Precalculus Formulas

Here you can find formulas for a precalculus Course.

1.1 Trigonometric functions
For the trigonometric ratios for a point p on the unit circle holds:
cos(φ) = x_p , sin(φ) = y_p , tan(φ) = y_p/x_p
sin² (x) + cos² (x) = 1 and cos⁻² (x) = 1 + tan² (x).
cos(a ± b) = cos(a) cos(b) ∓ sin(a) sin(b) ,
sin(a ± b) = sin(a) cos(b) ± cos(a)sin(b)
tan(a ± b) =( tan(a) ± tan(b) )/( 1 ∓ tan(a) tan(b) )
The sum formulas are:
sin(p) + sin(q) = 2 sin( 1 / 2 (p + q)) cos( 1 / 2 (p − q))
sin(p) − sin(q) = 2 cos( 1 / 2 (p + q)) sin( 1 / 2 (p − q))
cos(p) + cos(q) = 2 cos( 1 / 2 (p + q)) cos( 1 / 2 (p − q))
cos(p) − cos(q) = −2 sin( 1 / 2 (p + q)) sin( 1 / 2 (p − q))
From these equations can be derived! that
2 cos2 (x) = 1 + cos(2x) , 2 sin2 (x) = 1 − cos(2x)
sin(Ï€ − x) = sin(x) , cos(Ï€ − x) = − cos(x)
sin( 1 / 2 Ï€ − x) = cos(x) , cos( 1 / 2 Ï€ − x) = sin(x)
Conclusions from equalities:
sin(x) = sin(a)
⇒ x = a ± 2kÏ€ or x = (Ï€ − a) ± 2kÏ€, k ∈ N
cos(x) = cos(a)
⇒ x = a ± 2kÏ€ or x = −a ± 2kÏ€
tan(x) = tan(a)
⇒ x = a ± kÏ€ and x =/= Ï€ / 2 ± kÏ€
The following relations exist between the inverse trigonometric functions:
arctan(x) = arcsin(x/ √x² + 1)= arccos(1/√ x² + 1)

sin(arccos(x)) = √ 1 − x²

precalculus

Thin layer chromatography - Wikipedia, the free encyclopedia

Thin layer chromatography - Wikipedia, the free encyclopedia

Thin layer chromatography

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Thin layer chromatography
Separation of black ink on a TLC plate
Acronym	TLC
Classification	Chromatography
Other Techniques
Related	Agarose gel electrophoresis SDS-PAGE
This box: view • talk • edit

Thin layer chromatography (TLC) is a chromatography technique used to separate mixtures.^[1] It involves a stationary phase consisting of a thin layer of adsorbent material, usually si! lica gel , aluminium oxide, or cellulose immobilized onto a flat, inert carrier sheet. A liquid phase consisting of the solution to be separated is then dissolved in an appropriate solvent and is drawn up the plate via capillary action, separating the experimental solution based on the polarity of the components of the compound in question.

Its wide range of uses include

• assaying radiochemical purity of radiopharmaceuticals
• determination of the pigments a plant contains
• detection of pesticides or insecticides in food
• analysing the dye composition of fibers in fore! nsics, or
• identifying compounds present in a given substance

It is a quick, generic method for organic reaction monitoring.

Contents [hide] 1 Plate preparation 2 Technique 3 Analysis 4 Applications 5 References! 6 See also

[edit] Plate preparation

TLC plates are made by mixing the adsorbent, such as silica gel, with a small amount of inert binder like calcium sulfate (gypsum) and water. This mixture is spread as a thick slurry on an unreactive carrier sheet, usually glass, thic! k aluminum foil, or plastic, and the resultant plate is dried and activated by heating in an oven for thirty minutes at 110 °C. The thickness of the adsorbent layer is typically around 0.1–0.25 mm for analytical purposes and around 1–2 mm for preparative TLC. Every type of chromatography contains a mobile phase and a stationary phase.

[edit] Technique

Chromatogram of 10 essential oils coloured with vanillin reagent.

The process is similar to paper chromatography with the advantage of faster runs, better separations, and the choice between different stationary phases. Because of its simplicity and speed TLC is often used for monitoring chemical reactions and for the qualitative a! nalysis of reaction products.

A small spot of solution containing the sample is applied to a plate, about one centimeter from the base. The plate is then dipped in to a suitable solvent, such as ethanol or water, and placed in a sealed container. The solvent moves up the plate by capillary action and meets the sample mixture, which is dissolved and is carried up the plate by the solvent. Different compounds in the sample mixture travel at diff! erent rates owing to differences in solubility in the solvent,! and owi ng to differences in their attraction to the stationary phase

Separation of compounds is based on the competition of the solute and the mobile phase for binding places on the stationary phase. For instance, if normal phase silica gel is used as the stationary phase it can be considered polar. Given two compounds which differ in polarity, the most polar compound has a stronger interaction with the silica and is therefore more capable to dispel the mobile phase from the binding places. Consequently, the less polar compound moves higher up the plate (resulting in a higher Rf value). If the mobile phase is changed to a more polar solvent or mixture of solvents, it is more capable of dispelling solutes from the silica binding places and all compounds on the TLC plate will move higher up the plate. Practically this means that if you use a mixture of ethyl acetate and heptane as the mobile phase, adding more ethyl acetate results in higher Rf values for all compounds on th! e TLC plate. Changing the polarity of the mobile phase will not result in reversed order of running of the compounds on the TLC plate. If a reversed order of running of the compounds is desired, an apolar stationary phase should be used, such as C18-functionalized silica.

Tlc plate development sequence: a mixture of a red and blue compound is separated as the plate develops, i.e. as the light blue solvent moves up the plate.

The appropriate solvent in context of Thin layer chromatography will be one which differs from the stationary phase material in polarity. If polar solvent is used to dissolve the sample and spot is applied over polar stationary phase TLC, the sample spot will grow radially due to capillary action, which is not advisable as one spot may mix with the other. Hence, to restrict the radial growth of sample-spot, the solvent used for dissolving samples in order to apply them on plates should be as non-polar or semi-polar as possible when the stationary phase is polar, and vice-versa.

[edit] Analysis

As the chemicals being separated may be colorless, several methods exist to visualize the spots:

• Often a small amount of a fluorescent compound, usually manganese-activated zinc silicate, is added to the adsorbent that allows the visualization of spots under a blacklight (UV₂₅₄). The adsorbent layer will thus fluoresce light green by itself, but spots of analyte quench this fluorescence.
• Iodine vapors are a general unspecific color reagent
• Specific color reagents exist into which the TLC plate is dipped or which are sprayed onto the plate
• In the case of lipids, the chromatogram may be transferred to a PVDF membrane and then subjected to further analysis, for example mass spectrometry, a technique known as Far-Eastern blotting.

Once visible, the R_f value , or Retention factor, of each spot can be determined by dividing the distance traveled by the product by the total distance traveled by the solvent (the solvent front). These values depend on the solvent use! d, and the type of TLC plate, and are not physical constants.

[edit] Applications

In organic chemistry, reactions are qualitatively monitored with TLC. Spots sampled with a capillary tube are placed on the plate: a spot of starting material, a spot from the reaction mixture, and a "co-spot" with both. A small (3 by 7 cm) TLC plate takes a couple of minutes to run. The analysis is qualitative, and it will show if starting material has disappeared, product has appeared, and how many products are generated. Unfortunately, TLC! 's from low-temperature reactions may give misleading resu! lts, bec ause the sample is warmed to room temperature in the capillary. One such reaction is DIBALH reduction of ester to aldehyde.

As an example the chromatography of an extract of green leaves (for example spinach) in 7 stages of development. Carotene elutes quickly and is only visible until step 2. Chlorophyll A and B are halfway in the final step and lutein the first compound staining yellow.

Step 1	Step 2	Step 3	Step 4
Step 5	Step 6	Step 7

In one study TLC has been applied in the screening of organic reactions^[2] for example in the fine-tuning of BINAP synthesis from 2-naphtol. In this method the alcohol and catalyst solution (for instance iron(III) chloride) are place separately on the base line, then reacted and then instantly analyzed.

[edit] References

1. ^ Vogel's Textbook of Practical Organic Chemistry (5th Edition) (Hardcover) by A.I. Vogel (Author), A.R. Tatchell (Author), B.S. Furnis (Author), A.J. Hannaford (Author), P.W.G. Smith ISBN 0582462363
2. ^ TLC plates as a convenient platform for solvent-free reactions Jonathan M. Stoddard, Lien Nguyen, Hector Mata-Chavez and Kelly Nguyen Chem. Commun., 2007, 1240 - 1241, doi:10.1039/b616311d

• Hand book of Thin Layer Chromatography,Sherma, J.and Fried, B. (authors) 3^rd ed. Marcel Dekker, New York.

[edit] See also

• Gas chromatography
• Analytical chemistry
• Chromatography

polar solvents list

Online Math and Science Tutors from India

Using computer whiteboards and talking through the computer, many U.S. students are now finding that they can get knowledgeable math and science tutors from India.

Excerpt:"Both Growing Stars and Studyloft say that the majority of their tutors hold master's degrees or doctorates in their subjects; many also have degrees in education."

The idea is that tutors through the Internet relieve parents of commuting and scheduling, and cut the prices of established tutoring businesses like Sylvan Learning. If any of you have experiences with an online tutoring service(good or bad), please leave us a comment. Thanks in advance!

Online tutoring - The Boston Globe
Studyloft.com
Growingstars.com

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Math Help Online on www.TutorVista.com

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Factoring the time

I stumbled upon this comic that you might enjoy... factoring the time (from xkcd.com).

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Video Solution and Discussion of Twitter SAT Probability Question from 8-25-10

I decided to post a video solution of the Twitter problem I posted on 8-25-10:

4 red, 2 blue cards; 4 are chosen at random. What is the probability that 2 of the cards will be red? 

Because of the 140 character restriction on Twitter, the questions are often highly abbreviated and I actually consider it a "fun" challenge to write the question both concisely and clearly.  Of course, as we all know about human interpretation of word problems, "clear" is in the eye of the beholder!

There's no doubt that the question above needs some fleshing out and might appear on the SAT and other standardized tests something like this:

A set of six cards contains four red and two blue cards. If four cards are chosen at random, what is the probability that exactly two of these cards will be red?

I'm sure my astute readers can improve on this wording but we'll leave it at this.

A few questions naturally pop up:

(1) Could this really be an SAT/Standardized Test question? Well, as I state in the video below, a question quite similar to this appeared on the College Board website the other day as the Question of the Day.

(2) For whom is the video intended?  Everyone who happens upon it! I certainly wrote it to be helpful to students who will be taking the PSAT/SAT in the near future. Rather than simply presenting a single quick efficient solution, I demo'd 2-3 methods and indicated some important strategies and reviewed key pieces of knowledge to be successful on these harder probability questions. By the way, someone who is comfortable with probability will surely not find this question so formidable, but we're talking here about high school students or even undergraduates who struggle mightily with these.

(3) I'm hoping that the video will also serve as a catalyst for dialog in your math department. From the inception of this blog, I've never even intimated that a suggested way of explaining a concept, skill or a problem solution is in any way prescriptive. I encourage you to continue using whatever instructional methods have worked for you and to share these with our readers! However, for novice teachers or those who wish to see other approaches, I hope it will have some benefit. Of course, the video is not in a classroom. There are no students asking or being asked questions. There are no interruptions and I have a captive audience (except for my dogs who bark incessantly!).

SOME KEY STRATEGIES/TIPS/FACTS FOR PROBABILITY QUESTIONS

(1) It is highly recommended that students begin by listing 2-3 possible outcomes and to include at least one that is NOT one of the desired outcomes! This will help you to decide on a plan: organized list vs more advanced counting/probability methods. Further, you can ask yourself the key question in all counting/probability problems:  DOES ORDER COUNT!

(2) Although it appears difficult for most test-takers to be systematic when making a list under test-taking conditions, preparation is critical here. If one practices several of these in the weeks leading up to the test, the chances of success improve dramatically. Did I just suggest preparation and practice could make a difference!

Where do you find these problems? Any SAT/ACT review book or my Twitter Problems of the Day or my upcoming SAT Challenge Quiz book to name a few sources...

(3)  The basic definition of probability should always be in the forefront of your mind:

P(an event)  =  TOTAL NUMBER OF WAYS FOR THAT EVENT TO OCCUR DIVIDED BY TOTAL NUMBER OF OUTCOMES.

As indicated in the video, one can and should think of this ratio as TWO SEPARATE COUNTING PROBLEMS! Do the denominator first, i.e., the TOTAL number of possible outcomes.  In the Twitter problem it is 15 if order is disregarded.  Whether you arrive at 15 by listing/counting or by combinations methods, the denominator is 15 and is a completely separate question from  "How many ways are there to get 2 red and 2 blue cards?"

(4) Finally, there are other methods for solving this probability question using Laws of Probabilities and/or permutation methods. I was going to make a 2nd video but I'm not so sure about that now.

An important point about the video below: I used 4 Blue and 2 Red cards, the opposite of the original Twitter problem but that won't change the final result!








Look for my other videos on my YouTube channel MathNotationsVids.  Look for all of my Twitter SAT Problems on twitter.com/dmarain.  

As I develop my Facebook page further, I may start posting these questions there as well as my videos. Facebook allows up to 20 minutes videos, much less restrictive than YouTube's 10 minute limit.





"All Truth passes through Three Stages:
First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught" --from South Pacific

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free practice worksheets on www.math-drills.com

Much of the material we will be covering this session has worksheets that are free to download on www.math-drills.com This is an illustration of what that website looks like and how to manuver your way through it while avoiding all the ads which wish to sell you their services. Actually, these ads probably pay for this site to remain free, so you might be nice to them and read one or two even if you do not purchase their product.

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Math Mammoth Geometry 2

I have just finished writing the material for Math Mammoth Geometry 2 book. The material in it is suitable for grades 6-7. Download price is $5.80.

The main topics in the book include:

* angle relationships
* classifying triangles and quadrilaterals
* angle sum of triangles and quadrilaterals
* congruent transformations, including some in the coordinate grid
* similar figures, including using ratios and proportions
* review of the area of all common polygons
* circumference of a circle (Pi)
* area of a circle
* conversions between units of area (both metric and customary)
* volume and surface area of common solids
* conversions between units of volume (both metric and customary)
* some common compass-and-ruler constructions.

I've included several complete lessons from the book as samples (PDF). Feel free to download these and use with your students!

Angles in Polygons
Review: Area of Polygons, 1
Surface Area

Besides those, there are two other sample pages:

Area and Perimeter Problems
Basic Compass and Ruler Constructions, 1

What is next?

• See more information
• Purchase a download at Kagi
• Purchase a download at Currclick

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The Global Egyptian Museum

The Global Egyptian Museum

Directed by
Dirk van der Plas & Mohamed Saleh

Supported by
Comité Internal pour l'Egyptologie (CIPEG)
Committee of the International Council of Museums (ICOM-UNESCO)

Hosted by
Center for Documentation of Cultural and Natural Heritege (CULTNAT)

Preface

Preface to the first Edition

Credits

Rules, Conditions & Copyright

Participating collections

At a ro ugh estimate, over 2 million objects from ancient Egypt are kept in about 850 public collections, dispersed over 69 countries around the world. This website aims to collect them into a global virtual museum, which can be visited at any time, from any place. The Global Egyptian Museum is a long-term project, carried out under the aegis of the International Committee for Egyptology (CIPEG).

The Basic Mode, currently showcasing 1340 highlights, is geared to the interested public. A glossary of more than 400 items explains Egyptian terms and themes. Many objects are provided with audio comments and 3D-movies.

The Advanced Mode, equiped with a powerfull search and data entry engine, opens up the full database - presently 14975 objects - to professionals and amateurs.

Kids! offers information for children at the age of 8-12 years in an interactive way.

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Myth Debunking 1: What Are Aldehydes, How do Aldehydes Smell and Chanel No.5

Ask any aspiring perfumista about aldehydes and you will hear that they are synthetic materials first used in Chanel No.5, that thanks to them this is the first and the prototype of synthetic fragrances and that aldehydes themselves have a champagne-like, sparkly, fizzy odor that makes the fragrance fly off the skin. This is what they have been told time and again. Ask any seasoned perfumephile and you will hear that aldehydes have a waxy, citrusy or rosy aroma, like snuffed-out candles (Luca Turin was instrumental to that with his references appearing in “Emperor of Scent”). The truth? It’s a little more complicated than either!

It’s a fact that aldehydes became famous through their introduction in copious amounts into the formula of Chanel No.5 in 1921. However No.5 is definitely NOT the first fragrance to feature synthetic aldehydes :

~"Though Chanel No. 5 is recognized as the first Aldehydic fragrance, created in the mid-twenties, the truth is that the first Aldehydic fragrance was Rêve D'Or(Golden Dream), [1]created in 1905 by Armingeat." (sic[2]).

Nor is No.5 the first modern perfume to feature synthetic components (that honor belongs to Fougère Royale -Royal Fern- by Paul Parquet for Houbigant in 1882). Aldehydes themselves are organic compounds present in various natural materials (for instance natural citrus essences such as the one from orange rind, rose oil, pine essence, citronella and cinnamon bark ~they even appear in bovine heart muscle!). Several essence reconstitutions by chemists involved using aldehydes as various types can also be synthesized in the laboratory.!

Aldehydes (same as ketones) are organic compounds which incorporate a carbonyl functional group (that's C=O). The carbon atom of this group has two remaining bonds that may be occupied by hydrogen or alkyl or aryl substituents. If at least one of these substituents is hydrogen, the compound is an aldehyde. If neither is hydrogen, the compound is a ketone. The majority of aldehydes and ketones have strong odors. Ketones generally have a pleasant smell and they are frequently found in perfumes (e.g. muscone in musk-smelling colognes). They are also used in food flavorings. Aldehydes vary in smell ! with most of the lower molecular weight smelling bad (rotten fruits), yet some of the higher molecular weight aldehydes and aromatic aldehydes smell quite pleasant and are thus used in perfumery. Formaldehyde is the simplest aldehyde with a central carbon atom bound to two hydrogen atoms (H2C=O). Discovered in Russia by A. M. Butlerov in 1859 it is very reactive, used in dyes, medical drugs, insecticides and famously as a preservative and embalming fluid.
Aliphatic aldehydes possess intriguing smells outside the realm of simplistically nice: butyraldehyde for example smells of rancid butter (from βούτυρο/butyro which means "butter" in Greek)! Acetaldehyde is the name of the shortest carbon chain aldehyde and is one of the oldest known aldehydes (first made in 1774 by Carl Wilhelm Scheele). Its structure however was not completely understood until Justus von Liebig determined the constitution of acetaldehyde 60 years later, described its preparation from ethanol,! and baptised this chemical group “aldehydes”.

Hardly a fragrance exists without some kind of aldehyde in it, which incidentally makes insisting the greatness of No.5 is due to its synthetic materials comparable to saying that the Pyramids are monumental because of their shape alone. It is the cleverness of marketing and the propagation of a myth that No.5 was meant to evoke an unnatural smell (because supposedly Coco Chanel insisted that she wanted a perfume smelling of a woman and not of flowers ~"women do not want to smell of a bed of roses") which gave rise to this confusion. Chanel No.5 and No.22 later owe their vivid sprakle to a specific subgroup of aldehydes which are called “fatty”: strings of carbon atoms (between 8 and 13) coded in accordance to that number of atoms (ie.C8) with nomenclature deriving from Greek numerics, such as octanal from οκτώ/octo (=eight), in which each of the 8 carbon atoms is connected to two hydrogen atoms. The “! ;bouquet” of aldehydes C10, C11, and C12 in Chanel No.5 became so popular that all consequent “aldehydic fragrances” used that sequence of aldehydes, giving a fizzy perfume-y scent that is quite characteristic with the direct result of having the perfume lover confused as to how aldehydes themselves smell. Fatty aldehydes have a citrusy or floral note, and a pronounced fatty/waxy/soapy tone which is very apparent if you consider a modern fragrance that uses them in high ratio: Sicily by Dolce & Gabbana. The soapy feel is unmistakeable! As an exercise compare that smell with your Chanel No.5: you will pick up the soapy facets in that one as well. Another reason that they read as “soap” is exactly because they have been used in the production of soap for years to give that fresh lemony feel.

Most widely used aldehydes in perfumery are C7 (heptanal, naturally occuring in clary sage and pos! sessing a herbal green odour), C8 (octanal, orange-like), C9 (! nonanal, smelling of roses), C10 (decanal, powerfully evocative of orange rind; Citral, a more complicated 10-carbon aldehyde, has the odor of lemons), C11 (undecanal , “clean” aldehydic, naturally present in coriander leaf oil~also used is unsaturated C11 undecen-1-al), C12 (Lauryl aldehyde evocative of lilacs or violets), C13 (waxy, with grapefruit tone)and the infamous C14 peach-skin note of Mitsouko: technically not an aldehyde, but a lactone ~gamma undecalactone.

Often the compounds are patented under commercial names; therefore their true nature remains arcane even to perfume lovers who might have seen them mentioned. For instance Triplal, a patented molecule by IFF: its chemical name is 2,4-dimethyl-3-cyclohexene-1-carboxaldehyde. Its smell? Powerfully green and herbal, like crushing ligustra leaves between fingers. None of the characteristic aldehydes of Chanel No.5!
One interesting ingredient is phenylacetaldehyde which has a pr! onounced green note (top in natural narcissus and thus used to recreate a narcissus note in perfumery). The hydrocinnamic aldehydes are another family of materials from the manipulation of benzene and their odor profile resembles lily of the valley (muguet) and cyclamen. One of them is the famous Lilial (patented name for lily aldehyde; also known as Lilistralis), widely used in the replication of that elusive natural essence, lily of the valley. Another is Cyclamen aldehyde (usually produced with cumene as a starting material).
Aromatic aldehydes have very complex chemical structures but are the easiest to identify by smell. Anisaldehyde smells like licorice. Benzaldehyde on the other hand, has an odour profile of almonds and has several chemical constituents: cinnamaldehyde, amylcinnamic aldehyde, hexylcinnamic aldehyde. Condensation of benzaldehyde with other aldehydes gi! ves a series of α-substituted cinnamlaldehydes, the lowest me! mber of which is used in the production of cinnamyl alcohol, very important in the production of spicy perfumes (cinnamon note). Higher members, on the other hand, such as amylcinnamic aldehydes (ACA) and hexylcinnamic aldehyde (HCA) project a fatty jasmine impression despite their abscence from natural jasmine oils! Most synthetic jasmine perfumes today use one or both because they are inexpensive (Their fibre-substantive qualities also make them perfect candidates for laundry detergents and fabric conditionners). The hawthorn or aubépine note, rendered synthetically in perfumes for several decades, is produced via anisic aldehyde (p-methoxy benzaldehyde) and it has been sublimely woven into the gauzy cloth of Après L’Ondée by Guerlain (where it sings along with heliotropin). Additionally, the aldehyde vanillin! is a constituent in many vanilla-scented perfumes. So nothing is as simplistic as one might assume!

Aldehydic fragrances include (click links to read corresponding articles/reviews): Chanel No.5 and No.22, Lanvin Arpège, Guerlain Liu and Véga, Worth Je Revi! ens, Millot Crèpe de Chine , Balecianga Le ! Dix , Revillon Detchema, Caron Fleurs de Rocaille (not Fleur, singular), Infini and Nocturnes, Myrurgia Joya, Jean-Charles Brosseau Ombre Rose , Molyneux Vivre, Lancome Climat, Givenchy L’Interdit, Piguet Baghari, Madame Rochas and Mystère by Rochas, Rive Gauche by Yves Saint Laurent , Paco Rabanne Calandre, Estée Lauder Estée, White Linen, Pure White Linen, Van Cleef & Arpels First, Nina by Nina Ricci (the old formula in the ribbed bottle), E.Coudray Musc et Freesia, Bill Blass Nude and Amazing, Hermès Amazone, D& Sicily, Divine L’Ame Soeur, Serge Lutens La Myrrhe, Frederic Malle Iris Poudre, Ferré by Ferré, Agent Provocateur Maitresse, Annick Goutal Folavril, Le Labo Aldehyde 44.
Hermès Calèche is poised between floral aldehydic and floral chypre in some taxonomies.

[1]Bernand Chant, British Society of Perfumers 1982
[2]Armigeat is perfumer Pierre Armigeant (1874-1955) who composed Floramye and Azurea for L.T.Piver

Painting Gueridon 1913 by Georges Braques, courtesy of allposters.com. Chanel makeup ad via Bellasugar.

do the following have a ketone functional group?

Specific Suggestions to Implement the Concerns Raised in “A Plea for Critical Revisions to the Common Core State Standards for Mathematics” Developed by Susan Jo Russell and Steve Leinwand as a follow-up to a conversation with Bill McCallum, P

Following up on “A Plea for Critical Revisions to the Common Core State Standards in Mathematics”, we are pleased to supplement our general concerns with the following grade by grade suggestions for improving the elementary grades section of the Public Discussion Draft of the standards.  Our concerns, as discussed with you on April 5, 2010, fall into two broad categories that are addressed in the two charts that provide our specific suggestions for revision and improvement:

1)    Our belief that the Number-Base Ten domain is rushed and that nothing is lost by shifting the grade placement of some of the content to provide more time for the development of algorithmic mastery; and

2)    Our belief that the Number-Fraction domain in grades 3-5 can be strengthened – kept challenging, but made more reasonable.

Part I.  Number – Operations and Number – Base Ten

Grade            Domain            Suggested Revisions

K            NOP            Move to Grade 1:           

5. Understand that addition and subtraction are related. For example, when a group of 9 is decomposed into a group of 6 and a group of 3, this means not only 9 = 6 + 3 but also 9 – 3 = 6 and 9 – 6 = 3.        

                                

K&! nbsp;         &nb! sp; NBT            Move to Grade 1:

1. Understand that 10 can be thought of as a bundle of ones—a unit called a “ten.”

2. Understand that a teen number is composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

3. Compose and decompose teen numbers into a ten and some ones, e.g., by using objects or drawings, and record the compositions and decompositions in base-ten notation. For example, 10 + 8 = 18 and 14 = 10 + 4.

4. Put in order numbers presented in base-ten notation from 1 to 20 (inclusive), and be able to explain the reasoning.

5. Understand that a de! cade word refers to one, two, three, four, five, six, seven, e! ight, or nine tens.

6. Understand that the two digits of a two-digit number represent amounts of tens and ones. In 29, for example, the 2represents two tens and the 9 represents nine ones.

                          &! nbsp;        

1             NOP            Insert from Grade K           

5. Understand that addition and subtract! ion are related. For example, when a group of 9 is decomposed into a group of 6 and a group of 3, this means not only 9 = 6 + 3 but also 9 – 3 = 6 and 9 – 6 = 3.

                                   

1            NBT            Insert from Grade K

1. Understand that 10 can be thought of as a bundle of ones—a unit called a “ten.”

2. Understand that a teen number is composed of a ten and on! e, two, three, four, five, six, seven, eight, or nine ones.

3. Compose and decompose teen numbers into a ten and some ones, e.g., by using objects or drawings, and record the compositions and decompositions in base-ten notation. For example, 10 + 8 = 18 and 14 = 10 + 4.

4. Put in order numbers presented in base-ten notation from 1 to 20 (inclusive), and be able to explain the reasoning.

5. Understand that a decade word refers to one, two, three, four, five, six, seven, eight, or nine tens.

6. Understand that the two digits of a two-digit number represent amounts of tens and ones. In 29, for example, the 2represents two tens and the 9 represents nine ones.

                                

1            NBT                        Move to Grade 2:           

7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like units (tens and tens, ones and ones) and sometimes it is necessary to compose or decompose a higher value unit.

8. Given a two-digit numbe! r, mentally find 10 more or 10 less than the number, without h! aving to count.

9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and two-digit numbers.

10. Explain addition of two-digit numbers using concrete models or drawings to show composition of a ten or a hundred.

11. Add two-digit numbers to two-digit numbers using strategies based on place value,! properties of operations, and/or the inverse relationship between addition and subtraction; explain the reasoning used.

                                   

2            NBT            Insert from Grade 1:           

7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like units (tens and tens, ones and ones) and sometimes it is necessary to compose or decompose a higher value unit.

8. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.

9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and two-digit numbers.

10. Explain addition of two-digit numbers using concrete models or drawings to show composition of a ten or a hundred.

11. Add two-digit numbers to two-digit numbers using strategies based on place value, properties of operations, and/or the inverse relationship between addition and subtraction; explain the reasoning used.

                                

2            NBT            Move to grade 3:           

13. Compute sums of two three-digit numbers, and compute sums of three or four two-digit numbers, using the standard algorithm; compute differences of two three-digit numbers using the standard algorithm.

                                

2            NBT                 &n! bsp;      Move to grade 5 as culminating idea about algorithms:           

10. Understand that algorithms are predefined steps that give the correct result in every case, while strategies are

purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. For example, one might mentally compute 503 – 398 as follows! : 398 + 2 = 400, 400 +100 = 500, 500 + 3 = 503, so the answer ! is 2 + 1 00 + 3, or 105.

                                   

2          &! nbsp; NOP            New standard:        

Estimate sums and differences of two digit numbers using strategies based on rounding and place value; justify the estimates.

            !           &n! bsp;            

3            NBY            Insert from Grade 2:

13. Compute sums of two three-digit numbers, and compute sums of three or four two-digit numbers, using the standard algorithm; compute di! fferences of two three-digit numbers using the standard algorithm.

3            NBT        Revise #7:

Original #7. Understand that the distributive property is at the heart of strategies and algorithms for multiplication and division computations with numbers in base-ten notation; use the distributive property and other properties of operations to explain patterns in the multiplication table and to derive new multiplication and division equatio! ns from known ones. For example, the distributive property makes it possible to multiply 4 × 7 by decomposing 7 as 5 + 2 and using 4 × 7 = 4 × (5 + 2)= (4 × 5) + (4 × 2) = 20 + 8 = 28.        

REVISE (based on the fact that most of this appears, appropriately as 4-NBT 3a and 3b) as follows:

7. Understand that the distributive property is at the heart of strategies for multiplication computations with numbers in base-ten notation; use the distributive property and other properties of operations to explain patterns in the multiplication table and to derive new multiplication expressions from known ones. For example, the distributive property makes it possible to multiply 4 × 7 by decomposing 7 as 5 + 2 and using 4 × 7 = 4 × (5 + 2)= (4 × 5) + (4 × 2)! = 20 + 8 = 28.

3            NOP   Revise #7:           

Original #7. Solve word problems involving multiplication and division within 100, using an equation with a symbol for the unknown to represent the problem. This standard is limited to problems with whole-number quantities and whole-number quotients. Focus on situations described in the Glossary, T! able 2.        

REVISE (to  parallel 1 NOP-7) as follows: Solve word problems involving multiplication and division within 100, using objects, drawings and equations to represent the problem. This standard is limited to problems with whole-number quantities and whole-number quotients. Focus on situations described in the Glossary, Table 2.

3            NOP            New standard

Estimate sums and differences of numbers using strategies based on rounding and place value; justify the estimates.

3             NOP          Move to Grade 4           

9. Understand that multiplication and division can be used to compare quantities (see Glossary, Table 2); solve

multiplicative comparison problems with whole numbers (problems involving the notion of “times as much”).

4            NBT          Revise #6-8 to limit to 1 digit at grade 4, 2 digits at grade 5:           

ORIGINAL:

6. Compute products and whole number quotients of two-, three- or four-digit numbers and one-digit numbers, and compute products of two two-digit numbers, using strategies based on place value, the properties of operations, and/or the inverse relationship between multiplication and division; explain the reasoning used.

7. Explain why multiplication and division strategies and algorithms work, using place value and the properties of

operations. Include explanations suppo! rted by drawings, equations, or both. A range of reasonably ef! ficient algorithms may be covered, not only the standard algorithms.

8. Compute products of two-digit numbers using the standard algorithm, and check the result using estimation.

REVISED:

6. Compute products and whole number quotients of two-, three- or four-digit numbers and one-digit numbers, using strategies based on place value, the properties of operations, and/! or the inverse relationship between multiplication and division; explain the reasoning used.

7. Explain why multiplication strategies work, using place value and the properties of operations. Include explanations supported by drawings, equations, or both. A range of reasonably efficient algorithms may be covered, not only the standard algorithms.

4            NOP            New standard:           

8. Estimate products and quotients of two-, three- or four-digit numbers and one-digit numbers, using strategies based on rounding, compatible numbers and place value; justify the estimates.

!

4             NOP            Insert from grade 3:           

9. Understand that multiplication and division can be used to compare quantities (see Glossary, Table 2); solve

multiplicative comparison problems with whole numbers (problems involving the notion of “times as much”).

5            NBT            Insert, as revised, to focus on 2 digits, from Grade 4:           

6. Compute products of two two-digit numbers using strategies based on place value and the properties of operations; explain the reasoning used.

7. Explain why multiplication and division strategies and algorithms work, using place value and the properties of