Overview

Lesson Procedure

70-75 minutes of class time across 5 phases.

1. Warm-Up10 minutes
2. Direct Instruction20 minutes
3. Guided Practice15-20 minutes
4. Independent Practice15 minutes
5. Closure10 minutes

1. Warm-Up10 minutes

   Warm-Up Prompt

   "Look at a few grains of table salt with a hand lens (or at a photo). Many are tiny cubes. Why would particles far too small to see build a cube again and again? Write your best explanation in two sentences. Then read Kelvin's claim from 1893: 'Every crystal is a homogeneous assemblage of small bodies or molecules.' Which words would you need explained before you could test it?"

   Collect explanations and the words students flag (usually homogeneous and assemblage). Point out that Kelvin could not see molecules either: in 1893 no instrument could show how the particles of a crystal are arranged, so his sentence is a model, not an observation. Texts in the grades 11-CCR band often work this way, building an invisible structure out of words and reasoning. RST.11-12.10 asks students, by the end of grade 12, to read such texts alone and with real understanding. Today they learn four moves, watch them modeled on Kelvin, use them with a partner, and then alone.

2. Direct Instruction20 minutes

   Part 1: Four moves. Show Diagram 1. Build: make the model the words describe, as a sketch or with balls and sticks, and change it when the text adds a condition. Chain: trace the reasoning link by link, marking which sentences define, which set conditions, which report observations and which draw inferences (watch for hence, thus, therefore, must). Test: ask what holds up each link (an observation, a measurement, an analogy or pure geometry) and find the weakest one. Place: put the text in the history of its idea: what the writer could know then, whose work it answers, and what came later. The table shows how the supports come off.

   Supports for each source (Yes = given). Each source has fewer supports than the one before it.

   Support	Source 1 (modeled)	Source 2 (pairs)	Source 3 (alone)	Source 4 (cold read)
   Teacher reasons aloud	Yes	No	No	No
   Glossary	Yes	No	No	No
   Finished sketch (Diagram 2)	Yes	No	No	No
   Partner	No	Yes	No	No
   Prompts for each move	Yes	Yes	No	No
   Context note	Yes	Yes	Yes	No
   Move card with names only	Yes	Yes	Yes	No
   Supports given	6 of 7	4 of 7	2 of 7	0 of 7

   Part 2: The modeled text. Read the context note aloud: William Thomson, Lord Kelvin (1824-1907), was one of the leading physicists of the nineteenth century; the temperature scale that starts at absolute zero is named for him. He gave this lecture to a student science club at Oxford in May 1893, and it was printed the next year. X-rays were not discovered until 1895, and no one could yet measure how the particles in a crystal are placed. Kelvin therefore reasons from geometry alone. Glossary: tactics (here, arrangement, from the Greek for putting in order), homogeneous (the same throughout), assemblage (a collection of things placed together), converse (the statement with its two halves swapped), equi-distant (at the same distance), in virtue of (because of), rectangularly grouped (arranged in rows at right angles, like a checkerboard). Reason aloud through one move at a time, then hand out Diagram 2 as the finished sketch.

   1§ 1. My subject this evening is not the physical properties of crystals, not even their dynamics; it is merely the geometry of the structure—the arrangement of the molecules in the constitution of a crystal. Every crystal is a homogeneous assemblage of small bodies or molecules. The converse proposition is scarcely true, unless in a very extended sense of the term crystal (§ 20 below). I can best explain a homogeneous assemblage of molecules by asking you to think of a homogeneous assemblage of people. To be homogeneous every person of the assemblage must be equal and similar to every other: they must be seated in rows or standing in rows in a perfectly similar manner. Each person, except those on the borders of the assemblage, must have a neighbour on one side and an equi-distant neighbour on the other: a neighbour on the left front and an equi-distant neighbour behind on the right, a neighbour on the right front and an equi-distant neighbour behind on the left. His two neighbours in front and his two neighbours behind are members of two rows equal and similar to the rows consisting of himself and his right-hand and left-hand neighbours, and their neighbours’ neighbours indefinitely to right and left. In particular cases the nearest of the front and rear neighbours may be right in front and right in rear; but we must not confine our attention to the rectangularly grouped assemblages thus constituted. Now let there be equal and similar assemblages on floors above and below that which we have been considering, and let there be any indefinitely great number of floors at equal distances from one another above and below. Think of any one person on any intermediate floor and of his nearest neighbours on the floors above and below. These three persons must be exactly in one line; this, in virtue of the homogeneousness of the assemblages on the three floors, will secure that every person on the intermediate floor is exactly in line with his nearest neighbours above and below. The same condition of alignment must be fulfilled by every three consecutive floors, and we thus have a homogeneous assemblage of people in three dimensions of space. In particular cases every person’s nearest neighbour in the floor above may be vertically over him, but we must not confine our attention to assemblages thus rectangularly grouped in vertical lines.

   • Place: before reading closely

     What could Kelvin know in 1893, and what kind of text is this?

     Result: It is a lecture to students, printed a year later, by a physicist speaking in the first person ("My subject this evening"). He says at once that he is not discussing crystals' physical properties but "merely the geometry of the structure." Written before X-rays, it cannot report where molecules are; it describes what an orderly arrangement would have to be like.

   • Build: turning the analogy into a sketch

     Sketch one floor of the "homogeneous assemblage of people."

     Result: Draw dots in straight rows, all rows alike. Around one person, place three pairs of neighbours, each pair opposite and equally far away: one on each side, one at the left front with its partner behind on the right, and one at the right front with its partner behind on the left. Kelvin then warns that the rows need not meet at right angles, so the grid should be slanted. Diagram 2 is the finished sketch.

   • Chain: how the paragraph is built

     Label the links of the paragraph in order.

     Result: Claim: every crystal is a homogeneous assemblage. Limit: the converse (every homogeneous assemblage is a crystal) "is scarcely true." Definition by analogy: every person "equal and similar to every other." Conditions for one floor, then for many floors "at equal distances." Inference: three people on three floors "must be exactly in one line," and this "will secure" alignment for all. Conclusion: "a homogeneous assemblage of people in three dimensions of space."

   • Test: what holds up each link

     Which sentences rest on observation, and which on geometry?

     Result: None reports an observation. The opening claim about every crystal is a model, and the paragraph gives no evidence for it; the rest follows from the definition, so it is as certain as geometry but says nothing about whether real crystals obey it. The "must" in "These three persons must be exactly in one line" is a logical must: it follows from "homogeneousness," not from watching anything.

   • Test: the two warnings

     Why does Kelvin twice say "we must not confine our attention to" rectangular groupings?

     Result: Readers picture people in a theater or soldiers on parade, lined up at right angles. Kelvin's definition allows slanted rows and floors whose people are not directly above one another, and he wants the model to cover every such case. The warnings keep a familiar picture from narrowing a general idea: a reader who builds a checkerboard has built only a special case.

   Teacher tip. Build the model live: first draw a checkerboard, read the sentence about rectangular groupings, and redraw it slanted. Students should see that at this level the model is revised by the text, sentence by sentence, and that a sketch from memory of a theater would be wrong.

3. Guided Practice15-20 minutes

   Give pairs the move card with prompts and this context note, with no glossary and no finished sketch: A. E. H. Tutton (1864-1938) was a British crystallographer; his 1911 book explains the science for general readers, partly through its history. This passage describes the work of the Abbé René Just Haüy (1743-1822), a French priest and mineralogist, published in 1784 and 1801. Calcite is a common mineral that splits cleanly in three directions; to cleave a crystal is to split it along such a plane. Pairs try each move before asking for help.

   1The basis of Haüy’s conceptions was undoubtedly cleavage. He describes most graphically on page 10 of his “Essai” of 1784 how he was led to make the striking observation that a hexagonal prism of calcite, terminated by a pair of hexagons normal to the prism axis, similar to the prisms shown in Fig. 6 (Plate III.) except that the ends were flat, showed oblique internal cleavage cracks, by enhancing which with the aid of a few judicious blows he was able to separate from the middle of the prism a kernel in the shape of a rhombohedron, the now well-known cleavage rhombohedron of calcite. He then tried what kinds of kernels he could get from dog-tooth spar (illustrated in Fig. 7) and other different forms of calcite, and he was surprised to find that they all yielded the same rhombohedral kernel. He subsequently investigated the cleavage kernels of other minerals, particularly of gypsum, fluorspar, topaz, and garnet, and found that each mineral yielded its own particular kernel. He next imagined the kernels to become smaller and smaller, until the particles thus obtained by cleaving the mineral along its cleavage directions ad infinitum were the smallest possible. These miniature kernels having the full composition of the mineral he terms “Molécules Constituantes” in the 1784 “Essai,” but in the 1801 “Traité” he calls them “Molécules Intégrantes” as above mentioned. He soon found that there were three distinct types of molécules intégrantes, tetrahedra, triangular prisms, and parallelepipeda, and these he considered to be the crystallographic structural units.

   [...]

   2The essential difference between Haüy’s views and our present ones, which will be explained in Chapter IX., is that Haüy takes cleavage absolutely as his guide, and considers the particles, into which the ultimate operation of cleavage divides a crystal, as the solid structural units of the crystal, the unit thus having the shape of at least the molécule intégrante. Now every crystalline substance does not develop cleavage, and others only develop it along a single plane, or along a couple of planes parallel to the same direction, that of their intersection and of the axis of the prism which two such cleavages would produce, and which prism would be of unlimited length, being unclosed.

   3Again, in other cases cleavage, such as the octahedral cleavage of fluorspar, yields octahedral or tetrahedral molécules intégrantes which are not congruent, that is to say, do not fit closely together to fill space, as is the essence of Haüy’s theory. Hence, speaking generally, partitioning by means of cleavage directions does not essentially and invariably yield identical plane-faced molecules which fit together in contact to completely fill space, although in the particular instances chosen from familiar substances by Haüy it often happens to do so. Haüy’s theory is thus not adequately general, and the advance of our knowledge of crystal forms has rendered it more and more apparent that Haüy’s theory was quite insufficient, and his molécules intégrantes and soustractives mere geometrical abstractions, having no actual basis in material fact; but that at the same time it gave us a most valuable indication of where to look for the true conception.

   1. Build. Prompt: sketch what Haüy did to the calcite prism and what he imagined next. Teacher note: a six-sided prism with flat ends; cracks slanting through it; a slanted six-faced block (a rhombohedron) knocked out of the middle; then the same block imagined smaller and smaller until it is the smallest unit, the molécule intégrante.
   2. Chain. Prompt: label the observation, the inference and the new name in paragraph 1. Teacher note: observation: every form of calcite "yielded the same rhombohedral kernel," and each other mineral "its own particular kernel." Inference: keep cleaving and the kernels become "the smallest possible," the building units of the crystal. Naming: Haüy calls them molécules constituantes, later intégrantes, and sorts them into three shapes.
   3. Test. Prompt: find the two pieces of evidence Tutton uses against the theory. Teacher note: first, "every crystalline substance does not develop cleavage," and some cleave in only one or two directions, which cannot enclose a block; second, fluorspar cleaves into octahedra or tetrahedra, which "do not fit closely together to fill space." A theory built on cleavage fails where cleavage does not give space-filling blocks.
   4. Place. Prompt: what is Tutton's final verdict, and what does "our present ones" tell you about when he writes? Teacher note: the theory is "not adequately general" and its units "mere geometrical abstractions," yet "it gave us a most valuable indication of where to look for the true conception." Tutton writes from 1911, judging a theory of 1784 by the knowledge of his own day, one year before X-rays were first shown to be diffracted by crystals.

   Close with one question: Kelvin (Source 1) and Haüy both describe crystals as repeated identical units. Which of them reasons from real crystals, and which from a definition? (Haüy from calcite he broke with his own hands; Kelvin from the definition of homogeneous, with no crystal in sight.) Tell students that the next source comes with a context note and the move names only.

4. Independent Practice15 minutes

   Each student reads alone, with the names-only move card and no prompts, glossary or partner. Context note: William Henry Bragg (1862-1942) and his son William Lawrence Bragg shared the 1915 Nobel Prize in Physics for working out the arrangement of atoms in crystals with X-rays. This book prints lectures he gave to young audiences at the Royal Institution in London. Earlier in the chapter he explains how X-rays reflected from the layers of atoms in a crystal reveal how far apart the layers are. Students write one margin note for each move and a one-sentence statement of what the paragraph shows. The quiz uses this source.

   1First of all let us take the diamond, which is a prince among crystals. It is not only a beautiful and valuable gem, but in its structure it teaches us many things concerning the most fundamental truths of chemistry, particularly organic chemistry. Only one atom, that of carbon, goes to the building of the diamond; but that atom is of vital interest to us. It is a fundamental constituent of foods and fuels, dyes and explosives, of our own bodies and many other things. The structure of the diamond is remarkably simple, though, like all constructions in space, it is difficult to comprehend quickly. We are so accustomed to drawings on the flat, paper and pencil are so handy, that our minds easily grasp the details of a plane design. But we cannot draw in space; we can only construct models at much cost of time and energy, and so our power of conceiving arrangements in space is feeble from want of practice. A few have the natural gift, and some, being crystallographers, have trained themselves to think in three dimensions. Most of us find a great difficulty in our first efforts to realize the arrangements of the atoms and molecules of the crystal. Nevertheless, the diamond structure shown in Plate XIV a will become clear at the cost of a little consideration. The black balls represent carbon atoms, in respect to position only, not in any way as to size and form, of which we know very little. Every carbon atom is at the center of gravity of four others; these four lie at the corners of a four-cornered pyramid, or tetrahedron, and the first carbon atom is, of course, at the same distance from each of them. We have reason to believe that the ties between the atoms are very strong, and there is only one form of tie throughout the whole structure. In its uniform simplicity and regularity we can surely see the reason why the diamond is placed in the highest class on the scale of hardness. If it is pressed against any other crystal it is the atoms of the latter that must give way, not the atoms of the diamond. The diamond has a cleavage plane. In the figure it is parallel to the plane of the table on which the model stands; there are four such planes, one parallel to each face of the four-faced pyramid. The model can be turned over so as to rest on any one of the four faces, and looks exactly the same in each position. The distance between the centers of two neighboring carbons is 1.54 Ångstrom units; this unit is the hundred-millionth of a centimeter. It does not seem surprising that this particular plane should be the cleavage plane, because it cuts straight across the vertical connections between the horizontal layers that appear in the figure. Each of the layers may be described as a puckered hexagonal network.

5. Closure10 minutes

   Exit ticket: (1) Which move did Source 3 demand most, and at which sentence? (2) Kelvin in 1893 and Bragg in 1925 both describe an arrangement you cannot see. Name one thing Bragg can say that Kelvin could not, and why.

   Teacher note on the science and the sources. The passages are quoted exactly as printed in 1894, 1911 and 1925, with the writers' spellings. The science holds up, with three notes. First, Kelvin's "homogeneous assemblage" is what crystallographers now call a lattice, a pattern in which every point has identical surroundings; in a later section of the lecture he credits the French physicist Auguste Bravais, who showed in 1850 that there are only fourteen kinds in three dimensions. Second, Tutton wrote in 1911, one year before Max von Laue showed (1912) that crystals diffract X-rays; within a year the Braggs were using the effect to locate atoms, and diamond was among the first structures they solved, in 1913. Tutton's verdict on Haüy has lasted: the idea of a smallest repeating unit survives as the unit cell of modern crystallography. Third, Bragg's figures for diamond agree with modern measurements. The texts contain no offensive language; Kelvin's "his" for any person reflects the usage of 1893, and Tutton's "Abbé" is a French title for a priest.

   For homework: a cold read. Students read the passage below alone, with no card, no context note and no glossary, and keep a log of where they got stuck.

   1§ 4. The case in which each of the four faces of each of the tetrahedrons of § 3 is an equilateral triangle is particularly interesting. An assemblage fulfilling this condition may conveniently be called an ‘equilateral homogeneous assemblage,’ or, for brevity, an ‘equilateral assemblage.’ In an equilateral assemblage C’s twelve neighbours are all equi-distant from it. I hold in my hand a cluster of thirteen little black balls, made up by taking one of them and placing the twelve others in contact with it (and therefore packed in the closest possible order), and fixing them all together by fish-glue. You see it looks, in size, colour, and shape, quite like a mulberry. The accompanying diagram shows a stereoscopic view of a similar cluster of balls painted white for the photograph.

   2§ 5. By adding ball after ball to such a cluster of thirteen, and always taking care to place each additional ball in some position in which it is properly in line with others, so as to make the whole assemblage homogeneous, we can exercise ourselves in a very interesting manner in the building up of any possible form of crystal of the class called ‘cubic’ by some writers and ‘octahedral’ by others. You see before you several examples. I advise any of you who wish to study crystallography to contract with a wood-turner, or a maker of beads for furniture tassels or for rosaries, for a thousand wooden balls of about half an inch diameter each. Holes through them will do no harm and may even be useful; but make sure that the balls are as nearly equal to one another, and each as nearly spherical, as possible.

Differentiation Strategies

Assessment Guidance

1. Question 1 of 20 · Multiple Choice

   Place: why does Bragg begin his results with the diamond?

   AIts structure is simple, one kind of atom, and it shows basic truths about carbon BIt was the most valuable gem in the Royal Institution's collection of minerals CIt was the first crystal ever studied by anyone with the help of a microscope DIts structure is so complicated that it shows off the full power of X-ray methods

2. Question 2 of 20 · Multiple Choice

   Build: what do the black balls in Bragg's model show?

   AThe size and the shape of each carbon atom BOnly where each carbon atom is placed CThe electrons that hold the atoms together DCarbon atoms mixed with atoms of other kinds

3. Question 3 of 20 · Multiple Choice

   Build: which model matches the structure Bragg describes?

   AEach atom touches twelve others, six in its own layer and three above and below BEach atom sits at the center of a cube with a neighbor at every one of its corners CEach atom has four neighbors in a flat square around it, all in one plane DEach atom is the center of a tetrahedron, with its four neighbors at the corners

4. Question 4 of 20 · Multiple Choice

   Bragg writes that each carbon atom "is at the center of gravity of four others." What does center of gravity mean here?

   AThe point where the atom would land if it were dropped from a height BThe point inside the pyramid that is equally placed among the four CThe heaviest of the five carbon atoms in each group of neighbors DThe bottom face of the pyramid, on which the model rests

5. Question 5 of 20 · Multiple Choice

   Chain: which sequence matches Bragg's reasoning about hardness?

   AStrong ties of one kind in a regular structure, so the diamond is hardest, so others give way BOther crystals give way to it, so its ties must be strong, so its structure must be regular CThe diamond is a valuable gem, so it must be hard, so the ties between its atoms must be strong DThe diamond has a cleavage plane, so it is the hardest crystal, so its structure is regular

6. Question 6 of 20 · Multiple Choice

   Test: which link of the hardness argument does Bragg present as a belief rather than a measurement?

   AThat the diamond is made of carbon atoms only BThat the distance between neighboring carbons is 1.54 units CThat the ties between the atoms are very strong DThat there are four cleavage planes in the diamond

7. Question 7 of 20 · Multiple Choice

   "If it is pressed against any other crystal it is the atoms of the latter that must give way, not the atoms of the diamond." What does this sentence tell a reader?

   AA diamond scratches other crystals rather than being scratched by them BA diamond breaks apart when it is pressed hard against another crystal CThe atoms of the diamond move out of the way of the harder atoms in the other crystal DOther crystals are made of the same atoms as the diamond

8. Question 8 of 20 · Multiple Choice

   According to Bragg, how many cleavage planes does the diamond have, and how are they arranged?

   AOne, parallel to the table on which the model in the plate stands BFour, one parallel to each face of the four-faced pyramid CSix, one for each edge of the four-faced pyramid DThree, one for each direction of a cube's edges

9. Question 9 of 20 · Multiple Choice

   "The model can be turned over so as to rest on any one of the four faces, and looks exactly the same in each position." Why does Bragg add this?

   ATo show that the model is well made and does not wobble on any face BTo prove that the diamond has only one cleavage plane CTo show that the four cleavage directions are all equivalent DTo explain why diamonds are cut with many flat faces

10. Question 10 of 20 · Multiple Choice

    Bragg gives the distance between neighboring carbon centers as 1.54 Ångstrom units and says the unit is "the hundred-millionth of a centimeter." What is the distance in centimeters?

    A1.54 × 10^-6 cm B1.54 × 10^-8 cm C1.54 × 10^-10 cm D1.54 × 10⁸ cm

11. Question 11 of 20 · Multiple Choice

    One nanometer is 10^-7 cm. Using Bragg's figure, how far apart are neighboring carbon atoms in nanometers?

    A0.0154 nm B15.4 nm C1.54 nm D0.154 nm

12. Question 12 of 20 · Multiple Choice

    Chain: why, according to Bragg, is that particular plane the cleavage plane?

    AIt passes between the heaviest atoms in the crystal BIt cuts only the vertical ties between the horizontal layers CIt is the plane on which the model happens to stand DIt is the only flat surface to be found anywhere in the whole structure

13. Question 13 of 20 · Multiple Choice

    What does Bragg mean by "a puckered hexagonal network"?

    AA layer of six-sided rings that is not flat but rises and falls BA flat sheet of squares joined at their corners CA tangle of atoms with no regular pattern at all DA single ring of six atoms standing on its own, apart from any others

14. Question 14 of 20 · Multiple Choice

    Why does Bragg spend four sentences on how hard it is to picture arrangements in space?

    ATo warn readers that the diamond model is probably wrong BTo argue that only trained crystallographers should read on CTo admit that X-rays cannot show the positions of atoms DTo prepare readers for difficulty and urge them to persist

15. Question 15 of 20 · Multiple Choice

    Tutton (Source 2) faults Haüy for treating cleavage kernels as the solid units of a crystal. How does Bragg's model avoid that fault?

    AIt uses the cleavage of the diamond to find the shape of each carbon atom BIt claims only the positions of the atoms, not their size or shape CIt shows that the diamond has no cleavage at all DIt treats each carbon atom as a small tetrahedral kernel

16. Question 16 of 20 · Multiple Choice

    Which grade 12 student shows the kind of reading RST.11-12.10 describes?

    AJo reads a new paper on crystal defects alone, models it and judges each link of its argument BKai reads the same paper with a tutor who restates each paragraph before he goes on to the next CLee skims the paper's abstract and figure captions and writes a summary of them for class DMax copies every technical term in the paper and looks each one up in a science dictionary

17. Question 17 of 20 · Short Answer

    Build: from Bragg's description alone, write instructions for building a model of one carbon atom and its neighbors with balls and sticks, and name one thing your model would show that the real atom does not have, according to the text.

    Show model answer

18. Question 18 of 20 · Short Answer

    Test: Bragg writes that "we can surely see the reason why the diamond is placed in the highest class on the scale of hardness." Separate the evidence from the inference in his hardness argument, and explain what the word "surely" does.

    Show model answer

19. Question 19 of 20 · Short Answer

    Bragg says our "power of conceiving arrangements in space is feeble from want of practice." How does Kelvin in Source 1 deal with the same problem, and which approach worked better for you? Quote each source.

    Show model answer

20. Question 20 of 20 · Short Answer

    Reading log: which of the four moves did Source 3 demand most? Copy the words where you used it, explain what you did, and say what you understood afterward.

    Show model answer

These standards connect to RST.11-12.10: prerequisites to review first, parallel standards at the same level, and next steps that build on it.

Standard text: Common Core State Standards for English Language Arts & Literacy. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.