1.8: Anne Radcliffe, excerpt from The Mysteries of Udolpho (1794)
Anne Radcliffe, excerpt from The Mysteries of Udolpho (1794)
Jeanette A. Laredo
Chapter 26
I will advise you where to plant yourselves;
Acquaint you with the perfect spy o’ the time,
The moment on ‘t; for ‘t must be done to-night.
MACBETH
Emily was somewhat surprised, on the following day, to find that Annette had heard of Madame Montoni’s confinement in the chamber over the portal, as well as of her purposed visit there, on the approaching night. That the circumstance, which Barnardine had so solemnly enjoined her to conceal, he had himself told to so indiscreet an hearer as Annette, appeared very improbable, though he had now charged her with a message, concerning the intended interview. He requested, that Emily would meet him, unattended, on the terrace, at a little after midnight, when he himself would lead her to the place he had promised; a proposal, from which she immediately shrunk, for a thousand vague fears darted athwart her mind, such as had tormented her on the preceding night, and which she neither knew how to trust, or to dismiss. It frequently occurred to her, that Barnardine might have deceived her, concerning Madame Montoni, whose murderer, perhaps, he really was; and that he had deceived her by order of Montoni, the more easily to draw her into some of the desperate designs of the latter. The terrible suspicion, that Madame Montoni no longer lived, thus came, accompanied by one not less dreadful for herself. Unless the crime, by which the aunt had suffered, was instigated merely by resentment, unconnected with profit, a motive, upon which Montoni did not appear very likely to act, its object must be unattained, till the niece was also dead, to whom Montoni knew that his wife’s estates must descend. Emily remembered the words, which had informed her, that the contested estates in France would devolve to her, if Madame Montoni died, without consigning them to her husband, and the former obstinate perseverance of her aunt made it too probable, that she had, to the last, withheld them. At this instant, recollecting Barnardine’s manner, on the preceding night, she now believed, what she had then fancied, that it expressed malignant triumph. She shuddered at the recollection, which confirmed her fears, and determined not to meet him on the terrace. Soon after, she was inclined to consider these suspicions as the extravagant exaggerations of a timid and harassed mind, and could not believe Montoni liable to such preposterous depravity as that of destroying, from one motive, his wife and her niece. She blamed herself for suffering her romantic imagination to carry her so far beyond the bounds of probability, and determined to endeavour to check its rapid flights, lest they should sometimes extend into madness. Still, however, she shrunk from the thought of meeting Barnardine, on the terrace, at midnight; and still the wish to be relieved from this terrible suspense, concerning her aunt, to see her, and to sooth her sufferings, made her hesitate what to do.
‘Yet how is it possible, Annette, I can pass to the terrace at that hour?’ said she, recollecting herself, ‘the sentinels will stop me, and Signor Montoni will hear of the affair.’
‘O ma’amselle! that is well thought of,’ replied Annette. ‘That is what Barnardine told me about. He gave me this key, and bade me say it unlocks the door at the end of the vaulted gallery, that opens near the end of the east rampart, so that you need not pass any of the men on watch. He bade me say, too, that his reason for requesting you to come to the terrace was, because he could take you to the place you want to go to, without opening the great doors of the hall, which grate so heavily.’
Emily’s spirits were somewhat calmed by this explanation, which seemed to be honestly given to Annette. ‘But why did he desire I would come alone, Annette?’ said she.
‘Why that was what I asked him myself, ma’amselle. Says I, Why is my young lady to come alone? — Surely I may come with her! — What harm can I do? But he said “No — no — I tell you not,” in his gruff way. Nay, says I, I have been trusted in as great affairs as this, I warrant, and it’s a hard matter if I can’t keep a secret now. Still he would say nothing but —“No — no — no.” Well, says I, if you will only trust me, I will tell you a great secret, that was told me a month ago, and I have never opened my lips about it yet — so you need not be afraid of telling me. But all would not do. Then, ma’amselle, I went so far as to offer him a beautiful new sequin, that Ludovico gave me for a keep sake, and I would not have parted with it for all St. Marco’s Place; but even that would not do! Now what can be the reason of this?  No, ma’amselle, that he did not.’
Emily enquired who did, but Annette shewed, that she COULD keep a secret.
During the remainder of the day, Emily’s mind was agitated with doubts and fears and contrary determinations, on the subject of meeting this Barnardine on the rampart, and submitting herself to his guidance, she scarcely knew whither. Pity for her aunt and anxiety for herself alternately swayed her determination, and night came, before she had decided upon her conduct. She heard the castle clock strike eleven — twelve — and yet her mind wavered. The time, however, was now come, when she could hesitate no longer: and then the interest she felt for her aunt overcame other considerations, and, bidding Annette follow her to the outer door of the vaulted gallery, and there await her return, she descended from her chamber. The castle was perfectly still, and the great hall, where so lately she had witnessed a scene of dreadful contention, now returned only the whispering footsteps of the two solitary figures gliding fearfully between the pillars, and gleamed only to the feeble lamp they carried. Emily, deceived by the long shadows of the pillars and by the catching lights between, often stopped, imagining she saw some person, moving in the distant obscurity of the perspective; and, as she passed these pillars, she feared to turn her eyes toward them, almost expecting to see a figure start out from behind their broad shaft. She reached, however, the vaulted gallery, without interruption, but unclosed its outer door with a trembling hand, and, charging Annette not to quit it and to keep it a little open, that she might be heard if she called, she delivered to her the lamp, which she did not dare to take herself because of the men on watch, and, alone, stepped out upon the dark terrace. Every thing was so still, that she feared, lest her own light steps should be heard by the distant sentinels, and she walked cautiously towards the spot, where she had before met Barnardine, listening for a sound, and looking onward through the gloom in search of him. At length, she was startled by a deep voice, that spoke near her, and she paused, uncertain whether it was his, till it spoke again, and she then recognized the hollow tones of Barnardine, who had been punctual to the moment, and was at the appointed place, resting on the rampart wall. After chiding her for not coming sooner, and saying, that he had been waiting nearly half an hour, he desired Emily, who made no reply, to follow him to the door, through which he had entered the terrace.
While he unlocked it, she looked back to that she had left, and, observing the rays of the lamp stream through a small opening, was certain, that Annette was still there. But her remote situation could little befriend Emily, after she had quitted the terrace; and, when Barnardine unclosed the gate, the dismal aspect of the passage beyond, shewn by a torch burning on the pavement, made her shrink from following him alone, and she refused to go, unless Annette might accompany her. This, however, Barnardine absolutely refused to permit, mingling at the same time with his refusal such artful circumstances to heighten the pity and curiosity of Emily towards her aunt, that she, at length, consented to follow him alone to the portal.
He then took up the torch, and led her along the passage, at the extremity of which he unlocked another door, whence they descended, a few steps, into a chapel, which, as Barnardine held up the torch to light her, Emily observed to be in ruins, and she immediately recollected a former conversation of Annette, concerning it, with very unpleasant emotions. She looked fearfully on the almost roofless walls, green with damps, and on the gothic points of the windows, where the ivy and the briony had long supplied the place of glass, and ran mantling among the broken capitals of some columns, that had once supported the roof. Barnardine stumbled over the broken pavement, and his voice, as he uttered a sudden oath, was returned in hollow echoes, that made it more terrific. Emily’s heart sunk; but she still followed him, and he turned out of what had been the principal aisle of the chapel. ‘Down these steps, lady,’ said Barnardine, as he descended a flight, which appeared to lead into the vaults; but Emily paused on the top, and demanded, in a tremulous tone, whither he was conducting her.
‘To the portal,’ said Barnardine.
‘Cannot we go through the chapel to the portal?’ said Emily.
‘No, Signora, that leads to the inner court, which I don’t choose to unlock. This way, and we shall reach the outer court presently.’
Emily still hesitated; fearing not only to go on, but, since she had gone thus far, to irritate Barnardine by refusing to go further.
‘Come, lady,’ said the man, who had nearly reached the bottom of the flight, ‘make a little haste; I cannot wait here all night.’
‘Whither do these steps lead?’ said Emily, yet pausing.
‘To the portal,’ repeated Barnardine, in an angry tone, ‘I will wait no longer.’ As he said this, he moved on with the light, and Emily, fearing to provoke him by further delay, reluctantly followed. From the steps, they proceeded through a passage, adjoining the vaults, the walls of which were dropping with unwholesome dews, and the vapours, that crept along the ground, made the torch burn so dimly, that Emily expected every moment to see it extinguished, and Barnardine could scarcely find his way. As they advanced, these vapours thickened, and Barnardine, believing the torch was expiring, stopped for a moment to trim it. As he then rested against a pair of iron gates, that opened from the passage, Emily saw, by uncertain flashes of light, the vaults beyond, and, near her, heaps of earth, that seemed to surround an open grave. Such an object, in such a scene, would, at any time, have disturbed her; but now she was shocked by an instantaneous presentiment, that this was the grave of her unfortunate aunt, and that the treacherous Barnardine was leading herself to destruction. The obscure and terrible place, to which he had conducted her, seemed to justify the thought; it was a place suited for murder, a receptacle for the dead, where a deed of horror might be committed, and no vestige appear to proclaim it. Emily was so overwhelmed with terror, that, for a moment, she was unable to determine what conduct to pursue. She then considered, that it would be vain to attempt an escape from Barnardine, by flight, since the length and the intricacy of the way she had passed would soon enable him to overtake her, who was unacquainted with the turnings, and whose feebleness would not suffer her to run long with swiftness. She feared equally to irritate him by a disclosure of her suspicions, which a refusal to accompany him further certainly would do; and, since she was already as much in his power as it was possible she could be, if she proceeded, she, at length, determined to suppress, as far as she could, the appearance of apprehension, and to follow silently whither he designed to lead her. Pale with horror and anxiety, she now waited till Barnardine had trimmed the torch, and, as her sight glanced again upon the grave, she could not forbear enquiring, for whom it was prepared. He took his eyes from the torch, and fixed them upon her face without speaking. She faintly repeated the question, but the man, shaking the torch, passed on; and she followed, trembling, to a second flight of steps, having ascended which, a door delivered them into the first court of the castle. As they crossed it, the light shewed the high black walls around them, fringed with long grass and dank weeds, that found a scanty soil among the mouldering stones; the heavy buttresses, with, here and there, between them, a narrow grate, that admitted a freer circulation of air to the court, the massy iron gates, that led to the castle, whose clustering turrets appeared above, and, opposite, the huge towers and arch of the portal itself. In this scene the large, uncouth person of Barnardine, bearing the torch, formed a characteristic figure. This Barnardine was wrapt in a long dark cloak, which scarcely allowed the kind of half-boots, or sandals, that were laced upon his legs, to appear, and shewed only the point of a broad sword, which he usually wore, slung in a belt across his shoulders. On his head was a heavy flat velvet cap, somewhat resembling a turban, in which was a short feather; the visage beneath it shewed strong features, and a countenance furrowed with the lines of cunning and darkened by habitual discontent.
The view of the court, however, reanimated Emily, who, as she crossed silently towards the portal, began to hope, that her own fears, and not the treachery of Barnardine, had deceived her. She looked anxiously up at the first casement, that appeared above the lofty arch of the portcullis; but it was dark, and she enquired, whether it belonged to the chamber, where Madame Montoni was confined. Emily spoke low, and Barnardine, perhaps, did not hear her question, for he returned no answer; and they, soon after, entered the postern door of the gate-way, which brought them to the foot of a narrow stair-case, that wound up one of the towers.
‘Up this stair-case the Signora lies,’ said Barnardine.
‘Lies!’ repeated Emily faintly, as she began to ascend.
‘She lies in the upper chamber,’ said Barnardine.
As they passed up, the wind, which poured through the narrow cavities in the wall, made the torch flare, and it threw a stronger gleam upon the grim and sallow countenance of Barnardine, and discovered more fully the desolation of the place — the rough stone walls, the spiral stairs, black with age, and a suit of antient armour, with an iron visor, that hung upon the walls, and appeared a trophy of some former victory.
Having reached a landing-place, ‘You may wait here, lady,’ said he, applying a key to the door of a chamber, ‘while I go up, and tell the Signora you are coming.’
‘That ceremony is unnecessary,’ replied Emily, ‘my aunt will rejoice to see me.’
‘I am not so sure of that,’ said Barnardine, pointing to the room he had opened: ‘Come in here, lady, while I step up.’
Emily, surprised and somewhat shocked, did not dare to oppose him further, but, as he was turning away with the torch, desired he would not leave her in darkness. He looked around, and, observing a tripod lamp, that stood on the stairs, lighted and gave it to Emily, who stepped forward into a large old chamber, and he closed the door. As she listened anxiously to his departing steps, she thought he descended, instead of ascending, the stairs; but the gusts of wind, that whistled round the portal, would not allow her to hear distinctly any other sound. Still, however, she listened, and, perceiving no step in the room above, where he had affirmed Madame Montoni to be, her anxiety increased, though she considered, that the thickness of the floor in this strong building might prevent any sound reaching her from the upper chamber. The next moment, in a pause of the wind, she distinguished Barnardine’s step descending to the court, and then thought she heard his voice; but, the rising gust again overcoming other sounds, Emily, to be certain on this point, moved softly to the door, which, on attempting to open it, she discovered was fastened. All the horrid apprehensions, that had lately assailed her, returned at this instant with redoubled force, and no longer appeared like the exaggerations of a timid spirit, but seemed to have been sent to warn her of her fate. She now did not doubt, that Madame Montoni had been murdered, perhaps in this very chamber; or that she herself was brought hither for the same purpose. The countenance, the manners and the recollected words of Barnardine, when he had spoken of her aunt, confirmed her worst fears. For some moments, she was incapable of considering of any means, by which she might attempt an escape. Still she listened, but heard footsteps neither on the stairs, or in the room above; she thought, however, that she again distinguished Barnardine’s voice below, and went to a grated window, that opened upon the court, to enquire further. Here, she plainly heard his hoarse accents, mingling with the blast, that swept by, but they were lost again so quickly, that their meaning could not be interpreted; and then the light of a torch, which seemed to issue from the portal below, flashed across the court, and the long shadow of a man, who was under the arch-way, appeared upon the pavement. Emily, from the hugeness of this sudden portrait, concluded it to be that of Barnardine; but other deep tones, which passed in the wind, soon convinced her he was not alone, and that his companion was not a person very liable to pity.
When her spirits had overcome the first shock of her situation, she held up the lamp to examine, if the chamber afforded a possibility of an escape. It was a spacious room, whose walls, wainscoted with rough oak, shewed no casement but the grated one, which Emily had left, and no other door than that, by which she had entered. The feeble rays of the lamp, however, did not allow her to see at once its full extent; she perceived no furniture, except, indeed, an iron chair, fastened in the centre of the chamber, immediately over which, depending on a chain from the ceiling, hung an iron ring. Having gazed upon these, for some time, with wonder and horror, she next observed iron bars below, made for the purpose of confining the feet, and on the arms of the chair were rings of the same metal. As she continued to survey them, she concluded, that they were instruments of torture, and it struck her, that some poor wretch had once been fastened in this chair, and had there been starved to death. She was chilled by the thought; but, what was her agony, when, in the next moment, it occurred to her, that her aunt might have been one of these victims, and that she herself might be the next! An acute pain seized her head, she was scarcely able to hold the lamp, and, looking round for support, was seating herself, unconsciously, in the iron chair itself; but suddenly perceiving where she was, she started from it in horror, and sprung towards a remote end of the room. Here again she looked round for a seat to sustain her, and perceived only a dark curtain, which, descending from the ceiling to the floor, was drawn along the whole side of the chamber. Ill as she was, the appearance of this curtain struck her, and she paused to gaze upon it, in wonder and apprehension.
It seemed to conceal a recess of the chamber; she wished, yet dreaded, to lift it, and to discover what it veiled: twice she was withheld by a recollection of the terrible spectacle her daring hand had formerly unveiled in an apartment of the castle, till, suddenly conjecturing, that it concealed the body of her murdered aunt, she seized it, in a fit of desperation, and drew it aside. Beyond, appeared a corpse, stretched on a kind of low couch, which was crimsoned with human blood, as was the floor beneath. The features, deformed by death, were ghastly and horrible, and more than one livid wound appeared in the face. Emily, bending over the body, gazed, for a moment, with an eager, frenzied eye; but, in the next, the lamp dropped from her hand, and she fell senseless at the foot of the couch.
When her senses returned, she found herself surrounded by men, among whom was Barnardine, who were lifting her from the floor, and then bore her along the chamber. She was sensible of what passed, but the extreme languor of her spirits did not permit her to speak, or move, or even to feel any distinct fear. They carried her down the stair- case, by which she had ascended; when, having reached the arch-way, they stopped, and one of the men, taking the torch from Barnardine, opened a small door, that was cut in the great gate, and, as he stepped out upon the road, the light he bore shewed several men on horseback, in waiting. Whether it was the freshness of the air, that revived Emily, or that the objects she now saw roused the spirit of alarm, she suddenly spoke, and made an ineffectual effort to disengage herself from the grasp of the ruffians, who held her.
Barnardine, meanwhile, called loudly for the torch, while distant voices answered, and several persons approached, and, in the same instant, a light flashed upon the court of the castle. Again he vociferated for the torch, and the men hurried Emily through the gate. At a short distance, under the shelter of the castle walls, she perceived the fellow, who had taken the light from the porter, holding it to a man, busily employed in altering the saddle of a horse, round which were several horsemen, looking on, whose harsh features received the full glare of the torch; while the broken ground beneath them, the opposite walls, with the tufted shrubs, that overhung their summits, and an embattled watch-tower above, were reddened with the gleam, which, fading gradually away, left the remoter ramparts and the woods below to the obscurity of night.
‘What do you waste time for, there?’ said Barnardine with an oath, as he approached the horsemen. ‘Dispatch — dispatch!’
‘The saddle will be ready in a minute,’ replied the man who was buckling it, at whom Barnardine now swore again, for his negligence, and Emily, calling feebly for help, was hurried towards the horses, while the ruffians disputed on which to place her, the one designed for her not being ready. At this moment a cluster of lights issued from the great gates, and she immediately heard the shrill voice of Annette above those of several other persons, who advanced. In the same moment, she distinguished Montoni and Cavigni, followed by a number of ruffian-faced fellows, to whom she no longer looked with terror, but with hope, for, at this instant, she did not tremble at the thought of any dangers, that might await her within the castle, whence so lately, and so anxiously she had wished to escape. Those, which threatened her from without, had engrossed all her apprehensions.
A short contest ensued between the parties, in which that of Montoni, however, were presently victors, and the horsemen, perceiving that numbers were against them, and being, perhaps, not very warmly interested in the affair they had undertaken, galloped off, while Barnardine had run far enough to be lost in the darkness, and Emily was led back into the castle. As she re-passed the courts, the remembrance of what she had seen in the portal-chamber came, with all its horror, to her mind; and when, soon after, she heard the gate close, that shut her once more within the castle walls, she shuddered for herself, and, almost forgetting the danger she had escaped, could scarcely think, that any thing less precious than liberty and peace was to be found beyond them.
Montoni ordered Emily to await him in the cedar parlour, whither he soon followed, and then sternly questioned her on this mysterious affair. Though she now viewed him with horror, as the murderer of her aunt, and scarcely knew what she said in reply to his impatient enquiries, her answers and her manner convinced him, that she had not taken a voluntary part in the late scheme, and he dismissed her upon the appearance of his servants, whom he had ordered to attend, that he might enquire further into the affair, and discover those, who had been accomplices in it.
 Then, again, the dead form, which the curtain in the portal-chamber had disclosed, came to her fancy, and she uttered a groan, which terrified Annette the more, as Emily forbore to satisfy her curiosity, on the subject of it, for she feared to trust her with so fatal a secret, lest her indiscretion should call down the immediate vengeance of Montoni on herself.
Thus compelled to bear within her own mind the whole horror of the secret, that oppressed it, her reason seemed to totter under the intolerable weight. She often fixed a wild and vacant look on Annette, and, when she spoke, either did not hear her, or answered from the purpose. Long fits of abstraction succeeded; Annette spoke repeatedly, but her voice seemed not to make any impression on the sense of the long agitated Emily, who sat fixed and silent, except that, now and then, she heaved a heavy sigh, but without tears.
Terrified at her condition, Annette, at length, left the room, to inform Montoni of it, who had just dismissed his servants, without having made any discoveries on the subject of his enquiry. The wild description, which this girl now gave of Emily, induced him to follow her immediately to the chamber.
At the sound of his voice, Emily turned her eyes, and a gleam of recollection seemed to shoot athwart her mind, for she immediately rose from her seat, and moved slowly to a remote part of the room. He spoke to her in accents somewhat softened from their usual harshness, but she regarded him with a kind of half curious, half terrified look, and answered only ‘yes,’ to whatever he said. Her mind still seemed to retain no other impression, than that of fear.
Of this disorder Annette could give no explanation, and Montoni, having attempted, for some time, to persuade Emily to talk, retired, after ordering Annette to remain with her, during the night, and to inform him, in the morning, of her condition.
When he was gone, Emily again came forward, and asked who it was, that had been there to disturb her. Annette said it was the Signor- Signor Montoni. Emily repeated the name after her, several times, as if she did not recollect it, and then suddenly groaned, and relapsed into abstraction.
With some difficulty, Annette led her to the bed, which Emily examined with an eager, frenzied eye, before she lay down, and then, pointing, turned with shuddering emotion, to Annette, who, now more terrified, went towards the door, that she might bring one of the female servants to pass the night with them; but Emily, observing her going, called her by name, and then in the naturally soft and plaintive tone of her voice, begged, that she, too, would not forsake her. — —‘For since my father died,’ added she, sighing, ‘every body forsakes me.’
‘Your father, ma’amselle!’ said Annette, ‘he was dead before you knew me.’
‘He was, indeed!’ rejoined Emily, and her tears began to flow. She now wept silently and long, after which, becoming quite calm, she at length sunk to sleep, Annette having had discretion enough not to interrupt her tears. This girl, as affectionate as she was simple, lost in these moments all her former fears of remaining in the chamber, and watched alone by Emily, during the whole night.


10.2: Why Play?
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- Jennifer Paris, Kristin Beeve, & Clint Springer
- College of the Canyons
Play:
- Inspires imagination
- Facilitates creativity
- Fosters problem solving
- Promotes development of new skills
- Builds confidence and higher levels of self-esteem
- Allows free exploration of the environment
- Fosters learning through hands-on and sensory exploration
It is now understood that moments often discounted as “just play” or as “fiddling around” are actually moments in which children are actively learning (Hirsh-Pasek et al. 2009; Jones and Reynolds 2011; Zigler, Singer, and Bishop-Josef 2004; Elkind 2007.) While engaged in play, children explore the physical properties of materials and the possibilities for action, transformation, or representation. Children try out a variety of ways to act on objects and materials and, in so doing, experiment with and build concepts and ideas. This active engagement with the world of people and objects starts from the moment of birth.
This description of the young child as an active participant in learning informs the role of the teacher who works with young children birth to five. Early childhood teaching and learning begins with teachers watching and listening to discover how infants and young children actively engage in making sense of their everyday encounters with people and objects. When teachers observe and listen with care, infants and young children reveal clues about their thinking, their feelings, or their intentions. Children’s actions, gestures, and words illuminate what they are trying to figure out or how they attempt to make sense of the attributes, actions, and responses of people and objects. Effective early childhood teaching requires teachers to recognize how infants and young children actively search for meaning, making sense of ideas and feelings.
When teaching is viewed in this light, children become active participants alongside teachers in negotiating the course of the curriculum. Families who entrust their children to the care and guidance of early childhood teachers also become active participants in this process. Shared participation by everyone in the work of creating lively encounters with learning allows a dynamic exchange of information and ideas—from child to adult, from adult to child, from adult to adult, and from child to child. The perspective of each (child, family, teacher) informs the other, and each learns from the other. Each relationship (child with family, child with teacher, child with child, and family with teacher) is reciprocal, with each participant giving and receiving from the other, and each adding to the other’s learning and understanding. [2]
The Educators’ Guide to the Framework For School Age Care In Australia discusses the benefits of play:
Play is a valued process, not only for enjoyment and leisure, but also for learning. Through play, children develop a sense of identity and an understanding of their social and cultural worlds. Children use play to explore and understand cultures, communities and friendships. We gain a lot through playing, not just as children, but also as adults.
Recent brain research has heralded the benefits of a stimulating play-based environment in encouraging the brain to grow and develop (Diamond 1988). Low stress levels and high engagement combine to nourish neural development. Research by Vandell and others (2005) demonstrates how school-aged care environments achieve this through the combination of high intrinsic motivation and challenge, effort and enjoyment. Lester and Russell (2009) identified the flexibility and plasticity of the brain, which develops through play and increases potential for learning later in life.
The intellectual and cognitive benefits of playing have been well documented. Children who engage in quality play experiences are more likely to:
- Have well developed memory skills and language development,
- Have the ability to regulate their behavior, leading to enhanced adjustment to school and academic learning.
Play also provides children with an opportunity to just ‘be’. [4]
Pause to Reflect
What does it mean to just be?  What facilitated the opportunity? What feelings did you experience? What might this mean for young children?
Educators observe stages of play experiences children navigate in their programs. Educators use these observations of children to plan for environments, set individual objectives and create appropriate curricular experiences.
| Stage | Description |
|---|---|
| Functional Play | Exploring, inspecting, and learning through repetitive physical activity. [6] |
| Symbolic Play | The ability to use objects, actions, or ideas to represent other objects, actions, or ideas and may include taking on roles. [7] |
| Constructive Play | Involves experimenting with objects to build things; learning things that were previously unknown with hands on manipulations of materials. [8] |
| Games with Rules | Imposes rules that must be followed by everyone that is playing; the logic and order involved forms that the foundations for developing game playing strategy. [9] |
In addition to the stages of play described by Piaget, there are also a variety of types of play children use when interacting within our ECE Programs.
References
[1] Image by the California Department of Education is used with permission
[2] The Integrated Nature of Learning by the California Department of Education is used with permission (pg. 5);
Content by Clint Springer is licensed under CC BY 4.0
[3] Image by Ian Joslin is licensed under CC BY 4.0
[4] Australian Government Department of Education (n.d.) Educator My Time, Our Place. Retrieved from files.acecqa.gov.au/files/National-Quality-Framework-Resources-Kit/educators_my_time_our_place.pdf
[5] Cognitive and Social Types of Play (n.d.). Retrieved from https://groundsforplay.com/cognitive-and-social-forms-play
[6] Cognitive and Social Types of Play (n.d.). Retrieved from https://groundsforplay.com/cognitive-and-social-forms-play
[7] Play and Playground Encyclopedia (n.d.) Symbolic Play. Retrieved from https://www.pgpedia.com/s/symbolic-play
[8] Play and Playground Encyclopedia (n.d.) Constructive Play. Retrieved from https://www.pgpedia.com/c/constructive-play
[9] Play and Playground Encyclopedia (n.d.) Games with Rules. Retrieved from https://www.pgpedia.com/g/games-rules


7.2: Getting Started
To begin thinking about scheduling, let us consider an auto shop that is converting a car from gas to electric. A number of steps are involved. A time estimate for each task is given.
- Task 1: Remove engine and gas parts (2 days)
- Task 2: Steam clean the inside of the car (0.5 day)
- Task 3: Buy an electric motor and speed controller (2 days for travel)
- Task 4: Construct the part that connects the motor to the car’s transmission (1 day)
- Task 5: Construct battery racks (2 days)
- Task 6: Install the motor (0.5 day)
- Task 7: Install the speed controller (0.5 day)
- Task 8: Install the battery racks (0.5 day)
- Task 9: Wire the electricity (1 day)
Some tasks have to be completed before others – we certainly can’t install the new motor before removing the old engine! There are some tasks, however, that can be worked on simultaneously by two different people, like constructing the battery racks and installing the motor.
To help us visualize the ordering of tasks, we will create a digraph.
A diagraph is a graphical representation of a set of tasks in which tasks are represented with dots, called vertices, and arrows between vertices are used to show ordering.
For example, this digraph shows that Task 1, notated \(T_1\) for compactness, needs to be completed before Task 2. The number in parentheses after the task name is the time required for the task.

In scheduling jargon, the workers completing the tasks are called processors . While in this example the processors are humans, in some situations the processors are computers, robots, or other machines.
For simplicity, we are going to make the very big assumptions that every processor can do every task, that they all would take the same time to complete it, and that only one processor can work on a task at a time.
The finishing time is how long it will take to complete all the tasks. The finishing time will depend upon the number of processors and the specific schedule.
If we had only one processor working on this task, it is easy to determine the finishing time; just add up the individual times. We assume one person can’t work on two tasks at the same time, ignore things like drying times during which someone could work on another task. Scheduling with one processor, a possible schedule would look like this, with a finishing time of 10 days.
In this schedule, all the ordering requirements are met. This is certainly not the only possible schedule for one processor, but no other schedule could complete the job in less time. Because of this, this is an optimal schedule with optimal finishing time – there is nothing better.
An optimal schedule is the schedule with the shortest possible finishing time.
For two processors, things become more interesting. For small digraphs like this, we probably could fiddle around and guess-and-check a pretty good schedule. Here would be a possibility:
With two processors, the finishing time was reduced to 5.5 days. What was processor 2 doing during the last day? Nothing, because there were no tasks for the processor to do. This is called idle time .
Idle time is time in the schedule when there are no tasks available for the processor to work on, so they sit idle.
Is this schedule optimal? Could it have been completed in 5 days? Because every other task had to be completed before task 9 could start, there would be no way that both processors could be busy during task 9, so it is not possible to create a shorter schedule.
So how long will it take if we use three processors? About \(10/3 = 3.33\) days? Again we will guess-and-check a schedule:
With three processors, the job still took 4.5 days. It is a little harder to tell whether this schedule is optimal. However, it might be helpful to notice that since Task 1, 2, 6, and 9 have to be completed sequentially, there is no way that this job could be completed in less than \(2+0.5+0.5+1 = 4\) days, regardless of the number of processors. Four days is, for this digraph, the absolute minimum time to complete the job, called the critical time .
The critical time is the absolute minimum time to complete the job, regardless of the number of processors working on the tasks.
Critical time can be determined by looking at the longest sequence of tasks in the digraph, called the critical path


12.19: Later Crusades and Crusading's Ultimate Failure
After the Third Crusade, the re-established Crusader States managed to survive and even expand in power for the next several decades. Syria and Egypt were split between Saladin’s heirs, and the crusader kingdoms often enjoyed good relations with Ayyubid Egypt: indeed, a truce worked out between Holy Roman Emperor Frederick II and the Egyptian sultan al-Kamil in 1229 resulted in the city of Jerusalem itself returning to Christian rule.
In the 1240s, however, forces far from the Levant brought down the Kingdom of Jerusalem. As the Mongols gradually conquered Central Asia (see Chapter Eleven), the Khwarazmian Turks were driven from their realm in the steppes into Syria and northern Iraq. They ended up allying with Ayyubid Egypt against the Crusader Kingdom of Jerusalem, and, in 1244, the combined armies of Damascus and Jerusalem were defeated by an Egyptian/Khwarazmian army. Jerusalem fell under Muslim rule, under which it would remain until 1917.
In response to the fall of Jerusalem, Pope Innocent IV (r. 1243 – 1254) called a crusade that would be led by King Louis IX. While it had contingents from other Western European kingdoms, this effort was primarily an effort of the French crown. Although King Louis IX was able to manage the impressive logistical feat of organizing and equipping an army that seized the northern Egyptian port of Damietta, the effort to take all of Egypt was ultimately unsuccessful. Over the course of 1250, the French army was surrounded in the swamps of the Nile Delta outside of Cairo and forced to surrender, with Louis himself captured. The only lasting result in the Middle East was that, during the fighting, the Ayyubid sultan’s Mamluks launched a coup d’état and seized power in Egypt, thereby creating in Egypt a military power that would dominate the Levant for nearly three centuries (see Chapter Eight).
Indeed, in the four decades after Louis’s failure in the Nile Delta, the Mamluk sultans of Egypt would eventually conquer all of the Crusader States, with the last crusader stronghold in the Levant, the city of Acre, falling in 1291. Although Popes would still call crusades for military efforts against Muslim forces (and indeed, still make calls to retake the city of Jerusalem), crusading had failed. One reason for crusading’s ultimate failure was that, as Western European kings consolidated their power, they often had priorities other than crusading. England’s Edward I, for example, spent a few months fighting in the Levant in 1271; however, he spent most of his reign fighting to subdue England’s neighboring kingdoms of Wales and Scotland.
In the end, the Crusades failed, and their greatest long-term impacts were the destruction of the Byzantine Empire and the growth of the sea power of Genoa and Venice, whose ships and sailors had transported people and supplies between Europe and the Crusader States.


1.2: Geometry of Complex Numbers
The most fundamental new idea in the study of complex numbers is the “imaginary number” \(j\). This imaginary number is defined to be the square root of −1:
\[j = \sqrt{-1} \nonumber \]
\[j^2 = -1 \nonumber \]
The imaginary number \(j\) is used to build complex numbers \(x\) and \(y\) in the following way:
\[z = x + jy \nonumber \]
We say that the complex number \(z\) has “real part” \(x\) and “imaginary part” \(y\):
\[z = \mathrm{Re}[z] + j \mathrm{Im}[z] \nonumber \]
\[\mathrm{Re}[z] = x; \mathrm{Im}[z] = y \nonumber \]
In MATLAB, the variable \(x\) is denoted by
real(z)
, and the variable \(y\) is denoted by
imag
(z)
. In communication theory, \(x\) is called the “in-phase” component of \(z\), and \(y\) is called the “quadrature” component. We call \(z=x+jy\) the Cartesian representation of \(z\), with real component \(x\) and imaginary component \(y\). We say that the Cartesian pair \((x,y)\)codes the complex number \(z\).
We may plot the complex number \(z\) on the plane as in Figure. We call the horizontal axis the “real axis” and the vertical axis the “imaginary axis.” The plane is called the “complex plane.” The radius and angle of the line to the point z=x+jy are
\[r = \sqrt{x^2 + y^2} \nonumber \]
\[θ = \tan^{-1}{\left(\frac {y}{x}\right)} \nonumber \]
In MATLAB, \(\)r is denoted by
abs(z)
, and \(θ\) is denoted by
angle(z)
The original Cartesian representation is obtained from the radius \(r\) and angle \(θ\) as follows:
\[x = r\cos{θ} \nonumber \]
\[y = r\sin{θ} \nonumber \]
The complex number \(z\) may therefore be written as
\[z = x + jy \nonumber \]
\[z = r\cos{θ} + jr\sin{θ} \nonumber \]
\[z = r(\cos{θ} + j\sin{θ}) \nonumber \]
The complex number \(\cosθ+j\sinθ\) is, itself, a number that may be represented on the complex plane and coded with the Cartesian pair \((\cosθ,\sinθ)\). This is illustrated in Figure. The radius and angle to the point \(z=\cosθ+j\sinθ\) are 1 and \(θ\). Can you see why?
The complex number \(\cosθ+j\sinθ\) is of such fundamental importance to our study of complex numbers that we give it the special symbol \(e^{jθ}\)
\[e^{jθ} = \cosθ+j\sinθ \nonumber \]
As illustrated in the above Figure, the complex number \(e^{jθ}\) has radius 1 and angle \(θ\). With the symbol \(e^{jθ}\), we may write the complex number \(z\) as
\[z = re^{jθ} \nonumber \]
We call \(z = re^{jθ}\) a polar representation for the complex number \(z\). We say that the polar pair \(r∠θ\) codes the complex number \(z\). In this polar representation, we define \(|z|=r\) to be the magnitude of \(z\) and \(\mathrm{arg}(z)=θ\) to be the angle, or phase, of \(z\)
\[|z| = r \nonumber \]
\[/mathrm{arg}(z) = θ \nonumber \]
With these definitions of magnitude and phase, we can write the complex number \(z\) as
\[z = |z|e^{j\mathrm{arg}(z)} \nonumber \]
Let's summarize our ways of writing the complex number z and record the corresponding geometric codes
\[z = x + jy = re^{jθ} = |z|e^{j\mathrm{arg}(z)} \nonumber \]
In "Roots of Quadratic Equations" we show that the definition \(e^{jθ}=\cosθ+j\sinθ\) is more than symbolic. We show, in fact, that \(e^{jθ}\) is just the familiar function \(e^x\) evaluated at the imaginary argument \(x=jθ\). We call \(e^{jθ}\) a “complex exponential,” meaning that it is an exponential with an imaginary argument.
Prove \((j)^{2n}=(−1)^{n}\) and \((j)^{2n+1}=(−1)^{nj}\). Evaluate \(j^3,j^4,j^5\)
Prove \(e^{j[(π/2)+m2π]}=j,\; e^{j[(3π/2)+m2π]}=−j,\; e^{j(0+m2π)}=1\), and \(e^{j(π+m2π)}=−1\). Plot these identities on the complex plane. (Assume m is an integer.)
Find the polar representation \(z=re^{jθ}\) for each of the following complex numbers:
- \(z=1+j0\)
- \(z=0+j1\)
- \(z=1+j1\)
- \(z=−1−j1\)
Plot the points on the complex plane.
Find the Cartesian representation \(z=x+jy\) for each of the following complex numbers:
- \(z=\sqrt2 e^{jπ/2}\)
- \(z=\sqrt2 e^{jπ/4}\)
- \(z=e^{j3π/4}\)
- \(z=\sqrt2 e^{j3π/2}\)
Plot the points on the complex plane.
The following geometric codes represent complex numbers. Decode each by writing down the corresponding complex number \(z\):
- \((0.7,−0.1)\)
- \((−1.0,0.5)\)
- \(1.6∠π/8\)
- \(0.4∠7π/8\)
Show that \(\mathrm{Im}[jz]=\mathrm{Re}[z]\) and \(\mathrm{Re}[−jz]=\mathrm{Im}[z]\). Demo 1.1 (MATLAB). Run the following MATLAB program in order to compute and plot the complex number \(e^{jθ}\) for \(θ=i2π/360,i=1,2,...,360\):
j=
sqrt
(-1) n=360 for i=1:n,circle(i)=
exp
(j*2*pi*i/n);end; axis('square') plot(circle)
Replace the explicit for loop of line 3 by the implicit loop
circle=
exp
(j*2*pi*[1:n]/n);
to speed up the calculation. You can see from the Figure that the complex number \(e^{jθ}\), evaluated at angles \(θ=2π/360,2(2π/360),...,\) turns out complex numbers that lie at angle \(θ\) and radius 1. We say that \(e^{jθ}\) is a complex number that lies on the unit circle. We will have much more to say about the unit circle in Chapter 2.


Metallic Bonding
Skills to Develop
- Describe metallic bonding using MO theory
We previously discussed the "electron-sea model" of metallic bonding in the intro section and metal characteristics sections. This is a good time to review those sections, before we describe a better model. Why do we need a different model for metallic bonding than the electron-sea model? Although it explains some general properties of metals, like malleability and conductivity, it doesn't explain the relative properties of metals, like their hardness and melt points. These properties depend on how strong the bonding in the metals is. By the electron-sea model, we might think that more electrons makes the bonds stronger, so hardness and melt point would increase across the periodic table.
We can explain these properties using MO theory. In this case, we imagine combining many atomic orbitals (1 or more for each atom) to make and equal number of MOs that extend over the whole solid. Some MOs will have fewer nodes and be lower energy, while others will have more nodes and be higher energy. Each MO can hold no more than 2 electrons. Filling the lower energy MOs (bonding MOs) makes the bonds stronger, which is why alkali metals have low melt points and are soft (not many bonding MOs filled). Filling the higher energy MOs (antibonding MOs) makes the bonds weaker, which is why Cu, Ag, Au and Zn are soft and melt at low temperatures (Hg is a liquid at RT!).
When we have a really big number of MOs, some interesting things happen. One mole of Fe is about 7 mL or 7 cc (a pretty small amount). In each atom of Fe, we have 1 4s orbital, 5 3d orbitals, and 3 4p orbitals that can be involved in bonding. In a mole of Fe, we have 9 times Avogadro's number of AOs that can be involved in bonding. That's about 10 24 AOs, which means the same number of MOs. There just isn't space to have much difference in energy between all those MOs. The MOs have to have energies not too different from the AOs (they are definitely higher and lower, but not by too much), so they are limited to a relatively narrow range of energies. This means that the energies form bands rather than separate energy levels.
We can start to see how this happens by imagining a line of Li atoms. As we increase the number of Li atoms, the orbitals get closer together. When the number of Li atoms reaches infinity, the MOs become infinitely close together.
Band theory explains the conductivity of metals: for electrons to move, electrons have to be able to change MOs without gaining much energy. If a band is partly-filled with electrons, they can easily change states because there are empty states almost the same energy as the full states. We can tell that the bands from s, p and d orbitals must overlap in metals, because the alkaline earth metals are conductors. If there were an energy gap between the s band and the p or d band, they would not conduct because the s band would be full.


4.3: Contraception
In 2008, there were 6.4 million pregnancies to the 62 million women of reproductive age (15–44) in the United States. Sixty-six percent of these pregnancies resulted in live births and 19% in induced abortions. And, nearly half of pregnancies among American women—more than three million each year—are unintended.
Unintended Pregnancy
An unintended pregnancy is a pregnancy that is either mistimed or unwanted at the time of conception. It is a core concept in understanding the fertility of populations and the unmet need for contraception. Unintended pregnancy is associated with an increased risk of morbidity for women, and with health behaviors during pregnancy that are associated with adverse effects. For example, women with an unintended pregnancy may delay prenatal care, which may affect the health of the infant. Women of all ages may have unintended pregnancies, but some groups, such as teens, are at a higher risk.
Efforts to decrease unintended pregnancy include finding better forms of contraception, and increasing contraceptive use and adherence.
What is contraception?
Contraception, also known as birth control, is designed to prevent pregnancy. Some types of birth control include (but are not limited to):
- Barrier methods, such as condoms, the diaphragm, and the cervical cap, are designed to prevent the sperm from reaching the egg for fertilization.Intrauterine device, or IUD, is a small device that is inserted into the uterus by a health care provider. The IUD prevents a fertilized egg from implanting in the uterus. An IUD can stay in the uterus for up to 10 years until it is removed by a health care provider.
- Hormonal birth control, such as birth control pills, injections, skin patches, and vaginal rings, release hormones into a woman’s body that interfere with fertility by preventing ovulation, fertilization, or implantation.
- Sterilization is a method that permanently prevents a woman from getting pregnant or a man from being able to get a woman pregnant. Sterilization involves surgical procedures that must be done by a health care provider and usually cannot be reversed.
The choice of birth control depends on factors such as a person’s overall health, age, frequency of sexual activity, number of sexual partners, desire to have children in the future, and family history of certain diseases. A woman should talk to her health care provider about her choice of birth control method.
It is important to remember that even though birth control methods can prevent pregnancy, they do not all protect against sexually transmitted diseases or HIV.
Contraception Methods
There is no “best” method of birth control. Each method has its pros and cons.
All women and men can have control over when, and if, they become parents. Making choices about birth control, or contraception, isn’t easy. There are many things to think about. To get started, learn about birth control methods you or your partner can use to prevent pregnancy. You can also talk with your doctor about the choices.
Before choosing a birth control method, think about:
- Your overall health
- How often you have sex
- The number of sex partners you have
- If you want to have children someday
- How well each method works to prevent pregnancy
- Possible side effects
- Your comfort level with using the method
What are the different types of birth control?
You can choose from many methods of birth control. They are grouped by how they work:
Continuous abstinence
This means not having sex (vaginal, anal, or oral) at any time . It is the only sure way to prevent pregnancy and protect against sexually transmitted infections (STIs), including HIV.
Natural family planning/rhythm method
This method is when you do not have sex or use a barrier method on the days you are most fertile (most likely to become pregnant). You can read about barrier methods in the following chart.
A woman who has a regular menstrual cycle has about 9 or more days each month when she is able to get pregnant. These fertile days are about 5 days before and 3 days after ovulation, as well as the day of ovulation.
To have success with this method, you need to learn about your menstrual cycle. Then you can learn to predict which days you are fertile or “unsafe.” To learn about your cycle, keep a written record of:
- When you get your period
- What it is like (heavy or light blood flow)
- How you feel (sore breasts, cramps)
This method also involves checking your cervical mucus and recording your body temperature each day. Cervical mucus is the discharge from your vagina. You are most fertile when it is clear and slippery like raw egg whites. Use a basal thermometer to take your temperature and record it in a chart. Your temperature will rise 0.4 to 0.8° F on the first day of ovulation. You can talk with your doctor or a natural family planning instructor to learn how to record and understand this information.
 The sponge is effective for more than one act of intercourse for up to 24 hours. It needs to be left in for at least 6 hours after having sex to prevent pregnancy. It must then be taken out within 30 hours after it is inserted.
Only one kind of contraceptive sponge is sold in the United States. It is called the Today Sponge. Women who are sensitive to the spermicide nonoxynol-9 should not use the sponge.
Diaphragm, cervical cap, and cervical shield
These barrier methods block the sperm from entering the cervix (the opening to your womb) and reaching the egg.
- The diaphragm is a shallow latex cup.
- The cervical cap is a thimble-shaped latex cup. It often is called by its brand name, FemCap.
- The cervical shield is a silicone cup that has a one-way valve that creates suction and helps it fit against the cervix. It often is called by its brand name, Lea’s Shield.
The diaphragm and cervical cap come in different sizes, and you need a doctor to “fit” you for one. The cervical shield comes in one size, and you will not need a fitting.
Before having sex, add spermicide (to block or kill sperm) to the devices.  You can buy spermicide gel or foam at a drug store.
All three of these barrier methods must be left in place for 6 to 8 hours after having sex to prevent pregnancy. The diaphragm should be taken out within 24 hours. The cap and shield should be taken out within 48 hours.
Female condom
This condom is worn by the woman inside her vagina. It keeps sperm from getting into her body. It is made of thin, flexible, manmade rubber and is packaged with a lubricant. It can be inserted up to 8 hours before having sex. Use a new condom each time you have intercourse. And don’t use it and a male condom at the same time.
 Condoms can be made of latex, polyurethane, or “natural/lambskin”. The natural kind do not protect against STIs. Condoms work best when used with a vaginal spermicide, which kills the sperm. And you need to use a new condom with each sex act.
Condoms are either:
- Lubricated, which can make sexual intercourse more comfortable
- Non-lubricated, which can also be used for oral sex. It is best to add lubrication to non-lubricated condoms if you use them for vaginal or anal sex. You can use a water-based lubricant, such as K-Y jelly. You can buy them at the drug store. Oil-based lubricants like massage oils, baby oil, lotions, or petroleum jelly will weaken the condom, causing it to tear or break.
Keep condoms in a cool, dry place. If you keep them in a hot place (like a wallet or glove compartment), the latex breaks down. Then the condom can tear or break.
Hormonal methods
These methods prevent pregnancy by interfering with ovulation, fertilization, and/or implantation of the fertilized egg.
Oral contraceptives—combined pill (“The pill”)
The pill contains the hormones estrogen and progestin. It is taken daily to keep the ovaries from releasing an egg. The pill also causes changes in the lining of the uterus and the cervical mucus to keep the sperm from joining the egg.
Some women prefer the “extended cycle” pills. These have 12 weeks of pills that contain hormones (active) and 1 week of pills that don’t contain hormones (inactive). While taking extended cycle pills, women only have their period three to four times a year.
Many types of oral contraceptives are available. Talk with your doctor about which is best for you.
Your doctor may advise you not to take the pill if you:
- Are older than 35 and smoke
- Have a history of blood clots
- Have a history of breast, liver, or endometrial cancer
Antibiotics may reduce how well the pill works in some women. Talk to your doctor about a backup method of birth control if you need to take antibiotics.
Women should wait three weeks after giving birth to begin using birth control that contains both estrogen and progestin. These methods increase the risk of dangerous blood clots that could form after giving birth. Women who delivered by cesarean section or have other risk factors for blood clots, such as obesity, history of blood clots, smoking, or preeclampsia, should wait six weeks.
The patch
Also called by its brand name, Ortho Evra, this skin patch is worn on the lower abdomen, buttocks, outer arm, or upper body. It releases the hormones progestin and estrogen into the bloodstream to stop the ovaries from releasing eggs in most women. It also thickens the cervical mucus, which keeps the sperm from joining with the egg. You put on a new patch once a week for 3 weeks. You don’t use a patch the fourth week in order to have a period.
Women should wait three weeks after giving birth to begin using birth control that contains both estrogen and progestin. These methods increase the risk of dangerous blood clots that could form after giving birth. Women who delivered by cesarean section or have other risk factors for blood clots, such as obesity, history of blood clots, smoking, or preeclampsia, should wait six weeks.
Shot/injection
The birth control shot often is called by its brand name Depo-Provera. With this method you get injections, or shots, of the hormone progestin in the buttocks or arm every 3 months. A new type is injected under the skin. The birth control shot stops the ovaries from releasing an egg in most women. It also causes changes in the cervix that keep the sperm from joining with the egg.
Vaginal ring
This is a thin, flexible ring that releases the hormones progestin and estrogen. It works by stopping the ovaries from releasing eggs. It also thickens the cervical mucus, which keeps the sperm from joining the egg.
It is commonly called NuvaRing, its brand name.  You wear the ring for 3 weeks, take it out for the week that you have your period, and then put in a new ring.
Women should wait three weeks after giving birth to begin using birth control that contains both estrogen and progestin. These methods increase the risk of dangerous blood clots that could form after giving birth. Women who delivered by cesarean section or have other risk factors for blood clots, such as obesity, history of blood clots, smoking, or preeclampsia, should wait six weeks.
Implantable devices
These devices are inserted into the body and left in place for a few years.
Implantable rod
This is a matchstick-size, flexible rod that is put under the skin of the upper arm. It is often called by its brand name, Implanon. The rod releases a progestin, which causes changes in the lining of the uterus and the cervical mucus to keep the sperm from joining an egg. Less often, it stops the ovaries from releasing eggs. It is effective for up to 3 years.
 There are two types:
- Copper IUD The copper IUD goes by the brand name ParaGard. It releases a small amount of copper into the uterus, which prevents the sperm from reaching and fertilizing the egg. It fertilization does occur, the IUD keeps the fertilized egg from implanting in the lining of the uterus. A doctor needs to put in your copper IUD. It can stay in your uterus for 5 to 10 years.
- Hormonal IUD The hormonal IUD goes by the brand name Mirena. It is sometimes called an intrauterine system, or IUS. It releases progestin into the uterus, which keeps the ovaries from releasing an egg and causes the cervical mucus to thicken so sperm can’t reach the egg. It also affects the ability of a fertilized egg to successfully implant in the uterus. A doctor needs to put in a hormonal IUD. It can stay in your uterus for up to 5 years.
Sterilization implant (essure)
Essure is the first non-surgical method of sterilizing women. A thin tube is used to thread a tiny spring-like device through the vagina and uterus into each fallopian tube. The device works by causing scar tissue to form around the coil. This blocks the fallopian tubes and stops the egg and sperm from joining.
It can take about 3 months for the scar tissue to grow, so it’s important to use another form of birth control during this time. Then you will have to return to your doctor for a test to see if scar tissue has fully blocked your tubes.
Surgical sterilization
For women, surgical sterilization closes the fallopian tubes by being cut, tied, or sealed. This stops the eggs from going down to the uterus where they can be fertilized. The surgery can be done a number of ways. Sometimes, a woman having cesarean birth has the procedure done at the same time, so as to avoid having additional surgery later.
For men, having a vasectomy keeps sperm from going to his penis, so his ejaculate never has any sperm in it. Sperm stays in the system after surgery for about 3 months. During that time, use a backup form of birth control to prevent pregnancy. A simple test can be done to check if all the sperm is gone; it is called a semen analysis.
Emergency contraception
Used if a woman’s primary method of birth control fails. It should not be used as a regular method of birth control. Emergency contraception (Plan B One-Step or Next Step. It is also called the “morning after pill.”)
Emergency contraception keeps a woman from getting pregnant when she has had unprotected vaginal intercourse. “Unprotected” can mean that no method of birth control was used. It can also mean that a birth control method was used but it was used incorrectly, or did not work (like a condom breaking). Or, a woman may have forgotten to take her birth control pills. She also may have been abused or forced to have sex. These are just some of the reasons women may need emergency contraception.
Emergency contraception can be taken as a single pill treatment or in two doses. A single dose treatment works as well as two doses and does not have more side effects. It works by stopping the ovaries from releasing an egg or keeping the sperm from joining with the egg. For the best chances for it to work, take the pill as soon as possible after unprotected sex. It should be taken within 72 hours after having unprotected sex.
A single-pill dose or two-pill dose of emergency contraception is available over-the-counter (OTC) for women ages 17 and older.
Can all types of birth control prevent sexually transmitted infections (STIs)?
No. The male latex condom is the only birth control method proven to help protect you from STIs, including HIV. Research is being done to find out how effective the female condom is at preventing STIs and HIV. For more information, see Will birth control pills protect me from sexually transmitted infections (STIs), including HIV/AIDS?
How well do different kinds of birth control work? Do they have side effects?
All birth control methods work the best if used correctly and every time you have sex. Be sure you know the right way to use them. Sometimes doctors don’t explain how to use a method because they assume you already know. Talk with your doctor if you have questions. They are used to talking about birth control. So don’t feel embarrassed about talking to him or her.
Some birth control methods can take time and practice to learn. For example, some people don’t know you can put on a male condom “inside out.” Also, not everyone knows you need to leave a little space at the tip of the condom for the sperm and fluid when a man ejaculates, or has an orgasm.
Where to Get Birth Control
Where you get birth control depends on what method you choose.
You can buy these forms over the counter:
- Male condoms
- Female condoms
- Sponges
- Spermicides
- Emergency contraception pills (girls younger than 17 need a prescription)
You need a prescription for these forms:
- Oral contraceptives: the pill, the mini-pill
- Skin patch
- Vaginal ring
- Diaphragm (your doctor needs to fit one to your shape)
- Cervical cap
- Cervical shield
- Shot/injection (you get the shot at your doctor’s office)
- IUD (inserted by a doctor)
- Implantable rod (inserted by a doctor)
You will need surgery or a medical procedure for:
- Sterilization, female and male
Spermacides
You can buy spermicides over the counter. They work by killing sperm. They come in many forms:
- Foam
- Gel
- Cream
- Film
- Suppository
- Tablet
Spermicides are put in the vagina no more than 1 hour before having sex. If you use a film, suppository, or tablet, wait at least 15 minutes before having sex so the spermicide can dissolve.  You will need to use more spermicide each time you have sex.
Spermicides work best if used along with a barrier method, such as a condom, diaphragm, or cervical cap. Some spermicides are made just for use with the diaphragm and cervical cap. Check the package to make sure you are buying what you need.
All spermicides contain sperm-killing chemicals. Some contain nonoxynol-9, which may raise your risk of HIV if you use it a lot. It irritates the tissue in the vagina and anus, so it can cause the HIV virus to enter the body more freely. Some women are sensitive to nonoxynol-9 and need to use spermicides without it. Medications for vaginal yeast infections may lower the effectiveness of spermicides. Also, spermicides do not protect against sexually transmitted infections.
 This stops the sperm from going to the egg. “Pulling out” can be hard for a man to do. It takes a lot of self-control.
Even if you use withdrawal, sperm can be released before the man pulls out.  This fluid has sperm in it. So you could still get pregnant.
Withdrawal does not protect you from STIs or HIV.
Dental Dams
The dental dam is a square piece of rubber that is used by dentists during oral surgery and other procedures. It is not a method of birth control. But it can be used to help protect people from STIs, including HIV, during oral-vaginal or oral-anal sex.  You can buy dental dams at surgical supply stores.
For a handy summary of birth control information, see this Frequently Asked Questions Fact Sheet.
Find out the Real Cost of Contraceptives: “Choosing the most money-smart method isn’t as easy as crunching numbers because costs depend on a variety of factors, including how long you want the birth control to last, how often you need it, and how generous your insurance policy is.”
- Which birth control methods are the most affordable? Why?
Sources
Unintended Pregnancy : Unintended Pregnancy, CDC, www.cdc.gov/reproductivehealth/UnintendedPregnancy/index.htm
What is contraception?: What is Contraception? NLM, NIH, http://www.nlm.nih.gov/medlineplus/birthcontrol.html
Contraception Methods : Contraception Methods, NIH, womenshealth.gov/publications/our-publications/fact-sheet/birth-control-methods.cfm


1.4: Pivots and their influence on solution spaces
\(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\twovec}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix} } \) \(\newcommand{\threevec}[3]{\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \) \(\newcommand{\fourvec}[4]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \end{pmatrix} } \) \(\newcommand{\fivevec}[5]{\begin{pmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{pmatrix} } \) By now, we have seen several examples illustrating how the reduced row echelon matrix leads to a convenient description of the solution space to a linear system. In this section, we will use this understanding to make some general observations about how certain features of the reduced row echelon matrix reflect the nature of the solution space.
Remember that a leading entry in a reduced row echelon matrix is the leftmost nonzero entry in a row of the matrix. As we'll see, the positions of these leading entries encode a lot of information about the solution space of the corresponding linear system. For this reason, we make the following definition.
A pivot position in a matrix \(A\) is the position of a leading entry in the reduced row echelon matrix of \(A\).
For instance, in this reduced row echelon matrix, the pivot positions are indicated in bold:
\[
\left[\begin{array}{llll}
\mathbf{1} & 0 & * & 0 \\
0 & \mathbf{1} & * & 0 \\
0 & 0 & 0 & \mathbf{1} \\
0 & 0 & 0 & 0
\end{array}\right] .
\]
We can refer to pivot positions by their row and column number saying, for instance, that there is a pivot position in the second row and fourth column.
Preview Activity 1.4.1. Some basic observations about pivots.
a. Shown below is a matrix and its reduced row echelon form. Indicate the pivot positions.
\[
\left[\begin{array}{rrrr}
2 & 4 & 6 & -1 \\
-3 & 1 & 5 & 0 \\
1 & 3 & 5 & 1
\end{array}\right] \sim\left[\begin{array}{rrrr}
1 & 0 & -1 & 0 \\
0 & 1 & 2 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] .
\]
b. How many pivot positions can there be in one row? In a \(3 \times 5\) matrix, what is the largest possible number of pivot positions? Give an example of a \(3 \times 5\) matrix that has the largest possible number of pivot positions.
c. How many pivots can there be in one column? In a \(5 \times 3\) matrix, what is the largest possible number of pivot positions? Give an example of a \(5 \times 3\) matrix that has the largest possible number of pivot positions.
d. Give an example of a matrix with a pivot position in every row and every column. What is special about such a matrix?
When we have looked at solution spaces of linear systems, we have frequently asked whether there are infinitely many solutions, exactly one solution, or no solutions. We will now break this question into two separate questions.
When we encounter a linear system, we often ask
Existence
Is there a solution to the linear system? If so, we say that the system is consistent; if not, we say it is inconsistent.
Uniqueness
If the linear system is consistent, is the solution unique or are there infinitely many solutions?
These two questions represent two sides of a coin that appear in many variations throughout our explorations. In this section, we will study how the location of the pivots influence the answers to these two questions. We begin by considering the first question concerning the existence of solutions.
1.4.1 The existence of solutions
Activity 1.4.2.
a. Shown below are three augmented matrices in reduced row echelon form.
\[
\left[\begin{array}{lll|r}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -2 \\
0 & 0 & 0 & 0
\end{array}\right] \quad\left[\begin{array}{rrr|r}
1 & 0 & 2 & 3 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \quad\left[\begin{array}{rrr|r}
1 & 0 & 2 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\]
For each matrix, identify the pivot positions and determine if the corresponding linear system is consistent. Explain how the location of the pivots determines whether the system is consistent or inconsistent.
b. Each of the augmented matrices above has a row in which each entry is zero. What, if anything, does the presence of such a row tell us about the consistency of the corresponding linear system?
c. Give an example of a \(3 \times 5\) augmented matrix in reduced row echelon form that represents a consistent system. Indicate the pivot positions in your matrix and explain why these pivot positions guarantee a consistent system.
d. Give an example of a \(3 \times 5\) augmented matrix in reduced row echelon form that represents an inconsistent system. Indicate the pivot positions in your matrix and explain why these pivot positions guarantee an inconsistent system.
e. Write the reduced row echelon form of the coefficient matrix of the corresponding linear system in Item d? (Remember that the Augmentation Principle says that the reduced row echelon form of the coefficient matrix simply consists of the first four columns of the augmented matrix.) What do you notice about the pivot positions in this coefficient matrix?
f. Suppose we have a linear system for which the coefficient matrix has the following reduced row echelon form.
\[
\left[\begin{array}{rrrrr}
1 & 0 & 0 & 0 & -1 \\
0 & 1 & 0 & 0 & 2 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & -3
\end{array}\right]
\]
What can you say about the consistency of the linear system?
Let's summarize the results of this activity by considering the following reduced row echelon matrix:
\[
\left[\begin{array}{lll|l}
1 & * & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\]
In terms of variables \(x, y\), and \(z\), the final equation says
\[
0 x+0 y+0 z=0 .
\]
If we evaluate the left-hand side with any values of \(x, y\), and \(z\), we get 0 , which means that the equation always holds. Therefore, its presence has no effect on the solution space defined by the other three equations.
The third equation, however, says that
\[
0 x+0 y+0 z=1 \text {. }
\]
Again, if we evaluate the left-hand side with any values of \(x, y\), and \(z\), we get 0 so this equation cannot be satisfied for any \((x, y, z)\). This means that the entire linear system has no solution and is therefore inconsistent.
An equation like this appears in the reduced row echelon matrix as
\[
\left[\begin{array}{cccc|c}
\vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & 0 & 1 \\
\vdots & \vdots & \vdots & \vdots & \vdots
\end{array}\right] .
\]
The pivot positions make this condition clear: the system is inconsistent if there is a pivot position in the rightmost column of the corresponding augmented matrix.
In fact, we will soon see that the system is consistent if there is not a pivot in the rightmost column of the corresponding augmented matrix. This leaves us with the following
A linear system is inconsistent if and only if there is a pivot position in the rightmost column of the corresponding augmented matrix.
This also says something about the pivot positions of the coefficient matrix. Consider an example of an inconsistent system corresponding to the reduced row echelon form of the following augmented matrix
\[
\left[\begin{array}{lll|l}
1 & 0 & * & 0 \\
0 & 1 & * & 0 \\
0 & 0 & 0 & 1
\end{array}\right] .
\]
The Augmentation Principle says that that the reduced row echelon form of the coefficient matrix is
\[
\left[\begin{array}{lll}
1 & 0 & * \\
0 & 1 & * \\
0 & 0 & 0
\end{array}\right],
\]
which shows that the coefficient matrix has a row without a pivot position. To turn this around, we see that if every row of the coefficient matrix has a pivot position, then the system must be consistent. For instance, if our linear system has a coefficient matrix whose reduced row echelon form is
\[
\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right],
\]
then we can guarantee that the linear system is consistent because there is no way to obtain a pivot in the rightmost column of the augmented matrix.
If every row of the coefficient matrix has a pivot position, then the corresponding system of linear equations is consistent.
1.4.2 The uniqueness of solutions
Now that we have studied the role that pivot positions play in the existence of solutions, let's turn to the question of uniqueness.
Activity 1.4.3.
a. Here are the three augmented matrices in reduced row echelon form that we considered in the previous section.
\[
\left[\begin{array}{lll|r}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -2 \\
0 & 0 & 0 & 0
\end{array}\right] \quad\left[\begin{array}{rrr|r}
1 & 0 & 2 & 3 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \quad\left[\begin{array}{rrr|r}
1 & 0 & 2 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
\]
For each matrix, identify the pivot positions and state whether the corresponding linear system is consistent. If the system is consistent, explain whether the solution is unique or whether there are infinitely many solutions.
b. If possible, give an example of a \(3 \times 5\) augmented matrix that corresponds to a linear system having a unique solution. If it is not possible, explain why.
c. If possible, give an example of a \(5 \times 3\) augmented matrix that corresponds to a linear system having a unique solution. If it is not possible, explain why.
d. What condition on the pivot positions guarantees that a linear system has a unique solution?
e. If a linear system has a unique solution, what can we say about the relationship between the number of equations and the number of variables?
Let's consider what we've learned in this activity. Since we are interested in the question of whether a consistent linear system has a unique solution or infinitely many, we will only consider consistent systems. By the results of the previous section, this means that there is not a pivot in the rightmost column of the augmented matrix. Here are two possible examples:
\[
\left[\begin{array}{lll|r}
1 & 0 & 0 & 4 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 2
\end{array}\right] \quad\left[\begin{array}{lll|r}
1 & 0 & 2 & -2 \\
0 & 1 & 1 & 4 \\
0 & 0 & 0 & 0
\end{array}\right]
\]
In the first example, we have the equations
\[
\begin{array}{l}
x_1=4 \\
x_2=-1 \\
x_3=2
\end{array}
\]
demonstrating the fact that there is a unique solution \(\left(x_1, x_2, x_3\right)=(4,-1,2)\).
In the second example, we have the equations
\[
\begin{array}{rr}
x_1+2 x_3 & =-2 \\
x_2+x_3 & =4
\end{array}
\]
that we may rewrite in parametric form as
\[
\begin{array}{l}
x_1=-2-2 x_3 \\
x_2=4-x_3
\end{array} .
\]
Here we see that \(x_1\) and \(x_2\) are basic variables that may be expressed in terms of the free variable \(x_3\). In this case, the presence of the free variable leads to infinitely many solutions.
Remember that every column of the coefficient matrix corresponds to a variable in our linear system. In the first example, we see that every column of the coefficient contains a pivot position, which means that every variable is uniquely determined. In the second example, the column of the coefficient matrix corresponding to \(x_3\) does not contain a pivot position, which results in \(x_3\) appearing as a free variable. This illustrates the following principle.
Suppose that we consider a consistent linear system.
- If every column of the coefficient matrix contains a pivot position, then the system has a unique solution.
- If there is a column in the coefficient matrix that contains no pivot position, then the system has infinitely many solutions.
- Columns that contain a pivot position correspond to basic variables while columns that do not correspond to free variables.
When a linear system has a unique solution, every column of the coefficient matrix has a pivot position. Since every row contains at most one pivot position, there must be at least as many rows as columns in the coefficient matrix. Therefore, the linear system has at least as many equations as variables, which is something we intuitively suspected in Section 1.1 .
It is reasonable to ask how we choose the free variables. For instance, if we have a single equation
\[
x+2 y=4
\]
then we may write
\[
x=4-2 y
\]
or, equivalently,
\[
y=2-\frac{1}{2} x
\]
Clearly, either variable may be considered as a free variable in this case.
As we'll see in the future, we are more interested in the number of free variables rather than in their choice. For convenience, we will adopt the convention that free variables correspond to columns without a pivot position, which allows us to quickly identify them. For example, the variables \(x_2\) and \(x_4\) appear as free variables in the following linear system:
\[
\left[\begin{array}{rrrr|r}
1 & 0 & 0 & 2 & 3 \\
0 & 0 & 1 & -1 & 0
\end{array}\right]
\]
1.4.3 Summary
We have seen how the locations of pivot positions, in both the augmented and coefficient matrices, give vital information about the existence and uniqueness of solutions to linear systems. More specifically,
- A linear system is inconsistent exactly when a pivot position appears in the rightmost column of the augmented matrix.
- If a linear system is consistent, the solution is unique when every column of the coefficient matrix contains a pivot position. There are infinitely many solutions when there is a column of the coefficient matrix without a pivot position.
- If a linear system is consistent, the columns of the coefficient matrix containing pivot positions correspond to basic variables and the columns without pivot positions correspond to free variables.
1.4.4 Exercises
1. For each of the augmented matrices in reduced row echelon form given below, determine whether the corresponding linear system is consistent and, if so, determine whether the solution is unique. If the system is consistent, identify the free variables and the basic variables and give a description of the solution space in parametric form.
a.
\[
\left[\begin{array}{llll|r}
0 & 1 & 0 & 0 & 2 \\
0 & 0 & 1 & 0 & 3 \\
0 & 0 & 0 & 1 & -2
\end{array}\right] .
\]
b.
\[
\left[\begin{array}{llll|l}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .
\]
c.
\[
\left[\begin{array}{llll|l}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{array}\right] .
\]
d.
\[
\left[\begin{array}{lll|l}
1 & 0 & 0 & -3 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & -2
\end{array}\right]
\]
2. For each of the following linear systems, determine whether the system is consistent, and, if so, determine whether there are infinitely many solutions.
a.
\[
\begin{array}{rlr}
2 x_1-x_2+3 x_3 & = & 10 \\
-x_1+x_2+3 x_4 & =8 \\
2 x_2+2 x_3-x_4 & =-4 \\
3 x_1+2 x_2-x_3+x_4 & =10
\end{array}
\]
b.
\[
\begin{array}{rr}
2 x_1-x_2+3 x_3+3 x_4= & 8 \\
-x_1+x_2+2 x_4= & -1 \\
2 x_2+2 x_3+6 x_4= & 4 \\
3 x_1+2 x_2-x_3-3 x_4= & 1
\end{array}
\]
C.
\[
\begin{array}{rr}
2 x_1-x_2+3 x_3+3 x_4= & 8 \\
-x_1+x_2+2 x_4= & -1 \\
2 x_2+2 x_3+6 x_4= & 4 \\
3 x_1+2 x_2-x_3-3 x_4= & 0
\end{array}
\]
3. Include an example of an appropriate matrix as you justify your responses to the following questions.
a. Suppose a linear system having six equations and three variables is consistent. Can you guarantee that the solution is unique? Can you guarantee that there are infinitely many solutions?
b. Suppose that a linear system having three equations and six variables is consistent. Can you guarantee that the solution is unique? Can you guarantee that there are infinitely many solutions?
c. Suppose that a linear system is consistent and has a unique solution. What can you guarantee about the pivot positions in the augmented matrix?
4. Determine whether the following statements are true or false and provide a justification for your response.
a. If the coefficient matrix of a linear system has a pivot in the rightmost column, then the system is inconsistent.
b. If a linear system has two equations and four variables, then it must be consistent.
c. If a linear system having four equations and three variables is consistent, then the solution is unique.
d. Suppose that a linear system has four equations and four variables and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution.
e. Suppose that a linear system has five equations and three variables and that the coefficient matrix has a pivot position in every column. Then the linear system is consistent and has a unique solution.
5. We began our explorations in Section 1.1 by noticing that the solution spaces of linear systems with more equations seem to be smaller. Let's reexamine this idea using what we know about pivot positions.
a. Remember that the solution space of a single linear equation in three variables is a plane. Can two planes ever intersect in a single point? What are the possible ways in which two planes can intersect? How can our understanding of pivot positions help answer these questions?
b. Suppose that a consistent linear system has more variables than equations. By considering the possible pivot positions, what can you say with certainty about the solution space?
c. If a linear system has many more equations than variables, why is it reasonable to expect the system to be inconsistent?
6. The following linear systems contain either one or two parameters.
a. For what values of the parameter \(k\) is the following system consistent? For which of those values is the solution unique?
\[
\begin{aligned}
-x_1+2 x_2 & =3 \\
2 x_1-4 x_2 & =k
\end{aligned} .
\]
b. For what values of the parameters \(k\) and / is the following system consistent? For which of those values is the solution unique?
\[
\begin{array}{c}
2 x_1+4 x_2=3 \\
-x_1+k x_2=l
\end{array} .
\]
7. Consider the linear system described by the following augmented matrix.
\[
\left[\begin{array}{lll|l}
1 & 2 & 3 & 1 \\
4 & 5 & 6 & 4 \\
a & b & c & 9
\end{array}\right] .
\]
a. Find a choice for the parameters \(a, b\), and \(c\) that causes the linear system to be inconsistent. Explain why your choice has this property.
b. Find a choice for the parameters \(a, b\), and \(c\) that causes the linear system to have a unique solution. Explain why your choice has this property.
c. Find a choice for the parameters \(a, b\), and \(c\) that causes the linear system to have infinitely many solutions. Explain why your choice has this property.
8. A linear system where the right hand side of every equation is 0 is called homogeneous. The augmented matrix of a homogeneous system, for instance, has the following form:
\[
\left[\begin{array}{llll|l}
* & * & * & * & 0 \\
* & * & * & * & 0 \\
* & * & * & * & 0
\end{array}\right] .
\]
a. Using the concepts we've seen in this section, explain why a homogeneous linear system must be consistent.
b. What values for the variables are guaranteed to give a solution? Use this to offer another explanation for why a homogeneous linear system is consistent.
c. Suppose that a homogeneous linear system has a unique solution.
1. Give an example of such a system by writing its augmented matrix in reduced row echelon form.
2. Write just the coefficient matrix for the example you gave in the previous part. What can you say about the pivot positions in the coefficient matrix? Explain why your observation must hold for any homogeneous system having a unique solution.
3. If a homogeneous system of equations has a unique solution, what can you say about the number of equations compared to the number of variables?
9. In a previous math class, you have probably seen the fact that, if we are given two points in the plane, then there is a unique line passing through both of them. In this problem, we will begin with the four points on the left below and ask to find a polynomial that passes through these four points as shown on the right.
A degree three polynomial can be written as
\[
p(x)=a+b x+c x^2+d x^3
\]
where \(a, b, c\), and \(d\) are coefficients that we would like to determine. Since we want the polynomial to pass through the point \((3,1)\), we should require that
\[
p(3)=a+3 b+9 c+27 d=1 \text {. }
\]
In this way, we obtain a linear equation for the coefficients \(a, b, c\), and \(d\).
a. Write the four linear equations for the coefficients obtained by requiring that the graph of the polynomial \(p(x)\) passes through the four points above.
b. Write the augmented matrix corresponding to this system of equations and use the Sage cell below to solve for the coefficients.
c. Write the polynomial \(p(x)\) that you found and check your work by graphing it in the Sage cell below and verifying that it passes through the four points. To plot a function over a range, you may use a command like
\[
\operatorname{plot}\left(1+x-2 * x^{\wedge} 2, x \min =-1, x m a x=4\right) \text {. }
\]
1
d. Rather than looking for a degree three polynomial, suppose we wanted to find a polynomial that passes through the four points and that has degree two, such as
\[
p(x)=a+b x+c x^2
\]
Solve the linear system for the coefficients. What can you say about the existence and uniqueness of a degree two polynomial passing through these four points?
e. Rather than looking for a degree three polynomial, suppose we wanted to find a polynomial that passes through the four points and that has degree four, such as
\[
p(x)=a+b x+c x^2+d x^3+e x^4 .
\]
Solve the linear system for the coefficients. What can you say about the existence and uniqueness of a degree four polynomial passing through these four points?
f. Suppose you had 10 points and you wanted to find a polynomial passing through each of them. What should the degree of the polynomial be to guarantee that there is exactly one such polynomial? Explain your response.


2.15: Putting It Together- Internet Research
After learning about different ways to conduct research on the internet, you can now more confidently try searching for business information and even information to satisfy personal curiosity. The world is now open in a much larger way for you to discover ways to improve business management reports, presentations, conduct competitive analysis, and experience the variety of business information out on the web.
This module has covered quite a few methods for searching. As a quick refresher, watch this video that reiterates most of the aspects from this module.
Contributors and Attributions
CC licensed content, Original
- Putting It Together: Internet Research. Authored by : Sherri Pendleton. Provided by : Lumen Learning. License : CC BY: Attribution
CC licensed content, Shared previously
- Searching the Web. Authored by : The Vermont Council on Rural Development. Located at : https://www.youtube.com/watch?v=SkKC1BnuQsE . License : CC BY: Attribution


4.6: What is the purpose of school?
-
- Last updated
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BY: Jennifer Scarce
Learning Objectives
- Reader will recognize the social effects of school on a child's life.
- Reader will recognize the intellectual effects of school on a child's life.
- Reader will know when the first school came into existence.
WHAT IS SCHOOL?
1. an institution where instruction is given, esp. to persons under college age: The children are at school.
2. an institution for instruction in a particular skill or field.
3. a college or university.
4. a regular course of meetings of a teacher or teachers and students for instruction; program of instruction: summer school.
5. a session of such a course: no school today; to be kept after school.
6. the activity or process of learning under instruction, esp. at a school for the young: As a child, I never liked school.
7. one's formal education: They plan to be married when he finishes school.
8. a building housing a school.
9. the body of students, or students and teachers, belonging to an educational institution: The entire school rose when the principal entered the auditorium.
10. a building, room, etc., in a university, set apart for the use of one of the faculties or for some particular purpose: the school of agriculture
BACKGROUND
The first public school came into existence in the mid-nineteenth century. Its founders called it the “common” school. Common schools were funded by local property taxes, charged no tuition, open to all white children, governed by local school committees, and subject to a modest amount of state regulation(Tyack,2001). Students often went to the common school from ages six to fourteen, although this could vary widely. The duration of the school year was often dictated by the agricultural needs of particular communities, with children being off when they would be needed on the family farm(Katz, 1987). Typically, with a small amount of state oversight, each district was controlled by an elected local school board. Traditionally a county school superintendent or regional director was elected to supervise day-to-day activities of several common school districts. Since common schools were locally controlled, and the United States was very rural in the nineteenth century, most common schools were small one-room schools(Kaestle, 1983). Common schools had a single teacher (usually female) and all the students were taught together, regardless of age. Common schools typically taught reading, writing, arithmetic, history, geography, and math. Evaluation of students was very varied (from 0-100 grading to no grades at all), but an end-of-the-year recital was a common way that parents were informed about what their children were learning(Cremin,1980).
THE INTELLECTUAL IMPACT OF SCHOOL
What's the purpose of school anyway? This is a question that every child asks at some point in their adolescent years. After all, what is appealing to a child about going to bed early, getting up early, and sitting in a desk for six and a half hours a day? Children don't understand how important going to school really is, but the importance of education is quite clear. Education is the knowledge of putting one's potentials to maximum use. Training of a human mind is not complete without education. It tells one how to think and how to make decisions. The importance of education is that only through the attainment of education, man is enabled to receive information from the external world to acquaint himself with past history and receive all necessary information regarding the present(Katz, 1987). Knowledge of education is built upon a foundation that begins with pre-school, where a lifetime of learning begins. It is vital to have a solid foundation of learning because each year new applications are taught and learning is like building blocks, you continually build on what you were initially given. If you miss a piece of information along the way, you will never be able to reach the top.
GOOD CITIZENSHIP
Achieving a good education to further knowledge is not the only purpose of school. Today’s schools really focus on teaching good citizenship and good character. I have substituted for the past year and a half and I have to share an example of being a witness of following through with this application. I recently substituted at an elementary school that encourages children to be honest and that no good deed goes unnoticed. To follow through with this concept, each morning during announcements the principle recognizes children for picking up paper towels in the restroom or turning in money found on the floor, even if it is just a penny. The children are so proud of themselves when they are acknowledged and this has really become an epidemic throughout the entire school.
SOCIAL SKILLS
At a very young age children are taught to share and to be considerate of other people’s feelings. They are also taught a great deal about emotions, behavior, and the consequences of their actions. Being able to interact socially and have healthy relationships is a very important part of life. While in the classroom children are constantly interacting with others and whether they are aware of it or not, they are learning to co-exist in a very diverse world. This socialization helps children build critical thinking skills and develop good communication skills. Schools also provide lots of extracurricular activities that incorporate teamwork, good sportsmanship, and exercise into participation. The elementary school ages are considered to be the fundamental grades and stages of development. With this in mind, it is important that children are taught positive behaviors and habits early in life. By teaching children positive behaviors at a young age, kids are more able to understand and engage in long-term attitudes and actions that will guide them towards future success. To encourage students, and to teach all kids positive behaviors, elementary schools across the country have implemented positive behavior programs to improve student awareness, knowledge, and development(Wiggins, McTighe, 2008).
CONCLUSION
Even before they enter school, young children learn to walk, to talk, and to use their hands to manipulate toys, food, and other objects. They use all of their senses to learn about the sights, sounds, tastes, and smells in their environments. They learn how to interact with their parents, siblings, friends, and other people important to their world. When they enter school, children learn basic academic subjects such as reading, writing, and mathematics. They also continue to learn a great deal outside the classroom. They learn which behaviors are likely to be rewarded and which are likely to be punished. They learn social skills for interacting with other children. After they finish school, people must learn to adapt to the many major changes that affect their lives, such as getting married, raising children, and finding and keeping a job. Because learning continues throughout our lives and affects almost everything we do, the study of learning is important in many different fields. Teachers need to understand the best ways to educate children. Psychologists, social workers, criminologists, and other human-service workers need to understand how certain experiences change people’s behaviors. Employers, politicians, and advertisers make use of the principles of learning to influence the behavior of workers, voters, and consumers. The purpose of school is to provide a quality education so that all students have an equal opportunity to develop their full potential(Wiggins, McTighe, 2008).
Note
“The central job of schools is to maximize the capacity of each student.” ~Carol Ann Tomlinson
REFERENCES
Cremin, Lawrence (1980). American Education: The National Experience. New York: Harper Collins.
Kaestle, Carl (1983). Pillars of the Republic: Common schools and American society, 1780-1860. New York: Hill and Wang.
Katz, Michael (1987). Reconstructing American Education. Cambridge: Harvard.24-57.
Learning (2009). Merriam-Webster Online Dictionary. Retrieved February 20, 2009 from http://www.merriam-webster.com/dictionary/school
Wiggins, G., McTighe, J. (2008). Educational Leadership, Vol. 65 Issue 8, p36-41, 6p. Retrieved February 20, 2009 from Teachers Reference Journal.