Elementary School Mathematics Grades

Transcription

1 Key: = richly assessed at this grade level Critical Curriculum Areas for Grade 2 In Grade 2, instructional time should focus on four critical areas: (1) The extension of understanding of base-ten notation; (2) Building fluency with addition and subtraction; (3) The use of standard units of measure; and (4) The description and analysis of shapes. 1. Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand multidigit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones). 2. Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds. 3. Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with the understanding that linear measure involves an iteration of units. They recognize that the smaller the unit, the more iterations they need to cover a given length. 4. Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades Page 1

2 Critical Curriculum Areas for Grade 3 In Grade 3, instructional time should focus on four critical areas: (1) The developing understanding of multiplication and division, and strategies for multiplication and division within 100; (2) The developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) The developing understanding of the structure of rectangular arrays and of area; and (4) The description and analysis of two-dimensional shapes. 1. Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division. 2. Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators. 3. Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle. 4. Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole. Page 2

3 Critical Curriculum Areas for Grade 4 In Grade 4, instructional time should focus on three critical areas: (1) The development of understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) The development of an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; and (3) The understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. 1. Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context. 2. Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. 3. Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving symmetry. Page 3

4 Critical Curriculum Areas for Grade 5 In Grade 5, instructional time should focus on three critical areas: (1) The development of fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) The extension of division to 2-digit divisors, the integration of decimal fractions into the place value system, a developing understanding of operations with decimals to hundredths, a developing fluency with whole number and decimal operations, and (3) A developing understanding of volume. 1. Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.) 2. Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multidigit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately. 3. Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems. Page 4

5 Mathematics Standards and Benchmarks Standard 1: Number and Algebra: Learners understand that numbers and algebra represent and quantify our world and can be used to solve problems 1. Counting and Cardinality: Learners understand that numbers are a naming system. (CC) 2. Numbers Base Ten: Learners understand that the base ten place value system is used to represent numbers and number relationships. (NBT) 3. Operational Thinking and Algebra: Learners understand that numbers and algebra represent and quantify our world and can be used to solve problems. (OA) 4. Number and Operations Fractions: Learners understand that fractions and decimals are ways of representing whole-part relationships. (NF) Standard 2: Measurement and Data: Learners understand that objects and events have attributes that can be measured and compared using appropriate tools. Data analysis can help us interpret and make predictions about our world (MD) Standard 3: Geometry: Learners understand that geometry models and quantifies structures in our world and can be used to solve problems. (G) Standard 1: Number & Algebra 1.2 Numbers Base Ten: Learners understand that the base ten place value system is used to represent numbers and number relationships a. Understand the place value system b. Use place value understanding and properties of operations to add and subtract c. Generalize place value understanding for multi-digit numbers d. Use place value understanding and operations to perform multi-digit arithmetic e. Perform operations with multi---digit whole numbers and with decimals to hundredths 1.3 Operational Thinking and Algebra (OA): Learners understand that numbers and algebra represent and quantify our world and can be used to solve problems a. Represent and solve problems involving addition and subtraction b. Add and subtract within 20 c. Work with equal groups of objects to gain foundation for multiplication d. Represent and solve problems involving multiplication and division e. Understand properties of multiplication and the relationship between multiplication and division f. Multiply and divide within 100 g. Solve problems involving the four operations and identify and explain patterns in arithmetic h. Use the four operations with whole numbers to solve problems Page 5

6 i. Gain familiarity with factors and multiples j. Write and interpret numerical expressions k. Analyze patterns and relationships 1.4 Number and Operations Fractions (NF): Learners understand that fractions and decimals are ways of representing whole part relationships a. Develop understanding of fractions as numbers b. Extend understanding of fraction equivalence and ordering c. Build fractions from unit fractions by applying and extending previous understandings of operations and whole numbers d. Understand decimal notation for fractions, and compare decimal fractions e. Use equivalent fractions as a strategy to add and subtract fractions f. Apply and extend previous understandings of multiplication and division to multiply and divide fractions Standard 2: Measurement and Data (MP): Learners understand that objects and events have attributes that can be measured and compared using appropriate tools a. Measure and estimate lengths in standard units b. Relate addition and subtraction to length c. Work with time and money (S.A. Rand) d. Solve problems involving measurement and estimation of intervals of time, liquid volumes and masses of objects e. Understand concepts of area and relate area to multiplication and addition f. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures g. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit h. Convert like measurement units within a given measurement system i. Represent and interpret data Standard 3: Geometry (G): Learners understand that geometry models and quantifies structures in our world and can be used to solve problems a. Reason with shapes and their attributes b. Draw and identify lines and angles, and classify shapes by properties of their lines and angles c. Graph points on the coordinate place to solve real---world and mathematical problems d. Classify two---dimensional figures into categories based on their properties Grade 1 Exemplars Grade 2 Exemplars Grade 3 Exemplars Grade 4 Exemplars Grade 5 Exemplars Page 6

7 Mathematically proficient students... Mathematical Practices for Grade 2 1. Make Sense and Persevere in Solving Problems. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics 5. Use appropriate tools strategically. 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning. Explain to themselves the meaning of a problem and look for ways to solve it May use concrete objects or pictures to help them conceptualize and solve problems Make conjectures about the solution and plan out a problem-solving approach Recognize that a number represents a specific quantity Connect the quantity to written symbols Create a representation of a problem while attending to the meanings of the quantities Begin to know and use different properties of operations and relates addition and subtraction to length Construct arguments using concrete referents, such as objects, pictures, drawings and actions Explain their own thinking and listens to others explanations Decide if the explanations make sense and asks appropriate questions Experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Connect the different representations and explains the connections Is able to use all representations as needed Consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be better suited. For example, second grade students may decide to solve a problem by drawing a picture rather than writing an equation Use clear and precise language in their discussions with others Explain their reasoning Look for patterns. For example, they adopt mental math strategies based on patterns (making ten, fact families, doubles) Notice repetitive actions in counting and computation, etc. Look for shortcuts when adding and subtracting, such as rounding up and then adjusting the answer to compensate for the rounding Continually check their work by asking themselves, Does this make sense? Page 7

8 Mathematically proficient students Make Sense and Persevere in Solving Problems. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics 5. Use appropriate tools strategically. 6. Attend to precision 7. Look for and make use of structure Mathematical Practices for Grade 3 Explain to themselves the meaning of a problem and look for ways to solve it May use concrete objects or pictures to help them conceptualize and solve problems May check their thinking by asking themselves, Does this make sense? Listen to strategies of others and will try different approaches Will use another method to check their answers Recognize that a number represents a specific quantity Connect the quantity to written symbols and Create logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities May construct arguments using concrete referents, such as objects, pictures, and drawings Refine mathematical communication skills as they participate in mathematical discussions involving questions like How did you get that? and Why is that true? Explain their thinking to others and respond to others thinking Experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating equations, etc. Connect the different representations and explain the connections Evaluate their results in the context of the situation and reflect on whether the results make sense Consider the available tools (including estimation) when solving a mathematical problem and decise when certain tools might be helpful. For example, they may use graph paper to find all the possible rectangles that have a given perimeter Compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles Use clear and precise language in their discussions with others and in their own reasoning Are careful about specifying units of measure and state the meaning of the symbols they choose. For example, when figuring out the area of a rectangle they record their answers in square units Look closely to discover a pattern or structure. For example, students use properties of operations as strategies to multiply and divide (commutative and distributive properties) Page 8

9 8. Look for and express regularity in repeated reasoning. Notice repetitive actions in computation and look for more shortcut methods. Students may use the distributive property as a strategy for using products they know to solve products that they don t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2, then multiply 5 x 8, then 2 x 8 to arrive at or 56 Continually evaluate their work by asking themselves, Does this make sense? Mathematically proficient students Make Sense and Persevere in Solving Problems. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics Mathematical Practices for Grade 4 Know that doing mathematics involves solving problems an discussing how they solved them Explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or picture to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? Listen to the strategies of others and will try different approaches, They often use another method to check their answers Should recognize that a number represents a specific quantity Connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities Extend this understanding from whole numbers to their work with fractions and decimals. They write simple expressions, record calculations with numbers, and represent or round numbers using place value concepts May construct arguments using concrete referents, such as objects, pictures, and drawings Explain their thinking and make connections between models and equations Refine their mathematical communication skills as they participate in mathematical discussions involving questions like How did you get that? and Why is that true? Explain their thinking to others and respond to others thinking Experiment with representing problem situations in multiple ways including numbers, words (mathematical language) drawing pictures, using objects, making a chart, list or graph, creating equations, etc. Need opportunities to connect the different representations and explain the connections Page 9

10 5. Use appropriate tools strategically. 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning. Should be able to use all of these representations as needed. Should evaluate their results in the context of the situation and reflect on whether the results make sense Consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use a graph paper or a number line to represent and compare decimals and protractors to measure angles Use other measurement tools to understand the relative size of units within a system and express measurements given in larger units in terms of smaller units As mathematical communication skills develop, they try to use clear and precise language in their discussions with others and in their own reasoning Are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, they use appropriate labels when creating a line plot Look closely to discover a pattern or structure. For instance, students use properties of operations to explain calculations (partial products model) Relate representations of counting problems such as tree diagrams and arrays to the multiplication principal of counting Generate number or shape patterns that follow a given rule Should notice repetitive actions in computation to make generalizations Use models to explain calculations and understand how algorithms. For example, students use visual fraction models to write equivalent fractions Page 10

11 Mathematically proficient students Make sense and persevere in solving problems. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics Mathematical Practices for Grade 5 Solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers Solve problems related to volume and measurement conversions. They seek the meaning of a problem and look for efficient ways to represent and solve it May check their thinking by asking themselves, What is the most efficient way to solve a problem?, Does this make sense? and Can I solve this problem in a different way? Should recognize that a number represents a specific quantity Connect quantities to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities Extend this understanding from whole numbers to their work with fractions and decimals Write simple expressions that record calculations with numbers and represent or round numbers using place value concepts May construct arguments using concrete referents, such as objects, pictures, and drawings Explain calculations based upon models and properties of operations and rules that generate patterns Demonstrate and explain the relationship between volume and multiplication Refine their mathematical communication skills as they participate in mathematical discussions involving questions like How did you get that? and Why is that true? They explain their thinking to others and respond to others thinking Experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list or graph, creating equations, etc. Need opportunities to connect the different representations and explain the connections Should be able to use all of these representations as needed Should evaluate their results in the context of the situation and whether the results make sense Also evaluate the utility of models to determine which models are efficient to solve problems Page 11

12 5. Use appropriate tools strategically. 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning. Consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions Use graph paper to accurately create graphs and solve problems or make predictions from real world data Continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning Use appropriate terminology when referring to expressions, fractions, geometric figures, and coordinate grids Are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the volume of a rectangular prism they record their answers in cubic units Look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals Examine numerical patterns and relate them to a rule or a graphical representation Use repeated reasoning to understand algorithms and make generalizations about patterns Connect place value and their prior work with operations to understand algorithms to fluently multiply multi-digit numbers and perform all operations with decimals to hundredths Explore operations with fractions with visual models and begin to formulate generalizations Page 12

13 Glossary: Associative Appendices A method of combining two numbers or algebraic expressions is associative if the result of the combination of three objects does not depend on the way in which the objects are grouped. For example, addition of numbers is associative and the corresponding associative law is: for all numbers Multiplication is also associative: for all numbers but subtraction and division are not, because, for example, Alternate In each diagram below, the two marked angles are called alternate angles (since they are on alternate sides of the transversal). If the lines AB and CD are parallel, then each pair of alternate angles are equal. and Page 13

14 Angle An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The size of an angle Imagine that the ray OB is rotated about the point O until it lies along OA. The amount of turning is called the size of the angle AOB. A revolution is the amount of turning required to rotate a ray about its endpoint until it falls back onto itself. The size of 1 revolution is 360 o. A straight angle is the angle formed by taking a ray and its opposite ray. A straight angle is half of a revolution, and so has size equal to 180 o. Right angle Let AOB be a line, and let OX be a ray making equal angles with the ray OA and the ray OB. Then the equal angles AOX and BOX are called right angles. A right angle is half of a straight angle, and so is equal to 90 o. Page 14

15 Classification of angles Angles are classified according to their size. We say that An angle with size α is acute if 0 o < α < 90 o, An angle with size α is obtuse if 90 0 < α < 180 o, An angle with size α is reflex if < α < 360 o Adjacent angles Two angles at a point are called adjacent if they share a common ray and a common vertex. Hence, in the diagram, AOC and BOC are adjacent, and AOB and AOC are adjacent. Two angles that add to 90 o are called complementary. For example, 23 o and 67 o are complementary angles. In each diagram the two marked angles are called corresponding angles. Page 15

16 If the lines are parallel, then each pair of corresponding angles are equal. Conversely, if a pair of corresponding angles are equal, then the lines are parallel. Two angles that add to 180 o are called supplementary angles. For example, 45 o and 135 o are supplementary angles. Cartesian coordinate system Two intersecting number lines are taken intersecting at right angles at their origins to form the axes of the coordinate system. The plane is divided into four quadrants by these perpendicular axes called the x-axis (horizontal line) and the y-axis (vertical line). The position of any point in the plane can be represented by an ordered pair of numbers (x, y). These ordered are called the coordinates of the point. This is called the Cartesian coordinate system. The plane is called the Cartesian plane. The point with coordinates (4, 2) has been plotted on the Cartesian plane shown. The coordinates of the origin are (0, 0). Page 16

17 Congruent triangles The four standard congruence tests for triangles. Two triangles are congruent if: SSS: the three sides of one triangle are respectively equal to the three sides of the other triangle, or SAS: two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other triangle, or AAS: two angles and one side of one triangle are respectively equal to two angles and the matching side of the other triangle, or RHS: the hypotenuse and one side of one right angled triangle are respectively equal to the hypotenuse and one side of the other right angled triangle. Congruence Two plane figures are called congruent if one can be moved by a sequence of translations, rotations and reflections so that it fits exactly on top of the other figure. Two figures are congruent when we can match every part of one figure with the corresponding part of the other figure. For example, the two figures below are congruent. Matching intervals have the same length, and matching angles have the same size. Page 17

18 Commutative A method of combining two numbers or algebraic expressions is commutative if the result of the combination does not depend on the order in which the objects are given. For example, addition of numbers is commutative, and the corresponding commutative law is: for all numbers Multiplication is also commutative: for all numbers but subtraction and division are not, because, for example, and Complementary events Distributive Events A and B are complementary events, if A and B are mutually exclusive and Pr(A) + Pr(B) = 1. Multiplication of numbers is distributive over addition because the product of one number with the sum of two others equals the sum of the products of the first number with each of the others. This means that we can multiply two numbers by expressing one (or both) as a sum and then multiplying each part of the sum by the other number (or each part of its sum.) For example, This distributive law is expressed algebraically as follows: Data Data is a general term for a set of observations and measurements collected during any type of systematic investigation. Primary data is data collected by the user. Secondary data is data collected by others. Sources of secondary data include, web-based data sets, the media, books, scientific papers, etc. Univariate data is data relating to a single variable, for example, hair color or the number of errors in a test. Decimal Page 18

19 A decimal is a numeral in the decimal number system. For example, the decimal expansion of is. The integer part is and the fractional part is A decimal is terminating if the fractional part has only finitely many decimal digits. It is non-terminating if it has infinitely digits. For example, is a terminating decimal, whereas, where the pattern 16 repeats indefinitely, is non-terminating. Non-terminating decimals may be recurring, that is, contain a pattern of digits that repeats indefinitely after a certain number of places. For example, is a recurring decimal, whereas where the number of 0 s between the 1 s increases indefinitely, is not recurring. It is common practice to indicate the repeating part of a recurring decimal by using dots or lines as superscripts. For example, could be written as or The decimal number system is the base 10, place-value system most commonly used for representing real numbers. In this system positive numbers are expressed as sequences of Arabic numerals 0 to 9, in which each successive digit to the left or right of the decimal point indicates a multiple of successive powers (respectively positive or negative) of 10. For example, the number represented by the decimal Distributive is the sum Multiplication of numbers is distributive over addition because the product of one number with the sum of two others equals the sum of the products of the first number with each of the others. This means that we can multiply two numbers by expressing one (or both) as a sum and then multiplying each part of the sum by the other number (or each part of its sum.) For example, This distributive law is expressed algebraically as follows: Equally Likely outcomes Page 19

20 Equally likely outcomes occur with the same probability. For example, in tossing a fair coin, the outcome head and the outcome tail are equally likely. In this situation, Pr(head) = Pr(tail) = 0.5 Factorize To factorize a number or algebraic expression is to express it as a product. For example, is factorised when expressed as a product:, and is factorized when written as a product: Fraction The fraction (written alternatively as these parts. ), where is a non-negative integer and is a positive integer, was historically obtained by dividing a unit length into equal parts and taking of For example, refers to 3 of 5 equal parts of the whole, taken together. In the fraction the number is the numerator and the number is the denominator. It is a proper fraction if Frequencies and an improper fraction otherwise. Frequency, or observed frequency, is the number of times that a particular value occurs in a data set. For grouped data, it is the number of observations that lie in that group or class interval. An expected frequency is the number of times that a particular event is expected to occur when a chance experiment is repeated a number of times. For example, If the experiment is repeated n times, and on each of those times the probability that the event occurs is p, then the expected frequency of the event is np. For example, suppose that a fair coin is tossed 5 times and the number of heads showing recorded. Then the expected frequency of heads is 5/2. Page 20

21 This example shows that the expected frequency is not necessarily an observed frequency, which in this case is one of the numbers 0,1,2,3,4 or 5. A frequency table lists the frequency (number of occurrences) of observations in different ranges, called class intervals. The frequency distribution of the heights (in cm) of a sample of 42 people is displayed in the frequency table below Height (cm) Class interval Frequency 155-< < < < < < < <190 5 A frequency distribution is the division of a set of observations into a number of classes, together with a listing of the number of observations (the frequency) in that class. Frequency distributions can be displayed in tabular or graphical form. Frequency, or observed frequency, is the number of times that a particular value occurs in a data set. Page 21

22 For grouped data, it is the number of observations that lie in that group or class interval. Relative frequency is given by the ratio data set. Index, where f is the frequency of occurrence of a particular data value or group of data values in a data set and n is the number of data values in the Index is synonymous with exponent. The exponent or index of a number or algebraic expression is the power to which the latter is be raised. The exponent is written as a superscript. Positive integral exponents indicate the number of times a term is to be multiplied by itself. For example, Mean The arithmetic mean of a list of numbers is the sum of the data values divided by the number of numbers in the list. In everyday language, the arithmetic mean is commonly called the average. For example, for the following list of five numbers { 2, 3, 3, 6, 8 } the mean equals Median The median is the value in a set of ordered data that divides the data into two parts. It is frequently called the middle value. Where the number of observations is odd, the median is the middle value. For example, for the following ordered data set with an odd number of observations, the median value is five Where the number of observations is even, the median is calculated as the mean of the two central values. For example, in the following ordered data set, the two central values are 5 and 6, and median value is the mean of these two values, 5.5 Page 22

23 The median provides a measure of location of a data set that is suitable for both symmetric and skewed distributions and is also relatively insensitive to outliers. Mode The mode is the most frequently occurring value in a set of data. There can be more than one mode. When there are two modes, the data set is said to be bimodal. The mode is sometimes used as a measure of location. Number line A number line gives a pictorial representation of real numbers. Order of operations A convention for simplifying expressions that stipulates that multiplication and division are performed before addition and subtraction and in order from left to right. For example, in , the division is performed first and the expression becomes = 9. If the convention is ignored and the operations are performed in order, the incorrect result, 6.5 is obtained. Percentage A percentage is a fraction whose denominator is 100. For example, percent (written as ) is the percentage whose value is Similarly, 40 as a percentage of 250 is Point A point marks a position, but has no size. Sample A sample is part of a population. It is a subset of the population, often randomly selected for the purpose of estimating the value of a characteristic of the population as a whole. For instance, a randomly selected group of eight-year old children (the sample) might be selected to estimate the incidence of tooth decay in eight-year old children in South Africa (the population). Page 23

24 Square A square is a quadrilateral that is both a rectangle and a rhombus. A square thus has all the properties of a rectangle, and all the properties of a rhombus. Sum A sum is the result of adding together two of more numbers or algebraic expressions. Transversal A transversal is a line that meets two or more other lines in a plane. Variable Numerical variables are variables whose values are numbers, and for which arithmetic processes such as adding and subtracting, or calculating an average, make sense. A discrete numerical variable is a numerical variable, each of whose possible values is separated from the next by a definite gap. The most common numerical variables have the counting numbers 0,1,2,3, as possible values. Others are prices, measured in dollars and cents. Examples include the number of children in a family or the number of days in a month. Volume The volume of a solid region is a measure of the size of a region. For a rectangular prism, Volume = Length Width Height Page 24