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Concepts of Elementary Mathematics Enduring Understandings & Essential Questions Numbers and Operations Algebra Geometry Measurement Data Analysis and Probability NUMBER AND OPERATIONS How is math relevant to me? What do numbers convey? How can numbers be expressed, ordered, and compared? What are properties of whole numbers? What are different ways to count? What are efficient ways to count? NUMBER AND OPERATIONS In what ways can items be grouped? How do units within a system relate to each other? How does the position of a digit in a number affect its value? In what ways can numbers be composed and decomposed? How are place value patterns repeated in numbers? How can place value properties aid computation? What are strategies to make a reasonable estimate? NUMBER AND OPERATIONS What are different models of and models for addition and subtraction? What questions can be answered using addition and/or subtraction? What are efficient methods for finding sums and differences? What are different models of and models for multiplication and division? What questions can be answered using multiplication and division? What are efficient methods for finding products and quotients? NUMBER AND OPERATIONS How do the four operations related to one another? How are parenthesis used in numeric expressions? What computation tools are best suited to which circumstances? Fractions, decimals, and percents How can fractions be modeled, compared, and ordered? How are common fractions, decimals and percents alike and different? How is computation with rational numbers similar and different to whole number computation? ALGEBRA Where are patterns in nature, architecture, music, words, and numbers? What is the repeating and/or increasing unit in the pattern? What strategies can be used to continue a sequence? - express pattern as a rule (ex: frame & arrows, what¶s m y rule?) How does finding patterns help in counting and/or computation? How is an equation like a balance scale? How can relationships be expressed symbolically? Why are variables used? What strategies can be used to solve for unknowns? GEOMETRY How are geometric properties used to solve problems in everyday life? How can plane and solid shapes be described? How are geometric figures constructed? What strategies can be used to verify symmetry and congruency? How will a shape look when rotated, reflected, and/or translated? How is the location of a point on a grid described? How is this useful? What types of problems are solved with measurement? What are tools of measurement and how are they used? What is the purpose of standard units of measurement? How do units within a system relate to each other? When is an estimate more appropriate than an actual measurement? What strategies help estimate measurements? How can information be gathered, recorded, and organized? How does the type of data influence the choice of display? What aspects of a graph help people understand and interpret the data easily? What kinds of questions can and can not be answered from a graph? The expected outcome of an event How is the probability of an event determined and described? --------------------------------------- 1 Concepts of Elementary Mathematics Enduring Understandings & Essential Questions Numbers and Operations Algebra Geometry Measurement Data Analysis and Probability Above all, students should feel enchanted by the beauty of mathematics, empowered by their increasing abilities, and captivated by new concepts. --------------------------------------- 2 NUMBER AND OPERATIONS Enduring Understandings Essential Questions Numbers can represent quantity, How is math relevant to me? position, location, & relationships. - relate and apply concepts & skills to daily life What do numbers convey? - identify amount (cardinal) - name position (ordinal) - indicate location (nominal) How can numbers be expressed, ordered, and compared? - read and write positive numbers, decimals, and fractions - sequence numbers: zero through one billion - compare quantities as more, less, equal (>, <, =) Numbers can be classified What are properties of whole numbers? by attributes. - understand odd and even - identify prime and composite - create figurate numbers (ex: square, triangular) Counting finds out the answer What are different ways to count? to how manyŽ in objects/sets. - count all - count on - count back - skip count - count groups What are efficient ways to count? - count up (or back) from largest number - count sets of items - count to/using landmark numbers (ex: 10, 25, 50, 100) 2 --------------------------------------- 3 NUMBER AND OPERATIONS Enduring Understandings Essential Questions Grouping (unitizing) is a way to In what ways can items be grouped? count, measure, and estimate. - recognize common clusters (ex: tallies, dozens) - create groups for efficiency (ex: 10s, 100s, etc.) - make exchanges for unit of higher value (ex: 10 pennies for 1 dime) How do units within a system relate to each other? - understand relationships between and among: time: minute, hour, day, week, month, year, decade, century money: penny, nickel, dime, quarter, dollar length: inch, foot, yard, mile; millimeter, centimeter, decimeter, meter, kilometer volume: fluid ounce, cup, pint, quart, gallon weight: ounce, pound, ton; gram, kilo, metric ton Place value is based on How does the position of a digit in a number affect its value? groups of ten. - identify place value positions - add & subtract multiples of 10, 100, 1000 - multiply & divide by powers of 10 In what ways can numbers be composed and decomposed? - use expanded notation (ex: 349 = 300+40+9) - rename numbers (ex: 83 = 70+13, 8 tens + 3 ones = 7 tens + 13 ones) How are place value patterns repeated in numbers? - understand pattern of 10s as structure of our number system - identify patterns in digits and periods when numbers are sequenced How can place value properties aid computation? - apply single digit facts to multi-digit computation (ex: use 5x5 for 50x50) - regroup numbers for efficiency Estimation is a way to get What are strategies to make a reasonable estimate? an approximate answer. - determine possible range for answer - use anchoring, partitioning, and visualization techniques - round quantities to find approximate sum, difference, product, quotient 3 --------------------------------------- 4 NUMBER AND OPERATIONS Enduring Understandings Essential Questions Computation involves taking apart What are different models of and models for addition and subtraction? and combining numbers using a - join and/or separate items variety of approaches; - interpret diagrams: part-part-total, start-change-end, open number lines - understand and generate variety of number models Flexible methods of computation involve grouping numbers in What questions can be answered using addition and/or subtraction? strategic ways; - problem solve in context Proficiency with basic facts aids What are efficient methods for finding sums and differences? estimation and computation of - use identity property of addition and subtraction larger and smaller numbers. - apply tens partners; doubles & near-double neighbors; see 9, think 10 - demonstrate fluency with addition and subtraction facts within 20 - split addends or subtrahends for ease of multi-digit computation - regroup 1s, 10s, 100s for multi-digit addition and subtraction - add any series of five-digit addends - subtract any five-digit minuend with a smaller subtrahend - use and examine algorithms for efficiency: partial sum, regrouping, trade first What are different models of and models for multiplication and division? - make equal sets/groups, create fair shares - represent with objects, diagrams, arrays, area models - identify multiplication patterns - understand remainders What questions can be answered using multiplication and division? - problem solve in context What are efficient methods for finding products and quotients? - use identity and zero properties of multiplication and division - apply doubling/halving concepts to multiplication (ex: 16x5 is half of 16 x10) - demonstrate fluency with multiplication facts of factors 0-12 - identify factors/divisors of a number and multiples of a number - multiply any whole number by a two-digit factor - divide any whole number by a two-digit divisor and interpret the remainder - use and examine algorithms: partial product & partial quotient, and traditional 4 --------------------------------------- 5 NUMBER AND OPERATIONS Enduring Understandings Essential Questions How do the four operations related to one another? - construct fact families for addition/subtraction and multiplication/division - recognize inverse relationships: addition/subtraction and multiplication/division - understand whole number multiplication as repeated addition - comprehend whole number division as repeated subtraction - apply commutative, associative and distributive properties How are parenthesis used in numeric expressions? - indicate operation to be performed first in written expression containing more than two terms and different operations What computation tools are best suited to which circumstances? - determine when to use objects, number line, number chart, mental math, pencil/paper, calculator Fractions, decimals, and percents How can fractions be modeled, compared, and ordered? express a relationship between two - understand fractions as division of two whole numbers numbers. - identify fractions as part of a whole, part of a set, part of an area, and locations on the number line - read and write symbolic notation for unit fractions and common fractions - recognize and name equivalent fractions - order fractions, including improper fractions (ex:3/2) and mixed numbers (ex:1œ ) How are common fractions, decimals and percents alike and different? - compare fraction and decimal models using pictures, objects and numbers - identify and describe relationship between fractions and decimals - relate fractions and decimals to percents - recognize common equivalent fractions, decimals and percents How is computation with rational numbers similar and different to whole number computation? - add and subtract fractional numbers with like denominators - add and subtract decimals using standard algorithms 5 --------------------------------------- 6 ALGEBRA Enduring Understandings Essential Questions Patterns can be found in Where are patterns in nature, architecture, music, words, and numbers? many forms. - identify patterns in daily life and through integrated classroom studies Patterns can grow and repeat. What is the repeating and/or increasing unit in the pattern? - identify and continue numerical sequences - identify and continue figure patterns Patterns can be generalized. What strategies can be used to continue a sequence? - express pattern as a rule (ex: frame & arrows, what¶s m y rule?) - use a rule to determine unknowns (ex: complete a function table) How does finding patterns help in counting and/or computation? - problem solve in context Mathematical expressions How is an equation like a balance scale? represent relationships. - understand equal to mean both sides of the number model have an equivalent value How can relationships be expressed symbolically? - use and interpret notation: >, <, = , Number patterns and relationships Why are variables used? can be represented using variables. - express generalizations (ex: N+1= 1+N) - interpret formulas (ex: area = length x width) What strategies can be used to solve for unknowns? - apply models to determine what is known (ex: part-part-whole) - work backwards to identify missing number 6 --------------------------------------- 7 GEOMETRY Enduring Understandings Essential Questions Objects can be described How are geometric properties used to solve problems in everyday life? and compared using their - problem solve in context geometric attributes. How can plane and solid shapes be described? - understand properties of polygons (ex: closed, straight, flat/2-dimensional) Points, lines and planes - identify quadrilaterals (ex: square, rectangle, parallelogram, rhombus, trapezoid) are the foundation of - relate 3-dimensional objects to 2-dimensional counterparts (ex: cube to square) geometry. - classify angles as right, acute and obtuse - compare polyhedrons by edges, vertices, and shape of faces - identify parallel and perpendicular lines and figures How are geometric figures constructed? - use straight edge, ruler, compass and technology - identify and label points, lines, line segments, and rays - identify and label the vertex of an angle What strategies can be used to verify symmetry and congruency? - create model (ex: trace shape on paper: cut, fold, or place on top of shape) - measure - use mirror Transforming an object does not affect its attributes. How will a shape look when rotated, reflected, and/or translated? - practice slides, flips, and turns with variety of objects - predict the results of transforming two-dimensional shapes Ordered pairs show an exact How is the location of a point on a grid described? How is this useful? location on a coordinate plane. - find and record coordinates on grid - relate to mapping skills 7 --------------------------------------- 8 MEASUREMENT Enduring Understandings Essential Questions Objects have distinct attributes What types of problems are solved with measurement? that can be measured. - problem solve in context Standard units provide common What are tools of measurement and how are they used? language for communication - tell time with clocks measurements. - calculate elapsed time - describe temperature with thermometers The choice of measurement - determine length/height with rulers and measuring tapes tools depends on the measurable - measure weight with variety of scales attribute and the degree of - find area of rectangles precision desired. - calculate perimeter of polygons What is the purpose of standard units of measurement? - record with customary and metric units - communicate measurements How do units within a system relate to each other? - understand the relationships between and among units; - identify equivalent periods of unit within a system; - carry out conversions with units of: time: minute, hour, day, week, month, year, decade, century money: penny, nickel, dime, quarter, dollar length: inch, foot, yard, mile; millimeter, centimeter, decimeter, meter, kilometer volume: fluid ounce, cup, pint, quart, gallon weight: ounce, pound, ton; gram, kilo, metric-ton When is an estimate more appropriate than an actual measurement? - use of judgment in context What strategies help estimate measurements? - select appropriate aonr bcheonrchm ark for measurement - apply multiplicative reasoning (ex: anchor measurement x frequency of anchor) 8 --------------------------------------- 9 DATA ANALYSIS AND PROBABILITY Enduring Understandings Essential Questions The quality of the question used How can information be gathered, recorded, and organized? impacts the data collected and the - design investigate to address pertinent question or problem validity of the results. - determine wording and/or categories to generate fair, unbiased information - collect data via observations, surveys, samplings, experiments Graphs convey data in a How does the type of data influence the choice of display? concise way. - understand purpose of different graphs - create bar graph, chart/table, circle graph/pie chart, glyph line plot, line graph, pictograph, Venn Diagram What aspects of a graph help people understand and interpret the data easily? - read and write parts of a graph: title, labels on x-axis and y-axis, - understand different type of scales - create and interpret keys/legends What kinds of questions can and can not be answered from a graph? - understand data landmarks: minimum & maximum value, range - find measures of center: mode, median, mean - interpret salient points and key relationships in the data set The expected outcome of an event How is the probability of an event determined and described? is a prediction of what might happen - experience activities with random generators (ex: dice, spinners) in the long run. - test predictions - develop vocabulary to express probable outcomes - represent probability by a number from 0 to 1 - understand impact of law of large numobne arsny sin gle outcome Sources: About Teaching Mathematics K-8 (Burns, 2007) Everyday Mathematics (University of Chicago, 2007) Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) Teaching Mathematics Developmentally, Grades K-2 and Grades 3-5 (Van de Walle, 2005) 9 |