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CHEMISTRY ORGANIC 308 LECTURE III CHAPTER 15 ORGANIC CHEMISTRY 308 LECTURE III
CHAPTER 15
I. Benzene Our starting point. Where we see further consequences of conjugation
A. Historical Perspective. In the mid 19th century Benzene was a well known compound. Natural sources of benzene include volcanoes and forest fires. Benzene is also found in crude oil, gasoline, and cigarette smoke. It was recognized even then that there is something very unusual about benzene. 1. Its formula was known to be C6H6 indicating a high degree of unsaturation. But it did not behave like an alkene or an alkyne at all. For one thing it was observed to be much less reactive. 2. The details of its reactions were even more puzzling. For example when hexene reacts with bromine, the bromine readily adds. C6H12 + Br2 C6H12Br2
But when benzene reacts with Br2, it requires a Lewis acid catalyst such as FeCl3 for the reaction to occur. Only one product is obtained, but the product is a substitution product: C6H6 + Br2 C6H5Br + HBr
The product is called bromobenzene, which has many of the same properties as benzene itself. 3. Equally puzzling was the behavior of bromobenzene. When treated with Br2 and FeCl3 it also undergoes a substitution reaction: C6H5Br + Br2 C6H4Br2 + HBr
Just three isomeric products are obtained from this reaction.
1
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
4. The term aromatic was used to characterize benzene and its derivatives because they often had characteristic smells and to emphasize their differences from aliphatic compounds such as alkanes, alkenes etc. 5. First approach to a correct structure for benzene was made by Kekul in 1865 as the result of an opium dream: a snake with its tail in its mouth etc. It was further refined in 1872 by the addition of rapidly moving double bonds. 6. What Kekul and other 19th century scientists were trying to explain was the equivalence of all six carbon atoms (and as we now know all 6 carbon-carbon bonds. A modern description using sp2 hybrid orbitals and unhybridized p-orbitals very simply explains the equivalence.
Notice that 6 atomic orbitals are overlapping. Therefore there are six molecular orbitals spread over the six carbon atoms. Three of these MO's are bonding and three are antibonding. Since there are six electrons to be accommodated all six reside in bonding molecular orbitals. Similarly, a valence bond description, where benzene is described as a blend of two equivalent contributing structures also explains the equivalence of the bonds and atoms. These two contributing structures are often written as one structure with a circle instead of three double bonds.
2
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
B. Nomenclature 1. Monosubstituted benzenes are often encountered. They can be named by adding a substituent prefix to the name benzene and the names are easy to remember. Examples are chlorobenzene (or fluoro, bromo, or iodo), nitrobenzene (-NO2), and ethylbenzene etc. But many compounds of benzene were known in the 19th century and they have common names that must be memorized. You should know: phenol, aniline, toluene, anisole, styrene, benzaldehyde, acetophenone, and benzoic acid. (See page 668) 2. If there is more than one substituent on the benzene ring, each substituent is named separately in alphabetical order. Identical substituents take multiplying prefixes. For example a dibromochlorobenzene and a trinitrobenzene
OO N+ O N+
Br
-
OO N+ O
Br
Cl
or
3
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
3. The location of substituents on the ring can be indicated in one of two ways. a. By number following the usual rules of giving the lowest possible numbers 1-chloro-3-nitrobenzene not 1-chloro-5-nitro
Cl ON+ O
b. For Disubstituted benzenes we use a special system. There are three possibilities 1,2; 1,3; and 1,4. Each has a name: 1,2 is called ortho (o); 1.3 is meta (m); and 1,4 is para (p). These names just replace the numbers, keeping substituents in the same alphabetical order. The three chloronitrobenzenes are
Cl
-
Cl
-
O N+ O
O N+ O Cl N+
O
O-
o-chloronitrobenzene
m-chloronitrobenzene
p-chloronitrobenzene
c. If one of the substituents is usually designated by a common name (e.g. amino = aniline, methyl = toluene, aldehydic = benzaldehyde etc), i. the substituent is not identified separately but rather is identified by the common name and the compound is named as a 4
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
substituted common name. The common naming substituent gets the number 1. 2,4,6-trinitrotoluene and 4-hydroxy-3-methoxybenzaldehyde (vanillin)
OO N+ O N+ OO N+ OO O HO
4-hydroxy-3-methoxy-benzaldehyde
2,4,6-trinitrotoluene
ii. In the case of a disubstituted benzene, the o, m, p designations can be used. p-bromophenol and m-ethylaniline
Br
OH NH 2
p-bromophenol
m-ethylaniline
d. There are some common names on page 668. Know the xylenes, mesitylene, and cresols (not listed, but they are methyl phenols, o, m, and p).
HO OH
OH
o-cresol
m-cresol
p-cresol
5
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
e. We can also name radicals derived from benzene and substituted benzenes... The one from benzene itself is called phenyl (C6H5-) analogous to methyl or ethyl and is abbreviated Ph- or -. Any radical that has a benzene ring as part of its structure is called an aryl radical abbreviated Ar- (note analogy to alkyl R). The compound CH2CH2CH2 is most conveniently named 1,3-diphenylpropane. (Rule of maximum substituents on a single unit) Another common aryl radical is CH2- called the benzyl radical and the compound CH2OH is commonly called benzyl alcohol. C. Aromaticity 1. The unusual stability of benzene and its derivatives (aromatic compounds) has led to the term aromaticity to designate such unusual or extra stability, which results from cyclic delocalization. It is much more stability than what we expect from a simple resonance picture. This extra stability can be determined by measuring heats of reactions involving benzene and comparing the experimental values to values that we calculate based on simple conjugation, as shown in Figure 15.3. in which the reaction is hydrogenation. Note that the "1,3,5-Cyclohexatriene is an imaginary compound; it does not exist because it has no conjugation, although we can still calculate its heat of hydrogenation using bond energies. The comparisons of these H of hydrogenations is revealing because in all cases the product is the same, cyclohexane. Thus the differences reflect differences in the reactants.
6
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
The aromatic character of benzene manifests itself in many of its chemical and spectroscopic properties. We shall look at the chemistry later in this and the next chapter. 2. The most striking effect of the unusual electronic structure of the benzene ring is seen in the nmr spectrum of benzene and its derivatives. Protons attached to a benzene ring are very deshielded, much more than protons attached to ordinary carbon-carbon double bonds. Benzene derivatives are thus relatively easy to identify by looking at their nmr spectra. The rationale for this deshielding is shown in Figure 15.9.
When the benzene ring is placed in an external magnetic field it induces the 6 electrons to circulate around the ring, setting up what is called a ring current. Circulating electrons generate magnetic fields of their own. The direction of this induced magnetic field varies at different points in space. At the points where the aromatic protons are found the induced magnetic field and the external magnetic field are in the same direction. Thus the deshielding. The splitting patterns (and of course integration) in substituted benzenes are also useful. In a monosubstituted benzene (5 aromatic protons) the pattern is complex because all three kinds of protons couple with each other. In ortho and para disubstituted (same substituent) benzenes (4 aromatic protons), there are only two kinds of protons, (two of each kind) 7
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
which are ortho to each other and usually appear as two two-proton doublets. The splitting patterns are usually pretty clear because coupling of ortho protons is relatively large. Work out the patterns for meta disubstituted. 3. In recent years it has been recognized that the phenomenon of aromaticity (meaning extra stability) is not just limited to benzene and its derivatives. Other compounds or ions with certain structural features also display aromaticity. They all have the following features in common with benzene: 1. Cyclic 2. One unhybridized p-orbital on each atom of the ring 3. Planar (or nearly planar) so that there is continuous (or nearly continuous) overlap of all these p-orbitals As we shall see, these features, which describe a cyclic completely conjugated system, are necessary, but not sufficient. An additional feature is needed.
D. Examples of Aromatic Compounds 1. Just about anything with a benzene ring, no matter how it is substituted will be aromatic. The additional feature that makes benzene aromatic is that its three bonding molecular orbitals are completely filled by its 6 electrons. 2. There are many compounds that have a fused ring system made up of two or more benzene rings sharing atoms. They are called polycyclic aromatic compounds. A number of these ring systems are especially important and you should learn their names and structures.
8
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
3. It seems reasonable to wonder whether any species that has 6 electrons and meets the other three criteria for aromaticity will display the "extra" stability associated with this property. (Remember, to be aromatic a species does not have to be very stable, just more stable than expected.) The answer to this question is generally yes. Let's look at some examples 4. Heterocycles (From Chapter 25) a. A heterocycle is a compound with a ring in which one or more of the atoms of the ring is something other than carbon. The most common atoms that are found in heterocycles are nitrogen oxygen, and sulfur. Many heterocycles play key roles in the chemistry of living things. (See Chapter 25 of your text for more background on this topic) b. Nitrogen containing aromatic heterocycles are especially important biologically, particularly in the chemistry of proteins and nucleic acids. You have probably already encountered many examples of such compounds. We'll just present a couple at this point. i. Pyridine, has 6 electrons, five of the electrons are from the 5 C atoms of the ring and the 6th is from N (in the C=N bond), not from the lone pair. Its structure is
9
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
or
ii Pyrrole has a five membered ring with four carbon atoms, each of which contributes a electron. The other two electrons are the lone pair of the N which can be in an unhybridized p orbital if the N is sp2 hybridized.
or
iii. There are also many important nitrogen heterocycles that most of you have already encountered in Biology courses, in which there are two nitrogens in the ring. In six membered rings each N gives one pi electron to the double bond. A familiar example is pyrimidine.
10
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
In five membered rings one N gives a pi electron to the double bond and one gives a pair. A familiar example is imidazole
4, Ions. While an ion is not very stable compared to a neutral compound and also generally very reactive, we can observe ions that are aromatic because they are much more stable than regular ions. We can meet the requirements for aromaticity with 6 electrons in two ways.
a. An anion, that has the lone pair in a p orbital (as in pyrrole) and two double bonds. This ion is the cyclopentadienyl anion
The relatively low pKa of cyclopentadiene, (16) is compelling evidence for the extra stability of this anion. b. A cation whose vacant p-orbital becomes part of the continuous cyclic system. Example is the cycloheptatrienyl cation (also called tropyllium)
11
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
The ionic nature of cycloheptatrienyl bromide is compelling evidence for the extra stability of this cation. E. Aromatic compounds or ions with numbers of electrons other than 6. There are many species now known that are aromatic, but do not have 6 electrons. They must have the three features mentioned above; that is they must have a cyclic completely conjugated system. But they must also have a certain number of electrons in the cyclic conjugated system. That number is given by the expression 4n + 2, where is n any integer from 0 up. This observation is called Huckel's Rule and can be restated as: there must be an odd number of pi electron pairs for the compound to be aromatic. (for example, three in benzene etc) 1. Two pi electrons, one pair. (n = 0 in Huckel's rule) The bst known species that has continuous p-orbitals is the cyclopropenyl cation,
Evidence: While not as ionic as cycloheptatrienyl bromide, cycopropenyl halides can form ionic compounds when treated with Lewis acids.
Cl
+ SbCl5
+ SbCl6-
12
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
2. Ten or more electrons (n = 2 or more). There are cyclic hydrocarbons known that meet the three requirements for aromaticity and in fact display aromatic properties. The first one made has an 18-membered ring with 9 double bonds. For the sake of simplicity, completely conjugated cyclic polyenes are named in a special way as [n] annulenes where n is the number of carbon atoms. This compound is thus [18]-annulene and its structure is
This polyene behaves much more like benzene, both chemically and spectroscopically, than like an ordinary cyclic polyene. The reason that this compound of large ring size was easier to make than [10]annulene and [14]annulene, which also follow Huckel's Rule is related to strain from cross ring H-H interactions. 3. The stability associated with (4n + 2) electrons in a cyclic conjugated system is very great. In fact, it is possible to prepare doubly charged ions such as C8H82-, which has 10 electrons
13
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
or a substituted (tetramethyl) C4H42+, which has 2 electrons.
C+ C+
F. What about compounds that meet all the requirements for aromaticity except they have 2n electrons? Not only do such compounds not display aromaticity, but they don't even display the degree of stability we would expect from simple resonance. There appears to be a destabilization in such systems. We call it antiaromaticity. Let's look at some examples. 1. Four pi electrons: 1,3-Cyclobutadiene, (or [4]annulene) appears to be a perfectly normal molecule, a bit strained perhaps. But it is extremely difficult to make and can only be trapped at very low temperatures. The evidence is that the bonds are of unequal length, as in a noncyclic diene. The molecule is rectangular and puckers a bit. The two double bond structures are not resonance contributors, but isomers.
It also behaves as a diradical to some extent because the second bond is so poor. Similarly the cyclopropenyl anion and the cyclopentadienyl cation, which are both 4 electron systems, show no stability of any kind. 14
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
2. Eight electrons 1,3,5,7-Cyclooctatetraene (or [8]annulene) is an ordinary compound that is not aromatic, because it is nonplanar. (Figure 15-17)
It is nonplanar because the planar conformation is antiaromatic and also more strained than this "tub" conformation. Any resonance stabilization that might have resulted from planarity and overlap of p orbitals is offset by the antiaromatic nature of the 8 electron feature. G. Explanation 1. Simplest approach is to use MO theory to identify the molecular orbitals and their relative energies in these cyclic systems, and then put the electrons into the orbitals. 2. In general, if all the electrons can be accommodated by filling bonding MO's then the system will be relatively stable. On the other hand, if some electrons must go into nonbonding or antibonding orbitals that will be destabilizing and the system will be antiaromatic. In addition, even if all the electrons can be accommodated in bonding MO's, but there are not enough electrons to fill pairs of bonding orbitals of equal energy (we call them degenerate orbitals), the system will be antiaromatic.
15
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
3. There is a simple mnemonic called a Frost Circle [OH] that we can use to identify the orbitals and their energies. Figure 15.18 in your text shows some of the same ideas. a. Inscribe a polygon with the number of sides of the cyclic system in a circle with one vertex at the bottom of the circle. You can see from the geometry that except for the lowest energy bonding orbital and the highest energy antibonding orbital (in even numbered rings) the orbitals come in pairs of equal energy. Look at 4, 5, and 6 sided polygons (rings). b. Each vertex represents an MO that will hold two electrons. Its relative energy is shown by where it touches the circle. The lower down on the circle, the lower the energy of the orbital. c. The MO's below the horizontal diameter are bonding, those above are antibonding, and those on the line are nonbonding. Shown on the overhead projection.. d. Let's put electrons into the MO's. Show for four electrons in the 4 and 5 ring. Molecules have diradical and geometric distortion possibilities to avoid this unfavorable electronic arrangement. Thus cyclobutadiene, as we have seen, distorts its geometry. The cylopentadienyl cation exists as a triplet diradical. (It has only one double bond) e. Let's look at six electrons in the 5 and 6 ring. You can see why these systems are aromatic. All the electrons are in bonding MO's and they are filled. H. The chemistry of benzene and its derivatives is very different in many ways from that of the compounds we have looked at so far. The extra stability associated with the cyclic conjugated 6 electron system dominates the chemistry of these compounds. Let's begin by looking at the reaction of benzene with electrophiles. This reaction is probably the most important reaction of benzene and many substituted benzenes. 1. The overall result of the reaction is the substitution of the electrophile for one of the protons of the benzene ring. A typical overall reaction is 16
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
C6H6 + Br2 C6H5Br + HBr
In this case the electrophile is Br+. Needless to say the electrophile is never H+ because then there would be no overall change. Notice how different this reaction is from the reactions of alkenes with electrophiles. 2. The important point about this reaction and most reactions of aromatic compounds is that they form products in which the aromatic ring is still present. This observation is no surprise, since to lose the aromatic ring would be to raise the energy of the system substantially. 3. It is impossible to imagine a way in which H+ could be displaced from the benzene in one step by an electrophile. The mechanism is actually a two step process. Addition followed by elimination. As you can see, if the group that is eliminated is different from the group added, the overall result is a substitution. 4. Let's look at the general mechanism (reminder about the direction of the arrows) and consider some of its features.
a. Step 1. Addition of an electrophile Step 1 is the rate-determining step. Since the reaction destroys the aromaticity of the benzene ring you expect it to require vigorous conditions and/or a powerful electrophile, which it does. This requirement is the important aspect of these reactions and we shall focus on it for a bit. Still the reaction has some things going for it. Benzene has a lot of electrons readily 17
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
available to an electrophile. The intermediate ion formed, while not aromatic is stabilized by allylic resonance with two double bonds. b. Step 2. Loss of a proton This reaction is very favorable because it restores the aromaticity. There must be some kind of base in the system (no matter how weak) to accept the proton. Note it is the electron pair of the C-H bond that goes into the ring to reestablish the 6 electron aromatic system. This step is certainly favored over the alternative, which would be attack of a nucleophile on the cation. C. Let's survey some important electrophilic substitutions. The key issue will generally be how to get the electrophile and the conditions souped up enough to get that first step to go. 1. Halogenation produces a halobenzene, usually a chloro or bromo one. C6H6 + Br2 C6H5Br + HBr
Benzene does not normally react with these halogens because they are not electrophilic enough. But the halogen can be activated using a Lewis acid catalyst such as FeX3 or AlX3. The electrophile then is the equivalent of X+. a. Let's look at it for Br2. i. A complex is formed in which one Br donates an electron pair to the Lewis Acid thus making the other Br relatively more positive and electrophilic.
ii. This other Br attacks the benzene ring, while the FeBr4 group leaves.
-
18
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
iii. Finally this group acts as a base to accept the proton from the intermediate. In the process it regenerates the catalyst and produces HBr.
iv. The overall reaction is exothermic, making a C-X bond and an H-X and losing a C-H and an X-X bond, except in the case of I2. With F2 it is so exothermic that a catalyst is not needed. Note the trend, which parallels free radical halogenation for the same reason, the relative strengths of the H-X bond. 2. Nitration produces nitrobenzene. The overall reaction is C6H6 + HNO3 C6H5NO2 + H2O
a. The electrophile is nitronium ion, NO2+ and the challenge is the generation of the electrophile. Just using conc HNO3 isn't good enough. We generate the nitronium ion by adding conc H2SO4 to the HNO3.. The reaction is
19
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
b. Then the usual two steps take place, with the bisulfate from the generation of the nitronium acting as the base to accept the proton in the second step.
c. This reaction is the best way to introduce nitrogen onto a benzene ring. The NO2 can be readily converted to other functional groups such as NH2. 3. Sulfonation To produce a benzenesulfonic acid. a. Benzene does not react with conc H2SO4 at RT. We need fuming sulfuric, which is conc sulfuric with about 8% SO3 added, or we need elevated temperatures. The exact nature of the electrophile depends on conditions, but let's use SO3 for simplicity.
20
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
c. If the product is heated with dilute aqueous acid the reaction reverses because the SO3 is removed in a very favorable reaction with water, pulling everything back.
d. Benezenesulfonic acids and their derivatives are very important in the manufacture of detergents and dyes. Also an important group of drugs, the sulfa drugs, are compounds of this type. 4. Friedel-Crafts Alkylation is used to prepare alkyl benzenes. Basic idea is to use a carbocation as the electrophile to attack benzene, thus generating a C-C bond. a. Simplest version is to use an alkyl halide as the source of the carbocation or the carbocation equivalent and generate it using a Lewis acid, most commonly AlX3.
21
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
i For primary alkyl halides, the mechanism is analogous to that with X2, except the other X is the carbocation equivalent.
ii. For secondary and tertiary halides the free carbocations are more or less formed and the mechanism is like that with nitronium. b. The yields of these reactions are generally low because of a number of problems. i. The presence of an alkyl group on a benzene ring makes the ring more reactive in electrophilic substitution reactions. (Why, as compared to the other groups we have talked about so far?) Thus it is hard to stop the reaction after just one addition of R. We can try by using a large excess of benzene. ii. Rearrangements typical of carbocations occur. In the case of the primary ones, they occur even though the free ion is never formed. In fact even FC reactions of primary alkyl halides don't work very well.
5. Friedel Crafts Acylation. Compounds with the functional group C=O(X) are called acyl (or alkanoyl) halides and they can also lose halide under the influence of Lewis acids. The product of the overall reaction is a ketone. 22
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
a. The loss of halide from the acyl halide produces an acylium ion which reacts with benzene in the same way as the alkyl carbocations.
The ketone produced forms a strong complex with the Lewis acid.
23
ORGANIC CHEMISTRY 308 LECTURE III CHAPTER 15
b. Note that because the Lewis acid is complexed by the ketone it is not regenerated and is not a catalyst. So we need one mole of it for every mole of the acyl halide.
c. The acylation does not have the disadvantages of the alkylation. i. The electron withdrawing carbonyl makes the ring less reactive toward further substitution. ii. Because acylium ions are resonance stabilized, they do not rearrange.
24
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http:/dmoz.org/Science/Social_Sciences/Communication/ Introduction to Grounded Theory. By Steve Borgatti This article talks about the grounded theory, which lays down framework for making a good theory. The grounded theory is compiled of a set of ste
Penn State - CAS - 202
Caitlin Lovey CAS 202 Website Assignmenthttp:/eserver.org Orange Journal: A Study of Theories of Style in Technical Communication. By Lily SunThis article talks about what technical theory is and how its communicators use it. Technical communicat
Cornell - HD - 3430
HD 343After school programming Socialization and supportAgenda Journals and VOC/CV Readings Logs Upcoming class 4/9Clarification Journals Spirit of assignment Process- rough notes, typed notes (NT/NM), developed notes (2 shared field logs
Cornell - HD - 3430
Complete Schedule and About Yourself Select one option that fits your schedulePlacement Options* 1. Morning Mon/Wed AM (8-10,9-11, 10-12) Mon/Fri AM (8-10,9-11,10-12) Mon/Fri PM (11-1, 12-2, 1-3, 2-4, 3-5) Tues/Thurs AM (8-10,9-11,10-12) 2. Mid-day
Cornell - HD - 3430
Agenda Beginning placements (first reactions) Reading response discussion First paper (due 2/15) Break Beginning field logs (describe the placement environment) Assignment for next weekTaking an ethnographic approach to study the social world
Cornell - HD - 3430
Agenda Field logs/Traditions review Reading response and field log analysis Break Socialization: Processes and outcomes Relationships: Context and Impact (Paper Due 2/15)Field log practice Pair up Experience of note taking Thin or thick des
Cornell - HD - 3430
Through the perspectives of race and class, how can we view the complexities of childhood?Agenda Who Am I Review socialization methods and outcomes (Saras anecdote) Reading response discussion Two groups 1) Ben, Lareau: Chapters 2 and 3 Ashley M
Cornell - HD - 3430
Agenda Berns: Ecology of Teaching (Chapter 7) Field logs summary bankactivity.2.7.melissa lunchtime.2.13.rina 3 oclockactivity.2.15.ashley storytelling.2.4.merima Readings discussion Tobin and Perry- first reactions Sign up for next weekC
Cornell - HD - 3430
Home-School RelationshipsQuickTimeTM and a TIFF (Uncompressed) decompressor are needed to see this picture.Mesosystem influences School-family linkages1. Children from low SES and ethnically diverse families fare comparably to middle-class chil
Cornell - HD - 3430
HD 343Family Diversity Parenting and Socialization"You are the bows from which your children as living arrows are set forth." Kahil GibranAgenda Field log review Nutshell Discussion Full group (questions, ideas, and summary) Relate to other
Cornell - HD - 3430
Agenda Who am I revisited? Voice of the child/Community Voices- due ongoing Break Peer relationships in context (Berns) Readings and field log discussion Assignments for next weekWho are we? Who am I? (revisited) Round one: take on persona
UMBC - MATH - 150
Math 106 Suggested HW Problems Spring 2008Chapter 7 Sec 7.1: Sec 7.2: Sec 7.3: 7, 11, 15, 19, 25, 30, 35, 43, 49, 55, 60, 65, 69, 75, 79 7, 11, 17, 23, 23, 31, 35, 45, 51, 55, 67, 75 7, 11, 17, 23Chapter 4 Sec 4.1: Sec 4.2: Sec 4.3: Sec 4.4: Sec 4