Academic Standards for Mathematics
... sophistication that students are expected to achieve as they progress through school. With each standard divided into conceptual strands, this document avoids repetition of learned skills and makes an obvious progression across grade levels less explicit. Teachers shall expect that students know and ...
... sophistication that students are expected to achieve as they progress through school. With each standard divided into conceptual strands, this document avoids repetition of learned skills and makes an obvious progression across grade levels less explicit. Teachers shall expect that students know and ...
Is there beauty in mathematical theories?
... these distinctions and, even among mathematicians, there are widely different perceptions of the merits of this or that achievement, this or that contribution. On the other hand, I myself have a confidence in my own views with regard to mathematics that does not spring, at least not in the first ins ...
... these distinctions and, even among mathematicians, there are widely different perceptions of the merits of this or that achievement, this or that contribution. On the other hand, I myself have a confidence in my own views with regard to mathematics that does not spring, at least not in the first ins ...
Math 7 Scope and Sequence By
... The student is expected to: Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and ev ...
... The student is expected to: Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and ev ...
8. I can use place value and number facts to solve problems. 8. I can
... information using ‘many-toone’ in pictograms using simple ratios (2, 5 and 10). ...
... information using ‘many-toone’ in pictograms using simple ratios (2, 5 and 10). ...
MATHEMATICS INDUCTION AND BINOM THEOREM
... discussion is about positive integers sets Three steps to prove (using mathematics induction) that “P(n) is true for all n positive integers”: 1. Basic step: prove that P(1) is true 2. Inductive step: Assumed that P(k) is true, it can be shown that P(k+1) is true for all k 3. Conclusion: n P(n) i ...
... discussion is about positive integers sets Three steps to prove (using mathematics induction) that “P(n) is true for all n positive integers”: 1. Basic step: prove that P(1) is true 2. Inductive step: Assumed that P(k) is true, it can be shown that P(k+1) is true for all k 3. Conclusion: n P(n) i ...
Discrete Mathematics
... No odd perfect numbers are known. Even perfect numbers are easy to describe: they’re of the form 2p−1 (2p − 1) where p is a prime for which 2p − 1 is also a prime. The two examples above correspond to the cases p = 2 and p = 3. ...
... No odd perfect numbers are known. Even perfect numbers are easy to describe: they’re of the form 2p−1 (2p − 1) where p is a prime for which 2p − 1 is also a prime. The two examples above correspond to the cases p = 2 and p = 3. ...
AQA Foundation
... Construct and interpret a pictogram Construct and interpret a bar chart Construct and interpret a dual bar chart Interpret a pie chart Construct a pie chart Interpret a stem-and-leaf diagram Construct a stem-and-leaf diagram (ordered) Construct a frequency diagram Interpret a time series graph Draw ...
... Construct and interpret a pictogram Construct and interpret a bar chart Construct and interpret a dual bar chart Interpret a pie chart Construct a pie chart Interpret a stem-and-leaf diagram Construct a stem-and-leaf diagram (ordered) Construct a frequency diagram Interpret a time series graph Draw ...
The structure of the Fibonacci numbers in the modular ring Z5
... [3] Leyendekkers, J.V., A.G. Shannon. Geometrical and Pellian Sequences. Advanced Studies in Contemporary Mathematics. Vol. 22, 2012, No. 4, 507–508. [4] Leyendekkers, J.V., A.G. Shannon. On the Golden Ratio (Submitted). [5] Leyendekkers, J.V., A.G. Shannon. The Decimal String of the Golden Ratio (S ...
... [3] Leyendekkers, J.V., A.G. Shannon. Geometrical and Pellian Sequences. Advanced Studies in Contemporary Mathematics. Vol. 22, 2012, No. 4, 507–508. [4] Leyendekkers, J.V., A.G. Shannon. On the Golden Ratio (Submitted). [5] Leyendekkers, J.V., A.G. Shannon. The Decimal String of the Golden Ratio (S ...
Maths Challenge Semi-Final questions 2008
... The total length of the cubes is 19 x 13mm = 247mm 247mm = 24.7cm 24.7cm rounded to the nearest centimetre is 25cm ...
... The total length of the cubes is 19 x 13mm = 247mm 247mm = 24.7cm 24.7cm rounded to the nearest centimetre is 25cm ...
ratio and proportion
... How much is needed of each of the above ingredients for 30 cakes. There's nothing we can multiply 12 by to get 30, so we need to work out the amount of ingredients needed for a smaller amount first. Working out the ingredients for 6 works because it's easy to get from 12 to 6 (÷ 2) and easy to get f ...
... How much is needed of each of the above ingredients for 30 cakes. There's nothing we can multiply 12 by to get 30, so we need to work out the amount of ingredients needed for a smaller amount first. Working out the ingredients for 6 works because it's easy to get from 12 to 6 (÷ 2) and easy to get f ...
Connecting Repeating Decimals to Undergraduate Number Theory
... mathematics is that by and large, it is still dominated by the “chalk-and talk” paradigm, a carefully selected linear ordering of course content, and assessment which is heavily based on a final examination. The curriculum for undergraduates needs to be strengthened, broadened, and designed for more ...
... mathematics is that by and large, it is still dominated by the “chalk-and talk” paradigm, a carefully selected linear ordering of course content, and assessment which is heavily based on a final examination. The curriculum for undergraduates needs to be strengthened, broadened, and designed for more ...
Slide 1
... Ratios can be written in several different ways: •as a fraction; 3/4 •using the word "to“; “3 to 4 “ •or with a colon; 3 : 4 ...
... Ratios can be written in several different ways: •as a fraction; 3/4 •using the word "to“; “3 to 4 “ •or with a colon; 3 : 4 ...
Functions, Grade 11, University/College Preparation (MCF3M)
... Solving Problems Involving the Sine Law and the Cosine Law in Oblique Triangles Specific Expectations ...
... Solving Problems Involving the Sine Law and the Cosine Law in Oblique Triangles Specific Expectations ...
Lecture notes 2 -- Sets
... When these symbols are used to indicate di↵erent ideas, the distinction between ⇢ and ✓ is analogous to that between < and ; your understanding of subsets will benefit greatly from meditating on this analogy. NOTE 2: Many students first learning about the symbols 2 and ⇢ initially confuse them. Rem ...
... When these symbols are used to indicate di↵erent ideas, the distinction between ⇢ and ✓ is analogous to that between < and ; your understanding of subsets will benefit greatly from meditating on this analogy. NOTE 2: Many students first learning about the symbols 2 and ⇢ initially confuse them. Rem ...
What`s the error?
... 4. The scatter diagram above shows negative correlation. 5. In a bar chart for qualitative data, there must be gaps beetween the bars. 6. When drawing a pie chart, it is important to remember that the sum of the angles in the centre of the circle is 180°. 7. When drawing any histogram, the formula f ...
... 4. The scatter diagram above shows negative correlation. 5. In a bar chart for qualitative data, there must be gaps beetween the bars. 6. When drawing a pie chart, it is important to remember that the sum of the angles in the centre of the circle is 180°. 7. When drawing any histogram, the formula f ...
12.3 Geometric Sequences Series
... 7. A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height. a. What height will the ball bounce up to after it strikes the ground the 3rd time? b. What is its height after it strikes the ground the nth time? c. How many times does the ...
... 7. A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height. a. What height will the ball bounce up to after it strikes the ground the 3rd time? b. What is its height after it strikes the ground the nth time? c. How many times does the ...
Similar Right Triangles
... Name _________________________________________________________ Date _________ ...
... Name _________________________________________________________ Date _________ ...
Sociable Numbers - Ateneo de Manila University
... The number 6 has an interesting property. Its factors are 1, 2, 3, and 6. If we add all the factors of 6 that are not equal to itself, then we get 1 + 2 + 3 = 6, which is equal to itself – a happy coincidence! The number 6 is called a perfect number. Now if we take the number 220, and add all its fa ...
... The number 6 has an interesting property. Its factors are 1, 2, 3, and 6. If we add all the factors of 6 that are not equal to itself, then we get 1 + 2 + 3 = 6, which is equal to itself – a happy coincidence! The number 6 is called a perfect number. Now if we take the number 220, and add all its fa ...
Mathematics and art
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.Mathematics and art have a long historical relationship. Artists have used mathematics since the 5th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient times, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of proportion in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesberg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim.Mathematics has directly influenced art with conceptual tools such as linear perspective, the analysis of symmetry and mathematical objects such as polyhedra and the Möbius strip. The construction of models of mathematical objects for research or teaching has led repeatedly to artwork, sometimes by mathematicians such as Magnus Wenninger who creates colourful stellated polyhedra. Mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte, in engravings by M. C. Escher, and in computer art which often makes use of fractals, cellular automata and the Mandelbrot set. Controversially, the artist David Hockney has argued that artists from the Renaissance onwards made use of the camera lucida to draw precise representations of scenes; the architect Philip Steadman similarly argued that Vermeer used the camera obscura in his distinctively observed paintings.Other relationships include the algorithimic analysis of artworks by X-ray fluorescence spectroscopy; the stimulus to mathematics research by Filippo Brunelleschi's theory of perspective which eventually led to Girard Desargues's projective geometry; and the persistent view, based ultimately on the Pythagorean notion of harmony in music and the view that everything was arranged by Number, that God is the geometer of the world, and that the world's geometry is therefore sacred. This is seen in artworks such as William Blake's The Ancient of Days.