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Grade 9 math exam

Grade 9 math exam. Memory Aid Help. Projections and views. Area of common bases. Lateral and total surface area of solids. Pythagorean theorem. b 2 = c 2 - a 2 a 2 = c 2 - b 2 “c” must be the hypotenuse.

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Grade 9 math exam

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  1. Grade 9 math exam Memory Aid Help

  2. Projections and views

  3. Area of common bases

  4. Lateral and total surface area of solids.

  5. Pythagorean theorem • b2 = c2 - a2 • a2 = c2 - b2 • “c” must be the hypotenuse. • In a right triangle that has 30o and 60o angles, the longest side ( the hypotenuse) is always twice the length of the shortest side.

  6. REAL NUMBERS • Natural number: positive integers and no zero. • Example: 1,2,3,4,....89,.....756,.....1000000 • Whole number: natural + zero. • Example: 0,1,2,3.....76....3456.....282763.... • Integer: whole numbers and their opposites (no decimal) • Example: -45, -39, -8, 0, 123, 29874, 30000000 • Rational: number can be written as a ratio (fraction) of two integers. (in decimal form are terminating or repeating. • Example: ½ , 5.2222..., 0.19, -11/3, 2, -4.5, √25 • Terminating decimal numbers: 5/2 = 2.5, 5/8 = 0.625 • Repeating decimal numbers: 1/9 = 0.1111111...... or 0.1 • Irrational: number that cannot be written as a fraction of integers and whose decimal numbers are infinite and non-periodic (does not repeat). • Example: √2, √5, ∏

  7. Inverse variation function • Reverse x and y to get an inverse function • If x increases, y decreases and vice versa • When the product of each variables’ values is a constant you get an inverse variation function.

  8. Functional relation • a relation is a function when each value of the x-axis (abscissa) has one y-axis (ordinate) associated with it. x-axis (abscissa) = independent variable y-axis (ordinate) = dependent variable

  9. Intervals • [included] • ]excluded[ Intervals with infinity: infinity is never included. [-4, +∞[ = from -4 to positive infinity. ]- ∞, -1[ = negative infinity up to but excluding -1.

  10. Function properties • Domain (X): all x values from left to right. • Range (Y): all y values from down to up • Variation (X): it can increase, decrease or remain constant. • Extrema (Y): The minimum: smallest value of y. The maximum: largest value of y. • Sign (X): above x-axis is positive and below is negative. • X-intercept (zero) & y-intercept (initial value).

  11. Function property example • Domain: ]-∞,+ ∞[ • Range: ]- ∞,8] • Variation • Increasing: ]- ∞,-4] U [-1,3] • Decreasing: [-4,-1] U [3, + ∞[ • Constant: none • Extrema • Min: - ∞ • Max: 8 • Sign • Positive: [-6,-2] U [1,5] • Negative: ]- ∞,-6] U [-2,1] U [5,+ ∞[ • Zero: -6, -2, 1, 5 • Initial value: -2

  12. Variables • Variables are qualitative (words) or quantitative (numbers). • Discrete quantitative (counting numbers) • E.g. Dolls on a shelf • Continuous quantitative (all values included within an interval – can be decimal points) • E.g. Height

  13. Representative sampling • 1. simple random: by chance (from a hat) • 2. systematic: regular intervals from a list of the whole population (every 10th member) • 3. cluster: A random selection of clusters is chosen to represent the whole. Every individual within a selected cluster is selected. • 4. stratified: taking representative samples from each group.

  14. Cluster and stratified Percentage: 10% of 254 = 10/100 x 254 = 25.4

  15. Sources of bias • Sources of bias are different reasons that could lead researchers or survey people to draw the wrong conclusion from a survey or census. • There are 6 different sources of bias: • A non-representative sample of the population • A poorly formulated question • The attitude of the person doing the survey • Inadequate representation of the results • Large part of the sample is rejected • A processing error that occurs when compiling the data.

  16. Measures of central tendency – condensed (regular) data table • Median: is the number in the middle when values are placed in order. • Mode: the number that occurs most often in a distribution (list of numbers). • Mean: average of all numbers (sum of all values divided by the number of values). • Range: highest value – lowest value

  17. 2 different types of data tables: • Table of condensed data: mostly used when data values are repeated. • Table with data grouped into classes: data is grouped into intervals [a,b[ (included, excluded) – very few repeated values. • Need to determine the number of groups and how much data each one can carry (amplitude). • Amplitude = range/number of classes. • Amplitude of each interval must be the same!

  18. Measures of Central Tendency in grouped data • A) mode: class with highest frequency is called the modalclass. • Middle of modal class ≈ mode • B) median: the class that includes the median is called the median class. • Middle of median class ≈ median • C) mean: sum of midpoints of each class multiplied by its frequency divided by the number of data values. • D) range is a measure of dispersion • In condensed data: Highest value – lowest value • In grouped data: upper bound of highest group or class – lower bound of smallest group or data.

  19. Relative frequency Relative frequency Relative frequency is a percentage of a group within the total (how many red pens in a box full of colored pens)

  20. Rate of change or slope

  21. X and y • Independent = x values • Dependent = y values • ______y______ depends on ____x________. • Before starting a slope type word problem, figure out which variable is x and which is y.

  22. Determine the rule from 2 ordered pairs (table of values or graph) • 1. locate two ordered pairs (table or graph) • 2. find the rate of change (y2-y1)/(x2-x1) • 3. using the a you just found, substitute the variables of an ordered pair from your graph or table of values. • 4. solve for b. • 5. put a and b in the generic rule. • 6. y=ax+b

  23. Determine the rule from 1 ordered pair and “a” (table of values or graph) • 1. using the a you are given, substitute the variables of an ordered pair from your graph, table of values or description. • 2. solve for b. • 3. put a and b in the generic rule.

  24. Volume = area of base x height

  25. Cube root

  26. Similar solids • In 2 similar solids: correspondingangles are congruent and the measures of correspondingedges (sides) are proportional. • Ratio of similarity = measure of one edge of the mirror-image solid ÷ measure of corresponding edge of the initial solid

  27. Ratio of similarity area and volume • Ratio of areas = area of mirror image solid/area of initial solid • Ratio of volumes = volume of mirror image solid/volume of initial solid • In 2 similar solids: • Ratio of areas is equal to the square of the ratio of similarity • If ratio of area is 16, ratio of similarity is √16 = 4 • Ratio of volumes is equal to the cube of the ratio of similarity • If ratio of similarity is 4, ratio of volumes is 43 = 64

  28. am = a x a x a x ... x a (m times) a1 = a a0 = 1 a-m = a½ = √a a1/3 = ∛a am x an = am + n am ÷ an = am - n (ab)m = ambm (am)n = amn am = am b bm

  29. Negative exponents • With negative exponents we invert the number to the denominator. • If the denominator has a negative exponent, we send it to the numerator position. • = x3

  30. FOIL AND RAINBOW

  31. SCIENTIFIC NOTATION

  32. FACTORIZATION

  33. INEQUALITIES

  34. example • If dividing or multiplying both sides by a negative number you must switch the direction of the inequality sign. = -14a > 3 – 4 =-14a > -1 =-14a > -1 -14 -14 = a < 1/14 4 - 14a > 3

  35. System of equations • By comparison (Exam type) • Both equations are equal to each other • Solve for x • Then solve for y

  36. System of equations graph • Isolate y in equation so y = ........ • Give x random values and solve for y • Find two points for each equation • Plot points on graph and draw straight lines • Intersection = solution

  37. Box and whisker plots

  38. Box and whisker plots • Order values in increasing fashion • Find the median (n+1)/2 = position of median • Q2 • Find median of left and right • Q1 and Q3 • Draw number line with every number • Put lines at Q1, Q2 & Q3 and draw box • Whiskers go to min and max values • Interquartile range = Q3 – Q1

  39. Probabilities • Theoretical probability =

  40. and = multiply probabilities or = add probabilities.

  41. Permutation, arrangement and combination • Permutation = all values of set used, order important, formula = n! • Arrangement = subset of values of the set used, order important, formula is: • Combination = subset of values of the set used, order is not important, formula is: n = total number of values in the set r = number of ways to arrange them

  42. Geometric probability • One dimension = length • Two dimensions = area • Three dimensions = volume • Probability = favorable outcome/total outcomes • Example: probability that a point falls in circle is • Area of circle Area of square

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