams (metric notation)
The same figure may also be expressed as 0.267 milligrams (0.267 mg), although it is usually more common to see the significant digits represented as a figure greater than 1.
In recent years a new style of metric notation for electric quantities has emerged which seeks to avoid the use of the decimal point. Since decimal points (”.”) are easily misread and/or “lost” due to poor print quality, quantities such as 4.7 k may be mistaken for 47 k. The new notation replaces the decimal point with the metric prefix character, so that “4.7 k” is printed instead as “4k7”. Our last figure from the prior example, “0.267 m”, would be expressed in the new notation as “0m267”.
Review
- The metric system of notation uses alphabetical prefixes to represent certain powers-of-ten instead of the lengthier scientific notation.


9.3A: Colonialism and Neocolonialism
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Colonialism is the establishment, maintenance, acquisition, and expansion of colonies in one territory by people from another territory.
Learning Objectives
- Differentiate between dependency theory, world-systems theory, and the Marxist perspective on colonialism
Key Points
- The colonial period ranges from the 1450s to the 1970s, beginning when several European powers (Spain, Portugal, Britain, and France especially) established colonies in Asia, Africa, and the Americas.
- Decolonization took place after the First and Second World Wars as former colonies established independence from colonial powers.
- Neocolonialism refers to the unequal economic and power relations that currently exist between former colonies and former colonizing nations.
- Marx viewed colonialism as part of the global capitalist system, which has led to exploitation, social change, and uneven development.
- Dependency theory argues that countries have developed at an uneven rate because wealthy countries have exploited poor countries in the past through colonialism and today through foreign debt and trade.
- World-systems theory splits the world economic system into core, peripheral, and semi-peripheral countries.
Key Terms
- decolonization : The freeing of a colony or territory from dependent status by granting it sovereignty.
- Age of Discovery : A period in history starting in the early 15th century and continuing into the early 17th century during which Europeans engaged in intensive exploration of the world, establishing direct contact with Africa, the Americas, Asia, and Oceania and mapping the planet.
- Scramble for Africa : A process of invasion, occupation, colonization and annexation of African territory by European powers during the New Imperialism period, between 1881 and World War I in 1914.
- colonialism : the establishment, exploitation, maintenance, acquisition and expansion of territories (or colonies) in one geographic area by people from another area
Colonialism is the establishment, maintenance, acquisition, and expansion of colonies in one territory, imposed by people from another territory. It is a process whereby the metropole, or parent state, claims sovereignty over the colony, and the social structure, government, and economy of the colony are changed by colonizers from the metropole. Colonialism is a set of unequal relationships between the metropole and the colony, and between the colonists and the indigenous, or native, population.
History of Colonialism
Modern colonialism started with the Age of Discovery, during which Portugal and Spain discovered new lands across the oceans (including the Americas and Atlantic/South Pacific islands) and built trading posts. According to some scholars, building these colonies across oceans differentiates colonialism from other types of expansionism. These new lands were first divided between the Portuguese Empire and Spanish Empire, though the British, French, and Dutch soon acquired vast territory as well.
The 17 th century saw the creation of the French colonial empire, the Dutch Empire, and the English colonial empire, which later became the British Empire. It also saw the establishment of some Swedish overseas colonies and a Danish colonial empire.
The spread of colonial empires diminished in the late 18 th and early 19 th centuries, largely due to the American Revolutionary War and Latin American wars for independence. However, many new colonies were established after this period, including the German colonial empire and Belgian colonial empire. In the late 19 th century, many European powers were involved in the so-called Scramble for Africa, in which many African colonies were established.
Decolonization
After the First World War, the victorious allies divided up the German colonial empire and much of the Ottoman Empire according to League of Nations mandates. These territories were divided into three classes based on how quickly they would be ready for independence. Decolonization outside the Americas lagged until after World War II. In ideal cases, decolonized colonies were granted sovereignty, or the right to self-govern, becoming independent countries.
Neocolonialism
The term “neocolonialism” has been used to refer to a variety of contexts since the decolonization that took place after World War II. Generally, it does not refer to any type of direct colonization, but colonialism by other means. Specifically, neocolonialism refers to the theory that former or existing economic relationships—the General Agreement on Tariffs and Trade (GATT) and the Central American Free Trade Agreement—are used to maintain control of former colonies after formal independence was achieved. In broader usage, neocolonialism may simply refer to the involvement of powerful countries in the affairs of less powerful countries; this is especially relevant in modern Latin America. In this sense, neocolonialism implies a form of economic imperialism.
Colonialism and Neocolonialism in the World System
One approach sociologists take to colonialism and neocolonialism is a Marxist perspective. Marx viewed colonialism as part of the global capitalist system, which has led to exploitation, social change, and uneven development. He argued that it was destructive and produced dependency. According to some Marxist historians, in all of the colonial countries ruled by Western European countries, indigenous people were robbed of health and opportunities. From a Marxist perspective, colonies are considered vis-à-vis modes of production. The search for raw materials and new investment opportunities is the result of inter-capitalist rivalry for capital accumulation.
Dependency theory builds upon Marxist thought, blaming colonialism and neocolonialism for poverty within the world system. This theory argues that countries have developed at an uneven rate because wealthy countries have exploited poor countries in the past and today through foreign debt and foreign trade.
World-Systems Theory
The world-systems theory suggests that the aftermath of colonialism and the continuing practice of neocolonialism produces unequal economic relations within the world system. Sociologist Immanuel Wallerstein elaborated on these forms of economic inequality. In this theory, the world economic system is divided into a hierarchy of three types of countries: core, semiperipheral, and peripheral. Core countries (e.g., U.S., Japan, Germany) are dominant capitalist countries characterized by high levels of industrialization and urbanization. Peripheral countries (e.g., most African countries and low income countries in South America) are dependent on core countries for capital, and have very little industrialization and urbanization. Peripheral countries are usually agrarian and have low literacy rates and lack Internet connection in many areas. Semiperipheral countries (e.g., South Korea, Taiwan, Mexico, Brazil, India, Nigeria, South Africa) are less developed than core nations but are more developed than peripheral nations.


5: Pipelines
Student Learning Outcomes
After reading this chapter, you should be able to:
- Describe the different pipe materials and sizes
- Explain hydraulics and how water moves through pipes
- Identify the causes and results from water hammer and tuberculation
Throughout the history of water distribution, a critical component of getting water from the source to the customer is pipelines. Pipelines within a distribution system can be separated into three (3) main categories:
- Transmission Piping
- Distribution Piping
- Service Piping
These three (3) categories have very distinct characteristics and uses.
Transmission Piping
Transmission water mains (piping) are the largest in diameter of the three. They can range from as small as 16” in diameter up to more than 120” in diameter depending on the size of the distribution system, the location of the source water, and the number of customers within the distribution system service area. Smaller utilities with the sources of supply relatively close to the distribution system would have the smaller diameter pipes while larger utilities with the source water further from the distribution system would have larger diameter transmission mains. Transmission water mains convey large volumes of water from the source to areas within the main distribution system. They are commonly installed without any other connections to the pipe until it reaches the distribution system. However, in smaller distribution systems there can be additional pipes connected to a transmission main and there may even be service connections. Below is an example of a transmission water main.
Distribution Piping
Distribution system piping is simply that, piping that distributes water throughout the distribution system. Diameters of distribution system piping range from as small as 4” up to 24” and sometimes larger. The most common sizes of distribution system water mains are 8” – 12”. Distribution mains connect to transmission mains as these pipes enter a distribution system. These water mains branch off into roads within communities to bring water supplies to customers.
Service Main
In order to get water from the distribution main to a customer, a service main (lateral) is attached to the distribution main. In the diagram above, you can see this pipe between the distribution main and a black dot. The black dot represents the water meter or point of connection between the water utilities infrastructure and the customer’s piping. Service laterals typically range in size from 1” to 10” in diameter. Most single-family residential service laterals are 1” to 2” in diameter, while larger commercial services can range in size up to 10” in diameter and sometimes even larger. The size of a service lateral is dependent on the amount of water use from the customer.
Pipe Selection
There are a number of things, which dictate the type of pipe material a utility would select. As discussed above, the use is something utilities look at to determine the material to use. Other things include resistance to corrosion, interior smoothness, cost, ease of use, strength, and local conditions (i.e., soil type).
Pipeline Materials
There are various materials used for water mains. In the 1800s up to the early 1900s, wooden water mains were used. Wood was a usable material because hollowed-out wooden logs did not expand like metal pipes and the thickness provided insulation properties. However, as distribution systems became more sophisticated and pressures increased, wooden pipes were replaced with grey cast-iron pipes.
Grey cast-iron pipe (CIP) was easier to manufacture. They were strong and provided a long service life. One of the main drawbacks of this pipe was its brittleness. If the pipe wasn’t handled carefully, it would easily crack. Another major drawback was the lead poured joints. In order to properly connect sections of the pipe together a molten lead joint was poured around each connection point.
Ductile Iron Pipe
In the early 1970s, manufacturing plants moved away from CIP and started manufacturing ductile-iron pipe (DIP). DIP is even stronger than CIP, more versatile, not brittle, and does not use any lead. The only drawback to DIP is the susceptibility to corrosion from aggressive waters and soil. In order to protect the inside of DIP from corroding, they are commonly lined with cement mortar. If soils are corrosive, DIP is usually wrapped in plastic bags. The images below show examples of cement mortar lined (CML) DIP and DIP that has been “bagged” before installation. DIP typically comes in diameters of 4” – 64” and usually in lengths of 18’ – 20’. The pressure range for DIP is 150 psi to 350 psi.
Steel Pipe
Steel pipe has been used for well over a century. In the mid-1800s when water pressures were high, steel pipe was commonly manufactured for use. While steel pipe can be manufactured to small diameters (4”), the more common use of steel pipe is when large diameters are needed. Diameters of steel pipe ranges up to 120” and even larger if needed. Steel pipe is lighter in weight than CIP and DIP and can be fabricated for special needs. Some of the main disadvantages to steel pipe are both internal and external corrosion and the potential for a partial vacuum to collapse the pipe.
Asbestos-Cement Pipe
Asbestos-Cement Pipe (ACP) was first introduced in the US around 1930. It was commonly used in areas where metallic pipe was subject to corrosion. Pressure classes are not typically as high as pipe made from metal, but they come in pressure ranges up to 200 psi. ACP only comes in diameters ranging from 4” to 42” and the lengths (10’ – 13’) are shorter than their DIP counterpart. ACP is not subjected to aggressive waters as metal pipe, it is fairly lightweight and has a relatively smooth interior surface. It is fairly brittle and can crack if not properly handled. Special safety precautions must be taken in order to prevent employees from being exposed to asbestos fibers.
Polyvinyl Chloride Pipe
Polyvinyl chloride pipe (PVC) were first introduced around 1940. The durability, resistance to corrosion, lightweight, and cost-effectiveness soon became very attractive in the waterworks industry. Because these pipes are made with vinyl chloride, they must be tested and meet certain criteria to ensure no harmful chemicals leach out. In some instances, PVC pipe can also cause taste and odor problems. Another desirable characteristic of PVC piping is flexibility. When using PVC piping, special care must be taken when storing for long periods of time since plastic is susceptible and can become damaged from ultra-violet light.
The above-mentioned pipes are primarily used for transmission and distribution piping. The material used for service laterals are commonly made of either copper or PVC. However, in years past, galvanized pipes were also used. Copper is malleable and resistant to corrosion, which makes it a perfect material for piping between a distribution main and the point of connection to a customer. PVC is also used with the preferred type of PVC being polybutylene since it is not as ridged as schedule 40 or schedule 80.
Pipeline Connections
Water mains are generally connected together with either mechanical or push-on joints and sometimes flanged connections. The type of connection is determined by the installation. For example, above-ground pipeline installations, such as in pump stations, flanged connections are very common. A flanged connection provides a strong and ridged connection and allows for easy disassembly if a section of pipe or other appurtenance needs to be removed. Flanged connections are not as common in underground installations, primarily because of the exposed nuts and bolts of flanged fittings. Pipes connected with push-on joints are manufactured in a “bell” and “spigot” style. The bell end is a wider flared end and the spigot is narrower and tapered. See the example below.
The bell end has a gasket inside and the spigot end is “pushed” on to make the connection. Both ends of the pipe must be thoroughly cleaned and lubricated before making the connection. One of the main problems with push-on joints is the possibility of the joints separating under certain conditions. In addition, on slopes, push on joints must be installed with the bell end facing uphill. This will prevent the spigot end from sliding out of the bell. In some instances where additional support is needed, there are certain restraints that can be used. These types of connections are typically less expensive than other connections and are easier to install.
Mechanical joints, restrained joints, and flanged connections are also used to connect pipes and fittings together. Flanged connections are typically limited to above-ground installations because of the potential of corrosion and dirt filling in around the bolts.
Mechanical joints are similar to push-on joints. However, they have a means to “lock” the pipe and fittings together. Restrained joints are used with push-on fittings and require some type of restraining system. Below is an example of one such restraining system, mega lug rods. Tie rods are another type of restraining system. These help to hold two sticks of pipe together.
Sample Questions
-
Flanged pipeline connections are very common ___________.
- In below-ground installations
- Only for sewer systems
- In above-ground installations
- None of the above
-
Asbestos-cement pipe come in lengths ___________.
- Longer than DIP
- Shorter than DIP
- The same as DIP
- All of the above
-
Transmission water mains deliver ___________.
- Water directly to customers
- Water to treatment plants
- Carry large amounts of water
- Water directly from service laterals
-
Which of the following would be a correct order for how water gets to a customer?
- Source, distribution main, transmission main, service lateral
- Service lateral, distribution main, transmission main, source
- Source, transmission main, service lateral, distribution main
- Source, transmission main, distribution main, service lateral
-
Which of the following pipe material is most susceptible to collapse from partial vacuum?
- Steel
- DIP
- Plastic
- Concrete


1.3: Discussion 1- Introduce Yourself
Module 1 Discussion
Introductions
Introduction
Welcome to the class! It is important to feel engaged in this class – you shouldn’t feel isolated and alone at your computer. That is the purpose of these class discussions.
If you read the directions carefully, follow the listed suggestions, use proper grammar and spelling, make lots of meaningful replies, and post on time, you will have no trouble earning all of the required participation points in this class, and will likely earn lots of extra credit points, too!
For this discussion, you will write an intial post and reply to at least TWO of your classmates’ posts. Follow the guidelines for each posting below:
Basic Guidelines for Initial Post
(20 points, required)
Under the Discussion 1 topic, create a new thread. You need to include all of the following in your post:
- Put your full name somewhere in the subject line. For example, you could write “Hello from so and so…”.
- In the message box, start by telling us your preferred name, if you have one other than what the school assigned you, and let us know where you are currently located. It is fun to see who is located outside of Pima County, and it could help me to plan ahead for field trips.
- Share with your instructor and fellow students why you are taking this class. Is it to fulfill your transfer or AGEC requirements, or the requirements of a particular program at this or another college? Explain. This information is particularly helpful to me in terms of future planning for this online course. If you are willing to share, I would also like to know why you chose this particular course and format (online), rather than enroll in a different one.
- Tell us whatever else you want to share about yourself. You might consider including information about your major, current job, current classes, educational goals, family, recent accomplishments, etc. It is up to you what you want to include.
- Talk about your experience with taking online classes. Is this your first online class, or have you taken many of them? If you are new to online, share your fears or reservations about this class format. If you have taken other online courses, please share your tips for success. What do you think are your personal challenges to success in this class?
- Upload a picture of yourself. Be sure that it is large enough so that we can clearly see you, and tell us which person in the picture is you if there are multiple people shown. Being able to see what each other looks like will make us feel more like a community, and this class feel as much like a regular, face-to-face classroom as possible. Seeing each other helps a lot!
Basic Guidelines for Replies to Students
(10 points, required)
-
You are free to make replies to students with regard to either
- your own introductory post, or
- another student’s introduction (including mine).
- Each reply must contain 6 or 7 meaningful sentences responding to a student’s post. Any encouraging phrases and support are welcome!
- You must reply to at least 2 other students. (3, or more, for extra credit.)
- If your replies are too short, you will need to make more than 2 replies to reach your full sentence count for the discussion (2 replies x 6 or 7 sentences each = 12-14 sentences required in each discussion).
- Do not wait until the last day of the discussion to make your replies. You only receive credit for one reply (up to 10 points max) on the last day.
- Reply early and often, and earning your full discussion points will be easy!
Extra Credit
(5 points, optional)
Make a third reply (or more!) to a student following the guidelines above.
Learning Outcomes
As a result of your participation in the discussion, you will be introduced to the other students in the course and your instructor. It will also give you a chance to evaluate yourself as an online learner, as well as an opportunity to share successful study tips with your fellow classmates.