Cologne Academy. Mathematics Department Grade 6 Math. (Aligned Text: Holt McDougal Mathematics: Course 2) Core Knowledge Curriculum 100% Aligned

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1 Cologne Academy Mathematics Department Grade 6 Math (Aligned Text: Holt McDougal Mathematics: Course 2) Core Knowledge Curriculum 100% Aligned Adopted: 08/2014 Board Approved: 08/28/2014 Updated: 08/10/2017 AU Page 0

2 Table of Contents Math Department Lesson Plan Essentials... 2 Units and Pacing Charts Unit 1 Overview: Number Forms & Locations, Factors & Multiples... 3 Essential Vocabulary... 6 Pacing Chart... 7 Unit 2 Overview: Ratio, Variables, Patterns & PEMDAS... 9 Essential Vocabulary Pacing Chart Unit 3 Overview: Conversions, 2D, 3D, Angles & Time Essential Vocabulary Pacing Chart Unit 4 Overview: Data Analysis, Probability & Core Knowledge Essential Vocabulary Pacing Chart Highlighted items indicate overlap of MN State Standards and the Core Knowledge Sequence. Boxed items indicate content to be introduced post-mcas. Page 1

3 Math Department Lesson Essentials Topic: Title of lesson. *Objective: Academic goal for students to achieve by end of lesson. *Standard: MN State Standard or Core Knowledge Sequence reference. Agenda: Sequence of instruction and activities Closure: Brief summary/overview of lesson. May include formative assessment. Homework: Continued practice of lesson. May be used as formative assessment. *Indicates required co Note: The text has been as closely aligned with MN State Standards but additional resources may be required to include all skills (including within the Core Knowledge Sequence). Resources may be located on the s:drive under Mathematics Resources and by grade level or on the Cologne Academy intranet. Further research/exploration may be required to locate additional resources. Important Dates Pretest: August Interim 1: October Interim 2: January Interim 3: March Interim 4: May OLPA: 29 January 2 February MCA: April Dates may be subject to change. Page 2

4 Overview Strand(s): Number & Operation Unit 1: Number Forms & Locations, Factors & Multiples Approximate Duration of Study: 9 Weeks Between Interims. MNSS Knowledge Skills Review Graphing Read and write numbers (in digits and words) up to the trillions. Recognize place value up to hundred-billions. Round to the nearest ten, hundred, thousand, hundred thousand, and million. Orders of Operations (PEMDAS), with and without a calculator. Exponents Powers of Ten Write numbers in expanded notation using exponents (include example) Every rational number can be expressed as the ratio of two integers (where the denominator is not equal to 0). The sum of an integer and its opposite is 0. Every rational number can be expressed as a terminating decimal or repeating decimal. Classify numbers according the Real Number Set. o Rational, Integer, Whole, Natural. Locate positive rational numbers on a number line. Compare integers using <, >, =. Add and subtract positive and negative integers. Multiply and divide positive and negative integers. LCM & GCF Coordinate Grid is made up of a horizontal number line, the x- axis, and a vertical number line, the y-axis that intersects at one point, the origin. An Ordered Pair is a pair of numbers that can be used to locate a point on a coordinate plane. o The positive x-coordinate is units to the right of 0, the positive y-coordinate is units up from 0. The Greatest Common Factor (GCF) of two or more whole numbers is the greatest whole number that divides evenly into each number. The Least Common Multiple (LCM) of two or more whole numbers is the common multiple with the least value. Plot pairs of positive rational numbers on a coordinate grid. Identify ordered pairs on a coordinate plane. Plot points on a coordinate plane, using ordered pairs of positive and negative whole numbers. Use the terms origin, x-axis, and y-axis. Determine the GCF and LCM of given values. Find equivalent fractions. Add and subtract fraction with like and unlike denominators. Unit 1 Page 3

5 Percents Decimals & Fractions (Fractions & Decimals only) (Percents) A power consists of a base and its exponent. The exponent tells how many times to use the base as a factor. o 5 3 = A prime number is a whole number greater than 1 that has exactly two factors, one and itself. A composite number is a whole number that has more than two factors. Prime Factorization is written in ascending order according to the value of the base. Fractions can be written as decimals by dividing the numerator of a given fraction by the denominator. Per Centum: out of 100. A percent represents a ratio to 100 or a part out of 100. Read and evaluate numerical expressions with exponents. Find the square and square root of given values. Find the cube and cube root of given values. Determine whether a number is prime or composite. Show the prime factorization of given numbers. o 24 = 6 4 = = Convert between fractions and decimals. Compare positive rational numbers represented in various forms, using <, >, and =. 1 o > Compare and order whole numbers, mixed numbers, fractions and decimals, using <, >, =. Select among appropriate representations to solve problems. Add and subtract fractions and decimals, both positive and negative. Model percents. Write percents as equivalent fractions and decimals. o 75% corresponds to the ratio 75 to 100, which is equivalent to the ratio 3 to 4. Select among appropriate representations to solve problems. Unit 1 Page 4

6 The multiplication is the inverse operation of division and vice versa. o Just as 12 4 = 3 means 12 = 3 4, = 5 6 means = 2 3. Decimals must be removed from the divisor by multiplying by the least power of ten that will make the divisor an integer. The dividend must also be multiplied by the same power of ten to keep the problem equivalent. Multiply and divide decimals and fractions, positive and negative. Multiply multi-digit factors, with and without a calculator. Divide multi-digit dividends by up to three-digit divisors, with and without a calculator. Estimate a product and quotient. Multiply with integers, positive and negative. Equivalence When multiplying fractions, the numerator of the first fraction must be multiplied by the numerator of the second fraction; similarly for the denominators. To find the reciprocal of a fraction, switch the positions of the numerator and denominator. When dividing fractions, keep the first fraction, change the operation to multiplication and find the reciprocal of the second fraction and follow rules for multiplying fractions. Numbers (decimals/fractions) must be in the same form to perform multiplication and division. o = = o = 3 2 = 3 5 = 15 = Numbers can be represented in different, equivalent forms. Show different equivalent representations of a given positive, rational number Decimals, fractions and mixed numbers can be used to represent situations in the real world. Non-exact values of given numbers can be used to make predictions and check solutions when solving problems. Solve real world problems using decimals, fractions and mixed numbers. Estimate solutions to problems with whole number, fractions and decimals. Use estimates to assess reasonableness of results. Essential Vocabulary: Real Numbers: Rational, Integer, Whole, Natural, Terminating Decimal, Repeating Decimal, Ratio, Coordinate Grid (Plane), X-Axis, Y-Axis, Coordinate Pair, Ordered Pair, X-Coordinate, Y-Coordinate, Per Centum, Percent, Inverse Operation, Divisor, Dividend, Quotient, Reciprocal, Greatest Common Factor, Factor, Least Common Multiple, Multiple, Prime, Composite, Prime Factorization, Power, Base, Exponent, Ascending, Descending, Value, Equivalent. Prime Factorization. Interim 1 Unit 1 Page 5

7 Pacing Chart Unit 1: Number Forms & Locations, Factors & Multiples Time Frame Topic Suggested Activities/Assessments Resources & Text Alignment Week 1 Week 1 Week 2 Week 3 Week 4 Graphing LCM & GCF Pre-test & Review 9-7 Extension: Identifying and Graphing Irrational Numbers (focus: classification of numbers) S:MathematicsResources:Grade6:RationalNumberWS 2-11: Comparing and Ordering Rational Numbers 2-2: Adding Integers 2-3: Subtracting Integers 2-4: Multiplying and Dividing Integers 5-1: The Coordinate Plane Extension (Optional) 5-2: Interpreting Graphs 2-7: Greatest Common Factor 2-8: Least Common Multiple 9-7: Squares and Square Roots 2-6: Prime Factorization Week 5 Week 8 Percents Decimals & Fractions (Fractions & Decimals only) (Percents) : Equivalent Fractions and Mixed Numbers 2-10: Equivalent Fractions and Decimals 2-11: Comparing and Ordering Rational Numbers 3-1: Estimating with Decimals 3-2: Adding and Subtracting Decimals 3-3: Multiplying Decimals 3-4: Dividing Decimals 3-6: Estimating with Fractions Unit 1 Page 6

8 Week Equivalence Review Week 10 Interim 1 3-7: Adding and Subtracting Fractions 3-8: Adding and Subtracting Mixed Numbers 3-9: Multiplying fractions and Mixed Numbers 3-10: Dividing Fractions and Mixed Numbers 6-1: Percents 6-2: Fractions, Decimals, and Percents 6-3: Estimating with Percents Unit 1 Page 7

9 Overview Strand(s): Number & Operation, Algebra Unit 2: Ratio, Variables, Patterns & PEMDAS Approximate Duration of Study: 8 Weeks Between Interims. MNSS Knowledge Skills Ratios & Rates A ratio is the comparison of two quantities of the same unit. Ratios can be expressed as part-to-part, part-to-whole, or wholeto-part relationships. Comparing quantities using ratios is not the same as comparing quantities using subtraction. o In a classroom with 15 boys and 10 girls, compare the numbers by subtracting (there are 5 more boys than girls) or dividing (there are 1.5 times as many boys as girls). The comparison using division may be expressed as a ratio of boys to girls (3 to 2 or 3:2 or 1.5 to 1). Identify and use ratios to compare quantities. Identify ratios in various contexts and represent them in multiple ways; Use the relationships between ratios, equivalent fractions, and percents to solve a variety of problems, including those involving mixtures and concentrations. o If 5 cups of trail mix contains 2 cups of raisins, the ratio of raisins to trail mix is 2 to 5. This ratio corresponds to the fact that the raisins are 2 of 5 the total, or 40% of the total. o If one trail mix consists of 2 parts peanuts to 3 parts raisins, and another consists of 4 parts peanuts to 8 parts raisins, then the first mixture has a higher concentration of peanuts. A rate is ratio that compares to two quantities of different units. A unit rate is a rate whose denominator is 1. Determine the rate for ratios of quantities with different units. Determine unit rates. o 60 miles for every 3 hours is equivalent to 20 miles for every one hour (20 mph) Solve ratio and rate problems. o If 5 items cost $3.75, and all items are the same price, then 1 item cost 75 cents, so 12 items cost $9.00. Unit 2 Page 8

10 Variables and Patterns Coordinate Plane Core Knowledge Properties & PEMDAS Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. The relationship between two varying quantities can be represented with function rules, graphs and tables. The Coordinate Plane is flat plane divided into four quadrants by lines known as the x-axis and the y-axis. The x-axis runs horizontally, the y-axis runs vertically. Points, coordinate pairs, are plotted on the coordinate plane by a using a unique value of x and y: (x, y). The origin is the point at the very center of the coordinate plane; located at (0, 0). Properties of Algebra can be used to generate equivalent expressions. o Commutative: a b = b a; a + b = b + a o Associative: (a b) c = a (b c); (a + b) + c = a + (b + c) o Distributive: ab ac = a(b c); ab + ac = a(b + c) Evaluate algebraic expressions. o Find the value for an expression, given replacement values for the variables. For 7 example: what is when x = 2, y = 10? x y Use variables to represent an unknown quantity o If a student earns $7 an hour in a job, the amount of money earned can be represented by a variable and is related to the number of hours worked, which also can be represented by a variable. Evaluate algebraic expressions at specified values of their variables. o What is 7 c if c is 3.5? Translate between rules, graphs and tables. Derive formulas for given numerical and pictorial patterns. o Describe the terms in the sequence of perfect squares t = 1, 4, 9, 16, by using the rule t = n 2 for n = 1, 2, 3, 4, Graph simple functions (equations) and solve problems involving the use of a coordinate plane. Justify procedures used to simplify algebraic and numerical expressions. Simplify expression with variables by combining like terms. Use distributive property to multiply multi-digit factors. Understand the use of distributive property in variable expressions such as: 2x(2y + 3). Unit 2 Page 9

11 Apply the associative, commutative, and distributive properties to generate equivalent expressions and to solve problems involving positive rational numbers. 32 o 5 = = = = o (9 15 ) = = = = The Order of Operations dictates the order in which basic operations, including raising a positive rational number to wholenumber exponents, should be performed. Order of operations must be performed properly to simplify an expression when using a calculator or other technology. Properties of Equality are used to solve for the value of a variable. o Addition, Subtraction, Multiplication, and Division Properties of Equality. o Additive Inverse: a + (-a) = 0 o Multiplicative Inverse: a 1 = 1 a Addition and subtractive are inverse operations. Number sense, properties of equality and the idea of maintaining equality on both sides of an equal sign is used to solve equations. Apply order of operations to generate equivalent expressions and to solve problems involving positive rational numbers. Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions, and terminating decimals. Express powers as whole numbers solutions. Solve equations involving positive rational numbers. Recognize uses of variables and solve linear equations in one variable. Interpret a solution in the original context. Assess reasonableness of results. o A cellular phone company charges $0.12 per minute. If the bill was $11.40 in April, how many minutes were used? Key words in word problems determine whether an algebraic equation or inequality must be constructed and solved. Real-world & mathematical situations can be represented using equations and inequalities involving variables and positive rational numbers. Unit 2 Solve word problems with multiple steps. Solve word problems by assigning variables to unknown quantities, writing appropriate equations and solving them. Translate between words and mathematical sentences. Represent unknown quantities with a variable. Use given information to set up and solve an equation or inequality. o The number of miles m in a k kilometer race is represented by the equation m = 0.62k. Page 10

12 Ratio & Proportions Core Knowledge Percent Problems A ratio is a comparison between two quantities. Proportions can be set up and used to solve problems involving ratios. o A proportion is an equation stating that two ratios are equivalent. A proportion can be used to solve percent problems. o % (Part) = Part 100 (Whole) Whole Similar Figures have the same shape but not necessarily the same size. o Corresponding angles in similar figures have the same measure. o Corresponding sides in similar figures are proportional. Understand that a percent must be converted to a fraction or a decimal in order to multiply or divide; An equation can be created, to solve a percent problem, depending on the unknown value. The percent equation is p q = r. o If the percent is unknown, solve p q = r for p. o If the whole (quantity) is unknown, solve p q = r for q. o If the part (result) is unknown, solve p q = r. Use ratios and proportions to interpret map scales. Use ratios and proportions to interpret scale drawings. Solve proportions with one unknown. Solve word problems using proportions. Justify the use of cross-products to solve proportions. Set up and solve proportions from similar triangles. Calculate the percent of a number. Determine what percent one number is of another number. Solve problems involving percent increase and decrease. Find an unknown number when a percent of the number is known. Use expressions with a percent greater than 100% and less than 1% Solve percent problems in various contexts. Essential Vocabulary: Ratio, Rate, Unit Rate, Variable, Quantity, Evaluate, Algebraic Expression, Properties of Algebra, Expression, Commutative, Associative, Distributive, Order of Operations, Equivalent Expression, Terminating Decimal, Properties of Equality, Additive Inverse, Multiplicative Inverse. Interim 2 Unit 2 Page 11

13 Pacing Chart Unit 2: Ratio, Variables, Patterns & PEMDAS Time Frame Topic Suggested Activities/Assessments Resources & Text Alignment Week 11 Week 12 Week 13 Properties & PEMDAS Ratios & Rates : Order of Operations 1-5: Properties of Numbers 1-8: Simplifying Algebraic Equations 1-9: Equations and Their Solutions 1-10: Solving Equations by Adding or Subtracting 1-11: Solving Equations by Multiplying or Dividing 2-5: Solving Equations Containing Integers 3-5: Solving Equations Containing Decimals 3-11: Solving Equations Containing Fractions 12-1: Solving Two-Step Equations 12-2: Solving Multi-Step Equations 12-3: Solving Equations with Variables on Both Sides 1-7: Translating Words into Math 4-1: Ratios 4-2: Rates 4-3: Identifying and Writing Proportions 4-4: Solving Proportions 4-7: Dimensional Analysis Week 14 Week 15 Variables and Patterns : Variables and Algebraic Expressions 5-3: Functions, Tables and Graphs 5-4: Sequences Unit 2 Page 12

14 Week 16 Week 17 Week 18 Week 19 Coordinate Plane Core Knowledge Ratios & Proportions Core Knowledge Percent Problems Review Interim 2 5-5: Graphing Linear Functions Optional extension to: 5-6: Slope and Rates of Change 5-7: Slope-Intercept Form 4-8: Similar Figures and Proportions 4-9: Using Similar Figures 4-10: Scale Drawings and Scale Models 6-4: Percent of a Number 6-5: Solving Percent Problems 6-6: Percent of Change Unit 2 Page 13

15 Overview Strand(s): Geometry & Measurement Unit 3: Conversions, 2D, 3D, Angles & Time Approximate Duration of Study: 9 Weeks Between Interims. MNSS Knowledge Skills Weight/Capacity Conversion factors can be used to move from one unit of measure to an equivalent measure of a different unit. Use ratios to convert between measurement systems. o Weight, volume, time or measurement. Express measurements using appropriate units. Identify equivalent ratios within a measurement system. Solve real-world and mathematical problems involving conversion of units. Dimensional Analysis Core Knowledge The U.S. Customary System, and the Metric System are the two must commonly used measurement systems. Equivalents lengths are found within each measurement system. U.S. Customary System o 1 mile = 5,280 feet o 1 yard = 3 feet o 1 foot = 12 inches Metric System o 1 kilometer = 1,000 meters o 1 meter = 100 centimeters o 1 meter = 1,000 millimeters Converting from one unit to an equivalent unit using a conversion factor is Dimensional Analysis. Estimate weights, capacities and geometric measurements. o Estimate the height of a house by comparing to a 6-foot man standing nearby. Solve problems requiring conversion of units within the U.S. Customary System, and within the metric system. Associate prefixes used in the metric system with quantities. o kilo = one thousand units o hecto = one hundred units o deka = ten units o deci = tenth of a unit o centi = hundredth of a unit o milli = thousandth of a unit Unit 3 Page 14

16 Time Core Knowledge Time can be measured linearly. Solve problems on elapsed time. Express parts of an hour in fraction or decimal form. o 45 minutes = 3 hour = 0.75 hour 4 Convert between units of time. Angle Relationship Certain angles have special names and degree measures. o Straight: 180 O. o Right: 90 O. o Obtuse: > 90 O. o Acute: < 90 O. o Complementary: sum to 90 O. o Supplementary: sum to 180 O. o Vertical: opposite angle pairs; congruent. If two lines are parallel, any line perpendicular to one is also perpendicular to the other. Two lines perpendicular to the same line are parallel. A line that intersects a pair of parallel lines is called a transversal. Special angle relationships are formed by a transversal. o Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Vertical Angles, Same Side Interior Angles, Same Side Exterior Angles. Identify and measure the degrees in angles. Identify and use geometric symbols. o : congruent o ~: similar o : parallel o _ _: perpendicular Construct an angle congruent to a given angle. Construct a perpendicular bisector. Bisect an angle. Construct parallel lines and a parallelogram. Solve problems using the relationships between the angles formed by intersecting lines. o If two streets cross, forming four corners such that one of the corners forms an angle of 120 O, determine the o measures of the remaining three angles. Recognize that pairs of interior and exterior angles in polygons have measures that sum to 180 O. Identify straight and right angles and their measures. Identify complementary and supplementary angles and their measure sums. Calculate the measurement of angles created by a transversal and parallel lines. Unit 3 Page 15

17 Triangles & Angles Area Vertical angles are formed by a pair of intersecting lines, which cross at one point. Triangles are polygons with three sides and three interior angles. The interior angles of a triangle sum to Know terms by which triangles are classified. o By side length: equilateral, isosceles, scalene. o By angles: right, acute, obtuse. There is a relationship between angles in geometric figures. The sum of the interior angles of any triangles is 180 O. Polygons can be decomposed into triangles. Interior angle sum for an n-sided, regular polygon: 180(n 2) where n is the number of sides. A quadrilateral is any four-sided polygon. o Squares, rectangles, rhombuses, parallelograms, trapezoids and kites. Area applies to two-dimensional figures. Area is measured in square units. Area represents the number of square units needed to cover a given surface. Specific two-dimensional figures have formulas for calculating area. Area formulas can be derived by composing/decomposing two-dimensional figures. Perimeter represents the number of units measured around a figure. Identify intersecting lines and their vertical angles and their measures. Determine the measure of all angles in a pair of intersecting lines when given the measure of one angle. Construct different types of triangles. Determine missing angle measures in a triangle. Use models of triangles to show the interior angle sum of triangles is o Cut a triangle out of paper, tear of the corners and rearrange these corners to form a straight line. o Recognize that the measures of the two acute angles in a right triangle sum to Decompose polygons into triangles. Develop the formula for the interior angle sum of regular polygons. Use formulas to find the sums of the interior angles of any n- sided polygon. Calculate the area of quadrilaterals. Find the area and perimeter of plane figures. Given the area/perimeter find the missing dimension using formulas. o Rectangle: A = lw; P = 2(l + w) o Square: A = s 2 ; P = 4s o Triangle: A = 1 bh; P = s1 + s2 + s3 2 o Parallelogram: A = bh; P = 2(b + s) Unit 3 Page 16

18 The diameter is the distance across a circle, through its center point. Radius is half of the diameter (distance from center point to a point on the circle). Circumference is the distance around (perimeter) a circle. Pi π 3.14) Find the area of circle using the formula: A = πr 2. Solve problems involving application of formulas for finding the circumference of a circle. o C = πd; C = 2πr 3D Measures The surface area of a solid is equal to the area of its net. Volume is a measurement of the capacity of a threedimensional object. Justify the use of specific formulas for specific polygons. o The area of a kite is one-half the product of the lengths of the diagonals, and this can be justified by decomposing the kite into two triangles. Estimate the perimeter and area of irregular figures, on a grid, that cannot be decomposed into common figures. Estimate using correct units, such as cm and cm 2. Distinguish between surface area and volume. Calculate the surface area of prisms using appropriate units (units squared). Calculate the volume of prisms (rectangular solids) using appropriate units (units cubed) and formulas. o V = lwh o V = bh, where b is the area of the base. Given the volume, find a missing dimension. Justify the formulas used to calculate the surface area and volume of prisms. o Using decomposition, nets or other models. o The surface area of a triangular prism can be found by decomposing the surface into two triangles and three rectangles. Essential Vocabulary: Conversion Factor, Convert, Unit, U.S. Customary Units, Metric System, Dimensional Analysis, Kilo, Hecto, Deka, Deci, Centi, Milli, Equilateral, Isosceles, Scalene, Right, Acute, Obtuse, Quadrilateral, Polygon, Area, Two-Dimensional Figure, Square Unit, Formula, Perimeter, Surface Area, Net, Prism, Volume, Unit Cubed, Interior Angle, Interior Angle Sum Formula, Regular Polygon, Angle: Straight, Right, Acute, Obtuse, Complementary, Supplementary, Vertical; Intersecting Lines, Opposite Angles, Congruent, Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Vertical Angles, Same Side Interior Angles, Same Side Exterior Angles. Similar, Parallel, Perpendicular, Bisect, Perpendicular Bisector, Angle Bisector. Interim 3 Unit 3 Page 17

19 Pacing Chart Unit 3: Conversions, 2D, 3D, Angles & Time Time Frame Topic Suggested Activities/Assessments Resources & Text Alignment Week 20 Weight/Capacity : Customary Measurements 4-6: Metric Measurements Week 21 Dimensional Analysis Core Knowledge 4-7: Dimensional Analysis Supplement with additional resources to meet Core Knowledge Sequence Requirements. Week 21 Time Core Knowledge Angle Relationship Supplement with resources to meet Core Knowledge Sequence Requirements. 8-2: Classifying Angles 8-3: Line and Angle Relationships Week 22 Unit 3 Page 18

20 Week 23 Triangles & Angles : Classifying Triangles 8-8: Angles in Polygons Week 24 Week 25 Area : Perimeter and Circumference 9-3: Area of Parallelograms 9-4: Area of Triangles and Trapezoids 9-5: Area of Circles 9-6: Area of Irregular Figures Week 26 Week 27 3D Measures : Introduction to Three-Dimensional Figures 10-4: Surface Area of Prisms and Cylinders 10-5: Surface Area of Pyramids and Cones 10-2: Volume of Prisms and Cylinders 10-3: Volume of Pyramids and Cones Week 28 Review Week 29 Interim 3 Unit 3 Page 19

21 Overview Strand(s): Data Analysis & Probability & Core Knowledge Unit 4: Data Analysis, Probability & Core Knowledge Approximate Duration of Study: 6 Weeks Between Interims. MNSS Knowledge Skills Sample Spaces Probability A sample space of an experiment is a display of the set of all possible outcomes of that experiment. A sample space may be determined by the use of tree diagrams, tables or pictorial representations. o A 6 x 6 table with entries such as (1, 1), (1, 1), (1, 3),, (6, 6) can be used to represent the sample space for the experiment of simultaneously rolling two number cubes. A sample space may be used to estimate a population parameter (numerical characteristic, such as the mean). Larger samples provide a more stable estimate. The probability of an event is the chance (likelihood) that an event will occur; represented by a percent, decimal or fraction with a value that falls between 0 and 1. o The closer to one, the more likely the event will occur; the closer to 0, the less likely the event will occur. o Probability = Number desired events Total number of possible outcomes size of the event = size of the sample space A simple event has a single outcome. Determine and represent the sample space for a given experiment using tree diagrams, organized lists, tables, charts and pictures. Determine which members of the sample space are related to certain events. Determine the probability of an event. o Use terms to describe the likelihood of an event occurring: impossible, equally likely, likely, certain. Represent probabilities as percents, fractions and decimals between 0 and 1 inclusive. o Each outcome for a balanced number cube has a probability of 1, and the probability of rolling an 6 even number is 1 2. Unit 4 Page 20

22 Relative Frequency of an event is the ratio of the actual occurrences of the event to the total number of trials. Number of occurrences o Relative Frequency = Total number of trials Probability may not always match relative frequency of a data set. o The more trials, the closer relative frequency will be to probability. Perform experiments for which probabilities are known and collect data. Compare probability to relative frequency. o Heads and tails are equally likely when flipping a fair coin, but if several different students flipped fair coins 10 times, it is likely that they will find a variety of relative frequencies of heads and tails. Disjoint events are events that cannot occur simultaneously. They are mutually exclusive. o Turning left and right are disjoint events. The probability of disjoint events is the sum of their individual probabilities. Theoretical Probability represents what is expected occur. Experimental Probability represents what actually occurs during an experiment. Compute the probability of any one of a set of disjoint events. Calculate experimental probabilities from experiments. Represent experimental probability as a fraction, decimal and percent between 0 and 1 inclusive. Calculate theoretical probability. Compare experimental and theoretical probability. Use experimental probability to make predictions when theoretical probabilities are unknown. o Repeatedly draw colored chips with replacement from a bag with an unknown mixture of chips, record relative frequencies, and use the results to make predictions about the contents of the bag. Use probabilities to make decisions. Use probabilities to solve real world and mathematical problems. Data can be collected and represented in a variety of displays. o A histogram is used to display data in intervals. o A frequency chart is used to keep track of often a piece of data occurs. Construct a histogram using frequency chart. Solve problems requiring the interpretation and application of graphically displayed data. Unit 4 Page 21

23 Dispersion and Central Tendency Core Knowledge Corresponding Parts Core Knowledge Circles Core Knowledge Mean, median and mode are Measures of Central Tendency. o Mean: the average of a set of data. o Median: the middle number in a set of data. o Mode: the most frequent number in a set of data. Range is a Measure of Dispersion. o Range: In a set of data, the lowest value subtracted from the greatest value. The mean is used to describe the middle of a set that does not include an outlier. Also used to compare data but affected by outliers. The median describes the middle of a set that does have an outlier. Used to compare data and not affected by outliers. The mode is used to describe the most popular item. There may be more than one mode or, if not value repeats, all data in the set is considered the mode. Outliers do not affect this measure. Transformations may alter the size, position, or location of a figure on the coordinate plane. o Translation: slide o Reflection: flip o Rotation: turn Transformations result in congruent or similar figures. The line of symmetry is an imaginary line where an image can be folded and both resulting halves are exactly the same. A circle is a line forming a closed loop; every point on the line is of equal distance from the center point. An arc is a small portion of the circumference of a circle. A chord is any line that connects two points on a circle. Find the range and measures of central tendency given a set of data. Describe the differences among the measures of central tendency and when each might be used. Construct a figure congruent to a given figure by reflecting over a line of symmetry. Identify corresponding parts. Show how congruent plane figures can be made to correspond through reflection, rotation and translations. Identify congruent angles, sides and axes (lines) of symmetry in parallelograms, rhombuses, rectangles and squares. Identify arc, chord, radius (radii), and diameter. Use a compass to draw circles with a given diameter or radius. A compass is used to draw circles. Essential Vocabulary: Sample Space, Tree Diagram, Histogram, Frequency Chart, Table, Probability, Event, Simple Event, Relative Frequency, Theoretical Probability, Experimental Probability, Inclusive, Disjoint Event, Mutually Exclusive, Measure of Central Tendency, Mean, Median, Mode, Measure of Dispersion, Range, Coordinate Plane, Quadrants, X-Axis, Y-Axis, Coordinate Pair, Ordered Pair, Plot, Origin, Function, Transformation, Translation, Reflection, Rotation, Line of Symmetry, Circle, Arc, Chord, Compass. Interim 4 Page 22

24 Pacing Chart Unit 4: Data Analysis, Probability & Core Knowledge Time Frame Topic Suggested Activities/Assessments Resources & Text Alignment Sample Spaces : Sample Spaces 7-3: Bar Graphs and Histograms Week 30 Week 30 Week 31 Probability : Probability 11-2: Experimental Probability 11-4: Theoretical Probability 11-5: Making Predictions 11-6: Probability of Independent and Dependent Events Extend to Disjoint/Mutually Exclusive events. Week 32 Mathematics MCA Dispersion and Central Tendency 7-2: Mean, Median, Mode and Range Week 32 Core Knowledge Unit 4 Page 23

25 Week 32 Week 33 Corresponding Parts Core Knowledge 8-9: Congruent Figures 8-10: Translations, Reflections and Rotations 8-11: Symmetry Circles Core Knowledge 8-4: Properties of Circles Week 34 Week 35 Review Week 36 Interim 4 Unit 4 Page 24