# p. 1

aero mathematics standards grade 4 instructional focus in each grade level instructional time should focus on the following critical areas grade 3 1developing understanding of multiplication and division and strategies for multiplication and division within 100 2developing understanding of fractions especially unit fractions fractions with numerator 1 3developing understanding of the structure of rectangular arrays and of area 4describing and analyzing two-dimensional shapes grade 4 1 developing understanding and fluency with multi-digit multiplication and developing understanding of dividing to find quotients involving multi-digit dividends 2 developing an understanding of fraction equivalence addition and subtraction of fractions with like denominators and multiplication of fractions by whole numbers 3 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 grade 5 1 developing 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 extending division to 2-digit divisors integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths and developing fluency with whole number and decimal operations 3 developing understanding of volume problem solving 1.0 students will apply a wide variety of mathematical concepts processes and skills to solve a broad range of problems in various content areas and everyday situations analyze a problem to determine the questions to be answered identify necessary and extraneous information determine an efficient strategy verify interpret and evaluate the results with respect to the original problem try more than one strategy when the first strategy proves to be unproductive select and use strategies and procedures to find solutions to problems interpret and solve a variety of mathematical problems by paraphrasing check the reasonableness of a solution generalize and apply previous experiences and strategies to new problem solving situations use technology including calculators to develop mathematical concepts page 1 of 10 american school of madrid 2010-2011

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# p. 2

aero mathematics standards grade 4 reasoning and proof 2.0 students will apply mathematical reasoning skills to investigate evaluate justify and connect approaches and solutions to situations in mathematics and in other disciplines draw logical conclusions about mathematical problems follow a logical argument and judge its validity review and refine the assumptions and steps used to derive conclusions in mathematical arguments justify and explain the solutions to problems using manipulatives and physical models communication 3.0 students will accurately and clearly present and justify mathematical ideas in diverse formats use inquiry techniques to solve mathematical problems use a variety of methods to represent and communicate mathematical ideas through oral verbal and written formats identify and translate key words and phrases that imply mathematical operations use informal and mathematical language to explain why certain strategies or procedures were used to find a solution connections 4.0 students will develop the ability to use connections among mathematical ideas to build on one another when solving real-world problems and to interconnect ideas to produce an integrated coherent whole use mathematical ideas from one area of mathematics to explain an idea from another area of mathematics use physical models to explain the relationship between concepts and procedures apply mathematical thinking and modeling to solve problems that arise in other disciplines such as rhythm in music and motion in science identify explain and use mathematics in everyday life page 2 of 10 american school of madrid 2010-2011

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# p. 3

aero mathematics standards grade 4 numbers and operations standard benchmark 5.0 students will understand and apply numbers ways of representing numbers relationships among numbers and number systems 5.1 numbers and number sense students will understand and apply numbers ways of representing numbers relationships among numbers and number systems students will understand and demonstrate a sense of what numbers mean and how they are used grade 3 count by hundreds and thousands starting at any number from 1 to 9,999 name the number that is 10 more than or 100 more than any number from 0 through 9,999 and 10 less than or 100 less than any number from 100 through 10,000 identify read aloud and write numbers to 10,000 compare and order numbers from 0 to at least 10,000 using the words equal to greater than less than greatest or least when appropriate recording the results of comparisons with the symbols and when appropriate use symbols i.e and models to compare and order whole numbers through 9,999 grade 4 count by thousands and ten thousands starting at any number from 1 to 99,999 name the number that is 100 more than or 1000 more than any number from 0 through 99,999 and 100 less than or 1000 less than any number from 1000 through 100,000 read and write numbers to at least 100,000 grade 5 count by thousands ten thousands and hundred thousands starting at any number from 1 to 999,999 counting counting sets reading writing numbers ordering and comparing whole numbers read and write numbers to at least 1,000,000 compare and order numbers from 0 to at 1,000,000 using the words equal to greater than less than greatest or least when appropriate recording the results of comparisons with the symbols and when appropriate read write compare and order all whole numbers fractions mixed numbers and decimals using multiple strategies e.g symbols manipulatives number line and place value concepts identify and use place value positions of whole numbers and decimals to hundredths determine decimal equivalents or approximations of common fractions i.e ¼ ½ ¾ and 1 whole ordering and comparing numbers compare and order numbers from 0 to at 100,000 using the words equal to greater than less than greatest or least when appropriate recording the results of comparisons with the symbols and when appropriate use symbols i.e and models to compare and order whole numbers through 99,999 place value fractions and decimals model and identify place value of each digit utilizing standard and expanded form through 9,999 use concrete models and pictorial representations to demonstrate the meaning of fractions proper and improper as parts of a whole parts of a set and division by whole numbers through twelfths identify and interpret the place value for each digit in numbers through 99,999 use models to connect and compare equivalent fractions and decimals page 3 of 10 american school of madrid 2010-2011

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# p. 4

aero mathematics standards grade 4 equivalent identify name and use equivalent fractions fractions with denominators 2 4 and 8 compare and order fractions and decimals representing decimals compare and order fractions by using models benchmarks 0 ½ 1 or common numerators or denominators use numbers words pictures and physical objects to read write and represent decimal numbers to the tenths between 0 and 1 between 1 and 2 etc determine the equivalency among decimals fractions and percents e.g half 50 and ¼ 0.25 25 determine possible combinations of coins and bills to equal given amounts read write and use money notation equivalency decimals and fractions counting money money notation integers write a fraction equivalent to a given fraction using common multiples and simplify fractions using common factors compare and order positive fractions including positive mixed numbers and decimals on the number line in number sentences and in lists use numbers words pictures and physical objects to read write and represent decimal numbers to the hundredths between 0 and 1 between 1 and 2 etc determine the equivalency among decimals fractions and percents e.g 49/100 0.49 49 determine totals for monetary amounts in practical situations use money notation to add and subtract given monetary amounts illustrate the meaning of positive and negative integers using models such as the number line or colored chips and situations such as elevation or temperature make comparisons and describe quantitative relationships using ratios relate equivalent fractions and decimals with and without models including locations on a number line use models and drawings and find common denominators to compare fractions with unlike denominators round order and compare using symbols decimals to the tenths hundredths and thousandths place determine the equivalency between and among fractions decimals and percents in contextual situations determine totals differences and change due for monetary amounts in practical situations describe real-world situations using positive and negative numbers ratios percentages describe relationships between quantities using ratios benchmark computation whole numbers represent ratios and proportions and solve problems using models and pictures identify and represent ratios as comparisons of part-to-part and part-to-whole relationships and solve problems involving ratios 5.2 operations on numbers students will understand meanings of operations and how they relate to one another apply models of multiplication e.g equaladd and subtract whole numbers up to five add subtract multiply and divide with and sized groups arrays area models equal digit numbers without remainders using non-negative jumps on number lines and hundreds rational numbers charts and division e.g repeated subtraction partitioning and sharing to solve problems american school of madrid 2010-2011 page 4 of 10

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# p. 5

aero mathematics standards grade 4 meaning of apply the inverse relationship between operations multiplication and division e.g 5 x 6 30 30 ÷ 6 5 and the relationship between multiples and factors represent multiplication of up to four-digit by one digit numbers and describe how that representation connects to the related number sentence operations with fractions prime composite divisibility of numbers mental math properties of numbers use concrete models to add and subtract simple common fractions with the same denominator identify whole number factors and/or pairs of factors for a given whole number through 24 illustrate with manipulatives when a number is divisible by 2 3 5 or 10 add or subtract with numbers less than 100 using mental arithmetic use and explain the operations of multiplication and division including the properties e.g identity element of multiplication commutative property property of zero associative property inverse operations compare and order positive fractions including positive mixed numbers and decimals on the number line in number sentences and in lists identify factors of composite numbers less than 100 multiply four-digit numbers by two-digit numbers including whole numbers and decimals add subtract multiply and divide fractions and decimals to hundredths using concrete models or drawings and strategies based on place value model addition and subtraction of mixed numbers with and without regrouping and fractions with like and unlike denominators identify all whole number factors and pairs of factors for a given whole number through 144 model and distinguish between factor and multiple and prime and composite numbers recall from memory multiplying and dividing by 10 100 and 1,000 benchmark estimation rounding use divisibility concepts to classify numbers as prime or composite recall from memory multiplication facts for numbers from 1 to 10 apply models for multiplication e.g equalsized groups arrays area models equal intervals on the number line place value and properties of operations commutative associative and distributive find wholenumber quotients and remainders with up to four-digit dividends and one-digit divisors 5.3 numerical operations and estimation students will accurately calculate and use estimation techniques number relationships operation rules and algorithms they will determine the reasonableness of answers and the accuracy of solutions apply estimation skills rounding benchmarks estimate solutions to problems including use mental math and estimation strategies compatible numbers to solve and evaluate rounding benchmarks compatible numbers to predict the results of computations i.e reasonableness of an answer and evaluate the reasonableness of the whole numbers addition and subtraction of solution justify results fractions and to test the reasonableness of solutions round whole numbers through 10,000 to the round whole numbers to 1,000,000 to any nearest ten hundred and thousand and place value and round decimals to the nearest round fractions to the nearest whole number whole 10th or 100th place american school of madrid 2010-2011 page 5 of 10

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# p. 6

aero mathematics standards grade 4 problem generate and solve two step addition and solving subtraction problems and one step multiplication problems based on practical situations problem describe and show relationships between solving strategies and procedures for multiplying and dividing that involve addition and subtraction and explain strategies generate and solve addition subtraction multiplication and division problems using whole numbers in practical situations select and test algorithms used in computational situations that involve multiplication and division of whole numbers and explain strategies select sequence and use appropriate operations to solve multi-step word problems with whole numbers select and/or use an appropriate operations to show understanding of addition and subtraction of non-negative decimals and/or fractions measurement standard benchmark elapsed time 6.0 measurement student will use concepts and tools of measurement to describe and quantify the world 6.1 physical attributes students will demonstrate an understanding of units of measure and measurable attributes of objects use elapsed time in half-hour increments use elapsed time in quarter-hour increments beginning on the hour or half-hour to beginning on the quarter-hour to determine determine start end and elapsed time start end and elapsed time tell time to the nearest minute using digital use a.m and p.m appropriately in describing select and/or use an appropriate and analog clocks time operations to show understanding of addition and subtraction of non-negative decimals and/or fractions compare order and describe objects by estimate and convert units of measure for estimate and convert units of measure for various measurable attributes for area and length area and weight with the same weight and volume/capacity within the same volume/capacity measurement system metric estimate measurement system metric temperature in practical situations 6.2 systems of measurement students will identify and use units systems and processes of measurement select and use appropriate units of measure measure length area weight and measure volume and weight to a required and measure to a required degree of accuracy temperature to a required degree of accuracy degree of accuracy in metric systems to the nearest ½ unit in metric systems select and use the appropriate standard units use appropriate tools to determine estimate select and apply appropriate units for of measure abbreviations and tools for and compare units for measurement of measuring length mass volume and measuring length weight and capacity weight/mass area size of angle using the temperature in the metric system benchmark angles 45° 90° 180° 270° and 360 temperature length distance and volume in metric systems and time in real-life situations american school of madrid 2010-2011 time comparison estimation and conversion benchmark measuring units page 6 of 10

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# p. 7

aero mathematics standards grade 4 compare estimate and measure length using fractional order and parts to the nearest ½ unit in the metric estimation system metric systems formulas estimate and measure perimeter and area using links tiles grid paper geoboards and dot paper estimate and measure a given object to the nearest millimeter in the metric system convert capacity weight/mass and length within the metric system of measurement describe relationships of rectangular area to numerical multiplication estimate and measure length to nearest ½ millimeter in the metric system convert units within a given measurement system to include length weight/mass and volume develop compare and use formulas to estimate and calculate the perimeter and area of rectangles triangles and parallelograms patterns functions and algebra standard 7.0 patterns functions and algebra students will use various algebraic methods to analyze illustrate extend and create numerous representations words numbers tables and graphs of patterns functions and algebraic relations as modeled in practical situations to solve problems communicate reason and make connections within and beyond the field of mathematics 7.1 patterns relations and function students will recognize describe and develop patterns relations and functions create describe and extend growing and create describe and extend growing and identify a rule for a pattern involving repeating patterns with physical materials and repeating patterns with physical materials and addition subtraction or multiplication symbols including numbers symbols including numbers interpret and write a rule for a one operation function table ex adding 3 recognize and describe patterns using objects identify and describe patterns resulting from identify describe and represent patterns and numbers found in tables number charts operations involving even and odd numbers and relationships in the number system and charts such as even even even including triangular numbers and perfect squares 7.2 algebraic models students will represent and analyze mathematical situations and structures using algebraic symbols select and/or write number sentences select and/or write number sentences solve problems by finding the next term or equations to find the unknown in problemequations to find the unknown in problemmissing term in a pattern or function table solving contexts involving two-digit times one solving contexts involving two-digit by oneusing real world situations digit multiplication using appropriate labels digit division using appropriate labels determine the rule and explain how change in devise a rule for an input/output function one variable relates to the change in the table describing it in words and symbols second variable given an input/output model using two operations benchmark patterns identifying number patterns benchmark number sentences value of variables page 7 of 10 american school of madrid 2010-2011

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# p. 8

aero mathematics standards grade 4 number model explain and solve open number sentences sentences including addition subtraction and multiplication facts variables and create models for the concept of equality unknowns recognizing that the equal sign denotes equivalent terms such that 4+3=7 4+3=6+1 or 7=5+2 value of determine the value of missing quantities or variables variables within equations or number sentences and justify the process used number sentences expressions polynomials benchmark representing quantitative relationships benchmark change complete number sentences with the appropriate words and symbols model explain and solve open number sentences including addition subtraction multiplication and division select the solution to an equation from a given set of numbers find possible solutions to an inequality involving a variable using whole numbers as a replacement set determine the value of variables in equations and inequalities justifying the process solve equations with whole numbers using a variety of methods including inverse operations mental math and guess and check complete number sentences with the appropriate words and symbols including determine the value of variables in equations justify the process used to make the determination complete number sentences with the appropriate words and symbols ÷ x 7.3 algebraic representations students will develop and apply mathematical models to represent and understand quantitative relationships complete a chart or table to organize given create a chart or table to organize given draw conclusions and make predictions from information and to understand relationships information and to understand relationships models tables and line graphs and explain the results and explain the results 7.4 analysis of change students will analyze change in various contexts identify real situations and events that show identify describe and generalize relationships use a variety of methods to compare and change ex which day had the greatest change in which quantities change proportionally describe situations involving constant and/or in temperature varying rates of change geometry standard benchmark twodimensional figures 8.0 the student will develop an understanding of geometric concepts and relationships as the basis for geometric modeling and reasoning to solve problems involving one two and three-dimensional figures 8.1 geometric properties students will analyze characteristics and properties of 2 and 3 dimensional geometric shapes and develop mathematical arguments about geometric relationships describe compare analyze and classify twoanalyze and describe the similarities and analyze and describe the characteristics of dimensional shapes by sides and angles differences between and among two symmetry relative to classes of polygons dimensional geometric shapes figures and parallelograms triangles etc models using mathematical language american school of madrid 2010-2011 page 8 of 10

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# p. 9

aero mathematics standards grade 4 threeidentify describe and classify cube sphere analyze the relationship between threepredict and describe the effects of dimensional prism pyramid cone and cylinder in terms of dimensional geometric shapes in the form of combining dividing and changing shapes figures the number and shape of faces edges and cubes rectangular prisms and cylinders and into other shapes vertices their two-dimensional nets benchmark 8.2 transformation of shapes students will apply transformations and the use of symmetry to analyze mathematical situations congruency identify and create shapes that have lines of identify shapes that are congruent similar represent concepts of congruency similarity and similarity symmetry and/or symmetrical using a variety of and/or symmetry using a variety of methods methods including transformational motions including dilation and transformational motion transdescribe the transformational motions of compare figures to determine congruence predict the results of multiple formations geometric figures translation/slide using geometric transformations motions transformations on a geometric shape when reflection/flip and rotation/turn such as reflections flips rotations turns combinations of translation reflection and and translations slides rotation are used representing analyze the results of combining and represent the two-dimensional shapes predict and describe the effects of figures subdividing circles triangles quadrilaterals trapezoids rhombuses and parallelograms combining dividing and changing shapes pentagons hexagons and octagons and the three-dimensional shapes cubes into other shapesdescribe the characteristics rectangular prisms and cylinders of a right triangle triangles identify classify and draw triangles based on describe the characteristics of a right triangle their properties triangles determine the degrees of the interior angles of triangles lines angles analyze the results of combining and identify draw label and describe points line identify draw label and describe planes and their subdividing circles triangles quadrilaterals segments rays and angles parallel lines intersecting lines and properties pentagons hexagons and octagons perpendicular lines angles identify draw and classify angles including identify the attributes of an angle and draw straight right obtuse and acute angles using protractors benchmark 8.3 coordinate geometry students will specify locations and describe spatial relationships using coordinate geometry and other representational systems descriptions use coordinates to give or follow directions locate and label ordered pairs in the first graph coordinates representing geometric spatial from one point to another on a map or grid quadrant of the coordinate plane shapes in the first quadrant relationships benchmark 8.4 visualization and geometric models students will use visualization spatial reasoning and geometric modeling mathematical use the quantifiers all some and none to use the connectors and or and not to represent relationships using venn diagrams reasoning describe the characteristics of a set describe the members of a set logic page 9 of 10 american school of madrid 2010-2011

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# p. 10

aero mathematics standards grade 4 data analysis and probability standard 9.0 students will develop an understanding of data analysis and probability by solving problems in which there is a need to collect appropriately represent and interpret data to make inferences or predictions and to present convincing arguments and to model mathematical situations to determine the probability 9.1 data representation students will formulate questions that can be addressed with data and collect organize and display relevant data to answer them use graphical representations including use a variety of graphical representations use a variety of graphical representations number lines frequency tables and including frequency tables and plots to including line graphs stem-and-leaf plots pictographs to represent data organize and represent data histograms and box-and-whisker plots to organize and represent data pose questions that can be used to guide data pose questions that can be used to guide data pose questions that can be used to guide the collection organization and representation collection organization and representation collection of categorical and numerical data 9.2 data analysis students will select and use appropriate statistical methods to analyze data compare data and interpret quantities compare data and interpret quantities compare data and interpret quantities represented on tables and different types of represented on tables and graphs including represented on tables and graphs including graphs line plots pictographs and bar line graphs bar graphs frequency tables and line graphs stem-and-leaf plots histograms graphs make predictions and solve problems stem-and-leaf plots to make predictions and and box-and-whisker plots to make based on the information solve predictions and solve problems based on the information determine or calculate the mode collect and analyze data using mean median mean/average and range for a data set and mode to determine the best statistical measure 9.3 inferences and predictions students will develop and evaluate inferences and predictions that are based on data record results of activities involving chance make predictions and draw conclusions from make predictions and draw conclusions e.g coin flips dice rolls and make simple experiments based on data collected from a sample group reasonable predictions based upon data 9.4 probability students will understand and apply basic concepts of probability develop and conduct grade-appropriate design and conduct a simple probability construct a sample space and make a experiments using concrete objects e.g experiment using concrete objects examine hypothesis as to the probability of a real life counters number cubes spinners to and list all possible combinations using a tree situation over time test the prediction with determine the likeliness of events and list all diagram represent the outcomes as a ratio experimentation and present conclusions outcomes and present the results benchmark organizing and representing data formulating questions benchmark interpreting data central tendency benchmark inferences and predictions benchmark experimental probability page 10 of 10 american school of madrid 2010-2011

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