Elementary School Mathematics Grades Pre-K

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1 Key: = richly assessed at this grade level Critical Curriculum Areas for Pre-Kindergarten Pre-Kindergarten, the focus should be on bridging children s informal understanding of mathematics with more formal, school-based mathematics. That is, you design the learning environment by purposely placing mathematics materials in interest areas for child-initiated explorations and by intentionally introducing activities with a mathematical focus. You observe and listen as children interact with materials and their peers and then you use mathematical vocabulary to describe their actions and thinking. You ask questions as children investigate. You play logic games, create mathematical problem-solving stories, and include numerical and algebraic activities as part of the daily routine. (The Creative Curriculum for Preschool. Volume 4: Mathematics P. 739) Critical Curriculum Areas for Kindergarten In Kindergarten, instructional time should focus on two critical areas: (1) The representation, relation, and operation of whole numbers, initially with sets of objects; (2) The description and understanding of shapes and space. More learning time in Kindergarten should be devoted to number than to other topics. 1. Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as = 7 and 7 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away. 2. Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic twodimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as threedimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes. Page 1

2 Critical Curriculum Areas for Grade 1 In Grade 1, instructional time should focus on four critical areas: (1) The developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) The developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) The developing understanding of linear measurement and measuring lengths as iterating length units; and (4) Reasoning about attributes of, and the composing and decomposing of geometric shapes. 1. Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., making tens ) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction. 2. Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10. They compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative sizes. They think of whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed of a ten and some ones). Through activities that build number sense, they understand the order of the counting numbers and their relative magnitudes. 3. Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the mental activity of building up the length of an object with equal-sized units) and the transitivity principle for indirect measurement. 4. Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine shapes, they recognize them from different perspectives and orientations, describe their geometric attributes, and determine how they are alike and different, to develop the background for measurement and for initial understandings of properties such as congruence and symmetry. Page 2

3 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 1.1 Counting and Cardinality (CC): Learners understand that numbers are a naming system. PK K 1 a. Know number names and the count sequence. to 10 to 20 b. Count to tell the number of objects to 10 to 20 c. Compare numbers to 10 to Numbers Base Ten (NBT): Learners understand that the base ten place value system is used to represent numbers and number relationships. a. Work with numbers to gain foundations for place value. b. Extend the counting sequence c. Understand place value d. Use place value understanding and properties of operations to add and subtract. 1.3 Operational Thinking and Algebra (OA): Learners understand that numbers and algebra represent and quantify our world and can be used to solve problem a. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. b. Represent and solve problems involving addition and subtraction. c. Understand and apply properties of operations and the relationships between addition and subtraction. d. Add and subtract within 20. e. Work with addition and subtraction equations. Page 3

4 Standard 2: Measurement and Data (MP) : Learners understand that objects and events have attributes that can be measured and compared using PK K 1 appropriate tools. a. Describe and compare measureable attributes b. Measure lengths indirectly and by iterating length units. c. Tell and write time and money (S.A Rand). d. Sort objects and counts the number of objects in each category. e. 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. PK K 1 a. Identify, and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). b. Analyse, compare, create, and compose shapes. c. Reason with shapes and their attributes. Pre-K Exemplars K Exemplars Grade 1 Exemplars Page 4

5 Mathematical Practices for Pre- Kindergarten and Kindergarten Mathematically proficient students... - are fluent mathematicians able to develop and use these eight mathematical practices in alignment with the AISJ Standards and benchmarks 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. Use both verbal and nonverbal means, these students begin to explain to themselves and others the meaning of a problem, look for ways to solve it, and determine if their thinking makes sense or if another strategy is needed. Begin to use numerals to represent specific amount (quantity) Begin to draw pictures, manipulate objects, use diagrams or charts, etc. to express quantitative ideas such as a joining situation Begin to understand how symbols (+, -, =) are used to represent quantitative ideas in a written format. Begin to clearly express, explain, organize and consolidate their math thinking using both verbal and written representations. Begin to learn how to express opinions, become skillful at listening to others, describe their reasoning and respond to others thinking and reasoning. Begin to develop the ability to reason and analyze situations as they consider questions such as, Are you sure...?, Do you think that would happen all the time...?, and I wonder why...? Begin to experiment with representing real-life problem situations in multiple ways such as with numbers, words (mathematical language), drawings, objects, acting out, charts, lists, and number sentences. Begin to explore various tools and use them to investigate mathematical concepts. Experiment and use both concrete materials (e.g. 3- dimensional solids, connecting cubes, ten frames, number balances) and technological materials (e.g., virtual manipulatives, calculators, interactive websites) to explore mathematical concepts. Begin to express their ideas and reasoning using words. Begin to describe their actions and strategies more clearly, understand and use grade level appropriate vocabulary accurately, and begin to give precise explanations and reasoning regarding their process of finding solutions. Begin to look for patterns and structures in the number system and other areas of mathematics. Begin to notice repetitive actions in geometry, counting, comparing, etc. Page 5

6 Mathematically proficient students... Mathematical Practices for Grade 1 - are fluent mathematicians able to develop and use these eight mathematical practices in alignment with the AISJ Standards and benchmarks 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. Are willing to try other approaches 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. Construct arguments using concrete referents, such as objects, pictures, drawings, and actions. Explain their own thinking and listen to others explanations. Decide if the explanations make sense and ask 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 explain the connections. Decide when certain tools might be helpful when solving a mathematical problem. For example, first graders decide it might be best to use colored chips to model an addition problem. Use clear and precise language in their discussions with others and when they explain their own reasoning. Begin to discern a pattern or structure. For example, if students recognize = 15, then they also know = 15. (Commutative property of addition.) To add , the first two numbers can be added to make a ten, so = = 14. Notice repetitive actions in counting and computation, etc. Continually check their work by asking themselves, Does this make sense? Page 6

7 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 7

8 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 8

9 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 9

10 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 10

11 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 11

12 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 12

13 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 13

14 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 14

15 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 15

16 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 16

17 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 17

18 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 18