Math K-8 Common Core Alignment


Curriculum: K-8, Grade Level Vertical Alignment / Common Core Alignment

Kindergarten / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Oral Exercises: Number Recognition / One-to-one Correspondence / Counting / Enumeration: Count objects and recite number names in sequential order. Use cardinal numbers and ordinal numbers and understand the difference. Demonstrate cognitive awareness of one-to-one correspondence; conservation of quantity; and quantity-number permanence. Understand that for every number there is a number that is one larger; for every number there is one that is one smaller. Discriminate between odd and even numbers. Count up to ten then countdown to negative ten. Use everyday examples of negative numbers. In written form and oral presentation, demonstrate comprehension of the terms: greater than; less than; equal to. Seamlessly respond to the instructor’s “choral call” with a “choral response.” Discriminate between “whole group” and “individual” cues; Fluidly recognize the target (individual or group) and respond when nonverbally instructed to do so. Respond to choral calls including: Call any random number 1-50, respond next counting number (e.g., call “4” respond 5; call “2” respond “3”; call “7” respond “8” …); Call-any random number 1-50, respond next number 2 greater (e.g., call “5” respond “7”; call “2” respond “4”; call “9” respond “11” …); Call any random number 1-50,, respond next counting number, one less; Call random number 1-50, respond next counting number, two less. Understand the differences and similarities between numbers, numerals, and digits. Recognize and accurately name one and two digit numbers, 0-99 (see: 1-99 place value matrix). Understand that the one in the number 10 indicates that 0-9 has been counted through one time, 20 indicates that 0-9 has been counted through 2 times. Understand that 7 in the number 74 indicates that 0-9 has been counted through 7 times and the 4 indicates that the counting is 4 units (5digits) through the 70’s. Understand that the numeral “0” is an initializer representing an empty set. Understand that 80 represents that 0-9 has been counted through 8 times and 0 parts (fractional times) whereas 85 represents that 0-9 has been counted through 8 and one half times. COUNT: by ones 1-10; 1-20; 1-50; 1-100; 1-100. Skip count by 10’s to 100; by 100’s to 1000; by 1000’s to 20,000; by 5’s to 100; by 2’s to 100; by 4’s to 100; by 11’s to 99; by 9’s to 99; by 3’s to 99; by 6’s to 96; by 8’s to 96; by 7’s to 98. Using a number matrix (0-99), identify counting patterns for each of skip-counting (multiplication) exercise (e.g., 9’s count from bottom left diagonally to top right. 11’s count from top left diagonally to bottom right. etc.). Count forward and backward from any given number in sequence. From 10 count down to negative 10. Count up beginning with any given number between 0-100. Call-and-response: Call any number between 0-99, respond next higher number. Call-any number between 0-99, respond next lower number. Call any number between 0-99, respond next number two higher. Call any number between 0-99, respond next number two lower. Understand multiplication as repeated addition. Mastery recall single digit multiplication facts including the 5’s, 2’s, 1’s, 0’s, 10’s, 11’s, and 9’s. Call-and-respond (using numeric exercises) observations including: One times any quantity is that same quantity; 10 times any whole number is that same number with a zero; Zero times any quantity is zero; 2 times any quantity is that quantity added to its self; 11 times any single digit number is that digit repeated; 11 times any multi-digit whole number, is that same number added to itself with an offset; the product of 5 and any whole number ends in a zero or 5. During choral call-and-respond exercises: Call any number between 0-10, respond with the addend that sums to 10 (e.g., call “4: / respond “6.” Call any number between 0-100, respond with the addend that sums to 100 (e.g., call 43, respond 57). Read and write numbers from 0 to 99. Using the Hindu-Arabic place value numeral system, choral read numbers up to the hundred decillions. Decompose multi-period whole numbers into their place value equivalents. Using the Six Arithmetic Operations cognitive map, recite relationships between operations including: Inverse operations; Repeated operations; addition counts forward, subtraction counts in the reverse; multiplication counts faster than addition and exponentiation faster than multiplication; exponentiation is repeated, repeated addition, etc.

Numeric Operations: Arithmetic Operations: Understand addition as combining groups (combining sets). Understand subtraction as removing a smaller group (or set) from a larger group (or set). Represent addition and subtraction with concrete objects, mental constructions, and common experiences including: fingers, mental images, drawings, sounds, real-world situations, verbal explanations, numeric expressions, or equations K.OA.A.1. Solve addition and subtraction word problems. Add and subtract using objects or drawings to represent the problems K.OA.A.2. Decompose numbers into place value equivalents in more than one way K.OA.A.3. For any number 0-5, flash-recall the number it takes to add to 5. For any number 1-9, flash-recall the number it takes to add to ten. K.OA.A.4. Flash-recall add and subtract within 20 K.OA.A.5. Represent a quantity of objects with a written number 0-20. Understand the relationship between numbers and quantities. Understand differences and similarities between cardinal and ordinal numbers. Represent a number of objects with a written numeral, 0-20. Calculate all addition and subtractions facts with high fidelity. Add all permutations of single digit numbers in two rows (i.e., addends = 0-9, augends = 0-9). Subtraction Facts: Inverse Addition Facts; From the addition fact sums (0-9 + 0-9) subtract numbers 0-9 where the resulting differences are equal to or greater than 0. Add two rows of numbers, with and without regrouping. Solve simple addition and subtraction linear equations with one variable (e.g., a =b+ c, x = 20+5, 5 = x+3). Represent addition and subtraction with objects, fingers, mental images, drawings, sounds, verbal explanations, expressions, and equations. For simple addition and subtraction word problems, write a linear equation representing the unknown as a variable and solve. Identify groupings of objects of twos, fives, etc. FRACTIONS: Read, write, and identify simple common fractions, using drawings and physical objects, including: 1/2, 1/4, 2/4, 3/4, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8. Identify equivalent fractions in drawing and physical objects including: 2/4 = 1/2, 4/8 = 2/4 = 1/2, 2/8 = 1/4, 6/8 = 3/4. Add and subtract simple common fraction with like denominators.

Place Value / Base Ten Positional Notation / Hindu-Arabic numeral system: Read and write whole numbers: one digit numbers (0-9), two digit numbers (10-99); three digit numbers (100-999); and four through six digit numbers (1,000-999,999) using standard place-value nomenclature (see illustration). Read whole numbers to the hundred-decillions (decillion, nonillion, octillion, septillion, sextillion, quintillion, quadrillion, trillion, billion, million, thousand) with each period place filled with digits 1-9. Decompose (e.g., 1,357 = 1,000+300+50+7): two digit numbers (10-99); three digit numbers (100-999); and six digit numbers (1,000-9,999). Understand how to decompose any whole number to the decillions. Use objects, drawings, or a counting frame (abacus) to represent the concept of base ten, place -value. Pedagogical Note: Use a recursive-iteration instructional strategy to instill procedural-fixed-pattern-recall, high motivation and high confidence. Know period names (…octillion, septillion, sextillion …), and cyclic sub-period names (ones, tens, hundreds) in sequential order. Count by tens, hundreds, thousands, ten thousandsK.CC.A.1. Count forward beginning from a given number K.CC.A.2. From an instructor’s oral call, write numbers from 0 to 20; Write numbers from 0 to 999. Represent a quantity of objects with a written numeral K.CC.A.3. Count the number of objects in groups of 1, 2, and 5. Understand the relationship between numbers and quantities. Compare and contrast counting with cardinality K.CC.B.4. Demonstrate cognitive awareness of one-to-one correspondence: When counting objects, say the number names in sequential order, pairing each object with one and only one number name and each number name with one and only one object K.CC.B.4.a. Demonstrate cognitive awareness of conservation of quantity: the number of objects is the same regardless of arrangement or order counted K.CC.B.4.b. Understand that each successive number name refers to a quantity that is one larger K.CC.B.4.c. Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objectsK.CC.B.5. Compare numbers. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group. K.CC.C.6. Compare two numbers between 1 and 10 presented as written numeralsK.CC.C.7. Decompose numbers into hundreds, tens, and ones. Utilizing a standard addition matrix, compose given hundreds, tens, and ones into numbers. Understand that the value numeral is dependent on the place it occupies; this concept to the ones, tens, hundreds and so on. Understand that as place positions become filled the value of the number increases. Understand that each place value increases by a factor of ten. Apply the pattern of decomposition to progressively larger numbers. Decompose given numbers into addends; And Compose given addends into numbers by using objects or drawings. Record compositions and decompositions with drawings or equations K.NBT.A.1.

Systems of Measurement / Denominate Numbers: Describe and compare measurable attributes of one, two, and three dimensional objects, such as length (height, width, and depth), area, volume, weight, or temperature using SI and US customary units. Describe several measurable attributes of a single object K.MD.A.1. Compare two objects with a measurable attribute in common; determine which object has more or less of the attribute, and describe the difference. K.MD.A.2. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. Name numeric categories such as gross or dozen K.MD.B.3. Using common containers (milk cartons) identify liquid measurements orally: gallon, half-gallon, quart, pint, and cup (name number of: gallons, half-gallons, quarts, pints, cups; ounces in each). Exhibit a sense of length / approximate the linear measures (height, length, width) in terms of: feet, yards, inches, centimeters, meters, and kilometers. Measure height, width, depth of one, two, and three dimensional objects. Compare denominate measurements of objects. Demonstrate understanding of conservation of volume regardless of container shape. Recall: inches in a foot; feet in a yard; feet in a mile. Understand signed numbers in relationship to familiar uses (e.g., above and below freezing, above and below sea level, etc.) Demonstrate mastery-recall: Name the months of the year (i.e., January, February, March …); Name days of the week (i.e., Sunday, Monday, Tuesday …); Number of days in a; week, month, year; Number of seconds in a minute, minutes in an hour, hours in a day; days in a week; years in a century. Name the seasons; summer, fall, winter, spring in order. Using an analog clock discriminate between the second, minute, and hour hands. Tell time in hours, half-hours, and minutes using analog and digital clocks. Recognize and know the value of a: penny, nickel, dime, quarter, and dollar. Use the symbols: $ (dollars) and ¢ (cents.)

Geometry: Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using formal language to contrast their similarities and differences. Use standard nomenclature to refer to the first, second, and third dimension: line, shape, form, etc. Locate positive and negative whole numbers on a number line. Describe two dimensional shapes. Identify regular polygons using standard nomenclature, regardless of orientation or size (trigon, tetragon, pentagon, hexagon, heptagon, octagon, nonagon, and decagon). Identify generalized plane figures including: circles, triangles, squares, rectangles. Name relative positions of objects (above, below, beside, in front, behind, and next to). Identify three-dimensional generalizations such as: cylinder, sphere, cube, cone, etc. Identify shapes and forms of objects in the environment referring to geometric similarities and relative positions. K.G.A.1. Fold given shape patterns into polyhedra. Correctly name polygons and polyhedra regardless of orientation or size K.G.A.2. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”)K.G.A.3. Analyze, compare, create, and compose shapes. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/”corners”) and other attributes (e.g., having sides of equal length)K.G.B.4. Model shapes in the world by building shapes from cutout and drawn shapes K.G.B.5. Model internal star polygons by drawing straight lines connecting all vertices. Decompose regular polygons into simpler shapes. Compose simple shapes to form larger shapes K.G.B.6.

First Grade Students / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prior Knowledge / Prerequisite Knowledge / Skill Retention-reactivation: Prior to providing instruction in the following, the instructor shall review, assess, and reteach the Kindergarten Math Standards articulated above. To satisfy the First Grade Math Standards, Students must demonstrate competence and/or mastery in all objectives articulated in the Kindergarten Math Standards as well as all areas articulated in the First Grade Math Standards.

Oral Exercises / Counting / Mental Calculation Strategies/ Natural Numbers: Chunking addends: chunk addends of 10 (1+9, 2+8, 3+7, 4+6, 5+5, 6+4, 7+3, 8+2, 9+1); addends of 5 (1+4, 2+3, 3+2, 4+1). During choral exercises and in written problem solving exercises: Chunk columns of single digit numbers with six or more rows into 5’s then sum by skip counting; Chunk columns of single digit numbers with six or more rows into 10’s then sum by skip counting; Chunk columns of single digit numbers with six or more rows into 5’s with residuals then sum by skip counting; Chunk columns of single digit numbers with six or more rows into 10’s with residuals then sum by skip counting. Mentally add two numbers with four periods, arranged in a column, while simultaneously reading the sum with the correct place value names (without regrouping and with regrouping.) Mentally subtract two numbers with four periods, arranged in a column, while simultaneously reading the sum with the correct place value names (without regrouping and with regrouping.) Use counting on strategies with single column addition. Use counting up-to strategies with single column subtraction. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count. Add all permutations of one and two digit augends and addends within 100. Subtract all permutations of one and two digit minuends and subtrahends with differences zero or greater, within 100. Add, subtract (including subtrahends that are larger than minuends; positive and negative differences), and multiply with single digit multipliers and two digit multiplicands; with and without regrouping. Demonstrate concrete and abstract manipulation of mathematical reasoning and calculations; predict the outcome of mathematical actions (e.g., If a=b and b=c then a=c). Illustrate multiplication as repeated addition with abstract concepts and concrete objects including: familiar objects, polygons, drawings, and equations.) Demonstrate mastery recall of multiplication using single digit multipliers including tables: 0’s (Zero times is any quantity zero); 1’s (One times any quantity is that same quantity); 10’s (Ten times any number is that same number with a zero appended); 5’s (all end in 0 or 5, regular, first digit repeats. Read and write common fractions with single digit numerators and denominators. In whole group choral exercises, exhibit deductive and inferential mathematical reasoning. Add and subtract common fractions with the same denominators (e.g., 4/5 – 2/5 =1/5). Add and subtract common fractions with different denominators in which one denominator can be renamed to the other (e.g., 3/4 + 1/8 = 6/8 + 1/8). Using a place-value cognitive map, mastery-recall unit-period names. Call (0 times any quantity) response (is 0); Call (1 times any quantity) response is (that same quantity); Call (ten times and single digit number) response (the digit with a 0)… 11, 100, 1000.

Numeric Operations: Construct the Six Arithmetic Operations cognitive map including graphic and numeric representations of inverse operations and repeated operations (e.g., subtraction will undo addition and addition will … exponentiation is repeated multiplication) (See: 6-Arithmetic Operations Graphic: Kindergarten Objectives). Demonstrate relationships between addition and subtraction using strategies such as: a number-line, analog clock, relationships in a number sequence (such as: odd number sequence, even number sequence; sequence of alternating odd, even), etcetera. Using a various graphics, or equations demonstrate the relationships between: addition and subtraction; repeated addition and repeated subtraction; and repeated, repeated addition and repeated, repeated division, multiplication and division, exponentiation and root taking). Skip count forward and backward 1-11. Recognize and note numerical relationships between numbers and expressions using: <, =, and >. Use counting-on to solve subtraction problems. Demonstrate cyclical patterns of multiplication using the 0-99 number matrix. Flash-recall addition, subtraction, and multiplication facts, involving all permutations of single digit operands 0-9. Recognize numeric sequences using a conventional analog wall clock and numeric arrays including a multiplication matrix and the 1-99 place value matrix (See; 1-99 Place Value Matrix: Kindergarten Objectives.) Understand and apply properties of operations to problems involving addition (commutative, associative, additive identity, distributive, and reflexive) and subtraction (identity, equality, distributive, and reflexive). Apply properties of operations as strategies to add and subtract 1.OA.B.3. Understand subtraction as its inverse (addition) with and unknown addend (e.g., 5-2=X, X+2=5). Understand the mathematical logic in “counting-on” to solve subtraction problems. Understand addition as its inverse (subtraction) with and unknown subtrahend(3+2=X, X-2=5) 1.OA.B.4. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2) 1.OA.C.5. Demonstrating fluency with addition and subtraction. Add and subtract whole numbers consisting of any number of digits with and without regrouping, where the sums or differences are whole numbers. Use strategies such as counting on; making tens or fives, decomposing a number leading to a tens or fives, counting by fives or tens, strategies using multiplication to solve addition, chunking like pairs strategies, strategies of residuals, strategies of redistribution, strategies involving the relationship between addition and subtraction, and creating equivalent but easier or known sums 1.OA.C.6. Identify: even and odd numbers. Use and understand standard mathematical notation, nomenclature and symbols including: the equals bar, addition, subtraction, multiplication signs, >, <, and ?. Determine if equations containing addition, subtraction, and multiplication are true or false. Solve simple equations containing one variable and operations including: addition, subtraction, multiplication, and division. Understand basic concepts related to sets and use Venn diagrams to visually order sets of objects based on attributes.

WORD-PROBLEMS AND EQUATIONS: Represent the numeric relationships within word problems with objects, drawings, and equations. Translate simple word-problems into linear equations containing one variable that represent the relationships in the word-problem. Solve problems with unknown addends, subtrahends, products, and multiplicands. Use symbols (that are consistent with standard numeric conventions) to represent unknown numbers (variables). Solve word-problems and linear equations with one variable involving adding, subtraction, multiplication, and division. Solve word-problem equations with an unknown in all operand positions 1.OA.A.1. Solve word problems (individually and in group activities) that call for: addition, subtraction, multiplication, and division of whole numbers. Evaluate word problems that call for an operation to be carried out up to three times within the same problem. Evaluate word problems that call for one, two, or three different operations to be carried out in the same word-problem 1.OA.A.2. Work with addition and subtraction equations. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false 1.OA.D.7. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers 1.OA.D.8.

Place-Value / Numeration: Read and write numbers to the hundred decillions with random periods filled with 0’s (decillion, nonillion, octillion, septillion, sextillion, quintillion, quadrillion, trillion, billion, million, thousand, and units). Decompose any number from decillions to decillionths (e.g., 1,357 = 1,000+300+50+7). Identify the value of each place/period in whole numbers up to the hundred decillions, (e.g., in the number 632,547 there are 7 ones, 4 tens, 5 hundreds, 2 thousands, 3 ten-thousands, and 6 hundred-thousands) by decomposing them with subtraction and/or recomposing them with addition. Read decimal fractions with the proper place value up to thirty-five places to the right of the decimal point. Count by two’s to 50, three’s to 39, four’s to 48, five’s to 50, tens to 100, hundreds to 1,000. Count on starting with any number; by one’s and multiples. Read and write numerals representing any number of objects up to the decillions 1.NBT.A.1. Understand decimal place value notation, that is; the Hindu-Arabic base-10 positional notation system in common use today. Recognize that the system utilizes a decimal point marking the transition between the whole part of mixed number and its fractional part. Recognize that the numeral system utilizes ten unique glyphs (digits, 0-9) that cycle, without end, through positions of increasing value left to right by factors of ten. Recognize that in a two digit numeral the digit on the left represents the number of times 0-9 has been counted and the digit on the right represents the progress (in units of one) within 0-9. Recognize that in a three digit numeral, the two digits on the left represent the number of times 0-9 has been counted and in a four digit numeral the three digits on the left, represent the number of times 0-9 has been counted 1.NBT.B.2: Recognize that although 10 can be thought of as a bundle of ten ones — called a “ten;” a numeral system is properly described by the number of glyphs used (i.e., digits 0-9 equals ten digits); and by viewing zero as an initializer with the property of defining the counting space empty and defining the end of one cycle and beginning of another (See; 1-99 Place Value Matrix: Kindergarten Objectives) 1.NBT.B.2.a. Recognize that the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones 1.NBT.B.2.b. Recognize that the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones) 1.NBT.B.2.c. Compare numbers based on meanings of the hundreds, tens and one’s place value. Compare the place value of digits with symbols >, =, and < (e.g., 120>99) 1.NBT.B.3. Use understanding of place value and properties of operations to add and subtract. Add whole numbers with any number of digits in the addends and augends. Add two-digit numbers with one-digit numbers, and adding a two-digit numbers with multiples of 10. Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding three-digit numbers, the column on the left represents the addition of hundreds; the one in the middle represents addition of tens; and the column on the right represent addition of ones; And when the addition one column adds to a two digit number, the two digit number can be easily regrouped by writing the digit on the left above the next place value column and the number on the right beneath the column just added 1.NBT.C.4. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the numeric justification 1.NBT.C.5. Subtract multiples of 10 from multiples of 10 that have a whole number difference. Use concrete models, drawings and strategies based on place value, properties of operations, and/or the relationships between addition, subtraction, multiplication, and division to enhance understanding of these constructs; compose written descriptions of place value strategies and explain the numeric justification 1.NBT.C.6.

Systems of Measurement / Denominate Numbers: Order three objects by length; compare the lengths of two objects indirectly by using a third object. Measure lengths directly and indirectly. Measure the length of one object using a standard unit of measurement (e.g., inch, foot, yard, centimeter, and meter.) Order the measured object with two other objects by length; compare the lengths of two objects indirectly by using measured object. Express the length of an object as a whole number of length units one the basis of the object length as the standard and the denominate number as the standard; express both measures following standard conventions 1.MD.A.1. Visually estimate distances, heights of objects and lengths of objects in inches, feet, yards, meters, and centimeters. Compare the lengths of objects measure in SI and US customary units (e.g., feet with meters, inches with centimeters, etc.) Estimate the lengths of objects in both the SI and US customary units with reasonable accuracy. Informally convert between systems by “feel” by approximation (develop a sense for length in both systems). Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps 1.MD.A.2. Flash-recall inches in a foot; feet in a yard; feet in a mile; millimeters in a centimeter; centimeters in a meter; millimeters in a meter; meters in a kilometer. Ounces in a pound, pounds in a ton (short ton), grams in a kilogram, liters. Freezing-boiling point °F and °C room temp/ degrees above and below. Number of items in a: dozen (dz), half dozen, Baker’s dozen, pair, gross, great gross, small gross. Represent and interpret data. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another 1.MD.C.4.

TIME: Flash-recall: number of months in a year, number of days in a week, approximate number of weeks in a month, approximate number of days in a month, number of days in a year; weeks in a year; years in a century; years in a decade; decades in a century; years in a millennium; centuries in a millennium; decades in a millennium; seconds in a minute; minutes in an hour; hours in a day. Name: months of the year; days of the week in order. Name the seasons: summer; fall; winter; spring. Tell and write time (using common and standard notation) in terms of: AM, PM; half hours; quarter hours; hours-minutes-seconds using analog and digital clocks and laps-time stop watches 1.MD.B.3.

Geometry/ Graphing: Reason with shapes and identify their attributes. Distinguish between defining attributes (e.g., triangles are closed and three-sided) and non-defining attributes (e.g., color, orientation, overall size); build and draw shapes that exhibit defining attributes. Identify and name the five platonic solids (tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron.) Distinguish the defining attributes of regular polygons and platonic solids. Identify and name regular and irregular polygons. Identify two-dimensional shapes and their properties (e.g., radius, dimeter, circumference, diagonals, etc.) including: rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles. Identify three-dimensional shapes including: cubes, right rectangular prisms, right circular cones, and right circular cylinders. Locate positive and negative numbers on a number line (whole and fractional) 1.G.A.1. Draw two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape 1.G.A.2. Partition groups of circles, rectangles, and triangles into two, four, or more equal parts. Refer to the whole and fractional parts using standard common fraction nomenclature (e.g., numerator-denominator: four-fifths, two-thirds, one-fourth.) Name mixed numbers when presented with concrete examples (e.g., three quarters and two nickels; three and two-fifths quarters; eight and one-half dimes.) Understand the process of dividing whole units into subunits which can themselves represent whole units, which can themselves by further be decomposed (one day can be divided by hours, hours can be divided by minutes; one dollar can be divided by quarters, one quarter can be divided by nickels 1.G.A.3.

Second Grade / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prior Knowledge / Prerequisite Knowledge / Skill Retention-reactivation: Prior to providing instruction in the following, the instructor shall review, assess, and reteach the Kindergarten and/or First Grade Math Objectives if indicated. To meet the second grade math objectives, students must demonstrate facility/mastery in the objectives articulated in the Kindergarten, First Grade, and Second Grade Math Objectives.

Oral Exercises: Counting/ Choral Stream / Call and Response (C&R) / Choral Review: Count by 1/2’s, 1/3, 1/4’s, 1/5’s, 1/6’s 1/7’s, 1/8’s, 1/9’s to 5 (e.g., y, 1, 1y, 2, 2y … and ˆ, , , y, š, , , 1…). Name the parts of a common fraction (i.e., denominator, numerator, and vinculum.) Mentally, add, subtract, multiply, divide stings of terms in left to right order (e.g., 5 + 3 -6 * 9 ¸ 3). Mentally add from left-to-right, numbers containing three periods or more, arranged in columns and rows, while fluidly reading the sum using the correct place-value nomenclature (with regrouping). Recall from memory all products of all permutations of one-digit numbers (multiplication facts).Mentally subtract from left-to-right two numbers each containing three periods or more, while fluidly reading the correct place-value (with regrouping / positive differences only.) Identify perfect squares and perfect square roots to 100 with exponential and radical sign notation. Mentally calculate all single digits numbers with exponents of 2 (e.g., 1², 2², 3²…10²) mentally calculate their roots of their products (e.g., = 1, = 2, = 2 … = 10). Express the differences and similarities between numbers and numerals. Zero times any quantity, is zero. Any quantity times zero is zero. One times any quantity is that same quantity. Any quantity times one is that same quantity. Compare simple fractions with like and unlike denominators (unit fraction) using <, >, and =.

 gallon half-gallon quart pint cup cup 16 8 4 2 1 pint 8 4 2 1 1/2 quart 4 2 1 1/2 1/4 half-gallon 2 1 1/2 1/4 1/8 gallon 1 1/2 1/4 1/8 1/16 Natural Numbers/ Properties of numbers / Nomenclature / Notation: Discriminate between odd and even numbers. Write/ solve equations expressing: the sum or difference of two even numbers is an even number; the sum or difference of two odd numbers is an even number; the sum or difference of and even and an odd number is an odd number.) Using place value and the properties of operations, explain why addition and subtraction strategies work (Support with drawings and objects.) In oral presentation and written form, demonstrate comprehension of the terms and symbols including: greater than; less than; equal to, not equal to, greater than or equal to, less than or equal to, approximately equal to (>, <, =, ?, =, =, ˜).

Numeric Operations: Fluently add and subtract using mental strategies. Add and subtract whole numbers (right to left and left to right) with six digit or more; augends, addends, minuends, subtrahends; with and without regrouping; that have sums and differences that are whole numbers Multiply and divide whole numbers with up to six digit dividends and multiplicands and single digit divisors and multipliers with and without regrouping. Express remainders in “R” format, common fractional format, or decimal fraction format. Use knowledge of simple multiplication and skip counting to solve column addition problems. Flash-recall sums for all addend and augend permutations of all single digit numbers (i.e., know from memory all sums of two, one-digit numbers.) Flash-recall differences of simple subtraction facts using all the permutations of; two, one digit numbers added together; used as minuends and all one digit numbers used as subtrahends 2.OA.B.2. Flash-recall multiplication fact: products of all permutations of one digit multipliers and multiplicands. Flash-recall division facts (i.e., one digit divisors and quotients) where all the dividends are products of one digit multipliers and multiplicands (i.e., inverse multiplication facts), with division operators expressed as: a vinculum of an improper fractions (35/7 =5), as a vertical vinculum and horizontal equals bar (+- ); and as an obelus in linear form (35 ¸ 7 = 5). Flash recall basic multiplication facts 0-12 times tables. Flash-calculate multiplication with multipliers of 1, 10, 100, 1,000, 10,000 and multiplicands of any whole numbers. Understand, any quantity divided by one is that same quantity; any quantity multiplied by one is that same quantity; any quantity multiplied by zero is zero. Any quantity divided by zero is undefined. Divide quantities using the three remainder types (“R” remainder, common fractional remainder, decimal remainder).

ALPHANUMERIC EXPRESSIONS AND EQUATIONS: Write and solve alphanumeric equations. Translate one and two step word-problems, with and without “noise” (irrelevant information), containing one unknown (variable); into simple linear equations (representing relations between known quantities and variables) that involve simple addition, subtraction, multiplication, and division; And solve. Understand, in alphanumeric expressions, a variables is a letter that represents a quantity that can take a range of values; a constant is a number or letter that represents a fixed number; when letters are used in place of numbers, for constants, they often come from the beginning of the alphabet (a, b, c); when letters are used to represent variables, they usually come from the end of the alphabet (x, y, z); when a letter is used to represent an unknown quantity, it usually comes from the end of the alphabet (x, y, z). Use symbols that follow alphanumeric conventions. Solve word-problems with variables in all operand positions and that include multiple, different operations 2.OA.A.1. Determine the parity property of an integer; Recognize even and odd numbers. Understand that an even integer is evenly divisible by two; and odd integer is not. Understand that zero is an even number. Understand that fractions do not have parity. Determine whether a group of objects has an odd or even number of members. Write an equation to express that: two equal addends produce an even sum regardless of whether they are even or odd; two even numbers added, subtracted or multiplied result in an even number; two numbers, one even and the other odd added or subtracted produce an odd result; two odd numbers added or subtracted produce an even number; two numbers, one even and the other odd multiplied produce an even product; two odd numbers multiplied produce an odd product 2.OA.C.3. Use a matrix to demonstrate that multiplication is repeated addition and provide insight into division. Use addition to find the total number of objects arranged in rectangular arrays with rows and columns; write an equation to express totals as a sum of equal addends 2.OA.C.4. Understand and use problem solving strategies related to “order of operations” in simple linear expressions containing multiple operations. Read, write, and solve linear expressions that use parentheses and brackets. Use addition, subtraction, and multiplication and division to solve word problems involving measures that are given in different units. Solve two and three step word problems involving addition, subtraction, multiplication, division, simple inequalities, and single variables.

Place-value: Understand Hindu-Arabic base-10 positional notation. Using the standard place value nomenclature, read mixed numbers up to the hundred decillions with decimal fractions. Read decimal fractions with the standard place-value nomenclature up to twenty places to the right of the decimal point. Identify the value each place and period represents for whole and decimal numbers. Using the standard nomenclature, read and write common fractions with up to three digit numerators and denominators. Compare the relative magnitude of whole numbers, simple common fractions, decimal fractions, using symbols including >, =, and < . Understand and read Roman numerals using standard additive and subtractive forms from 1-1,000 (I, II, III, IV, V, VII, VIII, IX, X, L, C, D, M). Understand decimal place value notation, that is; the Hindu-Arabic base-10 positional notation system in common use today. Recognize that the system utilizes a decimal point marking the transition between the whole part of a mixed number and its fractional part. Recognize that the numeral system utilizes ten unique glyphs (digits, 0-9) that cycle, without end, through positions of increasing value left to right by factors of ten. Recognize that in a three digit numeral, the two digits to the left represent the number of times 0-9 has been counted and the digit on the right represents the progress (in units of one) within 0-9; And in a four digit numeral the three digits on the left, represent the number of times 0-9 has been counted and so on to infinity. Understand that the Hindu-Arabic place value system provides for infinitely larger numbers. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones (e.g., 706 equals 7 hundreds, 0 tens, and 6 ones) 2.NBT.A.1. Understand that although one can view “100 as a bundle of ten tens” as a “special case” It may be more productive to understand the special property of zero as an initializer, as the normal property of a cycle. It may be productive to view zero as an initializer with the property of defining the counting place empty such as by initializing (rendering the position empty) the beginning of each new cycle (N+0 cycles) with the counted value of the position “regrouped” by factors of 10 to the next positional value (See; 1-99 Place Value: Matrix Kindergarten Objectives) 2.NBT.A.1.a. Understand that the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones) 2.NBT.A.1.b. During Oral Activities, individually and whole group, skip-count by 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, 11s, 12s, 100s, 1000s’ 10000s 2.NBT.A.2. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form 2.NBT.A.3. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. Use place value understanding and properties of operations to add, subtract, multiply, divide, and exponentiate (base of two, positive exponents up to 10) 2.NBT.A.4. Fluently add and subtract using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction 2.NBT.B.5. Add numbers with any number of digits with regrouping, using strategies based on place value and properties of operations 2.NBT.B.6. Add and subtract using concrete models, abstract drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Describe strategies in written form with justifications that are numerically sound. Understand the place value justification for regrouping and recognize regrouping in addition, subtraction, multiplication, and division 2.NBT.B.7. Mentally add, subtract, multiply 10, 100, or 1000 to any given whole number. Multiply or divide any decimal number by and factor of ten buy repositioning the decimal point 2.NBT.B.8. Explain why addition, subtraction, multiplication, and division strategies work, using place value and the properties of operations 2.NBT.B.9.

Systems of Measurement / Denominate Numbers: Using standard units of measurement in SI and US customary units, measure the length; width and height (area); height, width, and depth (volume); and weight (mass) of objects with appropriate tools such as rulers, yardsticks, meter sticks, measuring tapes, digital scales, triple beam scale. Identify the dimensionality of each type of measurement. Calculate area, volume, surface area, weight, density and record using standard notation. Describe the differences and similarities in meaning between mass and weight. Compare measurements of the same object in in SI and US metrics 2.MD.A.1. Measure an object using different units in the same metric. Describe how the two measurements relate to one another and to the object 2.MD.A.2. Develop a “feel” for measurement. Estimate the length, area, volume, weight, and density of objects of different sizes and shapes using standard nomenclature. Compare estimated values with measured values and record the differences using standard notation 2.MD.A.3. Measure to determine how much more volume, surface area, weight, or density one object has than another, express the difference in terms of a standard units 2.MD.A.4. Compare and contrast the concepts of accuracy (how close a measurement is to the true value) with precision (the consistency of repeated measures). Measure objects to the nearest whole unit with that can be repeated measured; round to the nearest whole denominate unit. Average the results and differentiate between precision and accuracy relative to the tool and the act of measuring. Solve word-problems involving volume, surface area, area, length and weight measured in SI and US units 2.MD.B.5. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2… Represent whole-number sums and differences on a number line diagram 2.MD.B.6.

TIME: Flash-recall /Seconds in a: minute; hour / Minutes in an: hour; day / Hours in a: day; week / Days in a: week, each month; year (including leap year) /Weeks in a: month; year / Months in a: year; decade / Years in a: decade; century; millennium. Using analog and digital clocks, tell time and record in written form (using the standard notation and nomenclature) to the minute, and hour. Discriminate between a.m. and p.m. 2.MD.C.7. Name the seasons and climatic conditions relative to latitude and the northern and southern hemispheres. Demonstrate knowledge of astronomical phenomena that cause: the seasons (summer; autumn; winter; spring) in the Northern Hemisphere; a lunar month; day and night including variation in hours in day and night relative to latitude and time of the year. Represent and interpret data. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units 2.MD.D.9. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve word-problems using information presented in a bar graph 2.MD.D.10.

CURRENCY: Recognize and know relative values of: penny, nickel, dime, quarter, and dollar. Read and write the symbols: $ (dollars) and ¢ (cents). Use a decimal point to discriminate between whole dollars and fractions of a dollar. Viewing standard notion determine equivalent values in various denominations (e.g., $5.25 is the same value as 21 quarters, 52 and one-half dimes, 105 nickels, 525 pennies. Flash-recall: pennies in a: dollar, quarter, dime, nickel; nickels in a: penny, dollar, quarter, dime, nickel; all permutations (see matrix to the right.) Solve word problems involving US currency: dollars, quarters, dimes, nickels, pennies. Use standard denominate notation in answers 2.MD.C.8

Geometry / number lines / Cartesian Plane: Recognize specific shapes and forms, and their defining attributes. Use multiple names for shapes based on specific attributes including: right prisms and pyramids using multiple names (e.g., a regular tetrahedron is also called a right equilateral pyramid; a right equilateral triangular prism is also called a pentahedron; a cube is also called a square prism). Recognize and draw shapes having attributes, such as a given number of angles or a given number of equal faces. Draw and Identify triangles, quadrilaterals, pentagons, hexagons, and cubes 2.G.A.1. Partition a rectangle into same-size squares and count to find the total number 2.G.A.2. Partition circles and rectangles into two, three, or four equal shares, describe the shares using a variety of standard nomenclature. Describe a whole as fraction of unity: two-halves, three-thirds, four-fourths. Recognize that equal shares of identical wholes need not have the same shape 2.G.A.3. Represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1, 2… Represent whole-number sums and differences on a number line. Draw and identify properties of: points, lines, line segments, rays, angles (right, acute, obtuse, opposite, adjacent, etc.,) perpendicular and parallel lines; Identify in two-dimensional figures. Identify radii, diameter, similar and congruent figures. Recognize that there are two radii in on diameter. Know or be able to derive the formula for the area and perimeter for rectilinear shapes, and triangles. Know or be derive a formula for the volume of rectilinear forms. Plot common fractions on a number line. Define the three dimensions.

Third Grade / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prior Knowledge / Prerequisite Skills Retention-reactivation: Before providing instruction in the following objectives, the instructor shall assess students’ retention and facility with objectives articulated for prior grades and review or reteach where required. To satisfy mathematics requirements for the current grade, students must demonstrate mastery in objectives articulated for the prior grade along with those articulated for the current grade.

Choral Activities: Choral Stream / Call and Response (C&R) / Choral Review/ Mental Calculations: Mentally, add, subtract, multiply, divide, exponentiate, take roots: calculate the 0-10th power of base 2 (e.g., 2², 2³, 24 … 210); squares of bases 1-20 (e.g., 1², 2², 3²… 20²); square roots (e.g., = 2, = 3, = 4 … = 19, = 20); squares of bases that are multiples of 5, 5-100 (e.g., 5², 10², 15²… 100²); squares ¯10-10 (e.g., 1², 2², 3²… 20²); express results as common, and decimal fractions. C&R common fractions (Call) and their equivalent decimal fraction (Response) (e.g., 1/5 = .2, 3/4 = .75, 2/7 = .285614…, etc.) and vice versa (.125 = 1/8, .888… = 8/9, .8 = 4/5.) Zero divide by any number is zero. Zero times any quantity, is zero.

 

 

Numeric Operations: Recall standard nomenclature and use standard notation associated with: addition, subtraction, multiplication, division, exponentiation, and root taking (e.g., augend, term, divisor, operator, factor, coefficient, etc.). Recognize correspondence between notational symbols and nomenclature (e.g., divisor & denominator correspond; dividend & numerator correspond, etc.) Add, & subtract, numbers using chunking with residuals, and repeated addition; with and without regrouping. Multiply and divide numbers with any multi-digit dividends and multiplicands, any multi-digit divisors and multipliers, and dividends with regrouping. Divide using repeated subtraction strategies. Solve simple equations and work-problems representing all six arithmetic operations; with mixed numbers; with decimal fractions and common fractions. Multiply using partial products strategies and distribution strategies. Use estimation to approximate solutions. Use inverse operations to check solutions (e.g., use addition to check subtraction; subtraction to check addition; division to check multiplication; multiplication to check addition. Constituently estimate reasonableness of calculation outcomes. Know basic division, exponentiation facts. Master multiplication with multipliers and/or multiplicands of 1, 10, 100, 1,000, 10,000; decimal fractions; mixed numbers. Check division by multiplying and adding remainder. Divide quantities using the three remainder types (“R” remainder, common fractional remainder, decimal remainder). Read, write, and compare integers using inequality symbols. Locate positive and negative integers on a number line. Understand that for the set of real numbers and integers, numbers greater than “0” are positive and numbers less than “0” are negative, although “0” is an integer, real number, and signed number, it is not signed positive or negative. Understand and use standard integer notation and perform operations involving integers including addition and subtraction.

ALPHANUMERIC EXPRESSIONS AND EQUATIONS: Solve algebraic expressions and equations involving addition, multiplication, division, simple exponentiation (whole number exponents), and simple root taking (whole number roots). Represent mathematical relationships, expressed by algebraic expressions and equations, with Venn diagrams using standard set theory nomenclature and notation (e.g., Ç, È, É, Ì, Î, {}). Interpret the multiplier, in multiplication, as the number of sets of objects, the multiplicand as number of objects in a set, and the product as the total number of objects 3.OA.A.1. Understand division as the inverse operation of multiplication and division as repeated subtraction. Interpret division as a whole in terms of its parts. Understand both the partition concept of division (focus on the size of the parts) and the quotition concept (focus on the number of parts). Relate the concepts of partition and quotition to natural numbers and fractions 3.OA.A.2. Translate word-problems involving groups, arrays, and measurement quantities into equations representing the numeric relationships between constants and variables 3.OA.A.3 Transform equations and formulas, involving the six arithmetic operations (including division), using inverse operations, to solve for an unknown. Understand and preserve the balance of opposite sides of the equal sign. Determine the value for the unknown 3.OA.A.4. Understand that the inverse of multiplication is division and the inverse of division is multiplication. Understand the multiplication is repeated addition and division is repeated subtraction. Understand that division can undo multiplication and multiplication can undo division. Understand that multiplication can be used to check division and division can be used to check multiplication. Understand and apply the properties of multiplication including the: commutative property, associative property, multiplicative identity property, and distributive property. Understand and apply the properties of division including the: identity property, zero property, and equality property 3.OA.B.5. Understand that division can be expressed as multiplication with an unknown factor 3.OA.B.6. Flash-recall basic facts for addition, subtraction, multiplication, division, exponentiation (1-19 squared, 2 with exponents 0-10 with any whole exponent), and root taking (inverse of the exponentiation facts) 3.OA.C.7. Translate word-problems involving addition, multiplication, division, and exponentiation into equations using letters to represent unknown quantities. Solve word problems requiring multiple (different) operations including: word-problems containing “noise” (extraneous information), and interdependent word-problems (word-problems that require the use of information from other word-problems). Assess reasonableness of answers using mental computation, estimation strategies and rounding 3.OA.D.8. Identify patterns in the six arithmetic operations and explain them; utilizing properties of operations 3.OA.D.9. Understand and list the embedded organization (Russian Doll sequence) of the system of number

SYSTEMS OF NUMERALS, OPERATIONS AND PROPERTIES: Understand and explain: Commutative; associative; and distributive properties of multiplication; Factor; Prime numbers; Relative primes; Composite numbers; and Magnitude. Compare and contrast the Hindu-Arabic and Roman, numeral systems. Identify Roman numerals using standard additive and subtractive form from 1-1,000 (I, II, III, IV, V, VII, VIII, IX, X, L, C, D, M). Compare positional number systems to non-positional systems.

FRACTIONS: Identify fractions as numbers and recognize that their equivalent values can be expressed in several forms. Understand “undefined” as related to division by zero or common fractions with a denominator of zero. Understand that common fractions represent division with the vinculum serving as the operator and the numerator the operand. Understand that decimal fractions have an “understood” denominator that is multiple of ten. Understand that common fractions can be expressed as decimal numbers, and decimal number can be expressed as common fractions. Understand that improper common fractions represent values greater one and can be renamed to mixed numbers. Understand that decimal numbers include mixed numbers (values greater than one) and values less than one (proper fraction) and that decimal fractions can never be improper. Understand partition and quotition concepts as they apply to fractions. Understand a fraction as the quantity formed when a whole is partitioned into equal parts 3.NF.A.1. Locate decimal & common fractions on a number line 3.NF.A.2. Represent a fraction on a number line by defining the interval from 0 to 1 as a whole number; finer divisions as fractional parts of the whole 3.NF.A.2.a. Represent a fraction on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line 3.NF.A.2.b. Explain the equivalence of fractions and compare fractions by reasoning about their size 3.NF.A.3. Understand two fractions as equivalent (equal) if they occupy the same point on a number line 3.NF.A.3.a. Recognize and rename simple equivalent fractions (e.g., 1/2 = 2/4=3/6=4/8 = 5/10) Explain why the fractions are equivalent 3.NF.A.3.b. Express whole numbers and mixed numbers (decimal and common) as equivalent improper fractions (5=5/1, 4=32/8, 3.5=7/2). Recognize that improper fractions are equivalent to whole or mixed numbers 3.NF.A.3.c. Compare the relative value and rank order common fractions (with like and unlike denominators), decimal fraction, mixed numbers, and numbers with exponents. Use standard inequality notation 3.NF.A.3.d. Solve problems involving the six arithmetic operations and various forms of fraction notation including: decimal, vulgar, proper, improper, fractions of unity, percentages, equivalent, compound & complex fractions, fractions with like and unlike denominators, decimal and common fractions. Identify the inverse and direct relationships of the numerator and denominator to magnitude (e.g., The larger the denominator the smaller the quantity. The larger the numerator the larger the quantity.) Convert decimal fractions to: common fractions; percent. Read and/ write decimal fractions to the decillionths. Round common fractions: to the nearest whole half, third, fourth … tenth; decimal fractions to the nearest tenth, hundredth, thousandth, ten-thousandth, hundred-thousandth. Rewrite improper fractions to mixed numbers and mixed numbers to improper fractions. Rename: fractions simplest terms. Find LCD (lowest common denominator). Discriminate between unit fractions and fractions of unity. Solve word-problems involving fractional units (dominate quantities, using common and decimal fractions) to find area, surface area, length, perimeter, and volume.

Place-value / Properties of Numbers and Operations / Notation: Identify ordinal Position. Understand that ordinal numbers extend the set of natural numbers (N0). Compare ordinal with cardinal numbers. Using the standard place value nomenclature and notation, read and write mixed numbers up to the hundred decillions with decimal fractions to the hundred decillionths. Read and write decimal fractions with the proper place-value nomenclature up to thirty-five places to the right of the decimal point. Identify the value each place/period represents for whole and decimal numbers. Rewrite standard notation to expanded notation. Using the proper nomenclature, read and write common fractions with up to twelve digit numerators and denominators. Order and compare any set of magnitudes expressed with: common fractions, mixed numbers with common fractions, decimal fractions, mixed numbers with decimal fractions, p, e, (Euler’s number), f (the golden ratio), Ã, etc. using the signs <, >, ?, =, ˜ .Identify the correct place-value period names for powers of 10 in increments of three, from 0-36 (e.g., 100 = one, 103 = one thousand, 106 = one million, 1012 = one trillion … 1036). Master-recall standard SI prefixes for 101, 102, 103, 106 …1024. Read and write whole numbers up to the hundred decillions using engineering and normalized exponential (scientific) notation (6.721 x 107 = sixty-seven million, two hundred ten thousand = 62.71 x 106). Expand numbers written in scientific and engineering notation and read them using the proper place-value nomenclature. Read numbers on a calculator expressed in E-notation (normalized exponential notation). Round numbers to any increments of whole number (e.g., hundreds, tens, ones) and any decimal increment. Use place value understanding and properties of operations to perform multi-digit arithmetic. Use place value understanding to round whole numbers to the nearest 10 or 100 3.NBT.A.1. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction 3.NBT.A.2. Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations 3.NBT.A.3.

 PV SI 100 one - 101 ten deca- 102 hundred hector- 103 thousand kilo- 106 million mega- 109 billion giga- 1012 trillion tera- 1015 quadrillion peta- 1018 quintillion exa- 1021 sextillion zetta- 1024 septillion yotta- Systems of Measurement / Denominate Numbers: Develop a “sense” for measurement; provide reasonable estimates of an object for length, weight/mass, and volume in the SI and US metrics. Flash-recall /Seconds in a: minute; hour; day / Minutes in an: hour; day; week / Hours in a: day; week; month / Days in a: week, each month; year; decade; century / Months in a: year; decade; century / Years in: decade; century; millennium. Explain: The frequency of leap years; The meaning of AM and PM. Using an analog and digital clock, measure time intervals and record in written form (using the standard notation and nomenclature including AM, PM); to the second, minute, and hour. Solve word problems involving addition, subtraction, multiplication, and division of time intervals. Tell and write time to the nearest minute and measure time intervals in hours, minutes, and seconds. Solve word problems involving addition and subtraction of time intervals in minutes and seconds. Use standard notation for time. Represent and find the relative sequence and order of historical events on a visual time-line. 3.MD.A.1. Identify and name the phases of the moon; describe the physics involved. Name the seasons: summer; fall; winter; spring and explain variations in temperature over the year. Explain the variations in climate relative to latitude and altitude. Compare climate with weather. Solve word-problems involving addition and subtraction of time intervals including minutes, hours, days.

LENGTH, AREA, VOLUME: Measure and estimate liquid volumes and weights of objects using standard unit in the SI and US customary. Round numbers to the nearest whole number, ten, hundred, thousand, ten-thousand, hundred-thousand, million, ten-million, hundred-million, within denominate classifications. Mentally calculate running purchase price totals. Estimate/ Calculate conversions between the US customary and SI units (e.g., one meter is about 39 inches; one yard is about 91 centimeters; one mile is about 1.6 kilometers; one kilometer is about .62 miles; one inch is about 2.5 centimeters; 1 centimeter is about .4 of an inch; one pound is about .5 kilograms and one kilogram is about 2.2 pounds; one gallon is about 3.8 liters and one liter is about 1.06 quarts; etc.). Make linear measurements in yards, feet, and inches, miles centimeters, meters, and kilometers. Estimate linear measurements, measure to check. Add, subtract, multiply, or divide to solve word problems involving denominate numbers (within the same system and requiring conversion between systems). Measure and estimate liquid volumes and masses of objects using standard units of grams, kilograms, and liters. Estimate and measure liquid volumes and masses of objects using standard units. Use addition, multiplication, division, or exponentiation to solve multi-step word-problems including: word-problems containing “noise” (extraneous information), and interdependent word-problems (word-problems dependent other word-problems) involving masses or volumes that are given in the same units. Represent and interpret data 3.MD.A.2. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs 3.MD.B.3. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters 3.MD.B.4. Understand concepts of area and relate area to multiplication and to addition. Recognize area as an attribute of plane figures and understand concepts of area measurement 3.MD.C.5. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area 3.MD.C.5.a. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units 3.MD.C.5.b. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units) 3.MD.C.6. Relate area to the operations of multiplication and addition 3.MD.C.7. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths 3.MD.C.7.a. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning 3.MD.C.7.b. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning 3.MD.C.7.c. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems 3.MD.C.7.d. Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters 3.MD.D.8. Understand concepts including: frequency; period (the reciprocal of the frequency); hertz; cycles per second; revolutions per minute.

CURRENCY: Calculate tax and tips. Calculate change from purchases. Make change, using as few coins as possible. Add, subtract, multiply and divide amounts of money.

Statistics / Factorials / Probability / Odds / Graphs: Create, interpret information depicted on bar graphs, line graphs, and scatter plot. Solve simple problems using information in a bar graphs, line graphs, and scatter plots. Record outcomes for a series of events (coin flips, dice rolls, etc.) and display the results graphically. Understand factorial operation and notation, and compute factorials (e.g., n! 5! =120 6! = 720). Mentally calculate up to 8! Understand: factorial notation involves only non-negative integers; a factorial is the product of all positive integers less than and including a given number. Understand that the value of 0! =1 and 1! =1 and demonstrate provide and informal mathematical proof. Realize and explain that, there are always “n!” ways to arrange “n” distinct objects (e.g., three distinct things can be arrange six unique ways; ABC, ACB, BAC, BCA, BAC, CBA, CAB.) Solve real-world-problems, word-problems, and mathematical problems involving factorials, including arrangements and rearrangements. Add, subtract, multiply and divide factorials. Understand that probability is a measure of the likelihood than an event will occur and is measured with quantities 0-1 (0 = no probability and 1 = 100% probability.) Understand that the higher the probability that an event will occur the more nearly certain it is that that event will occur and vice versa. Understand the concept of bias. Understand that in a fair coin toss, the probability it will land on heads is nearly 50% (there is a very small probability that it will land on its edge).

Geometry: Identify and name triangles (trigons) on the basis of defining attributes. Describe and name two classification schemes that are each exhaustive, that is; all possible triangles are defined by each scheme on the basis of attributes: 1. Right angle attributes include, angle greater than, less than, or equal to a 90º.The right angle category comprises: right triangles, obtuse triangles, acute triangles. 2. Relative length of sides attribute and the category comprises: equilateral triangles, isosceles triangles, scalene triangles. Recognize that an equilateral triangle is also a regular polygon and the interior angles of all triangles add to 180º. Identify a classification scheme that sorts all possible convex quadrilaterals into categories based on attributes (see quadrilateral graphic). Parallelograms include rectangles and rhomboids, attributes of parallelograms include: opposite sides are parallel and of equal measure; the class of rectangles include a special case of rectangle, the square, and the general case, the rectangle. Attributes of the class of rectangles include, all angles are 90º and diagonals are of equal measure. The special case rectangle, the square, has an additional attribute, all four sides are of equal length. The class of rhomboids includes the special case of rhomboid, the rhombus, and the general case, the rhomboid. Attributes of the class of rhomboids include diagonals that are of a different measures, adjacent angles are of different measures and opposite angles are of an equal measure. The special case rhomboid, the rhombus, has an additional attribute, all four sides are of equal length. The trapezoids and trapezia makeup the last to classes. The class of trapezoids includes a special case, the isosceles trapezoid, and the general case, the trapezoid. The class of trapezia includes the special case, the kite, and the general case, the trapezium. Recognize that a square (tetragon) is also a regular polygon and all convex quadrilateral’s interior angles add to 360º. Identify categories and properties of triangles and quadrilaterals 3.G.A.1. Partition regular polyhedra into parts with equal areas. Express the area of each part as a unit fraction of the whole 3.G.A.2. Understand that square units, is a two dimensional measure, and cubic units is a three dimensional measure. Find the midpoint; perimeter; area; radius; length of side; apothem and angle measures (central, inside, outside) of regular polygons including: trigon, tetragon, pentagon, hexagon, heptagon, octagon, nonagon, decagon. Find side lengths of rectangles with the same perimeter and different areas, or with the same area and different perimeter. Transform formulas for the hypotenuse, area and perimeter for shapes including: circles, triangles, quadrilaterals. Know or derive formulas for surface area and volume of spheres, right pyramids (including right cones) with regular polygon bases, and right prisms with regular polygon bases. Plot points on a Cartesian Plane using ordered pairs. Given the co-ordinates, plot regular and irregular polygons on a Cartesian plane. Understand and use standard geometry nomenclature and apply concepts including: degrees in a circle, pi, angle, midpoint; regular polygons; Angles (adjacent, opposite, complementary, exterior, interior, and supplementary); Lines (parallel, segment, perpendicular, ray, transverse, congruent, similar, oblique, perpendicular. Solve word-problems involving perimeter, area, surface area, and volume with denominate measures

Fourth Grade / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prerequisite Knowledge / Skill Retention-reactivation: Before providing instruction in the following objectives, the instructor shall assess students’ retention / facility with objectives articulated for prior grades and review or reteach when indicated. To satisfy mathematics requirements for the current grade, students must demonstrate mastery in objectives articulated for prior grades along with those articulated for the current grade.

Six Arithmetic Operators Addition +, å Subtraction – Multiplication x, 2*3, of, 2•3, ab, 3a, 3(a+b) Division ??, ÷, •, ¾, %, .75, a:b Exponentiation 4², 42/1, 4.5, 4**2, 4^2, 4-² Root taking v, v4, 4½, 4.5, ?8 Choral activities: Choral Stream / Call and Response (C&R) Mental Calculations: Mastery-recall equivalent common fractions (single digit numerators and single digit denominators (excluding 0)), decimal fractions, and percent (e.g., 3/7 = .428561 … = .42 7/8 = 42 7/8 % 1/8 = .125 = 12.5% = 12 ½ %). Using the factorial cognitive map, mentally calculate factorial products 1-10. With and without the common fractions/decimal fractions cognitive maps mastery-recall relationships and nomenclature, give examples (e.g., All proper fractions are less than one, all improper fractions are greater than one, all fractions of unity are equal to one. Every digit to the right of the decimal point is less than one, every digit to the left of the …). Mastery-recall common operators for all Six Arithmetic Operations.

a^0= 1 a?+a? = 2a? a^m·a^(n )= a^(m+n) (a?)n=a?n a?/an = a??n (a/b)? =a?/b? a 1/n = v(n&a) a^(-n)=1/a^n Numeric Operations: Use a factor tree to divide a number into its prime factors. Identify; flash calculate or recall squares including: 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 202, 252, 302, 352, 402, 452, 502, 552, 602, 652, 702, 752, 802, 852, 902, 952, 1002, 11n. Calculate, and/or recall their square roots. Identify prime and composite numbers. Flash calculate or recall common cubes and cube roots. Add, subtract, multiply, divide, exponentiate, and take roots of number with decimals. Master left to right, vice versa add, subtract, and multiply (multi-digit) addends, subtrahends, multipliers. Effectively use strategy shifts with partial products and sums. Generalize place value understanding for multi-digit whole numbers. Recognize that in the Hindu-Arabic numeral decimal system, each digit positioned to left of another is greater by a factor of ten. Recognize that each digit to the right of another is smaller by a factor of ten. Recognize that this pattern continues without end both right and left; providing the Hindu-Arabic positional numeral system with the capacity to denote infinitely large and infinitely small numbers 4.NBT.A.1.Using the Hindu-Arabic numeral decimal system, read and write multi-digit whole numbers, multi-digit decimal fractions, and multi-digit mixed numbers from the decillions to the decillionths. Recognize that the Hindu-Arabic system is a base-10 cyclical system with ten unique digits 0-9. Recognize that the Hindu-Arabic system utilizes positional notation to indicate orders of magnitude. Recognize that period names count in revise order of the direction a number is read. Recognize that, while reading a place-value number, saying “and” indicates the location of the decimal point and signals that the part of the number about to be read is a decimal fraction. Compare two multi-digit numbers based on values of the digits and the position the digits occupy using >, =, and < symbols 4.NBT.A.2. Use understanding of positional notation, to round multi-digit numbers decimal numbers to any place. Use knowledge of place value and properties of operations to perform the six arithmetic operation with number of any digit length 4.NBT.A.3. Fluently add, subtract, multiply, and divide multi-digit whole numbers, multi-digit decimal fractions, and multi-digit mixed numbers using standard algorithms and situational specific strategies 4.NBT.B.4. Multiply multi-digit whole numbers, multi-digit decimal fractions, and multi-digit mixed numbers by multi-digit whole numbers, multi-digit decimal fractions, and multi-digit mixed numbers using strategies based on place value and the properties of operations. Illustrate and explain calculations with equations, rectangular arrays, and/or models 4.NBT.B.5. Find whole-number quotients and remainders using strategies based on place value, the properties of operations, and/or relationships between multiplication and division. Recognize and use the three forms of remainders (R, fractional, and decimal) and select the appropriate form for the given situation. Illustrate and explain calculations with equations, rectangular arrays, and/or models 4.NBT.B.6. Recognize and name common operators, grouping symbols, and mathematical notation including: ¸ (obelus), å (summation; upper case sigma), vinculum, brackets, parenthesis, braces, up caret, ¥ (lemniscate), asterisk and double asterisk, ellipsis. Discuss and numerically illustrate: multiplicative inverse; reciprocal; involution; Identify and name operands, and operators related to the six arithmetic operations including exponentiation, root taking, and logarithms.

OPERATIONS AND NUMBERS: Identify “properties of numbers” (field axioms) including: com mutative, associative, distributive, identity, symmetric (if a=b and b=a), multiplicative axiom (if a=b and c=d the n ac=bd); Axioms of equality: reflexive a=a, transitive (if a=b and b=c then a=c), substitution, partition (a quantity is equal to the sum of its parts), addition (if a=b and c=d then a+c=b+d), subtraction, multiplication, division; axioms of inequality; closure. Recognize / generate number series including: square, triangular, factorial, Fibonacci, etc.

Addition Multiplication Commutative a+b = b+a Commutative ab = ba Associative a+(b+c) = (a+b)+c Associative (ab)c = a(bc) Identity a+0 = a Identity a × 1 = a Inverse a +(–a) = 0 Distributive a(b+c) = ab+ac Inverse (Reciprocal) a x 1/a = 1 Distributive (a + b)c = ac + bc

INTEGERS: Understand and use standard integer notation. Identify an integer’s opposite; sum an integer and its opposite to zero. Add, subtract, multiply, divide, exponentiate, and take the roots of integers and signed numbers. Plot integers and signed numbers on a number line. Using graphic representations, diagrams, and algebraic variables to translate word-problems (including denominate numbers) into linear equations. Solve multistep word problems including: indirect word problems (information required to solve a problem is not given but must be derived from the given data); and word problems with extraneous information or “noise” (relevant information is scattered within jumbled data). Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Add, subtract, dived, exponentiate to solve real-world problems and word-problems containing integers, signed numbers, and fractions.

FRACTIONS, RATIOS: Add, subtract, multiply, divide common fractions and mixed numbers with like and unlike denominators. Exponentiate, and take roots of common and decimal fractions. Find the LCD (lowest common denominator), LCM (lowest common multiple), and GCF (greatest common factor) of common fractions with unlike denominators. Reduce fractions to simplest terms. Identify the reciprocal of fractions and recognize that the product of a fraction and its reciprocal is a fraction of unity (one). Understand that division by zero is undefined and the equivalent; any common fraction with a denominator of zero is undefined. Understand that the product of any multiplicand, and proper fraction multiplier, is a lesser quantity than the multiplicand. Find a percent of a quantity including: whole, numbers, fractions, denominate units. Identify fractions as representing division and recognize the equivalence of nomenclature of division operators and operands (e.g, devisor, and denominator; dividend and numerator; quotient, decimal ratio). Identify whole numbers as having implied denominators of “one”. Discriminate between fractions-of-unity and unit-fractions. Show relative size/value of common fractions, decimal fractions, and percentages; using <, =, >. Translate between: common fractions, decimal fractions, percentages, ratios. Use ratios to express denominate measure relationships such as; proportion, speed, velocity, and mass-density. Write and read ratio notation with confidence. Translate between decimal, common fraction, and ratio notation smoothly. Relate the concept of “odds” to ratios and simple probability. Recognize that decimal and common fractions, percentages, proportions and ratios are all expressions of the fourth arithmetic operation (division). From the conceptual basis of number-of-parts versus size-of-parts, understand and demonstrate equivalent fraction using a concrete model e.g., Fold a standard size sheet of paper in half, unfold to find the sheet is divided into two parts, refold it and fold it in half again, unfold and find the sheet divided into four parts, repeat and find the paper divided into eight parts and so on … note the exponential series (2, 4, 8, 16, 32, …) Fold a fresh sheet of paper in half and color one half of it with a marker. Repeat the previous series of folding. With every fold, take note of the evolving equivalent (1/2, 1/4, 1/8 …) Next: fold a fresh sheet to fourths and color only one-fourth … equivalent fractions of one-fourth. Repeat and color three-fourths …. Note that as the denominator becomes larger their sizes become smaller however the ratio between the numerator and denominator remain the same.) Understand and demonstrate equivalent fractions using a numeric illustration (e.g., any quantity divided by that same quantity is equivalent to one. Any quantity multiplied by one is that same quantity. All common fractions with numerators and denominators that represent the same quantity are fractions of unity. All fractions of unity are equal to one, therefore any fraction of unity times any quantity is the same quantity: , , ). Use these principles to Common Fractionsrecognize and generate equivalent fractions 4.NF.A.1. Using symbols >, =, or <, compare common fractions: with different numerators and the same denominator, with the same numerator and different denominators, with different numerators and different denominators. Use various problem-specific, cost-benefit strategies (e.g., Benchmark: because is greater than and is less than . Versions of “renaming unlike to like denominators:” because ; Cross multiplication because 3·61>2·71; Logic because 151<191; Prime factoring denominators + cross multiplication; Comparison of ratios because .6¯6¯6¯ > .6 = ) 4.NF.A.2. Understand that a unit fraction is a rational number and the reciprocal of a positive integer. Recognize that fractions of unity and unit fractions are not the same. Recognize that the reciprocal of a unit fraction is a whole number and the reciprocal of a whole number is a unit fraction. Recognize that the quotient of a common fraction, is that fraction’s equivalent decimal fraction; And that the reciprocal of a unit fraction’s decimal equivalent decimal equivalent of the unit fractions equivalent. Recognize that the product of a unit fraction and its reciprocal results in a product that is a fraction of unity 4.NF.B.3. Understand addition and subtraction of fractions as joining and separating parts 4.NF.B.3.a. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions 4.NF.B.3.b. Add, subtract, multiply, and divide mixed numbers with like and unlike denominators. Add, subtract, multiply, divide and exponentiate mixed decimal numbers 4.NF.B.3.c. Solve word problems involving addition, subtraction, multiplication, and division of common fractions with like and unlike denominators, and decimal fractions 4.NF.B.3.d. Multiply and divide common fractions and mixed numbers: by a whole number, by another fraction, and by a mixed number 4.NF.B.4. Understand a fraction a/b as a multiple of 1/b 4.NF.B.4.a. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number4.NF.B.4.b. Solve word-problems involving addition, subtraction, multiplication, and division of proper and improper fractions (with like and unlike denominators) 4.NF.B.4.c Rename common fractions to their decimal equivalents and rename decimal fraction to their common fraction equivalents. Use standard notation to express common and decimal fractions. Compare decimal fractions to common fractions. Express a fraction with denominator of 10 as an equivalent fraction with denominator 100, and use this technique to add common fractions with unlike denominators that are expressed as factors of ten 4.NF.C.5. Rename common fractions with denominators that are factors of ten to decimal fractions and rename decimal fractions to common fractions with denominators expressed in factors of ten. Use decimal notation for fractions with denominators 10- 1,000,000,000 4.NF.C.6. Compare decimals fractions, reasoning about their size (e.g., .1 >.09999999). Record the results of comparisons with the symbols >, =, or <, and justify the conclusions 4.NF.C.7.

 PV SI 100 one - 10-1 tenth deci- 10-2 hundredth centi- 10-3 thousandth milli- 10-6 millionth micro- 10-9 billionth nano- 10-12 trillionth pico- 10-15 quadrillionth femto- 10-18 quintillionth atto- 10-21 sextillionth zepto- 10-24 septillionth yocto- ALGEBRAIC EXPRESSIONS AND EQUATIONS: Know and use the properties of exponents in life-problems, word-problems, and numeric problems. Translate equations into verbal statements (1. An object’s speed is the distance it travels in an interval of time. 2. An object’s gain in speed (acceleration) is determined by the time it takes to gain additional speed. 3. The amount of energy used to do an amount of work determines how efficiently the work is done.) Represent verbal statements about numeric relationships with alphanumeric expressions (speed=distance / time, s=d/t; Acceleration=change in speed/time interval) 4.OA.A.1. Using word-problems, drawings, equations, Venn diagrams, or “number theory” graphics illustrate “comparisons” between and within the six arithmetic operations (See: number theory graphics illustrating “between” comparisons: Multiplication as repeated addition; Division as repeated subtraction; Exponentiation as repeated multiplication. Word-problem presenting a “within” multiplicative comparison: “Bob is a flashy fop who wears waistcoats and collects fobs. He has seven silver watch fobs flopped on each of four shelves, hidden beneath five loose floor boards. He has fivefold as many gold fobs as silver, folded in with his seventy-six socks, spread evenly between two boxes hidden beneath his four post bed. If half the fobs on two of the selves beneath his floor, hop off and float away, how many fobs will Bob the fop have in his fob collection?”) Solve word problems involving arithmetic and subtractive comparisons 4.OA.A.2. Translate word-problems involving addition, multiplication, division (with remainders), and exponentiation into linear equations with a letter standing for the unknown quantity. Solve word-problems that require multiple operations, contain “noise” (extraneous information), and that are indirect (digressive.) Assess reasonableness of answers using mental computation, estimation strategies and rounding4.OA.A.3. Construct factor trees (prime factor multi-digit whole numbers). Recognize that a whole number is a composite of its factors. Determine whether two given whole numbers have prime factors in common. Determine whether a given whole number is prime or composite 4.OA.B.4. Generate and analyze patterns. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself 4.OA.C.5.

Place Value: Identify the correct place-value period names for powers of 10 in increments of three, from 0 –36 (e.g., 100 = one, 10-3 = one thousandth, 10-6 = one millionth, 10-12 = one trillionth … 10-36). Master-recall standard SI prefixes for 10-1, 10-2, 10-3, 10-6 …10-24. Read and write whole numbers to the hundred decillionths using engineering and normalized exponential (scientific) notation (6.721 x 10-7 = sixty-seven millionths, two hundred ten thousandths = 62.71 x 10-6). Expand decimal fractions written in scientific and engineering notation and read them using standard place-value nomenclature. Read decimal fractions on a calculator expressed in normalized exponential notation.

 gallon half-gallon quart pint cup ounce ounces 128 64 32 16 8 1 cup 16 8 4 2 1 .128 pint 8 4 2 1 .5 .0625 quart 4 2 1 .5 .25 .03125 half-gallon 2 1 .5 .25 .128 .015625 gallon 1 .5 .25 .128 .0625 .0078125 Systems of Measurement / Denominate Numbers: Develop a “sense” for measurement; estimate and measure volumes, weights, and lengths of objects using SI and United States customary units. Solve problems involving measurement and conversion of measurements. Use ratios to convert between SI and US customary units. Know the approximate ratios to convert from SI to US: kilometer ˜ .62 mile, meter ˜ 1.09 yard, centimeter ˜ .39 inch, liter ˜ .26 gallon, kilogram ˜ 2.20 pounds. Convert within SI; know and understand the factors of ten relationship between SI measures of; Length: millimeter, centimeter, decimeter, meter, dekameter, hectometer, kilometer; Volume/capacity: milliliter, centiliter, deciliter, liter, dekaliter, hectoliter, kiloliter; Weight/mass: milligram, gram, kilogram. Convert within US customary units. Know the unit relationships for; Length: inches in a foot, feet in a yard, inches in a yard, feet in a mile; Weight: ounces in a pound, pounds in a ton; Volume/capacity: fluid ounces in a cup, cups in a pint, pints in a quart, quarts in a gallon 4.MD.A.1. Know that degrees Celsius is used to measure temperature by most countries in the world; with 100ºC being approximately equal to the boiling point of pure water at sea level (one standard atmosphere) and 0ºC being approximately each to the freezing point of pure water at one standard atmosphere. Know that degrees Fahrenheit is the official measure of temperature in the United States but is seldom used for scientific purposes. Know that the boiling point of pure water at one standard atmosphere is 212ºF freezing point of pure water at one standard atmosphere is 32ºF; and that there are exactly 180º between freezing and boiling.

TIME: Mastery-knowledge-explain: The frequency of leap years; daylight saving time; solar time; laps time; standard time; Greenwich Mean Time (GMT), Coordinated Universal Time (UTC), and Terrestrial Time (TT); leap year, leap seconds; time zones; the international date line; Milankovitch cycles, eccentricity, axial tilt, precession; AM and PM; AD, BC, Current Era (CE), Before Current Era (BCE). Using stopwatches record in written form (using the proper notation and nomenclature) to fractions of a second. Solve word problems involving addition and subtraction of time intervals including minutes, hours, and days, within and across time zones including the international dateline. Flash-recall: seconds in a minute, minutes in an hour, hours in a day, days in a week, approximate weeks in a month, weeks in a year, months in a year, years in a decade, decades in a century, centuries in millennium, years in a millennium. Use the arithmetic operations to solve word problems involving distances, intervals of time, liquid volumes, mass (weight), money, and temperature including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale 4.MD.A.2.

Geometry: Discriminate between right, acute, obtuse angles. Estimate and measure the degrees in an angle. Calculate the degrees in interior angles, exterior angles, and central angles of regular polygons (pentagons, hexagons, heptagons, octagons, nonagons, decagons, etc.). Identify the properties of: regular polygons. Know or derive formulas to find the area and perimeter of regular and common polygons including triangles and quadrilaterals. Know or derive a formula to find the area of triangles including: right, acute, obtuse, equilateral, isosceles, and scalene. Through geometric processes and knowledge of the shapes, derive formulas for each of the following quadrilaterals: square, rectangle, rhomboid, rhombus, isosceles trapezoid, trapezoid, kite, and trapezium. Recognize/ illustrate the special ratios relative to circles and right triangles. Understand that the Pythagorean theorem is a relationship between the sides of a right triangle. Know that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Use knowledge of the Pythagorean theorem solve geometric problems. Derive or know a formula to find the hypotenuse of a right triangle. Relate regular polygons to circles and triangles and derive formula to find interior, exterior, and central angles; perimeter; area; and apothem. Know or derive a formulas for the circumference and area of a circle through knowledge of triangles. Measure or calculate degrees of rotation, arc, and intersection. Know or derive formulas to find the surface area and volume of common polyhedra. Use an additive and/or subtractive strategies to find the surface area and volume of regular and irregular solids including: cones, pyramids, prisms, platonic solids, stellated platonic solids. Apply the area and perimeter formulas for rectangles in real-world and mathematical problems 4.MD.A.3. Represent and interpret data. Make a line plot to display a data set of measurements using unit fractions (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots 4.MD.B.4. Understand concepts of angle and measure angles. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement 4.MD.C.5. Understand that an angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. Understand than an angle that turns through 1/360 of a circle is called a “one-degree angle,” and is used to measure angles 4.MD.C.5.a. Understand that an angle that rotates n-degrees is said to have an angle measure of n degrees 4.MD.C.5.b. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure 4.MD.C.6. Recognize angle measure as additive; when an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve arithmetic problems to find unknown angles on a diagram in real-world and mathematical problems 4.MD.C.7. Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures 4.G.A.1. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles 4.G.A.2. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry 4.G.A.3. Use standard geometry nomenclature including: origin, slope, coordinate. Understand and use notation and nomenclature such as: lines- segments, rays, horizontal, vertical, perpendicular, parallel, tangent, intersection; angles- right, acute, obtuse, perpendicular, opposite, adjacent; polygons- trigon, tetragon, quadrilaterals, regular; solids, platonic solid.

Statistics / Factorials / Probability / Odds / Graphs: From a set of data calculate measures of the central tendency (mean, median, and mode). Plot the relative position of real numbers on a number line including: p, ??, ??, , Š, , 2.3, .23, 3!, etc. Plot linear equations on a Cartesian Plane using ordered pairs. Create, (from data)/ interpret/ analyze various types of graphs including: bar, pie, histogram, stem-and-leaf, and scatter. Plot data containing common and decimal fractions using scatter plot, line graph, bar graph etc. and evaluate. Identify quadrants, and lines of symmetry for two and three dimensional objects and graphs. Understand factorial operation and notation (e.g., 5! = 5x4x3x2x1, 0! =1, 1! =1). Understand that, there are “n!” ways to arrange “n” distinct objects. Solve real-world-problems, word-problems, and mathematical problems involving factorials and arrangements. Add, subtract, multiply and divide factorials with other factorials. Understand that “permutations” relate to the numbers of ways, a given number of items can be arranged, when taken at a given number at a time. Understand: With permutation, order matters; each different order of the same items is a unique permutation (e.g., Ordering the digits 1, 2, and 3: 123, 132, 213, 231, 312, 321 represent six permutations not one.) Understand and use standard permutation notation (i.e., n Pr = ) to solve real-world-problem, math-problems, and numeric problems (e.g., 9 P3 = = = 9·8·7 = 504). Understand: Probability is measured in quantities from “0” to “1” which is a measure of the likelihood that an event will occur. Understand that “0” represents impossibility and “1” represents certainty. Solve real-world-problems, word-problems, and mathematical problems involving probability (without replacement) and draw inferences about expected frequency. Construct a frequency distribution table using real-world data by recording event outcomes (e.g., total face value of two dice, for each roll, of fifty rolls). Create a histogram using such data. Relate empirical frequency distribution to the concept of a probability distribution of a continuous variable. Understand and use the probability formula, P (E) = where n(S) is the number of possible outcomes and n(E) is the number of favorable outcomes. Understand that empirical probability estimates probabilities from observation. Understand: Theoretical probability is the ratio of the number of times a specific event is expected to occur relative to the total number of possible outcomes. Create a theoretical frequency distribution and calculate theoretical probability base on the frequency distribution. Solve real-world-problems, word-problems, and numeric problems.

Fifth Grade / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prior Knowledge / Prerequisite Comprehension / Skill Retention-reactivation: Before providing instruction in the following objectives, the instructor shall assess students’ retention and facility with objectives articulated for prior grades and review or reteach where indicated. To satisfy mathematics requirements for the current grade, students shall demonstrate mastery in objectives articulated for prior grades along with those articulated for the current grade.

a/b+c/d=(ad+bc)/bd a/b-c/d=(ad-bc)/bd a/b×c/d=ac/bd a/b÷c/d=ad/bc (a/b)^n=a^n/b^n v(m&a/b) = v(m&a)/v(m&b) a^0= 1 a^(-n)=1/a^n a 1/n = v(n&a) a?+a? = 2a? a^m·a^(n )= a^(m+n) (a?)n=a?n (ab)?=a?b? (ab?)n=anb?n (??+??)²= a²+2????+??² a?/an = a??n (a?/bn)? =a??/bn? (a/b)? =a?/b? Numeric Operations: Understand the correspondence of notation between/ radical, logarithmic, and exponential notation (see illustration to the right). Understand exponential notation that includes exponents that are positive or negative, and exponents that are common or decimal fractions. Understand that every non-negative real number has a unique non-negative square root (the principal square root). Understand that every positive number has two square root, one positive, the other negative. Identify and name perfect squares (22, 32, 42…192). Use knowledge of notational correspondence between related operations to transform equations into equivalent forms. Demonstrate knowledge of order-of-operations along with grouping tools useful for altering operational-order including; vincula, parentheses, brackets, braces. Evaluate expressions containing these symbols. Understand and use exponents while solving real-world word-problems. Solve word-problems requiring the proportional scaling (up and down) of objects with given dimensions. Understand that a root of degree 2 is often called the “square root,” a root of degree 3 is often called the “cube root,” and a root of degree 4 is called the “fourth root,” and degree 5 the “fifth root…” Understand that there are two square roots for positive radicands (e.g., ±2) and similarly two roots for even radicands with even indexes. Recognize the equivalence of radical notation and exponential notation representing the operation of root taking (i.e., =) and switch between fluidly. Understand properties of radicals and exponential and radical equivalent notation including: fractional exponents (i.e., = = ); and decimal exponents (e.g., ) and negative exponents (i.e., = ). Understand numeric operations within radical notation including: addition; subtraction; multiplication ( = or =); and division. Understand that the “principal root” is the positive root of a positive number ( +3 is the principal root) but is sometimes also used to refer to the negative root of a negative radicand with an odd index ( is the principal and only root). Understand that in radical notation, even indexes and positive radicands imply too roots (positive and negative) and that exponents that are common fractions imply only the principal root. Understand that the “imaginary unite” refers to the solution for = ?1 = = ?. Prime factor multi-digit numbers.

FRACTIONS/ RATIOS/ PERCENTAGES/ PROPORTIONS: Recognize and use standard notation in reference to: common fractions, decimal fractions; ratios, percentages. Using standard algorithms and problem specific strategies: add, subtract, multiply, and divide common fractions and mixed numbers with like and unlike denominators. Exponentiate, and take roots of common fractions and decimal fractions. Use equivalent fractions as a strategy to add and subtract common fractions. Rename fractions with unlike denominators to equivalent fractions with like denominators 5.NF.A.1. Solve word problems involving addition and subtraction of common fractions and mixed numbers (with common fractions), including cases of like and unlike denominators. Use mental estimations to assess the reasonableness of answers 5.NF.A.2. Rename improper fractions to mixed numbers and mixed numbers to improper fractions. Rename common fractions to decimal fractions and decimal fractions to common fractions. Rename common fractions to equivalent common fractions. Reduce common fractions to simplest terms. Understand the concept of relative primes across denominators. Prime factor multi-digit denominators of common fractions as a strategy to find the LCD (lowest common denominator), LCM (lowest common multiple), and GCF (greatest common factor) of common fractions with unlike denominators. Reduce fractions to simplest terms. Identify the reciprocal of fractions and recognize that the product of a fraction and its reciprocal is a fraction of unity (one). Understand that division by zero is undefined and any fraction with a denominator of zero is undefined. Understand the meaning of undefined. Identify the parts of a common fraction: numerator, denominator, and vinculum. Identify fractions as an expression division: denominator (devisor), numerator (dividend) and quotient 5.NF.B.3. Understand the difference between fractions-of-unity and unit-fractions. Compare common fractions with unlike denominators relative to size (e.g., 5/17 to 20/71, 2/13 to 5/31, 3/37 to 6/73) and rank them on a number line. Use ratios to express denominate measure relationships such as; proportion, speed, velocity, and mass-density. Translate notation between decimal, common fraction, and ratio notation smoothly. Compute between decimal and common fractions, percentages, proportions and ratios are all expressions of the fourth arithmetic operation (division). Apply knowledge of equivalent notation, partial products, order of operations, prime factor trees, the double inverse and properties of operations to multiply a common fraction by a: common fraction, whole number, improper fraction, decimal fraction, mixed decimal number, variable. Solve problems involving the six arithmetic operations and fractions in which all numerators and denominators are expressed with indeterminates (variables) 5.NF.B.4. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ bContent 5.NF.B.4.a. Find the area of rectangles, trapezoids, trapezia, triangles, circles, irregular and regular polygons in which measures contain common fractions. Tile the polygons with square units and demonstrate by rearrangement that area can be expressed in square units for any polygon. 5.NF.B.4.b. Find the volume of solids (rectangular prisms; forms that have been created by adding rectangular prisms together; forms that have been constructed by removing rectilinear sections from a rectangular prism, and forms that have been created by slicing a rectangular prisms diagonally. Demonstrate by rearrangement that volume of polyhedra can be measure in cubic units. Recognize the scaling properties of multiplication (i.e., multiplying by a number that is less than one, produces a product that is scaled down; multiplying by a number that is greater than one, produces a product that is scaled up) 5.NF.B.5. Recognize that: The larger the denominator the smaller the quotient. The larger the numerator the larger the quotient 5.NF.B.5.a. Recognize that a given number multiplied by a proper fraction, produces a product that is smaller than the given number; And that a given number multiplied by an improper fraction, produces a product that is greater than the given number 5.NF.B.5.b. Using an equations Solve real world problems involving multiplication of fractions and mixed numbers by constructing an equation built from given information 5.NF.B.6. Solve word-problems involving: rate problems, e.g., speed, freefall (acceleration), mass density, markups and discounts, unit pricing, hourly wages, simple and compound interest, gratuities, commissions, etc.; gambling statistics; and probability arrays. Write equations which demonstrate the relationship between variables in the above examples. Construct tabular arrays graphs that reveal the relationships between numeric changes in variables (e.g., freefall; independent variable G) using intuitive design to facilitate understanding (e.g., slope of regression line slanting up to the right to express increase). Describe the difference between fractions of unity and unit fractions. Recognize that the reciprocal of a unit fraction is a rational integer. Recognize that the product of a unit fraction and its reciprocal results in a product that is a fraction of unity. Divide unit fractions by whole numbers and whole numbers by unit fractions 5.NF.B.7. Understand the mathematical logic involved in executing the double inverse (the inverse of division is multiplication and the reciprocal of a fraction closes the operation) while dividing whole numbers, mixed numbers, and fractions by common fractions. Interpret division of a unit fraction (1/n) by a non-zero whole number, and compute such quotients (i.e., Multiplication of two unit fractions produces a product that is a unit fraction, adding, subtracting, or dividing two unit fractions produces results that are not unit fractions 5.NF.B.7.a. Find the reciprocal of a unit fraction and recognize that the product of a fraction and its reciprocal is a fraction of unity (one). Identify a fraction’s opposite and recognize that the addition of a unite fraction and its opposite is zero. Recognize that division of a whole number by its reciprocal (a unit fraction) is that whole number’s perfect square. Interpret division of a whole number by a unit fraction, and compute such quotients 5.NF.B.7.b. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions 5.NF.B.7.c. Solve numeric problems involving all six arithmetic operations and only unit fractions

NUMBERS/ ABSOLUTE VALUE/ INEQUALITIES: Understand constructs related to integers, signed numbers, intervals, absolute value and inequalities/ use the appropriate notation relative to each, including: >, <, £, ³ , ¹, |x|, [x], (x), –¥, +¥, Ç, È. Understand that the absolute value of a real number is non-negative without regard to its sign and represents its distance from the origin in a coordinate system. Find the opposite of any given integer and recognize that 0 is its own opposite. Recognize that an interval may, or may not include the designated number. Perform arithmetic operations involving denominate numbers, integers, signed numbers, intervals, absolute value and inequalities. Discuss the properties of absolute value relative to arithmetic operations. Solve word-problems and graph equations and/or expressions involving integers, signed numbers, intervals, absolute value and inequalities. Determine the truth or false value of inequalities. Use standard notation for/ define subsets / understand numerical constructs such as: odd-even, integer, signed number, prime, composite, natural, algebraic, real, imaginary unit, rational, irrational, recursive, transcendental; and give examples of each: ; NÌ ZÌ Q Ì R; p, ??, ??; etc. Understand: prime numbers (P) and composite numbers are subsets of natural numbers (composite + prime = 2, 3, 4, …) Be aware that, under some classification schemes, natural numbers do not include “0” and under other schemes “0” is included; in which case the natural and whole number are identical; understand and use standard notation (N0). Discuss similarities and differences of each of aforementioned such as: rational and irrational numbers (e.g., Q: can be expressed as a ratio of two integers, decimal expansions end or repeat. Irrational numbers cannot be expressed as a ratios of two integers, decimal expansions do not end or repeat.) Give approximations for irrational numbers (e.g., p», 3.14; ??»2.718; » 1.414) and find their relative position on a number line.

ALGEBRAIC EXPRESSIONS AND EQUALITIES: Transform between linear equations in standard form ax+by=0, where a and b ? 0; and slope-intercept form y=mx+b, where m is the slope of line b. Solve systems of simple linear equations by: 1) Graphing each equation and identifying point of intersection; 2) Solving for one variable, then substituting one equation into the other; 3) Adding or subtracting equations to elimination variables; 4) Multiplying one or more equations, then eliminating variables by adding or subtracting. Determine if equations have infinite solutions, no solutions (the empty set), or unique solutions (intersection). Determine if two linear equations are: equivalent, independent, inconsistent. Graph linear equations in two variables. Write and interpret numerical expressions using order of operations conventions (i.e., Direction: left to right; stacked exponents top down, right to left; Operators: exponentiation and root taking; multiplication and division; addition and subtraction; Grouping: radical sign, vinculum, parentheses, brackets, braces) Understand the difference between (x-3) and (x¯3). Understand that the horizontal line, in the first example, is an operator for subtraction and means (x minus 3). Understand that the horizontal line in the second example is a property of the number 3, announcing that 3 is a signed number. The second example means (x times negative 3) Use notation conventions to enhance clarity 5.OA.A.1. Write alphanumeric expressions that record operators, coefficients, variables, constants and interpret 5.OA.A.2. Generate two numerical patterns using two given rules. Identify relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane 5.OA.B.3. Translate real-world (applied math concepts relevant to solving practical problems) word-problems into alphanumeric expressions including; representations of numeric relations between constants and variables; solution equations; and solution sets. Solve multistep word problems including: word-problems containing extraneous information or “noise” (relevant information must be discriminated from within a pool of information); indirect word-problems (needed information is not explicitly stated but can be derived from the given information); and interdependent word problems (needed information must be derived from solutions to other problems). Demonstrate competence solving real-world problems requiring conversion between systems of measurement. Understand, discuss, demonstrate inversely and directly proportional relationships (µ) ILLUSTRATED with numeric examples. Use numeric examples to demonstrate comprehension of constructs including: proportion, ratio, and percentage along with arithmetic operations involving these constructs. Solve word-problems involving the previously mentioned constructs.

 100 1 one deci- d 10-1 0.1 tenth centi- c 10-2 0.01 hundredth milli- m 10-3 0.001 thousandth micro- µ 10-6 0.000001 millionth nano- n 10-9 0.000000001 billionth pico p 10-12 0.000000000001 trillionth femto- f 10-15 0.000000000000001 quadrillionth atto- a 10-18 0.000000000000000001 quintillionth zepto- z 10-21 0.000000000000000000001 sextillionth yocto- y 10-24 0.000000000000000000000001 septillionth  1 110 = 1 Pascal’s Triangle 1 1 111 = 11 1 2 1 112 = 121 1 3 3 1 113 = 1331 1 4 6 4 1 114 = 14641 1 5 10 10 5 1 115 = 1 5 10 10 5 1 = 161051 1 6 15 20 15 6 1 116 = 1 6 15 20 15 6 1 = 1771561 1 7 21 35 35 21 7 1 117 = 1 7 21 35 35 21 7 1 = 19,487,171 17,000,000 + 2,100,000 + 350,000+35,000 + 2,100+71 = 19,487,171 Place Value: Understand the place value system. Read numbers using place period names the decillions to the decillionths. Write very large and very small numbers using scientific notation and engineering notation. Read/interpret/use/understand E-notation on a calculator. Rewrite scientific and engineering notation in standard notation. Add, subtract, multiply and divide numbers written in scientific and engineering notation involving denominate numbers. Identify significant figures when reading or writing numbers. Read and write using decimal notation. Understand that every digit to the right of the decimal point is less than one, every digit to the left of the Understand that digits to the left of the decimal point represent a whole number and digits to the right of the decimal point represent a fraction. Understand that in decimal notation: A number with digits only to the right of the decimal point, is a fraction; A number with digits only to the left of the decimal point, is a whole number; A number with digits both right and left of the decimal point, is a mixed number. Understand the “0” exception rule. Understand that digits to the left of the decimal point represent quantities one or greater. Understand that digits to the right of the decimal point represents quantities less than one. Recognize that in a multi-digit number, a digit in the ones place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left 5NBT.A.1. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10 5.NBT.A.2. Read, write, and compare decimals numbers to the decillionths 5.NBT.A.3. Read and write decimals to decillionths using base-ten numerals, number names, and expanded form 5.NBT.A.3.a. Compare two decimals to the hundred millionths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons 5.NBT.A.3.b. Use place value understanding to round decimals to any place 5.NBT.A.4. Understand that when a quantity is expressed in decimal notation: A number with digits only to the right of the decimal point, is a proper fraction; A number with digits only to the left of the decimal point, is a whole number; A number with digits both right and left of the decimal point, is a mixed number. Understand that decimal notation does not provide a means to express improper fractions; And all denominators are understood and are not explicitly stated. Understand that digits to the left of the decimal point represent quantities equal to one or greater. Understand that digits to the right of the decimal point represents quantities that are less than one. Understand the “0” exception rule. Fluently perform the six arithmetic operations on operands that are multi-digit mixed numbers, written in decimal notation, with the whole part and/or the fractional part consisting of any number of digits. Using standard algorithms and a variety of strategies specific to decimal numbers: multiply whole numbers with multi-digit multipliers and multiplicands 5.NBT.B.5. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models 5.NBT.B.6. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used 5.NBT.B.7.

Geometry: Identify geometric transformations (change in position and/or size) including: dilation (change in size, preserves shape); translations (slides, directional displaced); reflections (flips, mirror imagery), rotations (turns, displacement around a central axis). Using only a straightedge and compass construct: perpendicular lines; parallel lines; regular polygons with 1-6 sides; Goethe’s triangle; the proof-by-rearrangement, and “similar triangle proof” for the Pythagorean theorem. Draw two dimensional scaled elevations (top, front, side) from a model and written description. Create scaled three dimensional sketches (using linear perspective): from plans showing top, front, and side views; from written descriptions of objects; and from three dimensional rectilinear objects. Produce linear transformations (up, and down) of one and two dimensional constructions. Produce a uniformly scaled sketch that is proportionally scaled up (dilated) or down from another sketch or photograph. Understand and articulate in oral and written (essay) form, constructs including: Euclidean Plane; Euclidean Space; Dimensions (0-dimension, 1st dimension, 2nd dimension, 3rd dimension, 4th dimension). Recall and understand standard nomenclature used in geometry including: Angles (adjacent, opposite, complementary, exterior, interior, and supplementary, alternate,); Lines (parallel, segment, perpendicular, leg, ray, transverse, vertical, horizontal, congruent, similar, oblique, perpendicular, bisect, intersect); (similarity, congruence, coordinate, x-axis, y-axis, origin, ordered pair, linear transformation, translation, dilation, reflection, symmetry, scale factor) Use standard notation including: ?,, . Graph points on the coordinate plane to solve real-world and mathematical problems. Use a coordinate plane (Cartesian coordinate system) on which to plot ordered pairs (0,0) of points. Understand that the first number indicates displacement from along the x-axes and the second number indicates displacement along the y-axes. Use standard nomenclature that corresponds to the two axes and the coordinates (e.g., x-axis and x-coordinate, y-axis and y-coordinate)5.G.A.1. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation 5.G.A.2. Classify two-dimensional figures into categories based on their properties. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles 5.G.B.3. Classify two-dimensional figures in a hierarchy based on properties 5.G.B.4. Convert like measurement units within a given measurement system. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems 5.MD.A.1. Represent and interpret data. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots 5.MD.B.2. Understand concepts of volume. Recognize volume as an attribute of solid figures and understand concepts of volume measurement 5.MD.C.3. Recognize that a cube with side length of one unit, (unit cube,) fills a volume of “one cubic unit” that can be used to measure volume 5.MD.C.3.a. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units 5.MD.C.3.b. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units 5.MD.C.4. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume 5.MD.C.5. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication 5.MD.C.5.a. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems5.MD.C.5.b. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems 5.MD.C.5.c.

nCr = n!/r!(n-r)! = (nPr )/( r!)n Pr = n!/(n-r)! Statistics/ Graphing / Factorials, Permutations, Combinations, Probability: Understand/ recognize/ discriminate between variables including: qualitative, quantitative, discrete, continuous, confounding, dependent, Graph linear equations with one, two, and three variables (e.g., locate separate points along a one dimension. Construct an informal quantitative argument relative to an infinite data pool distribution curve and a Gaussian distribution (continuous probability distribution). Understand measures of the central tendency (e.g., mode, median, arithmetic mean.) Contrast the concept of the central tendency with the concept of dispersion. Relate a frequency distributions to permutations, combinations, and normal distributions. Understand that with “permutation” order matters and that with “combinations” order does not matter. Understand and use standard notation to represent and solve real-world-problem, math-problems, and numeric problems involving combinations and permutations with and without replacement. Understand: Solve real-world-problems, word-problems, and mathematical problems involving probability with and without replacement. Construct graphs, charts and tables using collected data. Read and interpret (through written essays and oral discussions) graphs, charts, and data arrays including: line graph, bar graphs, pictographs, flowcharts, pie charts, waterfall charts, stacked bar charts, box-and-whisker charts, scatterplot. Construct

Create a histogram using such data. Relate empirical frequency distribution to the concept of a probability distribution of a continuous variable Discriminate between: a theoretical distribution and an observed distribution; theoretical probability (ratio of number of ways an event is expected to occur to the total number of possible outcomes) and empirical probability (relative frequency, experimental probability; ratio of the number of times a specific event was observed to occur: to the total number of trials). Create a theoretical frequency distribution and calculate theoretical probability base on the frequency distribution. Understand the general concepts (surface ideas not rigorous definitions) of ideas including: bin (bucketing); probability density; cumulative frequency; sample space; bias. Understand that an unlikely event can occur at any time during an event cycle, even the first.

9C3 = 9!/3!(9-3)! = (9·8·7·6·5·4·3·2·1)/(3·2·1 (6·5·4·3·2·1)) = (9·8·7)/(3·2·1 ) = 84
9 P3 = 9!/(9-3)! = (9·8·7·6·5·4·3·2·1)/(6·5·4·3·2·1) = 9·8·7 = 504


9C3 = (9P3 )/( 3!) = (9·8·7)/(3·2·1 ) = 84

 

 

Sixth Grade / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prerequisite Knowledge / Skill Retention-reactivation: Before providing instruction in the following objectives, the instructor shall assess students’ retention / facility with objectives articulated for prior grades and review/ reteach where required. To satisfy mathematics requirements for the current grade, students must demonstrate mastery in objectives articulated for prior grades along with those articulated for the current grade.

a^0=2a^m a^m+a^m=2a^m a^n a^m=a^(n+m) a^m/a^n =a^(n-m) (a^m )^n=a^mn (ab)^m=a^m b^m (?ab?^m )^n=a^n b^mn (a+b)^2=a^2 ?+2ab+b?^2 v(n&a^m )=a^(m/n) (ab)^n=a^n b^n (a/b)^n=a^n/b^n (a^m/a^n )^n=a^mn/a^nn a^(1/n)=v(n&a) a^(-n)=1/a^n Numeric Operations: Apply knowledge of the six arithmetic operations, their repeated and inverse operations, properties of numbers, and properties of operations to understand notational equivalence in numeric expressions and to solve numeric problems. Demonstrate knowledge of order of operations including the use of grouping symbols including; vincula, parentheses, brackets, braces; and evaluate expressions containing these symbols. Solve word problems requiring the proportional scaling (up and down) of objects with given dimensions. Understand that radical notation with an even indexes and positive radicands imply both positive and negative roots and that exponents that are common fractions do not. Understand that the “imaginary unite” refers to the solution for = ?1 = = ?. Calculate and understand quotients of fractions. Solve word-problems involving addition of fractions to fractions, subtraction of fractions from fractions, multiplication of fractions by fractions, division of fractions by fractions 6.NS.A.1, exponentiation of fractions with fractional exponents (both improper and proper). Fluently use a variety of standard algorithms and problem specific strategies to add, subtract, multiply, divide, exponentiate, and take roots of: whole numbers that are multi-digit 6.NS.B.2, decimal numbers that are multi-digit, and numbers that have multi-digit whole parts and multi-digit decimal numbers 6.NS.B.3. Find common factors and multiples of multi-digit numbers. Prime factor multi-digit numbers. Use a variety of standard algorithms and problem specific strategies to add fractions to fractions, subtract fractions from fractions, multiply fractions by fractions, divide fractions by fractions, and exponentiation fractions with fractional exponents (both improper and proper) 6.NS.B.2. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor 6.NS.B.4. Apply knowledge of the six arithmetic operations, their repeated and inverse operations, properties of numbers, and properties of operations to understanding the system of rational numbers. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation 6.NS.C.5. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and on a plane with negative number coordinates 6.NS.C.6. Recognize opposite signs of numbers as indicating locations on opposite sides of the origin on a number line. Recognize that the opposite, of the opposite of a number is that number itself and that 0 is its own opposite 6.NS.C.6.a. Understand that signs of the numbers in an ordered pair, indicate which of the four quadrants (of a two dimensional coordinate plane) the point is located. Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes 6.NS.C.6.b. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane 6.NS.C.6.c. Understand ordering and absolute value of rational numbers 6.NS.C.7. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram 6.NS.C.7.a. Write, interpret, and explain statements of order for rational numbers in real-world contexts 6.NS.C.7.b. Understand the absolute value of a rational number as its distance from 0 on the number line. Interpret absolute value as magnitude for a positive or negative quantity in a real-world situation 6.NS.C.7.c. Distinguish comparisons of absolute value from statements about order 6.NS.C.7.d. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate 6.NS.C.8.

FRACTIONS/ RATIOS/ PERCENTAGES/ PROPORTIONS/ODDS: Add, subtract, multiply, divide common fractions (including complex and compound fractions) and mixed numbers with like and unlike denominators. Exponentiate, and take roots of common fractions and decimal fractions. Rename fractions with unlike denominators to fractions with like denominators. Reduce common fractions to simplest terms. Understand that division by zero is undefined ant that any common fraction with a denominator of zero is undefined. Explain why any given number multiplied by a common fraction, produces a product that is less than the given number. Recognize the inverse and direct relationships of the numerator and denominator to magnitude (e.g., The larger the denominator the smaller the quotient. The larger the numerator the larger the quotient.) Find the reciprocal of a fractions and recognize that the product of a fraction and its reciprocal is a fraction of unity (one). Identify a fraction’s opposite and recognize that the addition of a fraction’s opposite is zero. Discriminate between fractions-of-unity and unit-fractions. Read and/ write decimal fractions to the decillionths. Identify significant figures and understand conventions of rounding relative to writing and interpreting numbers. Use ratios to express denominate measure relationships such as; proportion, speed, velocity, mass-density, work, energy, power, pressure, buoyancy. Demonstrate equivalence of division notation, and translate between: percentages, common fractions, decimal fractions, quotients, and odds. Know that “Odds” are ratios and always represented by a pair of numbers; used in gambling and statistics. Know that “odds for” indicates the likelihood that a particular event will occur; and “odds against” indicates the likelihood that an event will not occur. Understand: that in gambling, odds are represented with ratios that indicates the amounts staked by parties to a wager (3:1 OR “3 to 1” odds, mean the first party stakes three times the amount that the second party stakes); that 3 to 1 odds mean that there are 3 possible ways an event cannot take place to each one way the event can occur (ratio of non-events to events). Find the percentage one quantity is of another; Find the percent of rational numbers including whole numbers, mixed numbers, fractions (common and decimal), denominate quantities (including ratios; i.e., 5% of the speed of light) 6.RP.A.3.c. Write equations expressing such relationships. Understand that: A ratio represents the number of times one number contains another (a ratio written a:b, or “a to b” where b?0, represents the number of times “a” contains “b”); Ratios are equivalent to fractions and can be reduced; Ratios can be expressed as common fractions, decimal fractions, or quotients 6.RP.A.1; If a=apples and b=bananas, a:b represents, “apples to bananas;” Ratios can represent relationships between quantities of the same or different units; Proportionality is an expression of the ratio concept; “Rate” is a ratio; The concept of “unit rate” is similar to the concept of a “unit fraction” (n/1); Unite rate and describes the number of units (of one kind of object) corresponds to one unit of another kind of object. Use standard “ratio” nomenclature and notation, 6.RP.A.2. Solve rate, ratio, and unit rate word-problems (including: unit pricing, velocity, acceleration, mass density, remuneration, momentum, efficiency, power, 6.RP.A.3.b) that: are indirect (relevant information must be derived); contain “noise” (extraneous information is included), and are interdependent (relevant information is external) 6.RP.A.3. Construct tabular arrays of equivalent ratios (e.g., solution tables: freefall- acceleration under varying “G” environments; velocity- displacement at varying time intervals) relating quantities with whole-number measurements, find missing values in tables, and graph or plot on the appropriate graphic format. Use tables to compare ratios6.RP.A.3.a.

ALPHANUMERIC EXPRESSIONS AND EQUATIONS: Apply knowledge of the six arithmetic operations and their inverse operation, along with properties of operations to understand algebraic expressions and transform them into equivalent expressions. Write and evaluate numeric and algebraic expressions with exponents that are: integers (positive and negative), vulgar fractions, and decimal fractions 6.EE.A.1. Transform linear equations into: standard form ax+by=0, where a and b ? 0; slope-intercept form y=mx+b, where m is the slope of line b; point-slope form y-y1=m(x-x1), where (x1, y1) is any point on a line; and intercept form + =1, where a and/or b ? zero. Translate real-world word-problems into alphanumeric expressions. Solve systems of linear equations by: 1) Solving for one variable, then substituting one equation into the other; 2) Adding or subtracting equations to elimination variables; 3) Multiplying one or more equations, then eliminating variables by adding or subtracting. Graph systems of linear equations and identify intersection points (if any). Evaluate systems of linear equations for: infinite solutions, no solutions; or unique solutions. Analyze equations for: equivalence, independence, inconsistency, and/or contradictions. Display mental dexterity solving multistep word-problems that: contains noise; are indirect or require prior knowledge (e.g., conversion between systems-of-measurement; translation between numerical expressions ( =.5 = = 50%); commonly known or derived formulas, V= pr2 h/3.); are interdependent (required information that is dependent on solutions from prior word problems); and require information to be found within given informational graphs, tables, or charts. Graph linear equations in two and three variables using technology. Understand and use standard nomenclature including: term, degree, variable, indeterminate, coefficient, operator, constant, expression, monomial, binomial, trinomial. Understand the how the commutative property relative to terms in polynomials. Know and use standard polynomial notation and ordering conventions. Identify the degree of polynomials. Use properties of operations to add, subtract, multiply, divide, and factor within linear equations. Solve multi-step, real-world and mathematical problems using equations containing integers, decimal and common fractions, percentages, inequalities, denominate metrics, irrational numbers. Assess the reasonableness of solutions to equations using mental computation and approximation strategies. Represent algebraic relationships, given in real-world and mathematical word-problems, with alphanumeric equations or inequalities. Read, write, and evaluate alphanumeric expressions involving the six arithmetic operations, in which quantities (numbers) can be substituted for variables (letters) 6.EE.A.2. Construct algebraic expressions and equations drawn from real-life observations (field observations), verbal descriptions, and written discussions of numeric relationships and physical properties (e.g., write and equation taken from a verbal description of the inverse square law) 6.EE.A.2.a. Using standard nomenclature, identify the distinct parts (components) of numeric expressions, alphanumeric expressions, algebraic expressions and equations such as: terms, grouping symbols (vinculum, brackets, parentheses, radicand), operands, operators (+, -, vinculum, etc.), positional operators (in the expression 2B the position of the operands is the multiplication operator; in the expression AB the superscript position of B is the exponentiation operator), sum, quotient, product, factor, root, coefficient, degree, etc.. Recognize, label, and understand the implications of hierarchical grouping schemes along with subordinate groupings (Russian Doll structures) in numeric expressions, alphanumeric expressions, algebraic expressions and equations 6.EE.A.2.b. Evaluate formulas (that include the use of the 6 arithmetic operations) utilized in real-world and hypothetical problems (such as Area: triangle A=½bh, square A=a2, rectangle A=wh, circle A=pr2, regular polygons, trapezoids, irregular shapes, etc. Volume and surface area: cube V=a3, rectangular prism V=lwh, cylinder V=pr2h, sphere V= 4p , cone V=p, regular dodecahedron V=¼ (15+7 ) a3; prisms, pyramids, spheres, etcetera) by substituting quantitative values for variables. Evaluate expressions, equations, formula, and functions at various structural levels of order-of-operations including: alphanumeric expression with no grouping symbols and the operator rule is followed (evaluation priority sequence – exponentiation and root taking, multiplication and division, addition and subtraction); vincula and radical signs alter the operator rule; parentheses alter the operator rule; parentheses alter the operator rule, the vincula rule, and the radical sign rule; and combinations of all including brackets.) Do not confuse (x-3) with (x¯3). The horizontal line, in the first example, is an operator for subtraction and means (x minus 3); order of operation rules apply. The horizontal line in the second example is a property of the number 3, meaning this three is a signed number – negative integer. The second example means (x times negative 3); order of operation rules do not apply.) Know how to apply notation conventions to enhance clarity; ¯3x means negative three time x 6.EE.A.2.c. Apply the properties-of-operations rules to generate equivalent expressions 6.EE.A.3. Identify when two expressions are equivalent 6.EE.A.4. View equations and inequality as a questions and their solutions as the process of answering those questions. Use substitution to determine which values from a given set, if any, make an equation or inequality, 6.EE.B.5. Solve real-world and mathematical problems by writing equations that express the numerical relationships between the known and unknown quantities. Use variables to represent any unknown or missing quantities. Understand that a variable can represent any known or unknown quantity 6.EE.B.6. Solve real-world and mathematical problems by writing and solving equations in the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers 6.EE.B.7. Write an inequality of the form x > c or x < c to represent a constraint or condition in the real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinite solutions; represent solutions of such inequalities on number line diagrams. Represent and analyze quantitative relationships between dependent and independent variables 6.EE.B.8.Use variables to represent two quantities that change in relationship; write an equation to express the dependent variable, in terms of the independent variable. Analyze relationships between the dependent and independent variables using graphs, tables, and equations 6.EE.C.9.

FUNCTIONS: Understand functions as relations between a set of inputs that correspond to a set of outputs. Represent and explain functions as an s (step-by-step instructions); a recipe, a rectangular array, graph, and/or an alphanumeric equation. Illustrate and explain concepts relative to the concept of functions using visual models or Venn diagrams to map a set of input elements to output elements. Understand that to determine an output value for a particular input value, the relation between them must be evaluated. Understand and use standard function nomenclature and notation including: argument, domain, codomain, range, input, output, relation, ordered pairs, and element. Recognize that for each input there is one and only one output. Evaluate simple linear and non-linear functions (e.g., ?(x) = x2) express inputs and outputs as ordered pairs (i.e., x, ?(x); x, x2). Graph with aid of technology. Translate formulas to function notation, evaluate, and chart, including concepts involving relationships such as: potential energy, kinetic energy, power, and pressure. Understand constructs related to functions including: A function is a relationship between inputs and outputs such that each input may map to one and only output, but one output may map to more than one input. Understand that Functions can be represented with formulas, diagrams, tables, or graphs. Understand nomenclature associated with functions including: domain, codomain, ordered pairs, arguments image, range, elements, mapping. Compare functions represented in various forms.

Systems of Measurement / Denominate Numbers: Use knowledge of ratios to convert units between and within systems of measurement (e.g., SI, US customary units). Manipulate and transform units when adding, subtracting, multiplying or dividing denominate numbers (simple and compound). Reduce to lower or higher denominations when appropriate. Recall US customary units (inch, foot, yard, mile; acre; ounce, cup, pint, quart, gallon; ounce, pound, short ton, degrees Fahrenheit) and SI International System of Units (SI: meter, kilogram, second, ampere, kelvin, candela, mole) Natural units (e.g., c, speed of light; G, gravitational constant) 6.RP.A.3.d. Within each system of measurement recall the number of sub-units contained in larger units (e.g., feet in a yard.) Solve word-problems containing denominate numbers. Understand and explain the following concepts and nomenclature: leap years; laps time; leap year, leap seconds; time zones; Milankovitch cycles, eccentricity, axial tilt, precession; AM, PM; AD, BC, Current Era (CE), Before Current Era (BCE). Solve word problems involving addition and subtraction of time intervals including minutes, hours, and days, within and across time zones including the international dateline. Solve real-world problem relative to the Geographic Coordinate System and Coordinated Universal Time system (UTC) and understand associated nomenclature including: latitude (?, phi), parallels (circles of latitude); longitude (? , lambda), meridian (line of longitude), elevation, parallels, ), Greenwich Mean Time (GMT), International Atomic Time (TAI), daylight saving time (DST), leap seconds, standard time, solar time, Greenwich Mean Time, nautical time, time offset international (IDL).

Geometry: Graph linear equations with one, two, and three variables on a two dimensional coordinate plane. Using models, compare and contrast the constructs of congruence and similarity: With lines used as the model (segments, intersections, parallel); With angles used as the model (e.g., right, 45°); With 2D-shapes as the model (e.g., regular polygons of various sizes, triangles of various size and angle measurements), And with 3D-forms used as models (e.g., platonic solids, pyramids, prisms of differing sizes). Validate that geometric similarity and congruency, of two dimensional constructions, are independent of orientation. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems 6.G.A.1. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems 6.G.A.2. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems 6.G.A.3. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems 6.G.A.4. Understand, recall and use (oral and written) standard nomenclature used in geometry including: Angles (adjacent, opposite, complementary, exterior, interior, and supplementary, alternate,); Lines (parallel, segment, perpendicular, leg, ray, transverse, vertical, horizontal, congruent, similar, oblique, perpendicular, bisect, intersect); (rotation, reflection, translation, dilation, linear transformation, scale factor, ); Congruence, Similarity.

Statistics, Probability, Factorials, Permutations, and Combinations: Understand that statistics is the collection, organization, and analysis of large amounts of data; And that probability deductions can be made about groups but not individuals. Recognize that a statistical question anticipates variability (in data related to the question) and accounts for it in the deductions 6.SP.A.1. Understand that a set of data collected to answer a statistical question, has a distribution which can be described by its center, spread, and overall shape. Understand that the probability density of distributions, such as a Gaussian distribution, can be described by measures of the central tendency (mean, median, and mode); spread (dispersion of measurements); and shape (kurtosis: leptokurtic, mesokurtic, Platykurtic; skewness: negative, positive) 6.SP.A.2. Understand that measures of the central tendency are “averages” of all the data points, and measures of dispersion (variance) describes the average distance the data points “spread” from the mean. Recognize that measures of the central tendency summarizes data values with a single number, measures of variation describe how much data values vary, summarized with a single number 6.SP.A.3. Display numerical data in plots on a number line, including dot plots, histograms, and box plots 6.SP.B.4. Design a statistical study: collect numerical data, organize it, summarize it in relation to context, and analyze the data 6.SP.B.5a. Write a report and give an oral presentation including: the number of observations 6.SP.B.5.a; a description of the subject under study; the method of data collection; the method of measurement ; the units of measure 6.SP.B.5.b; the method of assuring inter-rater reliability; measures of the central tendency, mean, median, mode; variance (standard deviation); density under the curve measures (kurtosis, skew); a discussion of the rationale behind the statistics used; a description of any unusual patterns 6.SP.B.5.c; and discussion of inferences suggested by the data and its analysis along with conclusions 6.SP.B.5.d. Understand constructs such as: random variables, stochastic processes and events, non-deterministic events, dynamic systems, and simple concepts relative to deterministic chaos or chaos theory. Understand that an unlikely event can occur at any time during an event cycle, even the first. Understand and discuss the concept, “Given infinite time and event cycles, any event that can occur, will occur.”

Seventh Grade / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prerequisite Knowledge / Skill Retention-reactivation: Before providing instruction in the following objectives, the instructor shall assess students’ retention / facility with objectives articulated for prior grades and review/ reteach where required. To satisfy mathematics requirements for the current grade, students must demonstrate mastery in objectives articulated for prior grades along with those articulated for the current grade.

Numeric Operations: Perform six arithmetic operations involving numbers containing positive and negative exponents, exponents that are common and decimal fractions. Solve real-world problems that require mastery knowledge of order of operations along with the mastery knowledge of grouping symbols including; vincula, parentheses, brackets, braces. Solve word problems requiring the proportional scaling (up and down) of objects with given dimensions. Solve numeric problems written in radical notation (with even and odd indexes and positive and negative radicands) Understand that the “imaginary unite” refers to the solution for = ?1 = = ?. Understand that a complex number contains a real part and imaginary part (a+b ?). Solve real-world, multistep word-problems including: Indirect word problems (information required to solve a problem is not given but must be derived); word-problems with “noise” (relevant information is embedded within extraneous information), and interdependent word-problems (problems that requires prior knowledge or information from previous word-problems). Add, subtract, multiply, divide, exponentiate, and take the root of rational numbers including: signed numbers, common fractions, decimal fractions, and mixed numbers. Represent addition and subtraction of rational and irrational numbers on a horizontal or vertical number line 7.NS.1. Solve real-world word-problems that involve vertical and horizontal number lines, and two and three dimensional Cartesian coordinate systems to reference physical and numeric conditions (e.g., altitude, sea-level, flight paths, temperature; orbits of the earth and moon, displacement of a vehicle traveling on a 30 mile circular path for an hour at 30 mph). Understand the constructs of negation, cyclical counting-with and with positional value, the additive inverse, identity and unity, multiplicative invers, zero-product property) 7.NS.1a. Understand that the sum of a real number and its opposite is zero. Understand that the absolute value of a real number can be described as that number’s distance from a fixed point on a horizontal or vertical number line, without regard to direction. In a real-world context recognize examples of numeric relationships including: absolute value, negative integers and signed numbers, negation, the additive inverse, the zero-product property 7.NS.1b. Understand that subtraction of rational numbers can be viewed as the inverse addition (additive inverse on a number line). Demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference. Apply this principle in a real-world context 7.NS.A.1.c. Understand properties of operations (including: commutative, associative, distributive, identity, symmetric, multiplicative, equality, reflexive, transitive, substitution, partition, distributive, etc.) and recognize their applicability while performing arithmetic operations (addition, subtraction, multiplication, division, exponentiation, root taking; their inverse and repeated operation) with rational numbers (including common fractions, decimal fractions, and complex fractions) and/or while transforming equations into equivalent equations Understand that properties-of-numbers and properties-of-operations apply to rational numbers including common fractions, decimal fractions. Apply properties-of-operations as strategies to add and subtract rational numbers 7.NS.A.1.d. Apply the properties-of-operations as strategies to multiply and divide rational numbers 7.NS.A.2. Understand and apply algebraic properties related to integers. Understand: like the set of natural numbers the set of integers is closed under the properties-of-operations for addition and multiplication (i.e., the sum or product of two integers, is an integer). Understand: integers are closed under subtraction and division. Understand: although natural numbers are closed under exponentiation, integers are not (= ). Know and mathematically justify the multiplication, division, exponentiation, and root taking rule signed numbers. Interpret products, quotients, and roots of rational numbers by recognizing and describing them relative to the real-world contexts 7.NS.A.2.a. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. Interpret quotients of rational numbers by describing real-world contexts 7.NS.A.2.b. Apply properties of operations as strategies to multiply and divide rational numbers 7.NS.A.2.c. Convert common fractions to decimal fractions using long division. Understand: rational number can be express in common fraction form; converting a common, proper fraction to a decimal fraction will result in a quotient that terminates or repeats indefinitely ( 2/3 = .6¯6¯6¯, 1/7 = .1¯4¯2¯8¯5¯6¯ ). Use standard notation including the repetend7.NS.A.2.d. Solve real-world and mathematical problems involving the six arithmetic operations with rational numbers 7.NS.A.3.

SYSTEMS OF NUMERALS, OPERATIONS AND PROPERTIES: Compare numeral systems that use position notation and those that do not (e.g., Hindu-Arabic with Roman). Compare and contrast binary, decimal, hexadecimal numeral systems. Using standard nomenclature and notation, illustrate the relationships between sets and subsets of numbers (e.g., N? Z? Q ? R ? C) including: odd and even numbers, integer and signed numbers, prime and composite numbers, real and imaginary numbers, concrete and abstract numbers, denominate numbers, rational and irrational numbers, transcendental and algebraic numbers, countable and uncountable numbers, natural numbers, ordinal and cardinal numbers, complex numbers. Discuss properties of each set of numbers (e.g., Irrational numbers cannot be expressed as a ratio of two integers, their decimal expansions do not end or repeat) and examples of each (e.g., transcendental numbers: p, ??; irrational numbers p, ??, , f ). Identify properties including: commutative, associative, distributive, identity, symmetric (if a=b and b=a), multiplicative axiom (if a=b and c=d then ac=bd); Axioms of equality: reflexive a=a, transitive (if a=b and b=c then a=c), substitution, partition (a quantity is equal to the sum of its parts), addition (if a=b and c=d then a+c=b+d). Apply knowledge of the six arithmetic operations to common fractions and mixed numbers comprised entirely of: numerals, a mix of numerals and variables, entirely of indeterminates and symbols representing constant quantities, and denominators (like denominators, unlike denominators in which one can be named to the other, unlike denominators which are relative primes, unlike denominators that contain primes in common, and unlike denominator that contain relative primes.) Solve real-world problems that require the application of this knowledge.

ALGEBRAIC EXPRESSIONS AND EQUATIONS: Describe polynomials (e.g., expressions containing only: variables, coefficients, non-negative integer exponents). Discuss similarities and differences between linear and nonlinear equations; a variable and indeterminate, and give examples. Understand and use standard nomenclature including: term, degree, variable, indeterminate, coefficient, operator, constant, expression, monomial (univariate), binomial, trinomial. Understand the commutative law relative to the ordering of terms in a polynomial. Know and use standard polynomial ordering and notation conventions (e.g., degree, indeterminate, use of lower/uppers case letters). Identify the degree of polynomials. Understand and use function notation involving polynomial equations (polynomial functions). Use properties of operations to transform equivalent expressions. Use properties of operations as strategies to add, subtract, multiply, divide, factor, and expand linear expressions with rational coefficients 7.EE.A.1. Understand that by transforming an expression into equivalent forms, obscured relationships between variables can be revealed 7.EE.A.2. Solve real-world and mathematical problems using numeric and alphanumeric expressions and equations. Solve multi-step real-life and mathematical problems containing positive and negative rational numbers in any form (whole and mixed numbers, common and decimal fractions). Apply properties of operations to calculate numbers in any of their forms; convert between forms as appropriate; and assess the reasonableness of answers using mental computation, approximation strategies, and estimation 7.EE.B.3. Construct simple equations and inequalities representing numeric relationships observed in the real-world. Construct equations and inequalities using letters near the end of the alphabet to represent variables quantities (or indeterminates), letters near the beginning of the alphabet to represent constant quantities, and arithmetic/algebraic notation to represent numeric relationships between the variables and constants. Solve real-world or numeric problem through mathematical reasoning (logical deduction, formal and informal inference, analysis, intuition) 7.EE.B.4. Use equations in the form ax + b = c and a(x + b) = c, where a, b, and c, are rational numbers; to solve real-word and word-problems. Compare algebraic to arithmetic solutions by identifying and contrasting the sequence of operations in each 7.EE.B.4.a.

Solve word-problems involving inequalities of the form ax + b > c or ax + b < c, where a, b, and c, are rational numbers; Graph the solution set and explain numeric relationship in oral and essay form 7.EE.B.4.b.

FRACTIONS/ RATIOS/ PERCENTAGES/ PROPORTIONS/ODDS: Know, and use standard notation and nomenclature to discuss: 1. Fractions (decimal, vinculum, common, vulgar, complex, compound, proper, improper, fractions of unity, unit fractions, equivalent fractions, relative primes, repetend, algebraic fractions, etc.); 2. Ratios (antecedent, consequent, factor, quotient, etc.); 3. Proportions (proportionality constant, directly proportional, inversely proportional, reciprocal factor, unit rate, coefficient of proportionality). Recognize that fractions, ratios, percentages, and proportions are expressions of the fourth arithmetic operation, division; And convert between such expressions. In conjunction with denominate numbers (i.e., unit measures for liquid volume, weight, time, atmospheric pressure, temperature, illumination, force, inertia, etc.), use knowledge of fractions, ratios, percentages, and proportions to solve multistep word problems related to real-world subjects such as: calculating monetary exchange rates, tax rates, velocity, acceleration, radioactive decay rate, interest rates; determining aspect ratios, dilution ratios, inverse-square law, freefall, :cost-benefit ratios, price-performance ratios, gradient ratios. Analyze proportional relationships and use them to solve real-world and mathematical problems. Utilizing a variety of “quotient-ratio” notation schemes, compute unit-rate-ratios in real-life word-problems of: lengths, areas; volumes; weights, forces; mechanical advantage; velocity; atmospheric-hydrostatic pressure; mass density; in various denominate systems of measure (e.g., US customary, SI); convert within (e.g., in. to feet) and between the SI and US customary) 7.RP.A.1. Recognize and represent proportional relationships between quantities 7.RP.A.2. Utilizing ratios, decide whether two quantities are in a proportional relationship. Evaluate for equivalent ratios using a table or by graphing on a coordinate plane. By testing for and equality of two ratios, or graphing and evaluating the attributes of the lines for slope and linearity. Understand and use standard notation (e.g., 7.RP.A.2.a. In tables, graphs, equations, diagrams, and verbal descriptions, determine if proportional relationships are present between variables (e.g., time speed, travel time, displacement), determine the relative directions of relationships (inversely proportional, directly proportional), and identify the coefficient-of-proportionality (proportionality constant), and/or unit rate 7.RP.A.2.b. Represent proportional relationships between variables with an equation 7.RP.A.2.c. On a graph of that represents a proportional relationship between two variables, explain the relevance of point (x, y) to the situation the graph represents with special attention to the points (0, 0) and (1, k) where k is the unit rate or the proportionality constant 7.RP.A.2.d. Use proportional relationships to solve multistep ratio and percent problems involving such things as: growth rate and age, birthrate and age, mortality and 7.RP.A.3.

Systems of Measurement / Denominate Numbers: Convert between and within systems of measurement (e.g., SI, US customary units). Recall the US customary units and SI units. Within each system recall the number of units contains of another (e.g., feet in a yard). Solve problems (including word-problems), containing denominate numbers. TIME: Mastery-knowledge-explain: The frequency of leap years; daylight saving time; solar time; laps time; standard time; Greenwich Mean Time (GMT), Coordinated Universal Time (UTC), and Terrestrial Time (TT); leap year, leap seconds; time zones; the international date line; Milankovitch cycles, eccentricity, axial tilt, precession; AM and PM; AD, BC, Current Era (CE), Before Current Era (BCE). Using stopwatches record in written form (using the proper notation and nomenclature) to fractions of a second. Solve word problems involving addition and subtraction of time intervals including minutes, hours, and days, within and across time zones including the international dateline. Solve word-problems involving plots and graphs of denominate numbers and fractional (decimal, common fractions, and sub-units). Discriminate between a scalar quantities (only magnitude) and vector quantities (direction and magnitude); give examples of each

Geometry: Use the “Pythagorean Triple” (3, 4, 5); one of its multiples (6, 8, 10); the Pythagorean relationships relative to isosceles right triangles (45°, 90°, 45°; 1,) and/ or equilateral triangles, and 30-60 right triangles (1, 2, ) to solve real-world problems. Use the Pythagorean Theorem to solve problems in two and three dimensions. Produce graphic proofs for the Pythagorean theorem. Using only a straightedge and compass, construct a “proof-by-rearrangement,” and a “similar-triangle” proof for the Pythagorean theorem. Identify geometric transformations (change in position and/or size) including: dilation (change in size, preserves shape); translations (slides, directional displaced); reflections (flips, mirror imagery), rotations (turns, displacement around a central axis). Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing. Redraw a scaled drawing at a different scale. Verbally and in essay form describe attributes of two and three dimensional rectilinear forms. In essay form describe a process of scaling that preserves proportions. Pictorially represent actual three dimensional figures on a two dimensional surface that are proportionally similar but scaled down (dilation). Use of linear perspective to produce a scaled, three dimensional sketch from a two dimensional elevation plan. Draft a scaled elevation plan from a three dimensional physical model. Draw two dimensional scaled plans to construct three dimensional geometric forms and construct the forms (e.g., platonic solids. In essay, describe the unique attributes of each constructed geometrical figures and the relationships between them. Solve problems involving scale drawings of geometric figures that require: Computation of lengths and areas from a scale drawing; scaling a drawing up or down 7.G.A.1. Draw (freehand, with ruler and protractor or with technology) specific geometric shapes and forms with given proportions. Using only a straightedge and compass, construct: perpendicular lines; parallel lines; regular polygons with up to ten sides; Goethe’s triangle7.G.A.2. Identify platonic solids (tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron) and discuss their properties. Identify regular polygons (trigons, tetragons, pentagons, hexagons, heptagons, octagons, nonagons, decagons) discuss their properties, and recall formulas and strategies to find their area, perimeter, apothem, circumradius, and interior angles. Recall nomenclature and classification schemes that sort all possible convex trigonal and tetragonal constructions into categories including: Quadrilaterals (rectangles, squares, rhomboids, rhombuses, isosceles trapezoids, trapezoids, kites, trapezia, parallelograms); triangles (right, obtuse, acute; equilateral, isosceles, scalene) discuss their properties and recall formulas and strategies to find the area, perimeter, diagonals, and interior angles. Identify and describe two dimensional figures obtained from the intersection of a two dimensional plane and three dimensional forms including: a plane and cone (conic sections), a plane and cylinder, a plane and rectilinear prisms and plane and pyramids. Discuss their attributes and use standard plane-geometry to identify and name the shapes such as: ellipse; parabola; hyperbola; circle; isosceles triangle; right triangle; rhomboid; skew pentagon 7.G.A.3 Describe the relationship between the circumference and area of a circle. Recall or derive formulas and use them to solve real-life and mathematical problems involving: the Pythagorean theorem; regular and irregular polygons and polyhedra; circles, triangles, quadrilaterals, angles, area and circumference of a circle, surface area, volume, in various metrics (US, SI) 7.G.B.4.

Solve multistep word-problems that contain noise, are interdependent, and are recursive (sounds like like) using knowledge about supplementary, complementary, vertical, and adjacent angles. Recall and transform formulas to solve for the unknown measures 7.G.B.5. Solve real-world and mathematical problems involving area, volume and surface area of two and three dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms Recall/derive formulas to find the area, diameter, radius, and circumference of a circle. Identify right prisms and pyramids having regular polygonal bases (including circular bases) and recall/derive formulas to find volume, surface area, dihedral angles, and edge length. Identify truncated and skew prisms and pyramids. Find the volume, surface area and radius of a sphere 7.G.B.6.

Know and use standard nomenclature and concepts including: degrees in a circle, pi, angle, midpoint; regular polygons; Angles (adjacent, opposite, complementary, exterior, interior, and supplementary); Lines (parallel, segment, perpendicular, ray, transverse, congruent, similar, oblique, perpendicular.

Statistics, Probability, Factorials, Permutations, and Combinations: Use random sampling to draw inferences about a population. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences 7.SP.A.1. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions 7.SP.A.2. Draw informal comparative inferences about two populations. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable 7.SP.B.3. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book 7.SP.B.4. Investigate chance processes and develop, use, and evaluate probability models. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event 7.SP.C.5. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times 7.SP.C.6. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy 7.SP.C.7. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected 7.SP.C.7.a. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies 7.SP.C.7.b? Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation 7.SP.C.8. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs 7.SP.C.8.a. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event 7.SP.C.8.b. Design and use a simulation to generate frequencies for compound events 7.SP.C.8.c. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linearly or nonlinearly). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Discriminate between: Causal and casual variables; dependent and independent variables; confounding variables and omitted variables, controlled and uncontrolled variable. Understand and discuss in verbally and written essay: line regression, derivatives, correlation, covariance, hypothesis, hypothesis testing, bias, ex post facto, error, variance, standard deviation, variance. Contrast and compare “descriptive statistics” with “inferential statistics.”

Eighth Grade / A.L.L. Mathematics Objectives, Aligned to Meet or Exceed Common Core Grade Level Standards / Mastered by or before year-end

Prerequisite Knowledge / Skill Retention-reactivation: Before providing instruction in the following objectives, the instructor shall assess students’ retention / facility with objectives articulated for prior grades and review/ reteach where required. To satisfy mathematics requirements for the current grade, students must demonstrate mastery in objectives articulated for prior grades along with those articulated for the current grade.

Numeric Operations: Know that numbers that are not rational are irrational. Understand that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number 8.NS.A.1. Use rational approximations of irrational numbers to compare the size of irrational numbers. Find the approximately of irrational numbers on a number line, and estimate the value of irrational numbers expressions (e.g., p2). Approximate the principal square root of nonnegative, real, two digit numbers, using a rough estimation strategy or standard algorithm (e.g., find square roots through repeated subtraction of odds; 8.NS.A.2.

ALGEBRAIC EXPRESSIONS AND EQUATIONS: Factor equations Write expressions in equivalent forms to solve factors, and coefficients problems. Factor a quadratic expression to reveal the zeros of the function it defines. Complete the square in (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Use the properties of exponents to transform expressions, Create equations and inequalities in one variable and use them to solve problems. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve quadratic equations in one variable. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Expressions and Equations Work with radicals and integer exponents. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27 8.EE.A.1. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that v2 is irrational 8.EE.A.2. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 8.EE.A.3. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology 8.EE.A.4. Understand the connections between proportional relationships, lines, and linear equations. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed 8.EE.B.5. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Analyze and solve linear equations and pairs of simultaneous linear equations 8.EE.B.6. Solve linear equations in one variable 8.EE.C.7. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers)8.EE.C.7.a. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms 8.EE.C.7.b. Analyze and solve pairs of simultaneous linear equations 8.EE.C.8. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously 8.EE.C.8.a. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection 8.EE.C.8.b. Solve real-world and mathematical problems leading to two linear equations in two variables 8.EE.C.8.c.

?:X?Y ? X?Y ?(x) = y ?¯¹(x) = y FUNCTIONS: Define, evaluate, and compare functions. Understand that a function is a rule that assigns to each input exactly one and only one output but one output may be produced by more than one input. Understand that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output 8.F.A.1. Compare properties of two functions each represented in a different way (algebraically, graphic ally, numerically in tables, or by verbal descriptions8.F.A.2. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Give examples of functions that are not linear 8.F.A.3. Use functions to model relationships between quantities. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values 8.F.B.4. Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally 8.F.B.5. Express linear functions in a variety of forms (e.g., tables, graphs) to model/compare/contrast varying relationships such as: Instantaneous velocity in freefall, at different time intervals, experienced by small objects of the same mass, near a variety of large bodies with different masses (gravitational force); Mass-density of objects of the same volume comprised of different elements; Pay-back amounts of equivalent loan-amounts with varying points, interest rates, payment-amounts, compound periods, repayment periods; Expected hourly wages relative to varying educational levels, return on investment associated with various universities; Expected scholarship/financial amounts relative to grades, SAT scores; Expected miles per gallon relative to speed, engine displacement, vehicle weight; Returns on investments relative to real-world variables. Express such relationships in simple algebraic form that can be applied in real-world situations.

Geometry: Compare and contrast the concept of congruency with the concept of similarity: Identify geometric figures that are congruent (figures that have the same shape, size, measures, or are exact mirror images of each other); identify geometric figures that are similar (figures that have the same shape and proportions but not the same size, including mirror images); and compare. Understand and illustrate non-rigid transformations (alterations in position and/or size) including: dilation (proportional scaling, preserves ratios between distances) and shear mapping (displacement occurs along parallel lines). Understand and demonstrate rigid transformations (displacement that preserves size and shape) including translations (every point is displaced a constant distance in the same direction); reflections (mirror images, every point maps across a reversed line of symmetry), rotations (every point is displacement around a central axis). Verify experimentally the properties of rotations, reflections, and translations 8.G.A.1. Verify the properties of rigid transformations and non-rigid transformations relative to: lines and line segments 8.G.A.1.a, angles 8.G.A.1.b, parallel lines 8.G.A.1.c. Understand that two, two-dimensional figures are congruent if one can be obtained from the other through a sequence of rotations, reflections, and translations. Using congruent polygons to demonstrate that congruency is independent of rigid transformation (a square remains a square even when it is turned on its corner although one may be call it a diamond) 8.G.A.2. Using coordinates, describe the effect of dilations, translations (slides), rotations (turns), and reflections (flips) on two-dimensional constructions 8.G.A.3. Understand that two, two-dimensional figures are similar if one can be obtained from the other through a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional constructions, describe a sequence of transformations that exhibit their geometric similarity 8.G.A.4. Demonstrate that: one dimensional constructs (e.g., rays, geometric vectors, plane-of-reference coordinate points); two dimensional constructs (e.g., interface planes, sectional planes) and polyhedra (three-dimensional forms) that are congruent; remain congruent through all rigid three-dimensional (inclination, rotation, radiation) displacements. Use informal arguments to establish facts about the sum of an internal angle and its exterior angle, and the angle sum and exterior angle of triangles. Use informal arguments to establish the relationship between the number of sides of a polygon (regular and irregular) and the sum of its interior angles. Use informal arguments to establish the relationship between the number of sides of a polygon (regular and irregular) and the sum of its exterior angles. Use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal line: including the similarity of opposite angles at each intersection, the sum of angles formed at each intersection, and relationships between each parallel lines and of angles formed by the transversal line. Use informal arguments to establish facts about the angle-angle criterion for similarity of triangles.8.G.A.5. Understand and apply the Pythagorean Theorem. Understand the geometric relationships stated by the Pythagorean theorem relative to right triangles. Recall the Pythagorean equation (a2+b2=h2) and solve for the each variable (h=, a=, b=). Recall and explain the “Pythagorean proof by rearrangement” and its converse (If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle) 8.G.B.6. Recall the “Primitive Pythagorean Triple” (3, 4, 5) and use it, along with its multiples (6, 8, 10; 9, 12, 15) to solve real-world and hypothetical problems. Understand Pythagorean relationships relative to isosceles right triangles (45°, 90°, 45°; 1,) equilateral triangles, and 30-60 right triangles (1, 2, ) and use this knowledge to solve word-problems. Use the Pythagorean Theorem to solve real-world problems in two and three dimensions (e.g., construct right angles, determine the side lengths of right triangles, determine the length of the hypotenuse, find elevations, etc.); and to find the distance between points on two-dimensional coordinate plane 8.G.B.7. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres 8.G.B.8. Recall the volume formulas for prisms, pyramids (with polygonal and circular bases) and spheres; use these formulas to solve hypothetical and real-world problems 8.G.C.9. Use and understand the following geometry nomenclature: abscissa, ordinate, correspondence, function, dysfunction,

Statistics, Probability, Factorials, Permutations, and Combinations: Recall and understand, commonly used statistical nomenclature and concepts including: variable (bivariate, dependent, independent, confounding regression analysis, descriptive statistics; Associations between variables (clustering, outliers, positive, negative, linear and nonlinear ) Construct and interpret scatter plots of bivariate data to expose patterns and/or associations (e.g., linear and nonlinear, positive or negative, clustering, outliers) if any 8.SP.A.1. Understand the concept of measurement error and the process of linear regression used to model relationships relative to scatter plots of bivariate data; visually search the displayed data for recognizable patterns 8.SP.A.2. For scatter plots that suggest a linear association, informally assess the data for closeness of fit to a straight line (draw a straight line dividing the distribution of points on the basis of density averages; scatter distance and quantity); And, write a slope-intercept equation representing the informally regressed line 8.SP.A.2. Use a linear equation (transformed into slope-intercept form) to model a bivariate data distribution. Make predictions and draw conclusions from simulated experiments 8.SP.A.3. Understand that patterns of association can also be seen in data using other types of displays such as frequencies tables, bar graphs, histograms, stem-and-leaf plots, correlation tables, and two-way tables. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables 8.SP.A.4.