Math 6: Year Review Concept Overview, Examples and Practice PATs Math 6 Review Package The math 6 review package contains multiple questions from old PATs that have been organized under headings pertaining to a particular unit that weve covered in grade 6 math this year. I have provided a detailed answer key that can be used to help students check their answers and/ or assist with areas of difficulty.
Math 6 Review Package Math 6 Review Package Key Give these tests a try! Try them yourself first by writing answers on a scrap piece of paper. Use the answer key to mark your test and/or to help you understand questions that you experienced difficulty with. Practice Provincial Achievement Tests (PATs)
2008 PAT 2008 PAT Answer Key 2010 PAT 2010 PAT Answer Key 2013 PAT 2013 PAT Answer Key Unit 1: Numeration and Place Value (2-3 weeks) Place Value vs Value (ex: 567 653) Place value of underlined digit: ten thousands place value or 10 000 Value of underlined digit: sixty thousand or 60 000 Large Numbers (ex: 5 767 349 205) Word Form (five billion seven hundred sixty-seven million three hundred fortynine thousand two hundred five) Standard Form (5 767 349 205) Expanded Form (5 000 000 000 + 700 000 000 + 60 000 000 + 7 000 000 +
300 000 + 40 000 + 9 000 + 200 + 5) Unit 1: Numeration and Place Value Ctd. Decimal Numbers (ex: 0.046 743) Word Form (forty six thousand seven hundred forty-three millionths) Standard Form (0.046 743) Expanded Form (0.04 + 0.006 + 0.0007 + 0.000 04 + 0.000 003) Combination of Whole number and Decimal (ex: 63 057.20343) Word Form (sixty-three thousand fifty seven and twenty thousand three hundred forty-three hundred-thousandths) Standard Form (63 057.20343) Expanded Form (60 000 + 3 000 + 50 + 7 + 0.2 + 0.003 + 0.0004 + 0.00003) Unit 1: Numeration and Place Value
(2-3 weeks) Ex: Three thousand six hundred eighty four and twenty six hundredths Unit 1: Numeration and Place Value Ctd. Rounding to the Nearest Place Value In order to round to a nearest place value (ex: tenths), you first need to examine the neighbor digit one to the right of that place value (ex: hundredths) to determine rounding. If the neighbor digit is 5 or higher, you round your place value digit up by 1. If the neighbor digit is less than 5, the place value digit stays the same. In either case, once rounded, all the digits following the rounded place value turn to zeros (or are dropped if after a decimal) Ex 1: Round 29.835 to the nearest tenth 29.835 (the 8 digit is in the tenths place and the neighbor digit is the 3 next to it on the right) *Since the 3 is less than 5, the 8 in the tenths place will remain the same. All the digits following are simply dropped (as they are after a decimal). Therefore 29.835 rounds to 29.800 = 29.8
Unit 1: Numeration and Place Value Ctd. Estimation (ex: 478 432) Round Round Round Round Round to
to to to to nearest nearest nearest nearest nearest ten (478 432 478 430) hundred (478 432 478 400) thousand (478 432 478 000) ten-thousand (478 432 480 000) hundred-thousand (478 432 500 000)
Estimation (ex: 3.4536) Round Round Round Round to to to to nearest
nearest nearest nearest one/whole number (3.4536 3) tenth nearest (3.4536 3.5) hundredth nearest (3.4536 3.45) thousandth nearest (3.4536 3.454) Unit 1: Numeration and Place Value Ctd. Understanding Place Value (Same number expressed in different ways)
Ex: Ex: Ex: Ex: 465 billion = 465 000 millions = 465 000 000 thousands 4 tenths = 40 hundredths = 400 thousandths = 4 000 ten-thousandths 0.4 = 0.40 = 0.400 = 0.4000 (digit still has the same place value) 3 = 03 = 003 = 0003 (digit still has the same place value) Unit 1: Numeration and Place Value Glossary Place Value: The value given to a digit based on its position in a number (ex: the number 2 in 327 represents 2 tens). Standard Form: Represents a number expressed in the usual form (ex: 375 375).
Word Form: Represents a number expressed in the form of writing (ex: 375 three hundred seventy-five). Expanded Form: Represents a number through addition of its individual place value values.(ex: 375 300 + 70 + 5). Unit 1: Numeration and Place Value Glossary Period: A group of hundreds, tens and ones in a numeral; there are 3 periods in a nine digit whole number (ex: 465 345 292) Unit 2: Multiples, Factors, Composite & Prime (4-5 weeks) Factors (Ex: Find the factors of 12)
1 x 12 = 12 2 x 6 = 12 3 x 4 = 12 The factors are 1, 2, 3, 4, 6, 12 Factor Rainbow 1 2 3 4 6 12 Unit 2: Multiples, Factors, Composite & Prime Ctd. Greatest Common Factor (GCF) Ex: Find the GCF of 24 and 36 24
1, 2, 3, 4, 6, 8, 12, 24 36 1, 2, 3, 4, 6, 9, 12, 18, 36 * Common Factors: 1, 2, 3, 4, 6, 12 GCF: 12 Unit 2: Multiples, Factors, Composite & Prime Ctd. Prime Factorization (Using Factor Trees) Factor until there is only prime numbers left prime *Remember:1 is not
Unit 2: Multiples, Factors, Composite & Prime Ctd. Multiples (Ex: Find the Multiples of 3) 1 2 3 3 x3=3 x3=6 x3=9
x 4 = 12 . 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. Unit 2: Multiples, Factors, Composite & Prime Ctd. Lowest Common Multiple (LCM) Ex: Find the LCM of 6 and 9 Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54.. Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72.. * Common Multiples of 6 and 9: 18, 36, 54. LCM: 18 Unit 2: Multiples, Factors, Composite & Prime Ctd. Prime Numbers Ex: Which of the following numbers are prime?
composite or prime 1, 2, 34, 41, 67, 77, 9, 21 number Answer: 2, 41, 67 * 1 is neither * 2 is the only even prime Composite Numbers Ex: Which of the following numbers are composite? 3, 4, 16, 15, 11, 13, 49, Answer: 4, 16, 15, 49 Unit 2: Multiples, Factors, Composite & Prime Glossary Factor(s): Any whole number(s) that can multiply to get a desired number (ex: 1, 2, 3, 4, 6 and 12 are all factors of 12).
Greatest Common Factor (GCF): The greatest factor that two or more numbers share (ex: The GCF of 36 and 12 is 12). Multiple: A number that is the product of two whole number factors (ex: 8 is a multiple of 4 because 2 x 4 = 8). Lowest Common Multiple (LCM): The lowest multiple that two or more numbers share (ex: The LCM of 4 and 5 is 20). Unit 2: Multiples, Factors, Composite & Prime Glossary Prime Number: A number that only has 2 factors, one and itself (ex: 11 1 x 11). Composite Number: A number that has more than 2 factors (ex: 15 1 x 15, 3 x 5). Prime Factorization: Represents the process of breaking a number down into its prime factors (ex: prime factorization of 30 is 3 x 2 x 5).
Unit 3: Integers and Order of Operations (2-3 weeks) Integers Ex: Is -43 greater than, less than or equal to 13? Use >, <, or = Answer: -43 < 13 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 12 13 14 15 Smaller Larger -3 -2 -1 0
1 2 3 4 5 6 7 8 9 10 11
Unit 3: Integers and Order of Operations Ctd. Order of Operations (BEDMAS - Brackets, Exponents, Division and Multiplication, Addition and Subtraction) Ex: 38 - (6 x 6) (12 - 3) + 10 38 - 36 (12 - 3) + 10 38 - 36 9 + 10 38 - 4 + 10 34 + 10 44 Unit 3: Integers and Order of Operations Glossary Whole numbers: All the counting numbers that begin at 0 and continue forever (ex: 0, 1, 2, 3, 4, 5). Integers: All the counting numbers (+1, +2, +3), zero (0), and all
the opposites of the counting numbers (-1, -2, -3). Order of Operations: A set of rules that are used when calculating so the answer is always the same (ex: BEDMAS). Unit 4: Ratio, Fraction, Decimal, Percent (3-4 weeks) Ratio (Part-to-Part, Part-to-Whole) Ex: A candy bag holds 5 gummy worms, 7 candy frogs, 4 sour keys, and 3 coca-cola candies. What is the ratio of the number of candy frogs to gummy worms? Answer: 7:5, 7/5, or 7 to 5 What is the ratio of the number of sour keys to candy frogs and coca-cola candies? Answer: 4:10, 4/10 or 4 to 10 (Lowest Terms - 2:5, 2/5 or 2 to 5) What is the ratio of gummy worms to total number of candies? Answer: 5:19, 5/19 or 5 to 19
Unit 4: Ratio, Fraction, Decimal, Percent (3-4 weeks) Creating Equivalent Ratios x3 Ex 1: 14= 42 x320 60 Ex 2: 6 24 :66 4:1
(Equivalent Ratio; good for solving proportions) (Lowest Terms; divide both values by their GCF) x20 Ex 3: 4= percent) x20 80 = 80% 5 100 (Equivalent Fraction out of 100; good for finding
Unit 4: Ratio, Fraction, Decimal, Percent Ctd. Converting between percent, fraction and decimal forms Ex: Convert 34% to fraction and decimal form Fraction form: 34/100 (Lowest terms 17/50)*Can divide the top and bottom by 2 to get lowest terms Decimal form: 0.34 (thirty-four hundredths) Ex: Convert the fraction 7/20 into percent and decimal form. *Multiply bottom by 5, then top by 5 to create an Percent form: 35%
equivalent fraction out of 100 to determine % Decimal form: 0.35 Convert the decimal 0.25 to percent and fraction form. Percent form: 25% Fraction form: 25/100 (Lowest terms 1/4) *Divide both top and bottom by 25 Unit 4: Ratio, Fraction, Decimal, Percent Glossary Ratio: A comparison of numbers or quantities that are measured in the same units. Part-to-Part Ratio: A ratio comparing one partial amount to another partial amount. Part-to-Whole Ratio: A ratio comparing one partial amount to the total quantity.
Equivalent Ratios: Ratios that can be created by multiplying or dividing all values in a ratio by the same number. Unit 4: Ratio, Fraction, Decimal, Percent Glossary Fraction: Numbers used to name part of a whole or part of a set (ex: ). Numerator: The number above the bar in a fraction; it tells the number of equal parts the fraction represents. Denominator: The number below the bar in a fraction; it tells the number of equal parts in one whole. Decimal: A way to describe fractions using place value; a decimal point separates the ones place from the tenths place (ex: 4.5). Unit 4: Ratio, Fraction, Decimal, Percent Glossary Percent: A part-to-whole ratio that compares a number to a whole
divided into 100 equal parts (ex: 25%) Equivalent Fractions: Fractions that represent the same part of a whole or the same part of a set (ex: is the same as 2/4). Unit 5: Improper Fractions and Mixed Number (2-3 weeks) Improper Fraction (fraction where the numerator (top) is larger than the denominator (bottom)) Examples: 4/3, 7/5, 21/10, 13/6 Mixed Number (Contains both a whole number and a regular fraction) Examples: 2 , 7 , 9 Convert between Mixed Numbers and Improper Fractions Ex: 7/4 = 1 Ex: 8 = 17/2
*The denominator (4) goes into the numerator (7) one whole time, with 3 out of 4 left over *Multiply the denominator (2) by the whole number (8) and add the numerator (1) to create the fraction 17 out of 2. *The denominator does not change after a conversion. It stays the same. Unit 5: Improper Fractions and Mixed Number Glossary Improper Fraction: A fraction with a numerator (top) greater than a denominator (bottom) (ex: 7/3). Mixed Number: A number consisting of a whole number part and a proper fraction part (ex: 3 ). Unit 6: Multiplication and Division of Decimals
(2-3 weeks) Multiplication of Decimals Ex: Larry purchased 4 milkshakes for him and his friends once a month for 6 months. If one milkshake costs $4.75, how much did Larry pay total over 6 months? Step 1 * Ignore the decimal Step 2 until the end. Put the Step 3 decimal in at the end. Step 4 Larry paid $114.00 over 6 months. Unit 6: Multiplication and Division of Decimals Ctd.
Division of Decimals Ex: Samantha ran a 5 km race with her school team. Samantha finished the race in 48.25 minutes. About how much time did it take Samantha to run each km? It took Samantha about 9.65 minutes to run the first km Unit 6: Multiplication and Division of Decimals Glossary Product: The result when you multiply. Dividend: The starting number in a division operation. Divisor: The number you divide by in a division operation. Quotient: The result you get when you divide. Unit 7: Patterns and Relations (3-4 weeks) Pattern Rules
Ex: Input (x) Output (y) 1 5 2 8 3 11
4 14 5 17 1st Column: Start at 1 and add 1 each time 2nd Column: Start at 5 and add 3 each time * But what about the relationship rule? Unit 7: Patterns and Relations Ctd. Relationship (between the input (x) and output (y)) Ex: Input (x)
Output (y) 1 5 2 8 3 11 4 14
5 17 Relationship rule: Multiply by 3, add 2 3x + 2 * If the amount increasing in the output is constant (ex: ), than that amount will most likely be the value that you will multiply by in your rule (*this only works if the value in the input column has a constant increase of +1). Unit 7: Patterns and Relations Ctd. Expressions with the use of variables Ex: m + 4 Add 4 to the number that m stands for.
Ex: 5n - 4 Subtract 4 from 5 times a number that n stands for. Ex: 12 - h 3 Subtract the number that h stands for from 12. Using tables with expressions Ex: Input (p) 1 2 3 4 Output (4p - 3)
1 5 9 13 Unit 7: Patterns and Relations Ctd. More Complex Tables *Sometimes you can see a pattern by identifying the change betweenFigure numbers 1
2 3 4 5 6 Ex 1: # of Shapes 4 7
13 22 ? ? +3 +6 +9 * The amount added each time increases by an additional 3 each time. SO...
Figure 5 6 # of Shapes 34 49 +12 +15
Unit 7: Patterns and Relations Ctd. Equivalent Expressions (Ex: Solve for c) c c c c 6c + 8 c c c c
c 2 c * Solving Visually c c c c c
c = 2c + 10 4c = Unit 7: Patterns and Relations Ctd. Equivalent Expressions (Ex: Solve for x) 5x - 3 = 12 the other * Solving Mathematically * What you do to one side, you must do to
5x - 3 + 3 = 12 + 3 * Add 3 to each side 5x = 15 5x 5 = 15 5 itself * Divide each side by 5 to get x by Unit 7: Patterns and Relations Glossary Pattern: Something that follows a rule while repeating or changing. Pattern rule: A description of how a pattern starts and how it continues (ex: For the pattern 6, 10, 14, 18, 22. the pattern rule
would be start at 6 and add 4 each time) Relationship rule: A description of the pattern between the input and output in a table of values Unit 7: Patterns and Relations Glossary Table of Values: A way to present numbers in columns and rows so you can see patterns and identify relationships; tables of values can be vertical or #horizontal of CDs 5(ex: 6 7 8 ) Cost ($) 40 45
50 55 Equation: A mathematical sentence in which the value of the left side is the same as the value of the right side (ex: 1 + 3 = 4) Expression: A phrase that uses operations with numbers and Unit 7: Patterns and Relations Glossary Variable: A letter or symbol used to represent an unknown number/quantity (ex: m). The value of a variable can change. Constant: A number that is on its own and has a fixed value (ex: 3). Coefficient: a numerical or constant quantity placed before and multiplying the variable in an algebraic variable expression constant 5n + 3
coefficient Unit 8: Angles, Triangles and Congruence (3-4 weeks) Types of Angles Acute (> 0, < 90) greater than 0, less than 90 Right ( = 90) Exactly 90 Obtuse (> 90, < 180) Greater than 90, less than 180 Straight (= 180) Exactly 180
Reflex (> 180, < 360) Greater than 180, less than 360 Measuring Angles using a Protractor Step 1: Place the center point of the protractor on top of the vertex of the angle Step 2: Line up the zero line of the protractor on one of the angle arms Step 3: Read and record the angle in degrees using the appropriate scale (inside/outside - whichever one is being used as the zero line) Unit 8: Angles, Triangles and Congruence Ctd. Classifying Triangles Classification according to side lengths Equilateral Triangle (All sides the same length) Isosceles Triangle (Two sides have the same length and one is a different length) Scalene Triangle (All three side lengths are different from each other)
Classification according to angles Acute Triangle (All angles in a triangle are >0, < 90) Right Triangle (One angle in a triangle = 90 exactly) Obtuse Triangle (One angle in a triangle > 90, < 180) Unit 8: Angles, Triangles and Congruence Ctd. Congruence Ex: Which of these triangles are congruent to one another? Unit 8: Angles, Triangles and Congruence Glossary Angle: A figure formed by two rays that have the same endpoint; the endpoint is called the vertex of the angle. (ex: ) Vertex (plural - vertices): The point at the corner of a shape or object where sides or edges meet (ex: A quadrilateral has 4 vertices)
Polygon: A closed 2-D shape with sides made from straight lines. Quadrilateral: A polygon with four straight sides and four vertices. Triangle: A polygon with three straight sides and three vertices. Unit 8: Angles, Triangles, Quadrilaterals and Congruence Glossary Degree: A unit of measurement for angles (ex: 90). Protractor: A tool used to measure angles in degrees. Congruent: Identical in size and shape. Unit 9: Perimeter, Area and Volume (2-3 weeks) Perimeter What is the perimeter of the following shape ?
6 cm ? 6 cm ? 9 cm 9+6+6+3+3+3 Answer: 30 cm Unit 9: Perimeter, Area and Volume Ctd. Area (Rectangles) Formula: A = l x w Area is measured in square units (Ex: cm 2) Ex: What is the area of the shaded shape? Large Rectangle
Small Rectangle A=lxw A=lxw 9m A=9x8 A= 5x4 2 2 A = 72 m A = 20 m 8m 5m Area of shaded shape = Large Area - Small Area Area of shaded shape = 72 - 20 Area of shaded shape = 52 m
2 4m Unit 9: Perimeter, Area and Volume Ctd. Volume (Rectangular Prisms) Formula: V = l x w x h Volume is measured in cubic units (ex: cm 3) Ex: What is the Volume of the following shape Volume of Rectangular Prism V=lxwxh 7 cm 20 cm
9 V = 1260 cm3 cm V = 20 x 9 x 7 Unit 9: Perimeter, Area and Volume Glossary Perimeter: The distance around a shape, often measured in cm or m. Area: The number of identical objects (area units) needed to cover a surface completely, often measured in cm2 or m2. Volume: The amount of space occupied by a 3-D object, often measured in cm3 or m3.
Unit 10: Data and Graphing (2-3 weeks) Questionnaires Ex: 1) What is your favorite ice cream flavor: (a) chocolate, (b) vanilla, (c) strawberry, (d) neapolitan or (e) other? 2) Which type of cone do you prefer: (a) waffle or (b) regular Databases Ex: 1) In which month was the maximum temperature 124F? Answer: August 2) What was the minimum F in Fort Valley? Answer: - 9F
Unit 10: Data and Graphing Ctd. Reading Bar Graphs Ex: 1) Which fruit is liked by the largest number of people? Answer: Blueberry 2) How many people total liked Apples and Kiwifruit? Answer: 60 people Unit 10: Data and Graphing Ctd. Reading Line Graphs Ex: 1) What is the difference in the number of cases between 1960 and 1985?
Answer: 250 2) In which year was the number of cases at 200? Answer: 1970 Unit 10: Data and Graphing Ctd. Ordered Pairs Represented in the form of (x, y) then y-axispairs for each of the Ex: List the ordered following letters. A (2, 6) B (3, 3) C (6, 4) D (6, 2)
E (7, 7) *x-axis first, Unit 10: Data and Graphing Glossary Data: Information gathered in a survey, in an experiment, or by observing; it can come in the form of words, numbers or even pictures. Bar Graph: A way to show data that uses horizontal or vertical bars. Database: An organized set of a large amount of information, often stored on a computer. Questionnaire: A set of questions used to gather information from people; this is sometimes used instead of interviewing people one- Unit 10: Data and Graphing Glossary Bar Graph: A way to show data that uses horizontal or vertical bars. Line Graph: A way to show data based on connecting plotted points
in a meaningful order; you use a line graph to show how data are changing. Origin: The point on a coordinate grid at which the horizontal (x) and vertical (y) axes intersect; the ordered pair of the origin is (0, 0). Coordinate Grid: A grid with each horizontal and vertical line numbered in order, starting at zero. Unit 10: Data and Graphing Glossary Title: A short phrase of text at the top of a graph, list, table or chart that describes what is been recorded or displayed. Axis (plural- axes): A horizontal or vertical line in a graph labelled with words or numbers to show what the bars or points in the graph mean. Ordered Pair: A pair of numbers that describes a point on a coordinate grid (ex: (7, 3) is an ordered pair) Unit 11: Probability
Theoretical Probability Represents what should happen Ex: 1) What is the theoretical probability of spinning a 3? Answer: 2) What is the theoretical probability of spinning a blue or red? Answer: 2/6 or Unit 11: Probability Ctd. Experimental Probability Represents what actually happens Ex: Greg rolled a number cube 50 times. 1) Based on his results, what is the experimental probability of rolling a 3? Answer: 4/50 or 2/25
2) Based on his results, what is the experimental probability of not rolling a 4 or greater? Answer: 25/50 or 1/2 Gregs Dice Experiment Unit 11: Probability Glossary Probability: How likely it is that an event will happen. Event: A set of one or more outcomes in a probability experiment (ex: the event of rolling an even number on a die includes the numbers 2, 4, or 6). Theoretical Probability: A fraction or ratio that compares the number of ways an event can happen to the number of equally likely outcomes; all probabilities range from 0 to 1 (ex: the theoretical probability of rolling an even number on a die is 3/6, which can be reduced to 1/2).
Unit 11: Probability Glossary Experimental Probability: A fraction or ratio that compares the number of times an outcome occurs in an experiment to the total number of times the experiment is done (ex: If a coin is flipped 10 times and heads is flipped 6 times, the experimental probability of flipping heads is 6/10). Unit 12: Transformations Slide/Translation Move an object in a straight line without changing shape, size or orientation Ex: Name and explain the following transformation from the grid. Answer: a) Slide/Translation b) Figure 1 translated 5 units right Unit 12: Transformations Ctd.
Flip/Reflection Flipping an object over a mirror line Ex: Name and explain the following transformation from the grid. Answer: a) Flip/Reflection b) Figure RST reflects over the mirror line x = 5 to create figure Unit 12: Transformations Ctd. Turn/Rotation Rotating an object around a point either clockwise or counterclockwise Ex: Name and explain the following transformation from the grid. Answer: a) Turn/Rotation
b) Rotate shape WXYZ 90 ccw about point P to create shape WXYZ 8 7 6 5 4 10 a) A to B Translation 6 units left and 5 units up. b) B to C Rotation 90 cw about point (3,7). c) C to D Reflection about the mirror line x = 6 3
Ex: Starting from image A, name, explain and draw the transformations that occur from: 2 Using a combination of slides, flips and turns to move an object/image 1 Combining Transformations 9 Unit 12: Transformations Ctd. Unit 12: Transformations Glossary
Transformation:The result of moving a shape according to a rule; translations, reflections and rotations are transformations. Translation/Slide: The result of sliding a shape along a straight line Reflection/Flip:The result of flipping a 2-D shape across a line of reflection; each point in a 2-D shape flips to the opposite side of the line of reflection, but stays the same distance from the line. Rotation/Turn: The result of turning a shape. Unit 12: Transformations Glossary Translation Rule: A way of describing a translation with pictures or numbers (ex: (R3, U4) translation of 3 right and 4 up). Line of Reflection: A line that falls exactly halfway between the points of a shape and the matching points of its mirror image. Centre of Rotation: A fixed point around which other points in a shape rotate in a clockwise (cw) or counterclockwise (ccw) direction; the centre of rotation may be inside or outside a shape.
Marketers' goal should be to exceed the customer's expectations, thereby creating strong customer satisfaction. Furthermore, it is not always possible to meet, much less exceed, expectations. Exaggerated expectations account for 75%of customer complaints.
Nathaniel Hawthorne & Dark Romanticism American Literature & Composition Ms. Villa Nathaniel Hawthorne Nathaniel Hawthorne (born Nathaniel Hathorne; July 4, 1804 - May 19, 1864) was an American novelist and short story writer.
What's the Point of a Financial Model? In physics you predict the future. In finance, you interpolate, extrapolate in the present. "Right" vs. "True"? 'Right' is consistent. 'True' is true. Accuracy? Ross: "Options pricing is the most successful theory, not...