DOCSLIB.ORG

Mathematics Glossary

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematics Glossary Intermediate School Level Mathematics Glossary English / Arabic Glossary Translation of Mathematics Terms Based on the Coursework for Mathematics Grades 6 to 8. Word-for-word glossaries are used for testing accommodations for ELL/LEP students Last Updated: 08/10/09 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 Education - P-16 Office of Elementary, Middle, Secondary, and Continuing Education and Office of Higher Education Office of Bilingual Education and Foreign Language Studies http://www.emsc.nysed.gov/biling/ THE UNIVERSITY OF THE STATE OF NEW YORK Regents of the University MERRYL H. TISCH, Chancellor, B.A., M.A., Ed.D. ....................................................................... New York MILTON L. COFIELD, Vice Chancellor, B.S., M.B.A., Ph.D. ......................................................... Rochester ROBERT M. BENNETT, Chancellor Emeritus, B.A., M.S. ............................................................... Tonawanda SAUL B. COHEN, B.A., M.A., Ph.D................................................................................................ New Rochelle JAMES C. DAWSON, A.A., B.A., M.S., Ph.D. ................................................................................ Plattsburgh ANTHONY S. BOTTAR, B.A., J.D. ................................................................................................... Syracuse GERALDINE D. CHAPEY, B.A., M.A., Ed.D. ................................................................................. Belle Harbor HARRY PHILLIPS, 3rd, B.A., M.S.F.S. ........................................................................................... Hartsdale JOSEPH E. BOWMAN, JR., B.A., M.L.S., M.A., M.Ed., Ed.D......................................................... Albany JAMES R. TALLON, JR., B.A., M.A. ............................................................................................... Binghamton ROGER TILLES, B.A., J.D................................................................................................................. Great Neck KAREN BROOKS HOPKINS, B.A., M.F.A........................................................................................ Brooklyn CHARLES R. BENDIT, B.A. ............................................................................................................. Manhattan BETTY A. ROSA, B.A., M.S. in Ed., M.S. in Ed., M.Ed., Ed.D..................................................... Bronx LESTER W. YOUNG, JR., B.S., M.S., Ed. D. ..................................................................................... Oakland Gardens CHRISTINE D. CEA, B.A., M.A., Ph.D. .......................................................................................... Staten Island WADE S. NORWOOD, B.A. ............................................................................................................. Rochester Interim President of the University and Commissioner of Education CAROLE F. HUXLEY Senior Deputy Commissioner of Education, P–16 JOHANNA DUNCAN-POITIER Associate Commissioner for Curriculum and Instructional Support JEAN STEVENS Coordinator, Office of Bilingual Education and Foreign language Studies PEDRO J. RUIZ Acknowledgements: The New York State Education Department Glossaries for English Language Learners were reviewed and updated during the 2008-2009 school year. We would like to thank in these efforts the New York State Education Department Language BETACs (Spanish, Asian and Haitian Bilingual Education Technical Assistance Centers), the NYS Office of Curriculum, Instruction and Instructional Technology; the New York City Department of Education Office of English Language Learners, and the NYC Department of Education Translation and Interpretation Unit. The State Education Department does not discriminate on the basis of age, color, religion, creed, disability, marital status, veteran status, national origin, race, gender, genetic predisposition or carrier status, or sexual orientation in its educational programs, services and activities. Portions of this publication can be made available in a variety of formats, including brailed, large print or audio tape, upon request. Inquiries concerning this policy of nondiscrimination should be directed to the Department’s Office for Diversity, Ethics, and Access, Room 530, Education Building, Albany, NY 12234. GRADES 6,7,8 COMPILED Arabic LIST ﻗﻴﻤﺔ ﻣﻄﻠﻘﺔ absolute value ﻋﻨﻮن اﻟﻌﻤﻞ ﺑﻄﺮﻳﻘﺔ ﺻﺤﻴﺤﺔ accurately label work واﺋﻢ، آﻴّﻒ adapt ﻣﻌﻜﻮس ﺟﻤﻌﻲ additive inverse ﻣﺘﺠﺎور adjacent ﺿﻠﻊ ﻣﺘﺠﺎور ﻟﻤﺜﻠﺚ adjacent side of a triangle اﻟﺠﺒﺮ algebra ﻣﻌﺎدﻟﺔ ﺟﺒﺮﻳﺔ algebraic equation ﺗﻌﺒﻴﺮ ﺟﺒﺮي algebraic expression ﻣﺘﺒﺎﻳﻨﺎت ﺟﺒﺮﻳﺔ algebraic inequalities ﻧﻤﻂ ﺟﺒﺮي algebraic pattern ﺣﻞ ﺟﺒﺮي algebraic solution ﺟﺒﺮﻳًﺎ algebraically زواﻳﺎ ﺧﺎرﺟﻴﺔ ﻣﺘﺒﺎدﻟﺔ alternate exterior angles زواﻳﺎ داﺧﻠﻴﺔ ﻣﺘﺒﺎدﻟﺔ alternate interior angles ارﺗﻔﺎع altitude ﺣ ﻠّﻞ analyze ﻣﻨﺼﱢﻒ اﻟﺰاوﻳﺔ angle bisector أزواج زواﻳﺎ angle pairs ﻃ ﺒّﻖ apply ﻃ ﺒّﻖ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻻﺳﺘﺮاﺗﻴﺠﻴﺎت apply a variety of strategies ﻣﺼﻄﻠﺤﺎت رﻳﺎﺿﻴﺔ ﻣﻨﺎﺳﺒﺔ appropriate mathematical terms ﺗﻘﺮﻳﺐ approximation ﻗﻮس arc ﻣﺴﺎﺣﺔ area ﻣﺴﺎﺣﺔ داﺋﺮة area of a circle ﺑﺮهﺎن، ﻓﺮﺿﻴﺔ argument ﺣﺴﺎﺑﻴًﺎ arithmetically ﺧﺎﺻﻴﺔ اﻟﺘﺠﻤﻴﻊ ﻓﻲ اﻟﺠﻤﻊ associative property of addition ﺧﺎﺻﻴﺔ اﻟﺘﺠﻤﻴﻊ ﻓﻲ اﻟﻀﺮب associative property of multiplication ﻗﺎﻋﺪة (ﻓﻲ اﻟﻤﺎﺋﺔ) (base (of percent ﻗﺎﻋﺪة ﺷﻜﻞ ﺛﻨﺎﺋﻲ اﻷﺑﻌﺎد base of a 2-dimensional shape ﻗﺎﻋﺪة ﺷﻜﻞ ﺛﻼﺛﻲ اﻷﺑﻌﺎد base of a 3-dimensional shape ﻗﺎﻋﺪة ﻧﻈﺎم اﻟﺘﺮﻗﻴﻢ اﻟﻌﺸﺮي base ten number system ذو ﺣﺪّﻳﻦ binomial ﻣﻨﺼﱢﻒ bisector أُﺣﺴﺐ calculate أُﺣﺴﺐ اﻟﻤﺴﺎﻓﺔ calculate distance أﺣﺴﺐ ﺛﻤﻦ اﻟﻮﺣﺪة calculate unit price أﺣﺴﺐ اﻟﺤﺠﻢ calculate volume ﻣﺌﻮي Celsius زاوﻳﺔ ﻣﺮآﺰﻳﺔ central angle وﺗﺮ chord داﺋﺮة circle رﺳﻢ ﺑﻴﺎﻧﻲ داﺋﺮي circle graph ﻣﺤﻴﻂ circumference أﺳﺌﻠﺔ ﺗﻮﺿﻴﺤﻴﺔ clarifying questions ﻣُﻌﺎﻣﻞ coefficient ﻣﺘﻤﺎﺳﻚ، ﻣﻨﺴﺠﻢ coherent ﺗﻌﺎون collaboration و ﺣّﺪ اﻟﺤﺪود اﻟﻤﺘﺸﺎﺑﻬﺔ combine like terms ﻋﻤﻮﻟﺔ، ﻟﺠﻨﺔ، ﻣﻬﻤﺔ commission ﺧﺎﺻﻴﺔ اﻹﺑﺪال ﻓﻲ اﻟﺠﻤﻊ commutative property of addition ﺧﺎﺻﻴﺔ اﻹﺑﺪال ﻓﻲ اﻟﻀﺮب commutative property of multiplication ﻗﺎرن اﻻﺳﺘﺮاﺗﻴﺠﻴﺎت compare strategies ﻗﺎرن أﺛﻤﻨﺔ اﻟﻮﺣﺪات compare unit prices ﺑﺮآﺎر، ﻓﺮﺟﺎر compass زاوﻳﺘﺎن ﻣﺘﺘﺎﻣﺘﺎن complementary angles أﺣﺪاث ﻣﺮآّﺒﺔ compound events إﻓﻬﻢ، إﺳﺘﻮﻋﺐ، ﺿﻤّﻦ comprehend إﺳﺘﻨﺘﺎج conclusion ﺗﺼﺮف، إدارة conduct ﻣﻨﻄﺒﻖ congruent ﺗﺨﻤﻴﻦ (conjecture (noun ﺧﻤﱢﻦ (conjecture (verb رواﺑﻂ connections أﻋﺪاد ﺻﺤﻴﺤﺔ ﻣﺘﺘﺎﻟﻴﺔ consecutive integers و ﺣّﺪ، ﺿﻢ، إﺟﻤﻊ consolidate ﺗﻘﻴﻴﺪات، ﺣﺼﺮ constraints إﻧﺸﺎء construction ﺣ ﻮّل اﻟﻌﻤﻠﺔ convert money ﺣ ﻮّل اﻟﺴﻌﺔ داﺧﻞ ﻣﻨﻈﻮﻣﺔ ﻣﻌﻴﻨﺔ convert capacity within a given system ﺣ ﻮّل اﻟﻜﺘﻠﺔ داﺧﻞ ﻣﻨﻈﻮﻣﺔ ﻣﻌﻴﻨﺔ convert mass within a given system ﺣ ﻮّل اﻟﺤﺠﻢ داﺧﻞ ﻣﻨﻈﻮﻣﺔ ﻣﻌﻴﻨﺔ convert volume within a given system ﺣ ﻮّل داﺧﻞ ﻣﻨﻈﻮﻣﺔ ﻣﻌﻴﻨﺔ convert within a given system هﻨﺪﺳﺔ إﺣﺪاﺛﻴﺔ coordinate geometry ﻣﺴﺘﻮى إﺣﺪاﺛﻲ coordinate plane زواﻳﺎ ﻣﺘﻨﺎﻇﺮة corresponding angles أﺿﻼع ﻣﺘﻨﺎﻇﺮة corresponding sides ﻣﺜﺎل ﻣﻀﺎد counterexample أﻋﺪاد ﺣﺴﺎﺑﻴﺔ (ﻃﺒﻴﻌﻴﺔ) counting (natural) numbers أﻋﺪاد ﺣﺴﺎﺑﻴﺔ counting numbers آﻮب cup وﺣﺪات ﻗﻴﺎس اﻟﺴﻌﺔ اﻟﻤﺘﻌﺎرف ﻋﻠﻴﻬﺎ customary units of capacity وﺣﺪات ﻗﻴﺎس اﻟﻜﺘﻠﺔ اﻟﻤﺘﻌﺎرف ﻋﻠﻴﻬﺎ customary units of mass ﻣﻌﻄﻴﺎت، ﺑﻴﺎﻧﺎت data ﻓﻚ اﻟﺮﻣﺰ decode داﻓﻊ، إﺛﺒﺖ defend درﺟﺔ اﻟﻤﺘﻌﺪد اﻟﺤﺪود degree of a polynomial آﺜﺎﻓﺔ density أﺣﺪاث ﺗﺎﺑﻌﺔ dependent events ﺻﻒ describe ﻃ ﻮّر ﺻﻴﻐ ﺎً، آﻮن ﻣﻌﺎدﻟﺔ develop formulas ﻗﻄﺮ diameter ﻣﻴﱢﺰ differentiate ﻣﺪﱢد dilate ﺗﻤﺪﱡد dilation أﺑﻌﺎد dimensions ﻧﺎﻗﺶ discuss ﻣﺴﺎﻓﺔ distance ﻣﻴّﺰ distinguish ﺧﺎﺻﻴﺔ اﻟﺘﻮزﻳﻊ distributive property ﻧﻄﺎق domain ﻣﺨﻄﻂ أو رﺳﻢ ﺑﻴﺎﻧﻲ ﺑﺄﻋﻤﺪة ﻣﺰدوﺟﺔ double bar graph ﻣﺨﻄﻂ أو رﺳﻢ ﺑﻴﺎﻧﻲ ﺑﺨﻄﻮط ﻣﺰدوﺟﺔ double line graphs أرﺳﻢ ﻣﺨﻄّﻄﺎً أو ر ﺳ ﻤ ﺎً ﺑﻴﺎﻧﻴﺎً draw a graph أرﺳﻢ ﺻﻮرة draw a picture ﺣﺪﱢد اﺳﺘﻨﺘﺎﺟﺎﺗﻚ draw conclusions ﻣﻌﺎدﻟﺔ equation ﻣﻌﺎدﻟﺔ ﺧﻂ equation of a line ﻣُﺘﺴﺎوي اﻷﺑﻌﺎد equidistant وﺣﺪات ﻗﻴﺎس اﻟﺴﻌﺔ اﻟﻤﺘﻌﺎرف ﻋﻠﻴﻬﺎ ا ﻟ ﻤُﻜﺎﻓﺌﺔ equivalent customary units of capacity آﺴﻮر ﻣُﻜﺎﻓﺌﺔ equivalent fractions ﺗﻌﺒﻴﺮات ﻋﺪدﻳﺔ ﻣُﻜﺎﻓﺌﺔ equivalent numerical expressions ﻧﺴﺐ ﻣُﻜﺎﻓﺌﺔ equivalent ratios ﻗﺪﱢر estimate ...اﻟﻤﺴﺎﺣﺔ area ...ﻣﺤﻴﻂ اﻟﺪاﺋﺮة circumference ...اﻟﺤﺠﻢ volume ﻗﻴﱢﻢ evaluate ﻗﻴﱢﻢ اﻟﺘﺨﻤﻴﻨﺎت evaluate conjectures ﻗﻴﱢﻢ اﻟﻜﻔﺎءة evaluate efficiency ﺟﺪول ﻗﻴﻤﺔ ﺗﺤﻮﻳﻞ اﻟﻌﻤﻼت exchange rate table ﻣﺼﺮوﻓﺎت expenses وﺿﱢﺢ explain إﺳﺘﻜﺸﻒ explore أس exponent ﺻﻴﻐﺔ أﺳﻴﺔ exponential form ﻣﺪّد extend زاوﻳﺔ ﺧﺎرﺟﻴﺔ exterior angle ﻃﺮﻓﺎ اﻟﺘﻨﺎﺳﺐ (extremes (of a proportion أﺳﻄﺢ وﻗﻮاﻋﺪ faces and bases ...اﻟﻤﺨﺮوﻃﺎت of cones ...اﻷﺳﻄﻮاﻧﺎت of cylinders ...اﻟﻤﻨﺸﻮرات of prisms ...اﻷهﺮام of pyramids ﻋﺎﻣﻞ factor درﺟﺔ ﻓﻬﺮﻧﻬﺎﻳﺖ Fahrenheit ﻧﺘﺎﺋﺞ ﻣﻨﺎﺳﺒﺔ favorable outcomes ﻣﺴﺎﻓﺔ ﻣﺤﺪّدة fixed distance ﺷ ﻜ ﻠ ﻴ ﺎً، ر ﺳ ﻤ ﻴ ﺎً formally ﺻﻴﻐﺔ، ﻣﻌﺎدﻟﺔ formula ﺻﻎ formulate ﺻﻎ أﺳﺌﻠﺔ رﻳﺎﺿﻴﺔ Formulate mathematical questions آﺴﺮ fraction ﺗﺮدد frequency ﺟﺪول اﻟﺘﺮدد frequency table داﻟﺔ function ﺗﺪوﻳﻦ اﻟﺪاﻟﺔ function notation ﻗﺎﻋﺪة اﻟﺪاﻟﺔ function rule ﻣﺒﺪأ اﻟﻌﺪ اﻷﺳﺎﺳﻲ fundamental counting principle ﻏﺎﻟﻮن gallon ﺗﻌﻤﻴﻤﺎت generalizations ﺷﻜﻞ هﻨﺪﺳﻲ geometric shape هﻨﺪﺳﺔ geometry ﺗﺨﻄﻴﻄﻴﺎً graphically ﻣﻜﺎﻓﺄة، ﺑﻘﺸﻴﺶ gratuity ﻋﺎﻣﻞ ﻣﺸﺘﺮك أآﺒﺮ greatest common factor ارﺗﻔﺎع height ﻣﺪرّج ﺗﻜﺮاري histogram وﺗﺮ اﻟﻤﺜﻠﺚ اﻟﻘﺎﺋﻢ اﻟﺰاوﻳﺔ hypotenuse أﻓﻜﺎر ideas ﻣﺘﻄﺎﺑﻘﺔ identity ﻋﻨﺼﺮ ﻣﺤﺎﻳﺪ identity element ﺧﺎﺻﻴﺔ اﻟﻌﺪد اﻟﻤﺤﺎﻳﺪ ﻓﻲ اﻟﺠﻤﻊ identity property of addition ﺧﺎﺻﻴﺔ اﻟﻌﺪد اﻟﻤﺤﺎﻳﺪ ﻓﻲ اﻟﻀﺮب identity property of multiplication ﻇﻞ image ﻧﺘﺎﺋﺞ ﻣﺴﺘﺤﻴﻠﺔ impossible outcomes دﺧﻞ income ﻏﻴﺮ ﻣﺒﺎﺷﺮ indirect اﺳﺘﺪﻻل اﺳﺘﻘﺮاﺋﻲ inductive reasoning ﻣﺘﺒﺎﻳﻨﺔ inequality ﻏﻴﺮ رﺳﻤﻲ informally ﻗﻴﻢ اﻟﻤﺪﺧﻞ input values ﻋﺪد ﺻﺤﻴﺢ integer ﻣﻌﺎﻣﻼت آﺎﻣﻠﺔ integer coefficients ﻣﺘﻜﺎﻣﻞ integral
Load more

Recommended publications
  • Arc Length. Surface Area
    Calculus 2 Lia Vas Arc Length. Surface Area. Arc Length. Suppose that y = f(x) is a continuous function with a continuous derivative on [a; b]: The arc length L of f(x) for a ≤ x ≤ b can be obtained by integrating the length element ds from a to b: The length element ds on a sufficiently small interval can be approximated by the hypotenuse of a triangle with sides dx and dy: p Thus ds2 = dx2 + dy2 ) ds = dx2 + dy2 and so s s Z b Z b q Z b dy2 Z b dy2 L = ds = dx2 + dy2 = (1 + )dx2 = (1 + )dx: a a a dx2 a dx2 2 dy2 dy 0 2 Note that dx2 = dx = (y ) : So the formula for the arc length becomes Z b q L = 1 + (y0)2 dx: a Area of a surface of revolution. Suppose that y = f(x) is a continuous function with a continuous derivative on [a; b]: To compute the surface area Sx of the surface obtained by rotating f(x) about x-axis on [a; b]; we can integrate the surface area element dS which can be approximated as the product of the circumference 2πy of the circle with radius y and the height that is given by q the arc length element ds: Since ds is 1 + (y0)2dx; the formula that computes the surface area is Z b q 0 2 Sx = 2πy 1 + (y ) dx: a If y = f(x) is rotated about y-axis on [a; b]; then dS is the product of the circumference 2πx of the circle with radius x and the height that is given by the arc length element ds: Thus, the formula that computes the surface area is Z b q 0 2 Sy = 2πx 1 + (y ) dx: a Practice Problems.
    [Show full text]
  • Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
    RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing ␪ by ⌬␪ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding ␾ fixed, then increasing ␾ by (1).
    [Show full text]
  • Art Concepts for Kids: SHAPE & FORM!
    Art Concepts for Kids: SHAPE & FORM! Our online art class explores “shape” and “form” . Shapes are spaces that are created when a line reconnects with itself. Forms are three dimensional and they have length, width and depth. This sweet intro song teaches kids all about shape! Scratch Garden Value Song (all ages) https://www.youtube.com/watch?v=coZfbTIzS5I Another great shape jam! https://www.youtube.com/watch?v=6hFTUk8XqEc Cookie Monster eats his shapes! https://www.youtube.com/watch?v=gfNalVIrdOw&list=PLWrCWNvzT_lpGOvVQCdt7CXL ggL9-xpZj&index=10&t=0s Now for the activities!! Click on the links in the descriptions below for the step by step details. Ages 2-4 Shape Match https://busytoddler.com/2017/01/giant-shape-match-activity/ Shapes are an easy art concept for the littles. Lots of their environment gives play and art opportunities to learn about shape. This Matching activity involves tracing the outside shape of blocks and having littles match the block to the outline. It can be enhanced by letting your little ones color the shapes emphasizing inside and outside the shape lines. Block Print Shapes https://thepinterestedparent.com/2017/08/paul-klee-inspired-block-printing/ Printing is a great technique for young kids to learn. The motion is like stamping so you teach little ones to press and pull rather than rub or paint. This project requires washable paint and paper and block that are not natural wood (which would stain) you want blocks that are painted or sealed. Kids can look at Paul Klee art for inspiration in stacking their shapes into buildings, or they can chose their own design Ages 4-6 Lois Ehlert Shape Animals http://www.momto2poshlildivas.com/2012/09/exploring-shapes-and-colors-with-color.html This great project allows kids to play with shapes to make animals in the style of the artist Lois Ehlert.
    [Show full text]
  • Solving Cubic Polynomials
    Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac.
    [Show full text]
  • Cones, Pyramids and Spheres
    The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 12 CONES, PYRAMIDS AND SPHERES A guide for teachers - Years 9–10 June 2011 YEARS 910 Cones, Pyramids and Spheres (Measurement and Geometry : Module 12) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. 2011. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 12 CONES, PYRAMIDS AND SPHERES A guide for teachers - Years 9–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 910 {4} A guide for teachers CONES, PYRAMIDS AND SPHERES ASSUMED KNOWLEDGE • Familiarity with calculating the areas of the standard plane figures including circles. • Familiarity with calculating the volume of a prism and a cylinder. • Familiarity with calculating the surface area of a prism. • Facility with visualizing and sketching simple three‑dimensional shapes. • Facility with using Pythagoras’ theorem. • Facility with rounding numbers to a given number of decimal places or significant figures.
    [Show full text]
  • Introduction Into Quaternions for Spacecraft Attitude Representation
    Introduction into quaternions for spacecraft attitude representation Dipl. -Ing. Karsten Groÿekatthöfer, Dr. -Ing. Zizung Yoon Technical University of Berlin Department of Astronautics and Aeronautics Berlin, Germany May 31, 2012 Abstract The purpose of this paper is to provide a straight-forward and practical introduction to quaternion operation and calculation for rigid-body attitude representation. Therefore the basic quaternion denition as well as transformation rules and conversion rules to or from other attitude representation parameters are summarized. The quaternion computation rules are supported by practical examples to make each step comprehensible. 1 Introduction Quaternions are widely used as attitude represenation parameter of rigid bodies such as space- crafts. This is due to the fact that quaternion inherently come along with some advantages such as no singularity and computationally less intense compared to other attitude parameters such as Euler angles or a direction cosine matrix. Mainly, quaternions are used to • Parameterize a spacecraft's attitude with respect to reference coordinate system, • Propagate the attitude from one moment to the next by integrating the spacecraft equa- tions of motion, • Perform a coordinate transformation: e.g. calculate a vector in body xed frame from a (by measurement) known vector in inertial frame. However, dierent references use several notations and rules to represent and handle attitude in terms of quaternions, which might be confusing for newcomers [5], [4]. Therefore this article gives a straight-forward and clearly notated introduction into the subject of quaternions for attitude representation. The attitude of a spacecraft is its rotational orientation in space relative to a dened reference coordinate system.
    [Show full text]
  • Lateral and Surface Area of Right Prisms 1 Jorge Is Trying to Wrap a Present That Is in a Box Shaped As a Right Prism
    CHAPTER 11 You will need Lateral and Surface • a ruler A • a calculator Area of Right Prisms c GOAL Calculate lateral area and surface area of right prisms. Learn about the Math A prism is a polyhedron (solid whose faces are polygons) whose bases are congruent and parallel. When trying to identify a right prism, ask yourself if this solid could have right prism been created by placing many congruent sheets of paper on prism that has bases aligned one above the top of each other. If so, this is a right prism. Some examples other and has lateral of right prisms are shown below. faces that are rectangles Triangular prism Rectangular prism The surface area of a right prism can be calculated using the following formula: SA 5 2B 1 hP, where B is the area of the base, h is the height of the prism, and P is the perimeter of the base. The lateral area of a figure is the area of the non-base faces lateral area only. When a prism has its bases facing up and down, the area of the non-base faces of a figure lateral area is the area of the vertical faces. (For a rectangular prism, any pair of opposite faces can be bases.) The lateral area of a right prism can be calculated by multiplying the perimeter of the base by the height of the prism. This is summarized by the formula: LA 5 hP. Copyright © 2009 by Nelson Education Ltd. Reproduction permitted for classrooms 11A Lateral and Surface Area of Right Prisms 1 Jorge is trying to wrap a present that is in a box shaped as a right prism.
    [Show full text]
  • Chapter 4 Complex Numbers Course Number
    Chapter 4 Complex Numbers Course Number Section 4.1 Complex Numbers Instructor Objective: In this lesson you learned how to perform operations with Date complex numbers. Important Vocabulary Define each term or concept. Complex numbers The set of numbers obtained by adding real number to real multiples of the imaginary unit i. Complex conjugates A pair of complex numbers of the form a + bi and a – bi. I. The Imaginary Unit i (Page 328) What you should learn How to use the imaginary Mathematicians created an expanded system of numbers using unit i to write complex the imaginary unit i, defined as i = Ö - 1 , because . numbers there is no real number x that can be squared to produce - 1. By definition, i2 = - 1 . For the complex number a + bi, if b = 0, the number a + bi = a is a(n) real number . If b ¹ 0, the number a + bi is a(n) imaginary number .If a = 0, the number a + bi = bi is a(n) pure imaginary number . The set of complex numbers consists of the set of real numbers and the set of imaginary numbers . Two complex numbers a + bi and c + di, written in standard form, are equal to each other if . and only if a = c and b = d. II. Operations with Complex Numbers (Pages 329-330) What you should learn How to add, subtract, and To add two complex numbers, . add the real parts and the multiply complex imaginary parts of the numbers separately. numbers Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company.
    [Show full text]
  • Geometric Modeling in Shape Space
    Geometric Modeling in Shape Space Martin Kilian Niloy J. Mitra Helmut Pottmann Vienna University of Technology Figure 1: Geodesic interpolation and extrapolation. The blue input poses of the elephant are geodesically interpolated in an as-isometric- as-possible fashion (shown in green), and the resulting path is geodesically continued (shown in purple) to naturally extend the sequence. No semantic information, segmentation, or knowledge of articulated components is used. Abstract space, line geometry interprets straight lines as points on a quadratic surface [Berger 1987], and the various types of sphere geometries We present a novel framework to treat shapes in the setting of Rie- model spheres as points in higher dimensional space [Cecil 1992]. mannian geometry. Shapes – triangular meshes or more generally Other examples concern kinematic spaces and Lie groups which straight line graphs in Euclidean space – are treated as points in are convenient for handling congruent shapes and motion design. a shape space. We introduce useful Riemannian metrics in this These examples show that it is often beneficial and insightful to space to aid the user in design and modeling tasks, especially to endow the set of objects under consideration with additional struc- explore the space of (approximately) isometric deformations of a ture and to work in more abstract spaces. We will show that many given shape. Much of the work relies on an efficient algorithm to geometry processing tasks can be solved by endowing the set of compute geodesics in shape spaces; to this end, we present a multi- closed orientable surfaces – called shapes henceforth – with a Rie- resolution framework to solve the interpolation problem – which mannian structure.
    [Show full text]
  • Circle Theorems
    Circle theorems A LEVEL LINKS Scheme of work: 2b. Circles – equation of a circle, geometric problems on a grid Key points • A chord is a straight line joining two points on the circumference of a circle. So AB is a chord. • A tangent is a straight line that touches the circumference of a circle at only one point. The angle between a tangent and the radius is 90°. • Two tangents on a circle that meet at a point outside the circle are equal in length. So AC = BC. • The angle in a semicircle is a right angle. So angle ABC = 90°. • When two angles are subtended by the same arc, the angle at the centre of a circle is twice the angle at the circumference. So angle AOB = 2 × angle ACB. • Angles subtended by the same arc at the circumference are equal. This means that angles in the same segment are equal. So angle ACB = angle ADB and angle CAD = angle CBD. • A cyclic quadrilateral is a quadrilateral with all four vertices on the circumference of a circle. Opposite angles in a cyclic quadrilateral total 180°. So x + y = 180° and p + q = 180°. • The angle between a tangent and chord is equal to the angle in the alternate segment, this is known as the alternate segment theorem. So angle BAT = angle ACB. Examples Example 1 Work out the size of each angle marked with a letter. Give reasons for your answers. Angle a = 360° − 92° 1 The angles in a full turn total 360°. = 268° as the angles in a full turn total 360°.
    [Show full text]
  • Modular Arithmetic
    CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range f0;1;:::;N ¡ 1g, and wraps around whenever you try to leave this range — like the hand of a clock (where N = 12) or the days of the week (where N = 7). Example: Calculating the day of the week. Suppose that you have mapped the sequence of days of the week (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday) to the sequence of numbers (0;1;2;3;4;5;6) so that Sunday is 0, Monday is 1, etc. Suppose that today is Thursday (=4), and you want to calculate what day of the week will be 10 days from now. Intuitively, the answer is the remainder of 4 + 10 = 14 when divided by 7, that is, 0 —Sunday. In fact, it makes little sense to add a number like 10 in this context, you should probably find its remainder modulo 7, namely 3, and then add this to 4, to find 7, which is 0. What if we want to continue this in 10 day jumps? After 5 such jumps, we would have day 4 + 3 ¢ 5 = 19; which gives 5 modulo 7 (Friday). This example shows that in certain circumstances it makes sense to do arithmetic within the confines of a particular number (7 in this example), that is, to do arithmetic by always finding the remainder of each number modulo 7, say, and repeating this for the results, and so on.
    [Show full text]
  • Arithmetic Algorisms in Different Bases
    Arithmetic Algorisms in Different Bases The Addition Algorithm Addition Algorithm - continued To add in any base – Step 1 14 Step 2 1 Arithmetic Algorithms in – Add the digits in the “ones ”column to • Add the digits in the “base” 14 + 32 5 find the number of “1s”. column. + 32 2 + 4 = 6 5 Different Bases • If the number is less than the base place – If the sum is less than the base 1 the number under the right hand column. 6 > 5 enter under the “base” column. • If the number is greater than or equal to – If the sum is greater than or equal 1+1+3 = 5 the base, express the number as a base 6 = 11 to the base express the sum as a Addition, Subtraction, 5 5 = 10 5 numeral. The first digit indicates the base numeral. Carry the first digit Multiplication and Division number of the base in the sum so carry Place 1 under to the “base squared” column and Carry the 1 five to the that digit to the “base” column and write 4 + 2 and carry place the second digit under the 52 column. the second digit under the right hand the 1 five to the “base” column. Place 0 under the column. “fives” column Step 3 “five” column • Repeat this process to the end. The sum is 101 5 Text Chapter 4 – Section 4 No in-class assignment problem In-class Assignment 20 - 1 Example of Addition The Subtraction Algorithm A Subtraction Problem 4 9 352 Find: 1 • 3 + 2 = 5 < 6 →no Subtraction in any base, b 35 2 • 2−4 can not do.
    [Show full text]