References
The First Steps Literacy resources have got everything covered when it comes to the interrelated strands of literacy learning (reading, writing, speaking and listening). This site provides resource books to map the development of your students with a range of teaching and learning experiences. A guide to how you can link assessment, teaching, and learning to maximise student success can also be found here and further supports effective classroom practices.
The First Steps Mathematics resources cover topic areas such as number, measurement, space, and chance and data. This resource provides a great range of teaching and learning experiences as well as diagnostic tasks designed to explicitly access the students’ ability in a chosen area of mathematics.
These resources may be of use to you for getting ideas for university assignments or for when you have your own classroom –
http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-literacy/
http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/
]]>Curated by Lefouque
]]>Everyday we all use different numbers, patterns everyday from telephone numbers, house/apartment numbers, time and many more. Once the child can recognise a number it is important to relate it back to their experience. For e.g. what is the number of our house? What is the floor for my work in the high rise building? What shape is the car wheels? What shape are the car windows?
Looking at shapes size, thickness, colour allows children to understand measurement, numeracy and foundations of counting. I would like to share a white paper which I came across on implementing mathematics at home, see below from early years the organisation for young children;
Maths in the Home
Maths is everywhere in the home. With the support of parents, children can grasp many mathematical concepts through their play.
Children will begin to:
Here are a few ways in which you can use play to learn mathematical concepts.
Physical Play
There are many opportunities for learning Maths through Play based and sensory play activities.
Synopsis: This is a collection of 20 science-fiction stories based on mathematical premises, all written since 1960. The stories are by different science fiction writers such as Isaac Asimov, Larry Niven and Frederik Pohl.
Published: January 1987 | ISBN-13: 978-0877958901
Editor’s Homepage: http://www.rudyrucker.com
[Image Credit: http://ecx.images-amazon.com/images/I/91W44P817BL.jpg ]
]]>Catch you on the flip side,
Hayley 🙊
I have many chances to experience guiding students through multiple units and on Monday we finished our 2D Shapes unit. The students took the assessment and based on what I observed, many of the students understood the content. There are still many students (about 6) who were unable to take the test because they were absent but I am eager to see their scores because I think they will show a good understanding of the content. One of the great things about teaching and planning math for the entire unit was that I am able to see the growth of my students and help address misconceptions and alter lesson plans to meet their needs.
For example, when teaching about equal parts, my students were confusing about whether or not certain squares were split into fourths. The students were able to recognize that four small squares or four rectangles that all looked the same are fourths but they were unable to make the connection when looking at a square split in half where the two halves are split in different ways.
In the top squares, each piece is the same size. (I apologize that the rectangular pieces are not perfect on the second square but I wanted to show a pictorial representation of a misconception that my students had.) The third square, however, looks as if the four parts are no equal. But each side of the square (shown below using colors) is half of the square. Since these two pieces are cut in half each of the four parts are equal.
This was a difficult concept for my students to grasp so I used actual paper that I cut up in front of the students along with drawings on the white board to help them understand. By providing this realia, I was able to bridge the gap in my students’ knowledge and allow many of them to grasp the idea. I felt like this was an important part to teach to help build their conceptual knowledge of splitting shapes into equal and unequal parts because my students made the assumption (which is a misconception) that when a 2D shape is split into parts of different shapes, that the parts are unequal.
For example the fourth shape is split into rectangles and squares as taken from the other shapes on top. The parts are all equal even though they are different shapes because they are all half of half of the original shape. Although this terminology seems confusing, this conceptual knowledge can be helpful in the future, especially when students have to discuss fractions using ½ and ¼ and multiplying fractions to find half of a half or ½ x ½ = ¼.
I am approaching the final days of my internship, which is exciting because of the new opportunities that await me in the future but also a bit sad because I will really miss my classroom and my students. I have made so much growth and progress over the course of this year and I just cannot believe everything that has happened to me throughout my experience in the Residency Program. I can definitely say I would not trade this for anything.
]]>A daring choice at the Stardate 46600 Sommelier Competition as Commander Sisko decides to pair a Denebian coq au vin with a tall glass of Strawberry Fanta Zero.
“The little dangly thing at the back of your throat has a smiley face on it.”
“Oh, please open up, Family Feud door, that we may compete against the stars of the hit CBS sitcom Dave’s World!
And that’s about it, except that the episode was — really! — nominated for an Emmy Award for Outstanding Individual Achievement in Hairstyling for a Series. Which is an actual thing, which is kind of wonderful.
Oh, also, just yesterday I had another bunch of mathematically-themed comic strips so if you wanted to read about those too I’d be glad. Thank you.
]]>There is nothing more beautiful than parabolic bridge cables at sunset.
– Anonymous
This view of the Mackinac Bridge in Michigan is sure to spark up some great conversation topics, including but not limited to:
Curated by Lefouque
]]>https://zoomcharts.com/en/gallery/all:piechart-donut-single-level-britney-spears
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https://zoomcharts.com/en/gallery/all:piechart-donut-single-level-britney-spears
ZoomCharts’ advanced data visualization line of software lets you efficiently view, analyze, and present high volumes of data. With the Pie Chart Donut, One Level tool, make your data presentation not only organized and easy to read, but visually pleasing, too.
Interactive data charts such as ZoomCharts’ Pie Chart Donut, One Level are used by clients in various educational fields like sciences and mathematics, including anatomy, biochemistry, ecology, microbiology, nutrition, neuroscience, physiology, zoology, chemical engineering, geochemistry, molecular biology, geology, paleontology, physics, astronomy, algebra, computer science, geometry, logic, and statistics, and the arts such as, music, dance, theatre, film, animation, architecture, applied arts, photography, graphic design, interior design, and mixed media.
– Interactive data representation lets you visualize large amounts of information
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– Single level chart view for visual ease
Step 1 Hover over a data series to display detailed information.
Step 2 Click on chart units to single out data information.
Check out ZoomCharts products:
Network Chart
Big network exploration
Explore linked data sets. Highlight relevant data with dynamic filters and visual styles. Incremental data loading. Exploration with focus nodes.
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Browse activity logs, select time ranges. Multiple data series and value axes. Switch between time units.
Pie Chart
Amazingly intuitive hierarchical data exploration
Get quick overview of your data and drill down when necessary. All in a single easy to use chart.
Facet Chart
Scrollable bar chart with drill-down
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http://www.zoomcharts.com
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https://zoomcharts.com/en/gallery/all:piechart-donut-single-level-britney-spears
ZoomCharts’ advanced data visualization line of software lets you efficiently view, analyze, and present high volumes of data. With the Pie Chart Donut, One Level tool, make your data presentation not only organized and easy to read, but visually pleasing, too.
Interactive data charts such as ZoomCharts’ Pie Chart Donut, One Level are used by clients in various educational fields like sciences and mathematics, including anatomy, biochemistry, ecology, microbiology, nutrition, neuroscience, physiology, zoology, chemical engineering, geochemistry, molecular biology, geology, paleontology, physics, astronomy, algebra, computer science, geometry, logic, and statistics, and the arts such as, music, dance, theatre, film, animation, architecture, applied arts, photography, graphic design, interior design, and mixed media.
– Interactive data representation lets you visualize large amounts of information
– Colored visuals allow for effortless data recognition
– Single level chart view for visual ease
Step 1 Hover over a data series to display detailed information.
Step 2 Click on chart units to single out data information.
Check out ZoomCharts products:
Network Chart
Big network exploration
Explore linked data sets. Highlight relevant data with dynamic filters and visual styles. Incremental data loading. Exploration with focus nodes.
Time Chart
Time navigation and exploration tool
Browse activity logs, select time ranges. Multiple data series and value axes. Switch between time units.
Pie Chart
Amazingly intuitive hierarchical data exploration
Get quick overview of your data and drill down when necessary. All in a single easy to use chart.
Facet Chart
Scrollable bar chart with drill-down
Compare values side by side and provide easy access to the long tail.
ZoomCharts
http://www.zoomcharts.com
The world’s most interactive data visualization software
In the beginning there was Maxwell
Now, light is an electromagnetic wave, so to describe it all we need are the following four equations:
These are the equations for the 6 vector components of the electric and magnetic fields, and includes dielectric materials through the factor (where is the refractive index). Light is a sinusoidal wave, so we know that we can assume that the only time dependence of these fields is a factor . Using this fact, one can show:
where is the light wavevector. If we further assume that gradients in refractive index are slow, we arrive at the Helmholtz equation:
where the refractive index has been absorbed into the definition of the material-dependent wavevector . It is possible to show that if a -function source is placed at the origin, the solution of the Helmholtz equation (the Green’s function) is:
Furthermore, in this limit of slow refractive index variations, it can be shown that all the components of the fields vary in the same way. We therefore only concentrate on one component, of the electric field say, and this is then scalar diffraction theory. Vector diffraction must be considered when calculating the field distribution of tightly-focussed laser pulses. The Green’s function simply tells us the phase of the light wave is constantly increasing with time at temporal frequency , and decreasing with propagation distance with spatial frequency .
The intensity of the light wave we consider is given by the Poynting vector
which scales with distance as as expected, and confirms that the Green’s function above represents the amplitude of an energy flux conserved over radial shells of area .
The setup
Now we’ve (rather rapidly) travelled from Maxwell’s equation to an expression for the wave emitted by a single point, we need to use this to figure out how a lens builds up an image. I’ll refer to the following conceptual setup in the derivation:
We have an object in plane , with coordinates . The light from this object travels a distance to the plane , just before the lens. The lens of focal length f applies some magic, and transforms to . Finally, this transformed wave travels a further distance to form an image at plane . Let’s get started.
Paraxial facts
Here on out we stipulate a further restriction: the light waves don’t travel too far from the optical axis, or where is the transverse size of the object/lens/image. This is the paraxial approximation, and makes things much simpler. In the real world the paraxial approximation is too simple, and instead lenses must take into account the fact that it isn’t perfectly valid. We’ll ignore this inconvenient fact here.
Suppose we propagate a light ray from a point to over a longitudinal distance . The total distance the ray travels is:
Substituting this expression into the Green’s function, in the paraxial approximation we have:
We have omitted a constant phase factor from the Green’s function as it is constant across the wave field, and when we consider physical intensities constant phase factors are physically irrelevant.
Getting to the lens
In the object plane at the wave amplitude is given by . From the Green’s function, we know that this point contributes to the point in the plane a factor . Adding up all of the waves emitted by the object we arrive at our first expression for the propagated wave, in this case just before the lens:
What the lens does
Let’s consider a simple lens, flat on one side and consisting of a spherical cap of radius of curvature on the other. The thickness of the lens as a function of position is then given by
Assuming the radius of curvature is large, we may expand the square root. If the refractive index of the lens is , the excess phase shift imposed by the lens is:
where again we have omitted a constant phase shift. You might recognise from the lensmakers equation the definition of the focal length of the lens
and so
The act of the lens is then to apply a phase shift to the incoming light which increases with the square of the radius, and which is proportional to the inverse of the focal length. The light field immediately after the lens is thus:
We’ve assumed the two fields at and occupy the same position, and so implicitly assume the lens is infinitely thin. Unsurprisingly this is known as the thin lens approximation, and is something else which doesn’t necessarily hold in the real world.
Making an image
We’re almost there. We’ve made it to and through the lens, the last step is to form an image. To avoid writing too many integral signs, let’s define a transfer function which dictates how the input field transforms into the output field:
We have most of the ingredients for above, we just need to apply a second propagation step over the distance . Doing this we end up with the slightly scary expression:
Let’s look at this term-by-term:
With these terms out the way, the expression is much more manageable. We have an inkling about the geometric properties of the problem now, so define the image magnification :
Let’s take the limit that – the geometric optics limit. This is equivalent to scaling the integration variables such that the limits go to infinity, and each integral becomes a function:
This transfer function tells us that the object and image planes are the same, as long as we change coordinates such that and . We therefore have a a new plane containing a perfect, inverted, scaled copy of the wave field at the object plane – also known as an image!
If we had included effects like the finite lens aperture, rather than a -function we would instead have an Airy function. The image would then be a convolution between the object and the Airy pattern, reducing resolution below the theoretically perfect one. If the imaging distances don’t fulfil the condition above, the transfer function is additionally broadened in a complex way which reduces resolution further.
The payoff
Well done for making it this far. If you were eagerly anticipating a fancy graphic or something, I’m afraid you’re out of luck. However, don’t despair! Now I’ve set the groundwork, the next post is pretty much all pretty GIFs. I’ll see you then, assuming you haven’t already unsubscribed.
]]>https://zoomcharts.com/en/gallery/all:piechart-donut-single-level-britney-spears
ZoomCharts’ advanced data visualization line of software lets you efficiently view, analyze, and present high volumes of data. With the Pie Chart Donut, One Level tool, make your data presentation not only organized and easy to read, but visually pleasing, too.
Interactive data charts such as ZoomCharts’ Pie Chart Donut, One Level are used by clients in various educational fields like sciences and mathematics, including anatomy, biochemistry, ecology, microbiology, nutrition, neuroscience, physiology, zoology, chemical engineering, geochemistry, molecular biology, geology, paleontology, physics, astronomy, algebra, computer science, geometry, logic, and statistics, and the arts such as, music, dance, theatre, film, animation, architecture, applied arts, photography, graphic design, interior design, and mixed media.
– Interactive data representation lets you visualize large amounts of information
– Colored visuals allow for effortless data recognition
– Single level chart view for visual ease
Step 1 Hover over a data series to display detailed information.
Step 2 Click on chart units to single out data information.
Check out ZoomCharts products:
Network Chart
Big network exploration
Explore linked data sets. Highlight relevant data with dynamic filters and visual styles. Incremental data loading. Exploration with focus nodes.
Time Chart
Time navigation and exploration tool
Browse activity logs, select time ranges. Multiple data series and value axes. Switch between time units.
Pie Chart
Amazingly intuitive hierarchical data exploration
Get quick overview of your data and drill down when necessary. All in a single easy to use chart.
Facet Chart
Scrollable bar chart with drill-down
Compare values side by side and provide easy access to the long tail.
ZoomCharts
http://www.zoomcharts.com
The world’s most interactive data visualization software
Other work
]]>Pythagoras of Samos, 560-BC To 480 BC, was a Greek philosopher and religious leader. Some doubt he ever existed; some doubt that the mathematical concepts, particular the Pythagorean Theorem, originated with him or his School; and some doubt that he ever proved the theorem. Proclus, an ancient commentator on Euclid, does not attribute the result to Pythagoras (Proclus and Morrow 1970) . Nevertheless, there exist accounts written after his death that say Pythagoas was born in Samos and migrated to Croton. Here he founded a philosophical and religious school that attracted many followers. Because no reliable contemporary records survive, and because the school practiced both secrecy and communalism, the contributions of Pythagoras himself and those of his followers cannot be distinguished. Pythagoreans believed that all relations could be reduced to number relations (“all things are numbers”). For the Pythagoreans, numbers meant integers of ratios of integers (rational numbers). Irrational numbers such as the square root of 2 were not allowed.
The Pythagoreans noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. They knew, as did the Egyptians and Babylonians before them, that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. The so-called Pythagorean theorem, that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, may have been known in Babylonia. It is reported that Pythagoras traveled in Egypt and Babylonia in order to gather the wisdom of priests and learned men. The so-called Plimptom 322 tablet (approximately 1700 BC) contains an impressive list of Pythagorean triplets. These are integer solutions to the a^{2} + b^{2} = c^{2} relation. This indicates that the Babylonians new of the relation. Whether they had a general proof or whether the assertion is a mere conjecture or belief is not entirely clear. However, that they knew the result held for all right triangles, not just those with integer legs, is testified to by their deep concern for the irrationality of the square root of two. In fact one story goes that when a disciple brought to Pythagoras’ attention, during a long sea voyage, that a right triangle with legs each equal one has a hypotenuse that is equal to the square root of two and this is an irrational (could not be represented by the ratio of two integers, the communicator was ordered thrown overboard. (Gray 1979).
Whereas much of the Pythagorean doctrine that has survived consists of numerology and number mysticism, the influence of the idea that the world can be understood through mathematics was extremely important to the development of science and mathematics.
The first undisputed proof of the theorem appears in Euclid’s Elements as Proposition 47. It is not the simplest possible proof and clearly depends on the troubling Postulate Five.
It is appropriate to review this most important of all mathematical truths because it subtly provides the genesis of our modern complexities. The circumstances where it is true justify Euclid’s geometry and his five postulates. It is known that its truth is synonymous with Euclid’s often challenged fifth postulate concerning the behavior of parallel lines. Euclid did not give much credit to Pythagoras. He gets around to proving the theorem only after 46 other proofs in Book I. Proofs of this theorem are many. There are some 367 separate proofs in the time between 900 BC and 1940 AD (Loomis, 1940). In fact, no one knows how or if Pythagoras proved the famous theorem. He probably learned of it during his stay in Egypt where it appears to have been used in practical applications. There is some evidence from tablets that it may have been known in a general way as far back as 1600 BC. One of these many proofs was hit upon by a former President of the United States, James A. Garfield (see Loomis Proof 231). Proofs fall into several categories, algebraic, spatial or geometric, vectors and matrices, and physical. Below is a compelling spatial-geometric proof. The center triangle is the right triangle of interest. We construct two large rectangles each with sides a and b. We construct within these three smaller squares within these, a by a on the a leg, b by b on the b leg, and c by on the c hypotenuse. It is clearly seen that the lower a by b contains four abc right triangles. It is also apparent that the upper a by b contains four abc triangles exterior to the c-square. These may be subtracted from both upper and lower a by b squares leaving a-square, b-square and c-square. The sum of the first two must equal the latter as equals subtracted from equals produce equals.
Algebraically we could describe this as the areas of each of the super squares as:
subtracting these reduces each super square to:
The extreme left is, of course, the familiar equation for the hypotenuse of the right triangle. It may be little known that this relationship was known in antiquity before Pythagoras and that it may be proved in multifarious ways. This theorem is true in any “flat” space. It is not true for example, on surface of a sphere, such the earth’s surface. However, any manifold can be embedded in a Euclidean (flat) space by looking in higher dimensions specified by the Whitney Embedding Theorem. See WIKII or Skopenkov, A. (2008), “Embedding and knotting of manifolds in Euclidean spaces”, In: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes. 347 (2): 248–342, arXiv:math/0604045
References:
Loomis, E. S. (1940). The Pythagorean Proposition. Washington, DC, National Council of Teachers of Mathematics.
Gray, J. (1979). Ideas of space : Euclidean, non-Euclidean, and relativistic. Oxford
New York, Clarendon Press ;
Oxford University Press.
Proclus and G. R. Morrow (1970). A commentary on the first book of Euclid’s Elements. Princeton, N.J., Princeton University Press.
]]>
“…it dawned on me that all the numbers we had been given to add up until that time had been kind of “cooked up” so you didn’t have to carry…; and I said to myself, “I wonder what else they’re holding back?”–Robert Brooks, Assistant Professor of Mathematics, University of Southern California
“Education courses are where you learn not to rock the boat.”–Charles Kalme, Assistant Professor of Mathematics, University of Southern California
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