Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Multilateration wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Steinitz's theorem wikipedia , lookup
Perceived visual angle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Elementary Education K-6 Mathematics Assessment of these competencies and skills will use real-world problems when feasible. http://subjectareatestprep.pds-hrd.wikispaces.net/file/view/saeMath.pptx SEE QUIA for additional support & tutorials http://www.math-drills.com http://www.coolmath.com/ 7 Knowledge of number sense, concepts, and operations "I write rhymes with addition and algebra, mental geometry." -- Ice-T 1. Associate multiple representations of numbers using word names, standard numerals, and pictorial models for real numbers (whole numbers, decimals, fractions, and integers). Term: Example: Whole Numbers Counting Numbers (1, 2, 1000) Factions a/b where a is the numerator and b is the denominator (2/3) Dividing Fractions;" The number you're dividing by, Turn upside-down and multiply " and integers). Decimals fractions written as powers of 10 (25/100 = 0.25) Integers +7, -10 Word Names One, Two, Five-Hundred Standard Numerals 1, Pictorial Models 2, 500 Using pictures to represent numbers Write an example for each real number: 11, 35, 8 Whole number Decimal Integer Word names Standard numerals Pictorial 2. Compare the relative size of integers, fractions, and decimals, numbers expressed as percents, numbers with exponents, and/or numbers in scientific notation. Fractions, Decimals, Percents Converting from fractions to decimals to percents: Operation Explanation Example Convert a decimal to a percent Move the decimal point 2 places to the right and add a percent (%) sign. If you need to, add a zero on the back to get the second decimal place. .123 = 12.3% Move the decimal point 2 places to the left. If you need to, put a zero on the front. 5% = .05 Divide the numerator by the denominator, using your calculator. 1/8 = .125 Convert a percent to a decimal Convert a fraction to a decimal Convert a percent to a fraction First turn the number into a decimal. Then turn 18% = .18 the number into a fraction by putting it over = 18/100 = 10, 100, 1000, or whatever number is big 9/50 enough to have enough zeroes for each place after the decimal. See http://www.marthalakecov.org/~math/fr_dec_pct.html for additional examples. Exponents Exponential notation is shorthand for repeated multiplications: 103, 36, and 18 as an exponent, in factor form and in standard form: Exponential Factor Standard Form Form Form 103 10 x 10 x 10 1,000 36 3x3x3x3x3x3 729 18 1x1x1x1x1x1x1x1 1 Rules for numbers with exponents of 0, 1, 2 and 3: Rule Example Any number (except 0) raised to the zero power is equal to 1. 1490 = 1 Any number raised to the first power is always equal to itself. 81 = 8 If a number is raised to the second power, we say it is squared. 32 is three squared If a number is raised to the third power, we say it is cubed. 43 is four cubed Scientific Notation Scientific notation is a way of writing very large and very small numbers more easily. 5,234 = 5.234 x 103 View this quick video on Scientific Notation from United Streaming. To login, use your B.E.E.P. username and password: Video: http://player.discoveryeducation.com/index.cfm?guidAssetId=8E41 5256-A36C-4D02-A45AD06E54030CC7&blnFromSearch=1&productcode=US 3. Apply ratios, proportions, and percents in real-world situations. Ratio is the relationship of one number to another to 4:3 Proportion refers to the relationship between two equivalent ratios Quick video on proportions: http://viewer.nutshellmath.com/?solution=67-33-71-41-77-31 Write each as a decimal, fraction, percent, exponent, in Scientific Notation and Ratio: 12, -240 Decimal Fraction 12.0 -240.0 Percent Exponent Scientific Notation Ratio Use Virtual Manipulatives to help solve or convert: http://nlvm.usu.edu/en/nav/topic_t_1.html 4. Represent numbers in a variety of equivalent forms, including whole numbers, integers, fractions, decimals, percents, scientific notation, and exponents. Equivalent Forms: Write the numbers 25, and -1090 in each equivalent form: Whole number Integer Fraction Decimal Percent 25 -1090 Scientific Notation Exponent Solve. In her science class, Leanne was doing an experiment that involved weighing small objects in grams and then converting those weights into ounces. Her teacher told her that each gram is equal to 0.035 ounces. Which is equivalent to 0.035? a. 7 20 b. thiry-five thousandths c. 35% d. 35 100 5. Recognize the effects of operations on rational numbers and the relationships among these operations (i.e., addition, subtraction, multiplication, and division). Rational numbers Rational numbers a real number that can be written as a ratio of two integers (fraction) excluding zero as a denominator a/b (a/0 is not allowed) a repeating or terminating decimal 0.33333 an integer -24 Operations: Addition (sum) Subtraction (difference) Multiplication (product) Division (quotient) Order of Operations PEMDAS (Please Excuse My Dear Aunt Sally) Parenthesis Exponents Multiplication* Division* Addition** Subtraction** When multiplying and dividing, solve whichever comes first from left to right. Example: 5x9/3x4 15 ÷ 5 / 3 x 4 45 / 3 x 4 3 / 12 45/12 When adding and subtracting, solve whichever comes first from left to right. Example: 5+9/3+4 9–5/3+4 14 / 3 + 4 4/7 14/ 7 Relationships of operations using integers: Adding Integers: Adding with Same signs (find the SUM): (+) + (+) = + (-) + (-) = - (+4) + (+5) = (+9) (-3) + (-6) = (-9) Adding with Different signs (find the DIFFERENCE): Find the difference (subtract) and take the sign of the larger number (+) + (-) = Find difference, take sign of larger number (+5) + (-7) = (-2) (-) + (+) = Find difference, take sign of larger number (-4) + (+6) = (+2) Subtracting Integers To subtract an integer, add it’s OPPOSITE 5-2=3 5 + (-2) = 3 (Change sign to +) - 3 – 4= -7 -3 + (-4) = -7 Multiplying Integers When you multiply two integers with the same signs, the result is always positive. + x x + = + (Same signs = Positive) - = + (Same signs = Positive) When you multiply two integers with different signs, the result is always negative. x + = - (Different Signs = Negative) Dividing Integers When you divide two integers with the same sign, the result is always positive. + / + = + (Same signs = Positive) 2=5 / - = + (Same signs = Positive) -10 / -2 = 5 When you divide two integers with different signs, the result is always negative. / + = - (Different Signs = Negative) -10 / 2 = -5 10 / -2 = -5 6. Select the appropriate operation(s) to solve problems involving ratios, proportions, and percents and the addition, subtraction, multiplication, and division of rational numbers. Ratios: A ratio is a comparison of two numbers. Example: Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange. 1) What is the ratio of books to marbles? Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4. Two other ways of writing the ratio are 7 to 4, and 7:4. 2) What is the ratio of videocassettes to the total number of items in the bag? There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total. The answer can be expressed as 3/15, 3 to 15, or 3:15. Proportions: A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal (3/4 = 6/8 is an example of a proportion). When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Variables are frequently used in place of the unknown number. Example: Solve for n: 1/2 = n/4. Using cross products we see that 2 × n = 1 × 4 =4, so 2 × n = 4. Dividing both sides by 2, n = 4 ÷ 2 so that n = 2. 7. Use estimation in problem-solving situations. Estimation: Estimation is determining an approximate or rough calculation based on rounding. Example: Problem: You ate at a restaurant and the bill you will pay is $28.90. You want to leave a tip of 15% of the bill. What is the equivalent amount of the 15% tip that you want to leave? Solution: 1) You can round off $28.90 to $30 since it is very close to that amount. 2) Using 10 as our reference number, divide $30/10 = $3. You will retain the $3. 3) Divide $3/2 = $1.50. 4) Then add $3 + $1.50 = $4.50, which is the estimated tip you will pay for a $28.90 bill. 5) If you use a calculator to compute the answer for the 15% tip from a $28.90 bill, you will get $4.33, which is close to our estimate of $4.50. 8. Apply number theory concepts (e.g., primes, composites, multiples, factors, number sequences, number properties, and rules of divisibility). Prime A number that has exactly two factors, 1 and itself Composite A number with more than 2 factors Multiples A number added to itself a number of times Factors number A whole number that divides exactly into another Number sequences term Numbers expressed in a pattern in the nth Number properties Commutative, Associative, Distributive, Rules of divisibility A number is divisible by two if it is even. A number is divisible by three if the sum of the digits adds up to a multiple of three A number is divisible by four if it is even and can be divided by two twice. A number is divisible by five if it ends in a five or a zero A number is divisible by six if it is divisible by both two and three. A number is divisible by nine if the sum of the digits adds up to a multiple of nine. This rule is similar to the divisibility rule for three. A number is divisible by ten if it ends in a zero. This rule is similar to the divisibility rule for five Write two examples for each category: Prim Composit Multiple Factor Number Number Rules of e e s s Sequenc e Propertie s Divisibilit y 9. Apply the order of operations. (Please Make Deliveries After Sunset) (Patrick Makes Delicious Apple Smoothies) Order of Operations The precedence set for solving math equations. In what order should the operations be performed in the expression below? Solve. 6 x (3 + 2) ÷ 3 - 1 a. +, x,Π,b. +, x,-,Π c. +,-, x,Π d. x,+,Π,- What is the value of the expression 2 + 5 X 32 ? 32 C.227 47 d. 441 http://subjectareatestprep.pdshrd.wikispaces.net/file/view/NUMBER+SENSE+%231.doc http://subjectareatestprep.pdshrd.wikispaces.net/file/view/NUMBER+SENSE+%232.doc K 1st 2nd 3rd 4th 5th add after* before between (M) add* addition (M) after* before* between* (M) addition* (M) after* count backward count forward addend*(M)* * array*(M) denominator* difference(M) addend*(M)* * array* calculate* cardinal billion calculate* cardinal number*(M) composite equal* even numbers* fourth(s)* group* (M) half (halves)* hundred(s )* count (on) difference (M) double (plus one) equal* estimate* (M) even numbers* fact family fourth(s)* count (on)* difference* (M) double (plus one)* equal* estimate* (M) even numbers* fact family* fourth(s)* digit* estimation*(M )** expanded form*(M) factor*(M)** fraction*(M)* * minuend* multiplicand* number(M) composite number*(M)* * consecutive* decimal number*(M)* * denominator * number*(M) ** consecutive* decimal number*(M) ** decimal point digit* dividend(M) less than* group * (M) fraction multiplication( digit* divisible* ** (M) more than (not) equal number* (M) number line*(M)** half (halves)* hundred(s)* less than* (M) more than* near (M) (not) equal* number* (M) number line* greater than (M) greatest group* (M) half (halves)* hundred(s)* less than* (M) near* (M) M) multiples*(M) ** multiplier* number line** numerator* place value* ** divisible* ** divisor** estimation(M )** expanded notation*(M) extraneous information* divisor* ** equivalent extraneous information* ** equivalent* greatest common odd number* one(s)* opposite order pattern* (M) ** odd number* one(s)* opposite* order* pattern*(M)** number* (M) number line* (M) ** odd number* one(s)* ordinal number product*(M)* * regroup* relative size*(M)** rounding*(M) * equivalent* factor(M)** fraction(M)** inverse operation** factor (GCF) inverse operation* ** least common (M) ** small(est) subtract rule skip-counting* small(est)* (M) pattern* (M) ** solution* subtrahend* sum(M)** million* denominator minuend* (LCD) multiples(M)* least ten(s)* zero* (M) subtract* subtract(ion)* (M) sum ten(s)* zero* (M) regroup rule* set (M) skip-counting* small(est)* subtract(ion)* (M) whole numbers*(M) * multiplicand* multiplier* natural numbers* numerator* operation* common multiple(LCM ) million* natural numbers* operation* sum* ten(s)* whole number (M) ** zero* (M) ** ordinal number*(M) percent*(M)* * perfect number* place value* ** prime ** ordinal number*(M) percent*(M) ** perfect number* prime factorization (M) number*(M)* prime * product*(M)* * quotient*(M) ** regroup* relative size*(M)** number*(M) ** quotient*(M) ** reduce remainder*( M) similarity*(M remainder*( M) rounding*(M) similarity*(M )** simplify* )** simplify* solution* subtrahend* value* whole numbers*(M) ** 8 Knowledge of measurement "We only think when confronted with a problem." -- John Dewey 1. Apply given measurement formulas for perimeter= 2xL+2xW , circumference =pi(3.14) x diameter, area= length x width, volume = length x width x height or displacement, and surface area = the sum of the areas of all sides in problem situations. Perimeter The distance around a shape. Circumference The distance around a circle. Area The size a surface takes up, measured in square units. Volume cubic Amount of space occupied by a 3D object, measured in units. Surface Area squared Total area of a surface of a 3D object, measured in units. Perimeter and Area: Find the area and perimeter of each rectangle. 7 ft wide, 37 ft long Perimeter = Area = 10 ft wide, 15 ft long Perimeter = Area = Sachi is building a brick patio and needs to determine its total area. The dimensions of the patio are shown in the diagram below. Circumference: The radius of a circle is 2 inches. What is the circumference? = 3.14 The diameter of a circle is 3 centimeters. What is the circumference? Volume: Find the volume of the rectangular prism to the nearest tenth. Length Width Height 8.8 feet 6 feet 11.5 feet F. 210.0 feet3 G. 380.3 feet3 H. 596.2 feet3 I. 607.2 feet3 Surface Area: a = 27 cm b = 32 cm c = 27 cm 2. Evaluate how a change in length, width, height, or radius affects perimeter, circumference, area, surface area, or volume. A company introducing a new cereal wants to use one of the two boxes shown. What is the volume, in cubic inches, of the box that will hold the greatest amount of cereal when full? 3. Within a given system, solve real-world problems involving measurement, with both direct and indirect measures, and make conversions to a larger or smaller unit (metric and customary). Solve. Tina was making punch for a party. She mixed 7 cups of apple juice, 4 cups of pineapple juice, 8 cups of ginger ale, and 1 cup of lemon juice. How many gallons of punch did she make? 4. Solve real-world problems involving estimates and exact measurements. At Smith Elementary School, the number of students in a kindergarten class ranges between 15 and 20. The school has 5 kindergarten classes. ESTIMATE the number of kindergarten students who attend Smith Elementary School. a. 25 c. 90 b. 75 d. 100 Exact Measurement: Miguel drives 310 miles on a tank of gas. When he refuels his car, it takes 12.4 gallons. How many miles per gallon did Miguel average on his last tank of gas? a. 25 miles per gallon b. 25.8 miles per gallon c. 297.6 miles per gallon d. 3,844 miles per gallon Tina was making punch for a party. She mixed 7 cups of apple juice, 4 cups of pineapple juice, 8 cups of ginger ale, and 1 cup of lemon juice. How many gallons of punch did she make? 5. Select appropriate units to solve problems. Measurement using customary-to-metric Chart: View the online Reference Sheet http://www.thefarm.org/charities/i4at/lib2/metric.htm Ethan lives at one end of Park Avenue. Brian lives at the other end of the avenue. It is 5.8 kilometers from one end of Park Avenue to the other. If Ethan walks 2.79 kilometers toward Brian's house, how many meters does he have to walk to get there? 1 Kilometer = 1,000 meters http://subjectareatestprep.pdshrd.wikispaces.net/file/view/MEASUREMENT+%232.doc http://subjectareatestprep.pdshrd.wikispaces.net/file/view/USING+LOGIC+TO+SOLVE+%23BB28B.doc K 1st 2nd 3rd 4th 5th big(gest) cent(s) clock (M) day * (M) dime* equal (parts)* estimate* first fourth* hour (hand)* balance big(gest)* cent(s)* clock* (M) coin (M) cup (M) day* (M) dime* dollar* equal (parts)* balance* calendar (M) coin* (M) cup* (M) date day* (M) dime* dollar* dollar sign equal (parts)* acute angle*(M) ** angle*(M) base*(M)* * closed figure* ** cone* congruent area*(M)** balance* capacity*( M) Celsius* centimeter *(M) compare clockwise* countercloc balance* capacity*( M) Celsius* cubic units(M) direct measure* Fahrenheit * (M) estimate* estimate* *(M)** kwise* mass*(M)* longer longest* measure minute(hand)*( M) money* (M) month* nickel* first* foot (M) fourth* gram half-dollar* half-hour* hour (hand)* (M) foot* (M) fourth* gram* half-dollar* half-hour* half-past hour (hand)* (M) cube*(M) cylinder*( M) face*(M)* * hexagon* obtuse angle*(M) customary units* ** degrees* direct measure* ** distance* * quantity* scale*(M)* * square unit* units of measure penny* second (hand) (M) shorter size* (M) small(est) inch (M) liter longer* longest* measure* minute(hand)* inch* (M) length (M)** liter* longest* minute(hand)*( M) ** octagon* open figure* pentagon* point* ** Fahrenheit * gram*(M) height* length* liter* volume*(M )** week* (M) year * (M) (M) money* (M) month* money* (M) month* nickel* reflection( flip)*(M)* * mass*(M)* * meter*(M) nickel* penny* pint pound (M) quart quarter second(hand)* penny* pint* pound* (M) quart* quarter* size* (M) temperature* rhombus* (M) right angle*(M) ** side* sphere*(M metric units*(M)* * minute* nonstandard units(M)** (M) shorter* size* (M) small (est)* temperature (M) thermometer time* (M) week* (M) year * (M) (M) thermometer* time* (M) week* (M) whole year* (M) ) straight angle* ** symmetry * ** translation (slide)* vertex (vertices)* ounce* perimeter* (M)** quantity* scale*(M)* * second* square unit* volume*(M )** weight* ** width*(M) 9 Knowledge of geometry and spatial sense "Geometry is just plane fun." 1. Identify angles or pairs of angles as adjacent, complementary, supplementary, vertical, corresponding, alternate interior, alternate exterior, obtuse(obese), acute(a-cute), or right. Identify angles or pairs of angles as adjacent, complementary, supplementary, vertical, corresponding, alternate interior, alternate exterior, obtuse, acute, or right. Adjacent Two angles are Adjacent if they have a common side and a common vertex (corner point). Complementary Angles Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other. Supplementary Angles Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other. Vertical Angles For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Corresponding Angles For any pair of parallel lines that are both intersected by a third line. Corresponding angles have the same degree measurement. Alternate Interior Angles For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same degree measurement. Alternate Exterior Angles For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same degree measurement. Obtuse Angles An obtuse angle is an angle measuring between 90 and 180 degrees. Acute Angles An acute angle is an angle measuring between 0 and 90 degrees. Right Angles A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are perpendicular. Interior Angles An Interior Angle is an angle inside a shape. 2. Identify lines and planes as perpendicular, intersecting, or parallel. Perpendicular lines Lines are perpendicular to each other if they form congruent adjacent angles (a T-shape). Intersecting lines Two or more lines that meet at a point are called intersecting lines. That point would be on each of these lines. Lines l and m intersect at Q. Parallel lines Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Always the same distance apart and never touching. The red line is parallel to the blue line: Solve: The map below shows a portion of downtown Washington, D.C. with named streets. Which of the following streets appear to be parallel to each other? 3. Apply geometric properties and relationships, such as the Pythagorean Theorem, in solving problems. "...The squaw on the hippopotamus is equal to the sum of the squaws on the other two hides!" (ie. hypotenuse² = base² + height²) This mnemonic is the punchline of a very old joke, but its serves its purpose extremely well! If you've never heard the joke before, there were 3 Indians who met to trade wives, and the first Indian told his wife to sit on a blanket made from the skin of a deer, while the second Indian placed his wife on a hide made of buffalo... (etc)! Pythagorean Theorem The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." Figure 1 According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue square, square C. Thus, the Pythagorean Theorem stated algebraically is: Given the lengths of two sides of a right triangle, use Pythagorean Theorem to find the length of the third side. 1. a = 3, b = 4, find c 2. a = 3, c = 4, find b 4. Identify the basic characteristics of, and relationships pertaining to, regular and irregular geometric shapes in two and three dimensions. Regular and irregular geometric shapes in two and three dimensions: Two-dimensional A two-dimensional figure, also called a plane or planar figure, is a set of line segments or sides and curve segments or arcs, all lying in a single plane. The sides and arcs are called the edges of the figure. The edges are onedimensional, but they lie in the plane, which is two-dimensional. Three-dimensional A three-dimensional figure, sometimes called a solid figure, is a set of plane regions and surface regions, all lying in three-dimensional space. These surface regions are called the faces of the figure. Each of them is twodimensional. The arcs of curves that are the edges of the faces of the figure are called the edges of the figure. They are one-dimensional. The endpoints of the edges are called its vertices. They are zero-dimensional. 5. Apply the geometric concepts of symmetry, congruency, similarity, tessellations, transformations, and scaling. Symmetry The "Line of Symmetry" (shown here in white) is the imaginary line where you could fold the image and have both halves match exactly. Congruency Same shape, same size Similarity Same shape, different size Tessellations Tessellations are the juxtaposition of shapes in a pattern. A pattern made of identical shapes. The shapes must fit together without any gaps. The shapes should not overlap. Activity: Try drawing each of these patterns by clicking on this site. Choose a starting polygon shape, then bend the lines until they look like the design you are recreating, then click the "tessellate" button. http://www.shodor.org/interactivate/activities/Tessellate/ Transformations Activity: Click on this site to change various geometric shapes using rotations, reflections, translations or enlargements - and combinations of these transformations. http://www.mathsnet.net/transform/index.html 6. Determine and locate ordered pairs in all four quadrants of a rectangular coordinate system. Coordinate System: A system for specifying points using coordinates measured in some specified way. Which point on the coordinate plane below best represents the location of the ordered pair (4, 3)? Identify each quadrant. Which statement describes a way in which one could move from point A to point B? a. Move left 3 units, then move c. Move up 2 units, then move down 2 units. b. Move up 3 units, then move right 4 units, then move down 2 units. right 3 units. d. Move right 4 units, then move up 3 units, then left 1 unit. http://www.teachertube.com/viewVideo.php?video_id=13972 http://www.teachertube.com/videoList.php?pg=videonew&tags=601 K 1st 2nd 3rd 4th 5th above* (M) above* (M) above* (M) acute acute acute behind (M) below * (M) beside* between (M) circle* (M) cone* corner* (M) behind* (M) below* (M) beside* between* (M) circle* (M) cone* below* (M) beside* circle* (M) cone* corner* (M) cube cup* cylinder* angle*(M)** angle*(M) base*(M)** closed figure* ** cone* congruent*( M)** angle*(M)** angle*(M)** base*(M)** circumference *(M) closed figure* ** cone* angle*(M)** circumference*( M) congruent* ** coordinates* dashed line* diagonal diameter* curves cylinder* in front of (M) inside (M) left* (M) corner* (M) cup curves* cylinder* in front of hexagon left* (M) line of symmetry* parallelogram plane shapes cube*(M) cylinder*(M) face*(M)** hexagon* obtuse angle*(M)** congruent* ** cube* cylinder*(M) dashed line* diameter* dotted line* dotted line* edge equilateral triangle horizontal*(M) intersection*(M) middle outside (M) rectangle* * (M) inside* (M) left* (M) rectangle* (M) rectangular octagon* open figure* pentagon* face*(M)** hexagon* horizontal*(M ** isosceles triangle (M) right* (M) shape* (M) slide* sort* sphere* square* (M) line of symmetry middle* outside* (M) rectangle* (M) prism* right*(M) shape*(M) side* slide* solid figure sort* point* ** reflection(fli p)*(M)** rhombus*(M ) right angle*(M)** ) intersection*( M)** line* ** line segment* ** obtuse line* ** line segment* ** obtuse angle*(M)** parallel lines* (M)** triangle* (M) under * (M) rectangular prism right* (M) shape* (M) side* slide* sort* sphere* square* (M) sphere* square* (M) trapezoid triangle* (M) turn* ** under* (M) side* sphere*(M) straight angle* ** symmetry* ** translation(s lide)* vertex (vertices)* angle*(M)** octagon* open figure* parallel lines*(M) ** parallelogram *(M) pentagon* perpendicular * ** parallelogram*( M) perpendicular* ** plane figure* ** polygon*(M)** quadrilateral*(M ) radius** ray** triangle* plane figure* rectangular (M) turn* ** under * (M) ** point* ** polygon*(M)* * quadrilateral* (M) rectangular prism*(M)** prism*(M)** right angle*(M)* rotation (turn)*(M)** scale model(M)** scalene triangle semi-circle reflection(flip) *(M)** rhombus*(M) right angle*(M)** rotation(turn) similar figures* ** similarity*(M)** solid figure*(M) straight angle* ** *(M)** side* ** similar figure* symmetry* ** tessellation(M) transformation* ** similarity*(M) ** solid figure* sphere* straight angle* ** ** translation (slide)*(M)** trapezoid* triangular pyramid two-dimensional symmetry* ** transformatio n* ** translation(sli de)*(M)** trapezoid* vertex(vertice s)* ** vertical*(M) threedimensional vertex(vertices) * ** vertical*(M) 10 Knowledge of algebraic thinking "Think! Think and wonder. Wonder and think. How much water can 55 elephants drink?" -- Dr. Seuss 1. Extend and generalize patterns or functional relationships. 2. Interpret tables, graphs, equations, and verbal descriptions to explain real-world situations involving functional relationships. 3. Select a representation of an algebraic expression, equation, or inequality that applies to a real-world situation. http://subjectareatestprep.pdshrd.wikispaces.net/file/view/ALGEBRAIC+THINKING.doc http://www.teachertube.com/viewVideo.php?video_id=140481 K 1st 2nd 3rd 4th 5th alike different match alike* different* match* alike* different* match* expressio n(M) equation equation*(M)** expression* ** inequality*(M)** equation*(M)** equivalent forms*(M)** input object output similar size sort* input* object* output similar* size* sort* input* object* output similar* size* sort* inequality (M) pattern(M ) rule solution* order of operations*(M)* * pattern*(M)** relationship*(M) ** rule* ** solution* symbol*(M) variable*(M)** evaluate** expression* ** inequality*(M)* * order of operations*(M)* * pattern*(M)** relationship*(M) * * rule* ** symbol*(M) variable*(M)** 11 Knowledge of data analysis and probability "Five out of four people have trouble with fractions." -- Steven Wright 1. Apply the concepts of range and central tendency; meAn(average), median(of the road), and mode(as in most). The mean is just the average of the numbers. Add up all the numbers, then divide by how many numbers there are. Example 1: What is the Mean of these numbers? 6, 11, 7 Add the numbers: 6 + 11 + 7 = 24 Divide by how many numbers: 24 ÷ 3 = 8 (Mean = 8) For each data set, find the mean, the median, the mode, and range. 1. Maddie scored 7,9,5,3,15 and 15 pionts in 6 basketball games. She wants to show that she is a valuable player. 2. Mr. and Mrs. Rodriquez collected donations of $50, $125, $10, $210, $50, $24, and $175 for charity. They wanted to show that they are good fundraisers. 3. Andy's phone calls lasted 20 min, 7min, 9min, 12min, and 7min. He wants to show how long a tpyical phone call is. 4. A golf team had scores of 383, 392, and 401 strokes in a tournament. The lowest score wins. The team wants to show that it played well.Range 2. Determine probabilities of dependent or independent events. Probability: Probability is the likelihood of something happening. For instance, when you roll a pair of dice, you might ask how likely you are to roll a seven. In math, "something happening" is called an "event." Solve: A cell phone company manufactures 1000 cell phones each day. Of those 1000, about 30 are returned each day needing adjustments or replacement parts. If you purchase a cell phone from this company, what is the probability that your cell phone will be defective? 3 1000 3 100 3 10 7 10 3. Determine odds for and odds against a given situation. Solve: Each customer at Blakely’s Hardware gets one chance to spin the arrow on the spinner shown below. The customer then receives the discount displayed on the section where the spinner arrow lands. If the sections on the spinner are congruent, what is the probability that the spinner arrow will land on a section for a 5% discount? F. 1 4 G. 1 2 H. 3 10 3 8 4. Apply fundamental counting principles such as combinations to solve probability problems. Probability = Number of ways and Event can occur Total number of possible Events Example: The probability of drawing a red marble at random is: number of red marbles 4 --------------------------- = --total marbles in jar 10 Since 4/10 reduces to 2/5, the probability of drawing a red marble where all outcomes are equally likely is 2/5. Expressed as a decimal, 4/10 = .4; as a percent, 4/10 = 40/100 = 40%. Solve: A game show has a large spinner that players spin during their turn. What is the likelihood of the pointer of the spinner landing on a slot worth more than $500? a. 3/16 b. 1/4 c. 3/13 d. 4/13 5. Interpret information from tables, charts, line graphs, bar graphs, circle graphs, box and whisker graphs, and stem and leaf plots. Box and Whisker Graph A box and whisker graph is used to display a set of data so that you can easily see where most of the numbers are. Example: Solve: Draw a box-and-whisker plot for the following data set: 4.3, 5.1, 3.9, 4.5, 4.4, 4.9, 5.0, 4.7, 4.1, 4.6, 4.4, 4.3, 4.8, 4.4, 4.2, 4.5, 4.4 Stem-and-Leaf Plot A stem-and-leaf plot, in statistics, is a device for presenting quantitative data in a graphical format. Solve: The numbers of points scored by some football teams in the first game of the season are shown in the stem-and-leaf plot below. What is the mode score? a. 4 c. 20 b. 14 d. 35 6. Make accurate predictions and draw conclusions from data. Mrs. Robertson, a teacher, saw the chart below in her local newspaper. She came to the conclusion that the number of failing students decreased from 1985 to 2005. Which is a true statement? a. The graph is not misleading. The number of failing students decreased. b. The graph is misleading because the number of years represented by each bar is not the same. This makes it look like the number of failing students has decreased. c. The graph is misleading because it is important to include students making Ds in a survey about failing students and this graph does not include students making Ds. d. The graph is not misleading because as a teacher at Oakwood School, Mrs. Roberson knows many students whose grades have improved in the past several years. 2. Determine probabilities of dependent or independent events. 3. Determine odds for and odds against a given situation. 4. Apply fundamental counting principles such as combinations to solve probability problems. 5. Interpret information from tables, charts, line graphs, bar graphs, circle graphs, box and whisker graphs http://www.worsleyschool.net/science/files/box/plot.html and stem and leaf plots. http://www.icoachmath.com/sitemap/problemslink.aspx?Search=stem-andleaf%20plot&grade=0 6. Make accurate predictions and draw conclusions from data. K 1st 2nd 3rd 4th 5th graph (M) table* (M) ** bar graph* certain chart** bar graph* certain* chart** axis(axes)* ** chance*(M) combinations( arrangeme nt* axis(axes)* arrangement * break* ** tally (mark)* tally equally likely graph* (M) impossible equally likely* impossible* less likely* M) coordinates data*(M)** ** (squiggle) break(squig certainty*(M gle)* ** ) (table)* less likely picture graph table* (M) ** tally (mark)* tally (table)* unlikely* pictograph picture graph* survey table* (M) ** tally (mark)* tally (table)* equally likely input* line graph(M) median*(M)** mode*(M)** ordered pair * ** certainty*( M) chance*(M) circle graph* ** combinatio ns chance*(M) circle graph* ** coordinates* ** data*(M)** data unlikely* output* pictograph*(M) ** point* ** range*( coordinates * ** data*(M)** data collection*( M) diagram*( M) double-bar graph*(M)* collection*(M ) diagram*(M) double-bar graph*(M)** equally likely* event* function*(M) ** * grid(plane)* equally likely* event* function*(M )** grid(plane)( M)** labels* ** ** increments input* intervals labels* ** likelihood* ** line likelihood* ** line graph*(M)* * mean*(M)* graph*(M)** location mean*(M)** ordered pair* ** organized * median*(M )** data* ** output* pie mode*(M)* * ordered pair* ** organized data* ** pie chart*(M) predict* probability*( M)** randomly chosen** ratio** chart*(M) point* ** predict* probability* (M)** range*(M)* * survey*(M) tree diagram* scale**(M) stem-andleaf plot** survey*(M) tree diagram* ** trend line** unorganized data** ** Venn Venn diagram*( M) x-axis* ** y-axis* ** diagram*(M) verify(M) xaxis* ** y-axis* ** 12 Knowledge of instruction and assessment "I never got a pass mark in math ... Just imagine --mathematicians now use my prints to illustrate their books." -- M.C. Escher 1. Identify alternative instructional strategies. Performance assessments, dramatic productions, key word alternative yet valid in relation to the content objectives. 2. Select manipulatives, mathematical and physical models, and other classroom teaching tools. Counters, marbles, cards, probability cubes, cards, bears, blocks, 3. Identify ways that calculators, computers, and other technology can be used in instruction. http://www.edutopia.org/multiple-intelligences-learning-stylesquiz?gclid=CNfzwIXv66ECFQifnAod6CW3JA 4. Identify a variety of methods of assessing mathematical knowledge, including analyzing student thinking processes to determine strengths and weaknesses. AREAS of opportunity!! Gardners 8 intellegences. Howard Gardner of Harvard has identified seven distinct intelligences. This theory has emerged from recent cognitive research and "documents the extent to which students possess different kinds of minds and therefore learn, remember, perform, and understand in different ways," according to Gardner (1991). According to this theory, "we are all able to know the world through language, logical-mathematical analysis, spatial representation, musical thinking, the use of the body to solve problems or to make things, an understanding of other individuals, and an understanding of ourselves. Where individuals differ is in the strength of these intelligences - the socalled profile of intelligences -and in the ways in which such intelligences are invoked and combined to carry out different tasks, solve diverse problems, and progress in various domains." Gardner says that these differences "challenge an educational system that assumes that everyone can learn the same materials in the same way and that a uniform, universal measure suffices to test student learning. Indeed, as currently constituted, our educational system is heavily biased toward linguistic modes of instruction and assessment and, to a somewhat lesser degree, toward logical-quantitative modes as well." Gardner argues that "a contrasting set of assumptions is more likely to be educationally effective. Students learn in ways that are identifiably distinctive. The broad spectrum of students - and perhaps the society as a whole - would be better served if disciplines could be presented in a numbers of ways and learning could be assessed through a variety of means." The learning styles are as follows: Visual-Spatial - think in terms of physical space, as do architects and sailors. Very aware of their environments. They like to draw, do jigsaw puzzles, read maps, daydream. They can be taught through drawings, verbal and physical imagery. Tools include models, graphics, charts, photographs, drawings, 3-D modeling, video, videoconferencing, television, multimedia, texts with pictures/charts/graphs. Bodily-kinesthetic - use the body effectively, like a dancer or a surgeon. Keen sense of body awareness. They like movement, making things, touching. They communicate well through body language and be taught through physical activity, hands-on learning, acting out, role playing. Tools include equipment and real objects. Musical - show sensitivity to rhythm and sound. They love music, but they are also sensitive to sounds in their environments. They may study better with music in the background. They can be taught by turning lessons into lyrics, speaking rhythmically, tapping out time. Tools include musical instruments, music, radio, stereo, CD-ROM, multimedia. Interpersonal - understanding, interacting with others. These students learn through interaction. They have many friends, empathy for others, street smarts. They can be taught through group activities, seminars, dialogues. Tools include the telephone, audio conferencing, time and attention from the instructor, video conferencing, writing, computer conferencing, E-mail. Intrapersonal - understanding one's own interests, goals. These learners tend to shy away from others. They're in tune with their inner feelings; they have wisdom, intuition and motivation, as well as a strong will, confidence and opinions. They can be taught through independent study and introspection. Tools include books, creative materials, diaries, privacy and time. They are the most independent of the learners. Linguistic - using words effectively. These learners have highly developed auditory skills and often think in words. They like reading, playing word games, making up poetry or stories. They can be taught by encouraging them to say and see words, read books together. Tools include computers, games, multimedia, books, tape recorders, and lecture. Logical -Mathematical - reasoning, calculating. Think conceptually, abstractly and are able to see and explore patterns and relationships. They like to experiment, solve puzzles, ask cosmic questions. They can be taught through logic games, investigations, mysteries. They need to learn and form concepts before they can deal with details. At first, it may seem impossible to teach to all learning styles. However, as we move into using a mix of media or multimedia, it becomes easier. As we understand learning styles, it becomes apparent why multimedia appeals to learners and why a mix of media is more effective. It satisfies the many types of learning preferences that one person may embody or that a class embodies. A review of the literature shows that a variety of decisions must be made when choosing media that is appropriate to learning style. Visuals: Visual media help students acquire concrete concepts, such as object identification, spatial relationship, or motor skills where words alone are inefficient. Printed words: There is disagreement about audio's superiority to print for affective objectives; several models do not recommend verbal sound if it is not part of the task to be learned. Sound: A distinction is drawn between verbal sound and non-verbal sound such as music. Sound media are necessary to present a stimulus for recall or sound recognition. Audio narration is recommended for poor readers. Motion: Models force decisions among still, limited movement, and full movement visuals. Motion is used to depict human performance so that learners can copy the movement. Several models assert that motion may be unnecessary and provides decision aid questions based upon objectives. Visual media which portray motion are best to show psychomotor or cognitive domain expectations by showing the skill as a model against which students can measure their performance. Color: Decisions on color display are required if an object's color is relevant to what is being learned. Realia: Realia are tangible, real objects which are not models and are useful to teach motor and cognitive skills involving unfamiliar objects. Realia are appropriate for use with individuals or groups and may be situation based. Realia may be used to present information realistically but it may be equally important that the presentation corresponds with the way learner's represent information internally. Instructional Setting: Design should cover whether the materials are to be used in a home or instructional setting and consider the size what is to be learned. Print instruction should be delivered in an individualized mode which allows the learner to set the learning pace. The ability to provide corrective feedback for individual learners is important but any medium can provide corrective feedback by stating the correct answer to allow comparison of the two answers. Learner Characteristics: Most models consider learner characteristics as media may be differentially effective for different learners. Although research has had limited success in identifying the media most suitable for types of learners several models are based on this method. Reading ability: Pictures facilitate learning for poor readers who benefit more from speaking than from writing because they understand spoken words; self-directed good readers can control the pace; and print allows easier review. Categories of Learning Outcomes: Categories ranged from three to eleven and most include some or all of Gagne's (1977) learning categories; intellectual skills, verbal information, motor skills, attitudes, and cognitive strategies. Several models suggest a procedure which categorizes learning outcomes, plans instructional events to teach objectives, identifies the type of stimuli to present events, and media capable of presenting the stimuli. Events of Instruction: The external events which support internal learning processes are called events of instruction. The events of instruction are planned before selecting the media to present it. Performance: Many models discuss eliciting performance where the student practices the task which sets the stage for reinforcement. Several models indicate that the elicited performance should be categorized by type; overt, covert, motor, verbal, constructed, and select. Media should be selected which is best able to elicit these responses and the response frequency. One model advocates a behavioral approach so that media is chosen to elicit responses for practice. To provide feedback about the student's response, an interactive medium might be chosen, but any medium can provide feedback. Learner characteristics such as error proneness and anxiety should influence media selection. Testing which traditionally is accomplished through print, may be handled by electronic media. Media are better able to assess learners' visual skills than are print media and can be used to assess learner performance in realistic situations. from "The Distance Learning Technology Resource Guide," by Carla Lane Competency 12. Knowledge of instruction and assessment Identify alternative instructional strategies. Alternative instructional strategies: http://spacegrant.nmsu.edu/NMSU/fac_dev/grasp_alternative.pdf Select manipulatives, mathematical and physical models, and other classroom teaching tools. Virtual Manipulatives: http://nlvm.usu.edu/en/nav/topic_t_1.html Math Manipulatives: http://math.about.com/gi/o.htm?zi=1/XJ/Ya&zTi=1&sdn=math&cdn=educat ion&tm=6&f=22&tt=14&bt=0&bts=1&zu=http%3A//mathcentral.uregina.ca/ RR/database/RR.09.98/loewen2.html 3. Identify ways that calculators, computers, and other technology can be used in instruction. An Article on calculators, computers, and other technology used in instruction. http://www-users.math.umd.edu/~dac/650/bowespaper.html 4. Identify a variety of methods of assessing mathematical knowledge, including analyzing student thinking processes to determine strengths and weaknesses. This article gives good information about “ASSESSING TO SUPPORT MATHEMATICS LEARNING” http://www.nap.edu/openbook.php?record_id=2235&page=67 Other Online Resources: Math Reference Sheet: http://fcat.fldoe.org/pdf/fc8miref.pdf Virtual Manipulatives: http://nlvm.usu.edu/en/nav/topic_t_1.html A Maths Dictionary for Kids (This is a great interactive site for kids and adults): http://www.teachers.ash.org.au/jeather/maths/dictionary.html Fractions, Decimals, Percents: See http://www.marthalakecov.org/~math/fr_dec_pct.html for additional examples. Scientific Notation from United Streaming. To login, use your B.E.E.P. username and password: Video: http://player.discoveryeducation.com/index.cfm?guidAssetId=8E415256A36C-4D02-A45A-D06E54030CC7&blnFromSearch=1&productcode=US Proportions from United Streaming: http://viewer.nutshellmath.com/?solution=67-33-71-41-77-31 Gallon Man http://www.twogetherexpress.com/gallon%20man2.htm