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Michigan Curriculum Matrix for Mathematics

Copyright © 2008 International Center for Leadership in Education Math –

Michigan Mathematics Strands/Grade Level Content Expectations

Grade 3

National Essential Skills Study (NESS)Rankings

RankMEAP NESS Priority

NUMBER & OPERATIONSUnderstand and use number notation and place valueN.ME.03.01 Read and write numbers to 10,000 in both numerals and words, and relate them to the quantities they represent, e.g., relate numeral or written word to a display of dots or objects.

M35

Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value H M H

N.ME.03.02 Recognize and use expanded notation for numbers using place value to 10,000s place, e.g., 2,517 is 2 thousands, 5 hundreds, 1 ten, and 7 ones; 4 hundreds and 2 ones is 402. Identify the place value of a digit in a number, e.g., in 3,241, 2 is in the hundreds place.

M35

Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value. H M H

N.ME.03.03 Compare and order numbers up to 10,000. M35

Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value.

M M M

Count in steps, and understand even and odd numbersN.ME.03.04 Count orally by 6’s, 7’s, 8’s and 9’s starting with 0, making the connection between repeated addition and multiplication.

M1Perform operations fluently with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results L H M

M35 Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value.

N.ME.03.05 Know that even numbers end in 0, 2, 4, 6,or 8; name a whole number quantity that can be shared in two equal groups or grouped into pairs with no remainders; recognize even numbers as multiples of 2. Know that odd numbers end in 1, 3, 5, 7, or 9, and work with patterns involving even and odd numbers.

M35

Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value.

M M M

Add and subtract whole numbersN.FL.03.06 Add and subtract fluently two numbers: up to and including two-digit numbers with regrouping and up to four-digit numbers without regrouping.

M1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. H H H

N.FL.03.07 Estimate the sum and difference of two numbers with three digits (sums up to 1,000), and judge reasonableness of estimates.

M1Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. H H H

M35 Use the properties of real(rational and irrational) numberAnd demonstrate understanding of ordering and absolute value.



Grade 3



N.FL.03.08 Use mental strategies to fluently add and subtract two-digit numbers. M1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. L H M


Multiply and divide whole numbersN.MR.03.09 Use multiplication and division fact families to understand the inverse relationship of these two operations, e.g., because 3 x 8 = 24, we know that 24 ÷ 8 = 3 or 24 ÷ 3 = 8. Express a multiplication statement as an equivalent division statement.


M35Use the properties of real(rational and irrational) numberAnd demonstrate understanding of ordering and absolute value.

N.MR.03.10 Recognize situations that can be solved using multiplication and division including finding “How many groups?” and “How many in a group?” and write mathematical statements for those situations.

M1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. H H H

N.FL.03.11 Find products fluently up to 10 x 10; find related quotients using multiplication and division relationships.

M1Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results.

H H H

N.MR.03.12 Find solutions to open sentences such as 7 x = 42 or 12 ÷ = 4, using the inverse relationship between multiplication and division.


M H H

N.FL.03.13 Mentally calculate simple products and quotients up to a three-digit number by a one-digit number involving multiples of 10, e.g., 500 x 6, or 400 ÷ 8.

M1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. L H M

N.MR.03.14 Solve simple division problems involving remainders, viewing remainder as the “number left over” (less than the divisor), e.g., 4 children per group; we have 25 children; there are 6 groups with 1 child left over; interpret based on problem context.

M1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results.

M H H

Problem-solving with whole numbersN.MR.03.15 Given problems that use any one of the four operations with appropriate numbers, represent with objects, words (including “product” and “quotient”), and mathematical statements; solve.

M1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. M H H

Understand simple fractions, relation to the whole, and addition and subtraction of fractions



Grade 3



N.ME.03.16 Understand that fractions may represent a portion of a whole unit that has been partitioned into parts of equal area or length; use the terms “numerator” and “denominator.”

M35Use the properties of real(rational and irrational) numberAnd demonstrate understanding of ordering and absolute value. H M H

N.ME.03.17 Recognize, name and use equivalent fractions with denominators 2, 4, and 8, using strips as area models.

M1Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. M H H


N.ME.03.18 Place fractions with denominators of 2, 4, and 8 on the number line; relate the number line to a ruler; compare and order up to three fractions with denominators 2, 4, and 8.

M35

Use the properties of real(rational and irrational) numberAnd demonstrate understanding of ordering and absolute value. M M M

N.ME.03.19 Understand that any fraction can be written as a sum of unit fractions, e.g., 3/4 = 1/4 + 1/4 + 1/4.

M1Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. M H H


N.MR.03.20 Recognize that addition and subtraction of fractions with equal denominators can be modeled by joining and taking away segments on the number line.



Understand simple decimal fractions in relation to moneyN.ME.03.21 Understand the meaning of $0.50 and $0.25 related to money, e.g., $1.00 shared by two people means $1.00 ÷ 2 = 1/2 dollar = $0.50.


H H H

MEASUREMENTMeasure and use units for length, weight, temperature and timeM.UN.03.01 Know and use common units of measurements in length, weight and time. M8

Solve problems using units of metric measure and convert between metric and English/customary units. H H H

M.UN.03.02 Measure in mixed units within the same measurement system for length, weight and time: feet and inches, meters and centimeters, kilograms and grams, pounds and ounces, liters and milliliters, hours and minutes, minutes and seconds, years and months.

M8

Solve problems using units of metric measure and convert between metric and English/customary units.

H H H



Grade 3



M.UN.03.03 Understand relationships between sizes of standard units, e.g., feet and inches, meters and centimeters.

M13Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real-world situations (e.g., scale drawings).

H H H

M.UN.03.04 Know benchmark temperatures such as freezing (32ºF, 0ºC); boiling, (212ºF, 100ºC); and compare temperatures to these, e.g., cooler, warmer.

M8 Solve problems using units of metric measure and convert between metric and English/customary units

H H HS13

Measure or estimate physical properties using dimensional quantities (e.g., time, length, mass, pressure, volume, acceleration, temperature) and use significant figures correctly when estimating, measuring, and calculating these quantities

Understand meaning of area and perimeter and apply in problemsM.UN.03.05 Know the definition of area and perimeter and calculate the perimeter of a square and rectangle given whole number side lengths. M9

Compute the perimeter and area of two-dimensional figures.

M H H

M.UN.03.06 Use square units in calculating area by covering the region and counting the number of square units.

M9 Compute the perimeter and area of two-dimensional figures.

H H HM13

Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real –world situations (e.g., scale drawings).

M.UN.03.07 Distinguish between units of length and area and choose a unit appropriate in the context.

M13Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real –world situations (e.g., scale drawings).

M H H

s23

Measure properties of the environment using dimensional quantities such as time, length, mass, pressure, volume, acceleration, etc. Compare, estimate and predict measurements.

M.UN.03.08 Visualize and describe the relative sizes of one square inch and one square centimeter. M13

Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real –world situations (e.g., scale drawings).

M H H

Estimate perimeter and areaM.TE.03.09 Estimate the perimeter of a square and rectangle in inches and centimeters; estimate the area of a square and rectangle and square inches and square centimeters.

M9 Compute the perimeter and area of common two-dimensional figures.

M H HM12

Understand accuracy and precision of measurement, round off numbers according to the correct number of significant figures, and determine percent error.

Solve measurement problemsM.PS.03.10 Add and subtract lengths, weights and times using mixed units within the same measurement system.

M8Solve problems using units of metric measure and convert between metric and English/customary units. M H H



Grade 3



M.PS.03.11 Add and subtract money in dollars and cents. m1

Perform operations fluently with positive and negative numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results.

H H H

M.PS.03.12 Solve applied problems involving money, length and time. M8 Solve problems using units of metric measure and convert

between metric and English/customary units.

M H HM10

Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems.

M.PS.03.13 Solve contextual problems about perimeters of rectangles and areas of rectangular regions.


M H HM10


GEOMETRYRecognize the basic elements of geometric objectsG.GS.03.01 Identify points, line segments, lines and distance. M19

Compute the distance between two points on a coordinate plane (length of a line segment) and determine the midpoint of a line segment between two points.

M M M

M41

Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship with intuitive concepts (i.e., collinear points, coplanar points, opposite rays, and parallel lines).

G.GS.03.02 Identify perpendicular lines and parallel lines in familiar shapes and in the classroom.

M4Understand the characteristics of parallel, perpendicular, and intersecting lines in problem-solving situations. M H H

G.GS.03.03 Identify parallel faces of rectangular prisms, in familiar shapes and in the classroom. M4 Understand the characteristics of parallel, perpendicular, and

intersecting lines in problem-solving situations M H HM26 Know the classification

Name and explore properties of shapesG.GS.03.04 Identify, describe, compare and classify two-dimensional shapes, e.g., parallelogram, trapezoid, circle, rectangle, square and rhombus, based on their component parts (angles, sides, vertices, line segment) and the number of sides and vertices.

m10Understand the properties of circles (e.g., radius, arc, diameter, chord, secant, tangent, etc.).

H H H

m27Understand the properties and classification of quadrilaterals by orientation (e.g., parallelogram, rectangle, rhombus, square, and trapezoid).

G.SR.03.05 Compose and decompose triangles and rectangles to form other familiar two-dimensional m14 Understand the angle relationships in triangles (i.e., acute,

obtuse, right, interior, and exterior). H H H



Grade 3



shapes, e.g., form a rectangle using two congruent right triangles, or decompose a parallelogram into a rectangle and two right triangles.


m49 Apply transformation concepts to understand and create congruent and similar figures.

Explore and name three-dimensional solidsG.GS.03.06 Identify, describe, build and classify familiar three-dimensional solids, e.g., cube, rectangular prism, sphere, pyramid, cone, based on their component parts (faces, surfaces, bases, edges, vertices).

m29

Know the classification and properties of solid figures such as prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres. H H H

G.SR.03.07 Represent front, top, and side views of solids built with cubes. m29

Know the classification and properties of solid figures such as prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres.

H H H

DATA & PROBABILITYUse bar graphsD.RE.03.01 Read and interpret bar graphs in both horizontal and vertical forms. m5

Understand the best procedures for statistical data collection, organization, and display including making estimates and predictions and drawing inferences.

H H H

D.RE.03.02 Read scales on the axes and identify the maximum, minimum, and range of values in a bar graph.

m5Understand the best procedures for statistical data collection, organization, and display including making estimates and predictions and drawing inferences.

H H H

D.RE.03.03 Solve problems using information in bar graphs including comparison of bar graphs. m5


H H H



Grade 4

Curriculum Survey of Essential SkillsNational Rankings

RankMEAP Curriculum

Survey Priority

NUMBER & OPERATIONUnderstand and use number notation and place valueN.ME.04.01 Read and write numbers to 1,000,000; relate them to the quantities they represent; compare and order.

m19Understand the definitions and properties of rational and irrational numbers. H H H

N.ME.04.02 Compose and decompose numbers using place value to 1,000,000’s, e.g., 25,068 is 2 ten thousands, 5 thousands, 0 hundreds, 6 tens, and 8 ones.

m19

Understand the definitions and properties of rational and irrational numbers. H H H

N.ME.04.03 Understand the magnitude of numbers up to 1,000,000; recognize the place values of numbers, and the relationship of each place value to the place to its right, e.g., 1,000 is 10 hundreds.

m19


Use factors and multiplesN.ME.04.04 Find all factors of a whole number up to 50, and list factor pairs. m22

Understand factoring a composite number into its prime factors, and how to find the largest monomial factor of a polynomial to write the polynomial as the product of the monomial and a polynomial.

H H H

N.ME.04.05 List the first ten multiples of a given one-digit whole number; determine if a whole number is a multiple of a given one-digit whole number and if a one-digit number is a factor of a given whole number.

m19 Understand the definitions and properties of rational and irrational numbers.

H H Hm22


N.MR.04.06 Know that some numbers including 2, 3, 5, 7, and 11 have exactly two factors (1 and the number itself) and are called prime numbers.


H H Hm22


N.MR.04.07 Solve problems about factors and multiples, e.g., since 100 = 4 x 25, and 200 = 2 x 100, then 200 = 2 x 4 x 25 = 8 x 25. m22


H H H

Add and subtract whole numbersN.FL.04.08 Add and subtract whole numbers fluently. m1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions.

H H H

Multiply and divide whole numbersCopyright © 2008 International Center for Leadership in Education Math –


Grade 4


RankMEAP Curriculum

Survey Priority

N.ME.04.09 Multiply two-digit numbers by 2, 3, 4, and 5, using the distributive property, e.g., 21 x 3 = (1 + 20) x 3 = (1 x 3) + (20 x 3) = 3 + 60 = 63

m3Understand basic algebraic properties (i.e., commutative: ab = ba; associative: ab(c) = a(bc); and distributive: a(b+c) = (ab)+(ac)).

H H H

N.FL.04.10 Multiply fluently any whole number by a one-digit number, and a three-digit number by a two-digit number; for a two-digit by one-digit multiplication, use distributive property to develop meaning for the algorithm.

m1Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions. H H H

m3Understand basic algebraic properties (i.e., commutative: ab = ba; associative: ab(c) = a(bc); and distributive: a(b+c) = (ab)+(ac)).

N.FL.04.11 Divide numbers up to four digits by one-digit numbers and by 10. m1


H H H

N.FL.04.12 Find unknowns in equations such as a ÷ 10 = 25; 125 ÷ b = 25. m7

Understand the use of variables in expressions such as 4x, x+2, and 2x-1, solve for the variable, and know how to represent expressions such as "twice the number" or "four more than the number" using variables.

H H H

N.MR.04.13 Use the relationship between multiplication and division to simplify computations and check results, e.g., 6840 ÷ 20 = (6840 ÷ 10) ÷ 2 = 684 ÷ 2 = 342.

m1Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions.

L H M

m11

Use addition and multiplication to simplify an algebraic expression by identifying the order of operations and techniques necessary to carry out the operations (e.g., 5-3(x-2) = 5-3x+6 = 11-3x).

N.FL.04.14 Solve applied problems involving whole number multiplication and division. m1


H H Hm7


m8 Understand the correct order of operations for performing algebraic computations.

Read, interpret and compare decimal fractionsN.ME.04.15 Read and interpret decimals up to two decimal places; relate to money and place value decomposition.

m1 Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions. H H H





Grade 4


RankMEAP Curriculum

Survey Priority

N.ME.04.16 Know that terminating decimals represents fractions whose denominators are 10, 10 x 10, 10 x 10 x 10, etc., e.g., powers of 10.

m19Understand the definitions and properties of rational and irrational numbers.

N.ME.04.17 Locate tenths and hundredths on a number line. m19 Understand the definitions and properties of rational and

irrational numbers. H H H

N.ME.04.18 Read, write, interpret, and compare decimals up to two decimal places. m19 Understand the definitions and properties of rational and


N.MR.04.19 Write tenths and hundredths in decimal and fraction forms, and know the decimal equivalents for halves and fourths.


Understand fractionsN.ME.04.20 Understand fractions as parts of a set of objects. m19 Understand the definitions and properties of rational and


N.MR.04.21 Explain why equivalent fractions are equal, using models such as fraction strips or the number line, for fractions with denominators of 12 or less, or equal to 100.

m19


N.MR.04.22 Locate and compare fractions on the number line, including improper fractions and mixed numbers with denominators of 12 or less.


N.MR.04.23 Understand the relationships among halves, fourths and eighths and among thirds, sixths and twelfths.


N.MR.04.24 Know that fractions of the form where m/n, is greater than n, are greater than 1 and are called improper fractions; locate improper fractions on the number line; express as mixed numbers.

m19


N.MR.04.25 Write improper fractions as mixed numbers, and understand that a mixed number represents the number of “wholes” and the part of a whole remaining, e.g., 5/4 = 1 + 1/4= 1 1/4



N.MR.04.26 Compare and order up to three fractions with denominators 2, 4, and 8, and 3, 6, and 12, including improper fractions and mixed numbers.


Add and subtract fractionsN.MR.04.27 Add and subtract fractions less than 1 with denominators 12 or less and including 100, in cases where the denominators are equal or when one denominator is a multiple of the other, e.g., 1/12 + 5/12= 6/12; 1/6 + 5/12 = 7/12; 3/10 - 23/100 = 7/100.

m1


H H H



Grade 4


RankMEAP Curriculum

Survey Priority

N.FL.04.28 Solve fraction problems involving sums and differences for fractions where one denominator is a multiple of the other (denominators 2 through 12, and 100).

m1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions. H H H

N.MR.04.29 Solve for the unknown in equations such as: 1/8+ x = 5/8 or 3/4 - y =1/2. m1 Perform operations with signed (positive and negative)

numbers, including decimals, ratios, percents, and fractions.

H H Hm7


Multiply fractions by whole numbersN.MR.04.30 Multiply fractions by whole numbers, using repeated addition and area or array models. m1 Perform operations with signed (positive and negative)

numbers, including decimals, ratios, percents, and fractions. H H H

Add and subtract decimal fractionsN.MR.04.31 Use mathematical statements to represent problems that use addition and subtraction of decimals with up to two-digits; solve.

m1 Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions.

H H Hm7


N.FL.04.32 Add and subtract decimals up to two decimal places. m1 Perform operations with signed (positive and negative)


Multiply and divide decimal fractionsN.FL.04.33 Multiply and divide decimals up to two decimal places by a one-digit whole number where the result is a terminating decimal, e.g., 0.42 ÷ 3 = 0.14, but not 5 ÷ 3 = 1. 6

m1


H H H

EstimateN.FL.04.34 Estimate the answers to calculations involving addition, subtraction, or multiplication. m1


H H H

N.FL.04.35 Know when approximation is appropriate and use it to check the reasonableness of answers; be familiar with common place-value errors in calculations.



N.FL.04.36 Make appropriate estimations and calculations fluently with whole numbers using mental math strategies.

m1 Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions.

H H H



Grade 4


RankMEAP Curriculum

Survey Priority


Problem-solvingN.MR.04.37 Solve applied problems using the four basic arithmetic operations for appropriate fractions, decimals, and whole numbers.


H H Hm7



MEASUREMENTMeasure using common tools and appropriate unitsM.UN.04.01 Measure using common tools and select appropriate units of measure.

m33

Use the technique of dimensional analysis to convert units of measure (e.g., convert km/hr to m/min) including drawing to scale and applying ratios. Understand and use various techniques for estimating, making and converting measure; and using these to perform dimensional analysis. H H H

s23


M.PS.04.02 Give answers to a reasonable degree of precision in the context of a given problem.

m33

Use the technique of dimensional analysis to convert units of measure (e.g., convert km/hr to m/min) including drawing to scale and applying ratios. Understand and use various techniques for estimating, making and converting measure; and using these to perform dimensional analysis.

H H H

s23


s81 Know how to round off numbers according to the correct number of significant figures.

M.UN.04.03 Measure and compare integer temperatures in degrees.

m33 Use the technique of dimensional analysis to convert units of measure (e.g., convert km/hr to m/min) including drawing to scale and applying ratios. Understand and use various techniques for estimating, making and converting measure; and using these to perform dimensional analysis.

H H H



Grade 4


RankMEAP Curriculum

Survey Priority

s23


s31Know the concepts of temperature and absolute temperature. Know the temperature scales (Fahrenheit, Celsius, and Kelvin) and their relationships.

M.TE.04.04 Measure surface area of cubes and rectangular prisms by covering and counting area of the faces.

m13 Compute the perimeter and area of two-dimensional figures.

H H Hm33


Convert measurement unitsM.TE.04.05 Carry out the following conversions from one unit of measure to a larger or smaller unit of measure: meters to centimeters, kilograms to grams, liters to milliliters, hours to minutes, minutes to seconds, years to months, weeks to days, feet to inches, ounces to pounds (using numbers that involve only simple calculations).

m33


H H H

Use perimeter and area formulasM.TE.04.06 Know and understand the formulas for perimeter and area of a square and a rectangle; calculate the perimeters and areas of these shapes and combinations of these shape using the formulas.

m13

Compute the perimeter and area of two-dimensional figures. H H H

M.TE.04.07 Find one dimension of a rectangle given the other dimension and its perimeter or area. m13 Compute the perimeter and area of two-dimensional

figures. H H H


M.TE.04.08 Find the side of a square given its perimeter or area. m13 Compute the perimeter and area of two-dimensional

figures. H H H


m13 Compute the perimeter and area of two-dimensional figures. H H H



Grade 4


RankMEAP Curriculum

Survey Priority

M.PS.04.09 Solve contextual problems about perimeter and area of squares and rectangles in compound shapes.


Understand right anglesM.TE.04.10 Identify right angles and compare angles to right angles. m4

Understand the characteristics and terminology of angles, e.g., degree measure, classification of angles by measure (acute, right, obtuse, and straight), supplementary and complementary angles, and vertical angles.

H H H

Problem-solvingM.PS.04.11 Solve contextual problems about surface area. m13 Compute the perimeter and area of two-dimensional

figures. H H H

GEOMETRYUnderstand perpendicular, parallel, and intersecting linesG.GS.04.01 Identify and draw perpendicular, parallel, and intersecting lines using a ruler and a tool or object with a square (90º) corner.

m2 Understand the characteristics of parallel, perpendicular, and intersecting lines.

H H Hm28

Use geometric methods (e.g., an unmarked straightedge and compass) to complete basic geometric constructions (e.g., perpendicular bisector of a line segment, angle bisector, etc.).

Identify basic geometric shapes and their components, and solve problemsG.GS.04.02 Identify basic geometric shapes including isosceles, equilateral and right triangles, and use their properties to solve problems.

m16Understand the properties and classification triangles by sides (i.e., scalene, isosceles, and equilateral). H H H

G.SR.04.03 Identify and count the faces, edges, and vertices of basic three-dimensional geometric solids including cubes, rectangular prisms, and pyramids; describe the shape of their faces.

m29

Know the classification and properties of solid figures such as prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres. H H H

Recognize symmetry and transformationsG.TR.04.04 Recognize plane figures that have line symmetry. m55

Understand the concepts of symmetry and transformations and graphically apply line reflections, rotation, translations, and dilation.

H M H

G.TR.04.05 Recognize rigid motion transformations (flips, slides, turns) of a two-dimensional object. m55

Understand the concepts of symmetry and transformations and graphically apply line reflections, rotation, translations, and dilation.

H M H

DATA & PROBABILITYRepresent and solve problems for given dataD.RE.04.01 Construct tables and bar graphs from given data. m5


H H H



Grade 4


RankMEAP Curriculum

Survey Priority

D.RE.04.02 Order a given set of data, find the median, and specify the range of values. m5


H H Hm15 Understand the characteristics of measures of central tendency (i.e., mean, median, and mode).

m36Understand the characteristics of measures of dispersion (i.e., range, mean deviation, variance, and standard deviation).

D.RE.04.03 Solve problems using data presented in tables and bar graphs, e.g., compare data represented in two bar graphs and read bar graphs showing two data sets.

m5

Understand the best procedures for statistical data collection, organization, and display including making estimates and predictions and drawing inferences. H H H



Grade 5


RankMEAP Curriculum

Survey Priority

NUMBER & OPERATIONSUnderstand division of whole numbersN.MR.05.01 Understand the meaning of division of whole numbers, with and without remainders; relate division to fractions and to repeated subtraction.



N.MR.05.02 Relate division of whole numbers with remainders to the form a = bq + r, e.g., 34 ÷ 5 = 6 r 4, so 5 • 6 + 4 = 34; note remainder (4) is less than divisor (6).

m1


H H H

N.MR.05.03 Write mathematical statements involving division for given situations. m1 Perform operations with signed (positive and negative)


H H Hm7


Multiply and divide whole numbersN.FL.05.04 Multiply a multi-digit number by a two-digit number; recognize and be able to explain common computational errors such as not accounting for place value.



N.MR.05.05 Solve applied problems involving multiplication and division of whole numbers. m1 Perform operations with signed (positive and negative)


H H Hm7



N.FL.05.06 Divide fluently up to a four-digit number by a two-digit number. m1 Perform operations with signed (positive and negative)


Find prime factorizations of whole numbersN.MR.05.07 Find the prime factorization of numbers between 1 and 50, express in exponential notation, e.g., 24 = 233 x 311, and understand that every whole number can be expressed as a product of primes.

m22


M H H

Understand meaning of decimal fractions and percentages



Grade 5


RankMEAP Curriculum

Survey Priority

N.ME.05.08 Understand the relative magnitude of ones, tenths, and hundredths and the relationship of each place value to the place to its right, e.g., 1 is 10 tenths, one tenth is 10 hundredths.

m19

Understand the definitions and properties of rational and irrational numbers. M H H

N.ME.05.09 Understand percentages as parts out of 100, use % notation, and express a part of a whole as a percentage.

m19Understand the definitions and properties of rational and irrational numbers. M H H

Understand fractions as division statements; find equivalent fractionsN.ME.05.10 Understand a fraction as a statement of division, e.g., 2 ÷ 3 = 2/3 using simple fractions and pictures to represent.


N.ME.05.11 Given two fractions, express them as equivalent fractions with a common denominator, but not necessarily a least common denominator, e.g., 1/2 = 4/8 and 3/4 = 6/8; use denominators less than 12 or factors of 100.

m19

Understand the definitions and properties of rational and irrational numbers.

M H H

Multiply and divide fractionsN.FL.05.12 Find the product of two unit fractions with small denominators using area model. m1


H H H

N.FL.05.13 Divide a fraction by a whole number and a whole number by a fraction, using simple unit fractions.


H H H

Add and subtract fractions using common denominatorsN.FL.05.14 Add and subtract fractions with unlike denominators of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 100, using the common denominator that is the product of the denominators of the 2 fractions, e.g.,3/8 + 7/10 = (3 x 10) + (7 x 8)= 30 + 56 80= 86/80


H H H

m8

Understand the correct order of operations for performing algebraic computations.

Multiply and divide by powers of tenN.MR.05.15 Multiply a whole number by powers of 10: 0.01, 0.1, 1, 10, 100, 1,000; and identify patterns.

m1Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions. L H M

m24 Understand the basic properties and laws of exponents and scientific notation.



Grade 5


RankMEAP Curriculum

Survey Priority

N.FL.05.16 Divide numbers by 10’s, 100’s, 1,000’s, using mental strategies. m1

Perform operations with signed (positive and negative) numbers, including decimals, ratios, percents, and fractions. H H H


N.MR.05.17 Multiply one-digit whole numbers by decimals up to two decimal places. m1


H H H

Solve applied problems with fractionsN.FL.05.18 Given an applied situation involving addition and subtraction of fractions, write mathematical statements describing the situation.


H H H

m7


N.MR.05.19 Solve word problems that involve finding sums and differences of fractions with unlike denominators using knowledge of equivalent fractions.



N.FL.05.20 Solve applied problems involving fractions and decimals; include rounding of answers and checking reasonableness; use examples involving money.



N.MR.05.21 Solve for the unknown in such equations as: ¼ + x = 7/12. m1


H H H

m7


Express, interpret, and use ratios; find equivalencesN.MR.05.22 Express fractions and decimals as percentages and vice versa. m1


M H H



Grade 5


RankMEAP Curriculum

Survey Priority

N.ME.05.23 Express ratios in several ways given applied situations, e.g., 3 cups to 5 people, 3 : 5, 3/5; recognize and find equivalent ratios.


M H H

MEASUREMENTKnow, and convert among, measurement units within a given systemM.UN.05.01 Recognize the equivalence of 1 liter, 1.000 ml and 1000 cm3 and include conversions among liters, milliliters, and cubic centimeters. m33


H H H

M.UN.05.02 Know the units of measure of volume: cubic centimeter, cubic meter, cubic inches, cubic feet, cubic yards, and use their abbreviations (cm3, m3, in3, ft3, yd3).

m33


s23


M.UN.05.03 Compare the relative sizes of one cubic inch to one cubic foot, and one cubic centimeter to one cubic meter. m33


H H H

M.UN.05.04 Convert measurements of length, weight, area, volume, and time within a given system using easily manipulated numbers. m33


H H H

Find areas of geometric shapes using formulasM.PS.05.05 Represent relationships between areas of rectangles, triangles, and parallelograms using models.



M.TE.05.06 Understand and know how to use the area formula of a triangle: A = 1/2 bh (where b is length of the base and h is the height), and represent using models and manipulatives.


m16 Understand the properties and classification triangles by sides (i.e., scalene, isosceles, and equilateral).



Grade 5


RankMEAP Curriculum

Survey Priority

M.TE.05.07 Understand and know how to use the area formula for a parallelogram: A = bh, and represent using models and manipulatives.


H H Hm27

Understand the properties and classification of quadrilaterals by orientation (e.g., parallelogram, rectangle, rhombus, square, and trapezoid).

Understand the concept of volumeM.TE.05.08 Build solids with unit cubes and state their volumes. m29

Know the classification and properties of solid figures such as prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres.

H H H

m33


M.TE.05.09 Use filling (unit cubes or liquid), and counting or measuring to find the volume of a cube and rectangular prism.

m29Know the classification and properties of solid figures such as prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres.

H H Hm33


s23


M.PS.05.10 Solve applied problems about the volumes of rectangular prisms using multiplication and division and using the appropriate units.

m17Compute the volume of three-dimensional figures (solids).

H H H

GEOMETRYKnow the meaning of angles, and solve problemsG.TR.05.01 Associate an angle with a certain amount of turning; know that angles are measured in degrees; understand that 90°, 180°, 270°, and 360° are associated, respectively, with 1/4, 1/2, and 3/4 and full turns.

m4


H H H

G.GS.05.02 Measure angles with a protractor, and classify them as acute, right, obtuse, or straight. m4


H H H



Grade 5


RankMEAP Curriculum

Survey Priority

G.GS.05.03 Identify and name angles on a straight line and vertical angles. m4


H H H

G.GS.05.04 Find unknown angles in problems involving angles on a straight line, angles surrounding a point and vertical angles.

m4


H H H

G.GS.05.05 Know that angles on a straight line add up to 180°and angles surrounding a point add up to 360°; justify informally by “surrounding” a point with angles.

m4


H H H

G.GS.05.06 Understand why the sum of the interior angles of a triangle is 180°and the sum of the interior angles of a quadrilateral is 360°, and use these properties to solve problems.

m14 Understand the angle relationships in triangles (i.e., acute, obtuse, right, interior, and exterior).

H H Hm27

Understand the properties and classification of quadrilaterals by orientation (e.g., parallelogram, rectangle, rhombus, square, and trapezoid).

Solve problems about geometric shapesG.GS.05.07 Find unknown angles using the properties of: triangles, including right,isosceles, and equilateral triangles; parallelograms, including rectangles and rhombuses; and trapezoids.

m4


H H Hm14 Understand the angle relationships in triangles (i.e., acute, obtuse, right, interior, and exterior).


DATA & PROBABILITYConstruct and interpret line graphsD.RE.05.01 Read and interpret line graphs, and solve problems based on line graphs, e.g., distance-time graphs, and problems with two or three line graphs on same axes, comparing different data.

m5


D.RE.05.02 Construct line graphs from tables of data; include axis labels and scale. m5


H H H

Find and interpret mean and mode for a given set of dataD.AN.05.03 Given a set of data, find and interpret the mean (using the concept of fair share) and mode. m15 Understand the characteristics of measures of central

tendency (i.e., mean, median, and mode). H H H



Grade 5


RankMEAP Curriculum

Survey Priority

D.AN.05.04 Solve multi-step problems involving means. m15 Understand the characteristics of measures of central

tendency (i.e., mean, median, and mode). H H H



Grade 6


RankMEAP Curriculum

Survey Priority

NUMBER & OPERATIONSMultiply and divide fractionsN.MR.06.01 Understand division of fractions as the inverse of multiplication, e.g., if 4/5 ÷ 2/3 = , then 2/3 x = 4/5, so = 4/5 • 2/3 = 12/10.


H H H

N.FL.06.02 Given an applied situation involving dividing fractions, write a mathematical statement to represent the situation.


H H H

m7


N.MR.06.03 Solve for the unknown in equations such as: 1/4÷ = 1 3/4 ÷ = 1/4 and 1/2 = 1 • . m1


H H H

N.FL.06.04 Multiply and divide any two fractions, including mixed numbers, fluently. m1


H H H

Represent rational numbers as fractions or decimalsN.ME.06.05 Order rational numbers and place them on the number line. m19 Understand the definitions and properties of rational and

irrational numbers. M H H

N.ME.06.06 Represent rational numbers as fractions or terminating decimals when possible, and translate between these representations.


N.ME.06.07 Understand that a fraction or a negative fraction is a quotient of two integers, e.g., -8/3 is -8 divided by 3.


Add and subtract integers and rational numbersN.MR.06.08 Understand integer subtraction as the inverse of integer addition; add and subtract integers using integers from 10 to -10.


M H H

N.FL.06.09 Add, subtract, multiply, and divide integers between -10 and 10; use number line and strip models for addition and subtraction.


H H H

N.FL.06.10 Add, subtract, multiply and divide positive rational numbers fluently. m1


H H H

Find equivalent ratios



Grade 6


RankMEAP Curriculum

Survey Priority

N.ME.06.11 Find equivalent ratios by scaling up or scaling down. m1


M H H

m33


Solve decimal, percentage and rational number problemsN.FL.06.12 Calculate part of a number given the percentage and the number. m1


H H H

N.FL.06.13 Solve word problems involving percentages in such contexts as sales taxes and tips, and involving positive rational numbers.


H H H

N.FL.06.14 For applied situations, estimate the answers to calculations involving operations with rational numbers.



N.FL.06.15 Solve applied problems that use the four operations with appropriate decimal numbers. m1


H H H

Use exponentsN.ME.06.16 Understand and use integer exponents, excluding powers of negative numbers; express numbers in scientific notation.

m24Understand the basic properties and laws of exponents and scientific notation. M H H

Understand rational numbers and their location on the number lineN.ME.06.17 Locate negative rational numbers (including integers) on the number line; know that numbers and their negatives add to 0, and are on opposite sides and at equal distance from 0 on a number line.

m19


M H H

N.ME.06.18 Understand that rational numbers are quotients of integers (non-zero denominators), e.g., a rational number is either a fraction or a negative fraction.

m19

Understand the definitions and properties of rational and irrational numbers. M H H



Grade 6


RankMEAP Curriculum

Survey Priority

N.ME.06.19 Understand that 0 is an integer that is neither negative nor positive. m19 Understand the definitions and properties of rational and


N.ME.06.20 Know that the absolute value of a number is the value of the number, ignoring the sign, or is the distance of the number from 0.


ALGEBRACalculate ratesA.PA.06.01 Solve applied problems involving rates including speed, e.g., if a car is going 50 mph, how far will it go in 3 ½ hours? m33


H H H

Understand the coordinate planeA.RP.06.02 Plot ordered pairs of integers and use ordered pairs of integers to identify points in all four quadrants of the coordinate plane. m23

Know the components and properties of the rectangular coordinate system, (i.e., x - y axis, origin, quadrants, abscissa (x-coordinate) and ordinate (y-coordinate), and the general representation of a point (x,y)).

H H H

Use variables, write expressions and equations, and combine like termsA.FO.06.03 Use letters, with units, to represent quantities in a variety of contexts, e.g., y lbs., k minutes, x cookies. m7


H H H

A.FO.06.04 Distinguish between an algebraic expression and an equation. m7


H H H

A.FO.06.05 Use standard conventions for writing algebraic expressions, e.g., 2x + 1 means “two times x, plus 1” and 2(x + 1) means “two times the quantity (x + 1).”

m7


H H H

A.FO.06.06 Represent information given in words using algebraic expressions and equations. m7


H H H

A.FO.06.07 Simplify expressions of the first degree by combining like terms, and evaluate using specific values.

m7 Understand the use of variables in expressions such as 4x, x+2, and 2x-1, solve for the variable, and know how to represent expressions such as "twice the number" or "four more than the number" using variables.

H H H



Grade 6


RankMEAP Curriculum

Survey Priority

m11


Represent linear functions using tables, equations, and graphsA.RP.06.08 Understand that relationships between quantities can be suggested by graphs and tables. m40

Understand appropriate terminology used to define relations and functions and their properties (e.g., domain, range, function composition, inverses, etc.).

H M H

A.PA.06.09 Graph and write equations for linear functions of the form y = mx, and solve related problems, e.g., given n chairs, the “leg function” is f(n) = 4n; if you have 5 chairs, how many legs?; if you have 12 legs, how many chairs?

m40Understand appropriate terminology used to define relations and functions and their properties (e.g., domain, range, function composition, inverses, etc.). H M H

m64Know how to express a linear function (e.g., y = 1/3x+5) using the functional notation f(x) = 1/3x+5, and determine the ordered pairs.

A.RP.06.10 Represent simple relationships between quantities, using verbal descriptions, formulas or equations, tables, and graphs, e.g., perimeter-side relationship for a square, distance-time graphs, and conversions such as feet to inches.



Solve equationsA.FO.06.11 Relate simple linear equations with integer coefficients to particular contexts, and solve, e.g., 3x = 8 or x + 5 = 10. m7

Understand the use of variables in expressions such as 4x, x+2, and 2x-1, solve for the variable, and know how to represent expressions such as "twice the number" or "four more than the number" using variables. H H H

m11


A.FO.06.12 Understand that adding or subtracting the same number to both sides of an equation creates a new equation that has the same solution. m11


L H M

A.FO.06.13 Understand that multiplying or dividing both sides of an equation by the same non-zero number creates a new equation that has the same solutions.

m11


M H H



Grade 6


RankMEAP Curriculum

Survey Priority

A.FO.06.14 Solve equations of the form ax + b = c, e.g., 3x + 8 = 15 by hand for positive integer coefficients less than 20, using calculators otherwise, and interpret the results.

m11

Use addition and multiplication to simplify an algebraic expression by identifying the order of operations and techniques necessary to carry out the operations (e.g., 5-3(x-2) = 5-3x+6 = 11-3x). L H M

m35

Find the solution of linear equations and inequalities where the variable appears on both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x+2(x-3) = -4x+5 for x).

MEASUREMENTConvert within measurement systemsM.UN.06.01 Convert between basic units of measurement within a single measurement system, e.g., square inches to square feet. m33


H H H

Find volume and surface areaM.PS.06.02 Draw patterns (of faces) for a cube and rectangular prism that, when cut, will cover the solid exactly (nets).

m29Know the classification and properties of solid figures such as prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres.

H H H

m33


M.TE.06.03 Compute the volume and surface area of cubes and rectangular prisms given the lengths of their sides using formulas.


m17 Compute the volume of three-dimensional figures (solids).GEOMETRYG.GS.06.01 Understand and apply basic properties of lines, angles, and triangles, including:—triangle inequality—relationships of vertical angles, complementary angles, supplementary angles—congruence of corresponding and alternate interior angles when parallel lines —are cut by a transversal, and that such congruencies imply parallel lines—locate interior and exterior angles of any triangle, and use the property that an exterior —angle of a triangle is equal to the sum of the remote (opposite) interior angles

m2 Understand the characteristics of parallel, perpendicular, and intersecting lines.

H H H

m4




m27 Understand the properties and classification of quadrilaterals by orientation (e.g., parallelogram, rectangle, rhombus, square, and trapezoid).



Grade 6


RankMEAP Curriculum

Survey Priority

—know that the sum of the exterior angles of a convex polygon is 360º. m49 Apply transformation concepts to understand and create

congruent and similar figures. Understand the concept of congruence and basic transformationsG.GS.06.02 Understand that for polygons, congruence means corresponding sides and angles have equal measures.

m49Apply transformation concepts to understand and create congruent and similar figures. H M H

G.TR.06.03 Understand the basic rigid motions in the plane (reflections, rotations, translations), relate these to congruence, and apply them to solve problems.

m49

Apply transformation concepts to understand and create congruent and similar figures. H M H

G.TR.06.04 Understand and use simple compositions of basic rigid transformations, e.g., a translation followed by a reflection.

m55Understand the concepts of symmetry and transformations and graphically apply line reflections, rotation, translations, and dilation.

H M H

Construct geometric shapesG.SR.06.05 Use paper folding to perform basic geometric constructions of perpendicular lines, midpoints of line segments and angle bisectors; justify informally.

m28


H H H

DATA & PROBABILITYUnderstand the concept of probability and solve problemsD.PR.06.01 Express probabilities as fractions, decimals or percentages between 0 and 1; know that 0 probability means an event will not occur and that probability 1 means an event will occur.

m25

Determine the probability of single and compound events using the basic premise that the probability of an event is equal to the number of ways it can occur divided by the total number of outcomes.

M H H

D.PR.06.02 Compute probabilities of events from simple experiments with equally likely outcomes, e.g., tossing dice, flipping coins, spinning spinners, by listing all possibilities and finding the fraction that meets given conditions.

m20

Understand the characteristic differences between theoretical and empirical probability (e.g., the theoretic probability of rolling a six and a die is 1/6; empirical probability is derived from repeated experimentation or accumulated statistics).

M H H



Grade 7


RankMEAP Curriculum

Survey Priority

NUMBER & OPERATIONSUnderstand derived quantitiesN.ME.07.01 Understand derived quantities such as density, velocity, and weighted averages. s23


M H H

N.FL.07.02 Solve problems involving derived quantities.

m33


s23


Understand and solve problems involving rates, ratios, and proportionsN.FL.07.03 Calculate rates of change including speed. m1


H H H

m33


N.MR.07.04 Convert ratio quantities between different systems of units such as feet per second to miles per hour. m33


M H H

N.FL.07.05 Solve simple proportion problems using such methods as unit rate, scaling, finding equivalent fractions, and solving the proportion equation a/b = c/d; know how to see patterns about proportional situations in tables.

m33


H H H

m52 Find the solution of proportions with monomial monomial and binomial terms (e.g., x/(x-2) = 6/5, therefore, x = 12).

Recognize irrational numbersN.MR.07.06 Understand the concept of square root and cube root, and estimate using calculators. m19 Understand the definitions and properties of rational and




Grade 7


RankMEAP Curriculum

Survey Priority

Compute with rational numbersN.FL.07.07 Solve problems involving operations with integers. m1 Perform operations with signed (positive and negative)


N.FL.07.08 Add, subtract, multiply and divide negative rational numbers. m1 Perform operations with signed (positive and negative)


N.FL.07.09 Estimate results of computations with rational numbers. m1 Perform operations with signed (positive and negative)

numbers, including decimals, ratios, percents, and fractions. H H Hm19 Understand the definitions and properties of rational and

irrational numbers. ALGEBRAUnderstand and apply directly proportional relationships and relate to linear relationshipsA.PA.07.01 Recognize when information given in a table, graph, or formula suggests a proportional or linear relationship.

m40Understand appropriate terminology used to define relations and functions and their properties (e.g., domain, range, function composition, inverses, etc.).

H M H

A.RP.07.02 Represent directly proportional and linear relationships using verbal descriptions, tables, graphs, and formulas, and translate among these representations.

m7


H H Hm33




A.PA.07.03 Given a directly proportional or linear situation, graph and interpret the slope and intercept(s) in terms of the original situation; evaluate y = kx for specific x values, given k, e.g., weight vs. volume of water, base cost plus cost per unit.

m34 Know the equation for the slope of a line and compute slope given the coordinates of two points.

H H Hm45Know the equation of a line and interpret graphically using the slope-intercept form (i.e., y = mx+b), and the point-slope form (i.e., y-b = m(x-a)).




Grade 7


RankMEAP Curriculum

Survey Priority

A.PA.07.04 For directly proportional or linear situations, solve applied problems using graphs and equations, e.g., the heights and volume of a container with uniform cross-section, height of water in a tank being filled at a constant rate, degrees Celsius and degrees Fahrenheit, distance and time under constant speed.

m33


H H Hm34 Know the equation for the slope of a line and compute slope given the coordinates of two points.

m45Know the equation of a line and interpret graphically using the slope-intercept form (i.e., y = mx+b), and the point-slope form (i.e., y-b = m(x-a)).


A.PA.07.05 Understand and use directly proportional relationships of the form y = mx, and distinguish from linear relationships of the form y = mx + b, b non-zero; understand that in a directly proportional relationship between two quantities one quantity is a constant multiple of the other quantity.


H H Hm40Understand appropriate terminology used to define relations and functions and their properties (e.g., domain, range, function composition, inverses, etc.).


Understand and represent linear functionsA.PA.07.06 Calculate the slope from the graph of a linear function as the ratio of “rise/run” for a pair of points on the graph, and express the answer as a fraction and a decimal; understand that linear functions have slope that is a constant rate of change.


M H Hm40Understand appropriate terminology used to define relations and functions and their properties (e.g., domain, range, function composition, inverses, etc.).


A.PA.07.07 Represent linear functions in the form y = x + b, y = mx, and y = mx + b, and graph, interpreting slope and y-intercept.


H H Hm45Know the equation of a line and interpret graphically using the slope-intercept form (i.e., y = mx+b), and the point-slope form (i.e., y-b = m(x-a)).


A.FO.07.08 Know that the solution to a linear equation corresponds to the point at which its graph crosses the x-axis.


L M M



Grade 7


RankMEAP Curriculum

Survey Priority

Understand and solve problems about inversely proportional relationshipsA.PA.07.09 Recognize inversely proportional relationships in contextual situations; know that quantities are inversely proportional if their product is constant, e.g., the length and width of a rectangle with fixed area, and that an inversely proportional relationship is of the form y = k/x where k is some non-zero number.


L M M

m73

Understand inverse functions as the set of ordered pairs obtained by interchanging the first and second elements of each pair belonging to a one-on-one function. Use one-on-one functions to create symmetric figures consisting of the graphs of a function and its inverse function.

A.RP.07.10 Know that the graph of y = k/x is not a line; know its shape; and know that it crosses neither the x nor the y-axis.


L M M

Apply basic properties of real numbers in algebraic contextsA.PA.07.11 Understand and use basic properties of real numbers: additive and multiplicative identities, additive and multiplicative inverses, commutativity, associativity, and the distributive property of multiplication over addition.

m3Understand basic algebraic properties (i.e., commutative: ab = ba; associative: ab(c) = a(bc); and distributive: a(b+c) = (ab)+(ac)). H H H


Combine algebraic expressions and solve equationsA.FO.07.12 Add, subtract, and multiply simple algebraic expressions of the first degree, e.g., (92x + 8y) – 5x + y, or – 2x (5x – 4), and justify using properties of real numbers.

m11

Use addition and multiplication to simplify an algebraic expression by identifying the order of operations and techniques necessary to carry out the operations (e.g., 5-3(x-2) = 5-3x+6 = 11-3x). H H H


A.FO.07.13 From applied situations, generate and solve linear equations of the form ax + b = c and ax + b = cx + d, and interpret solutions. m35


L H M

GEOMETRYDraw and construct geometric objectsG.SR.07.01 Use a ruler and other tools to draw squares, rectangles, triangles and parallelograms with specified dimensions. m28


H H H



Grade 7


RankMEAP Curriculum

Survey Priority

G.SR.07.02 Use compass and straightedge to perform basic geometric constructions: the perpendicular bisector of a segment, an equilateral triangle, and the bisector of an angle; understand informal justifications.

m28


H H H

Understand the concept of similar polygons, and solve related problemsG.TR.07.03 Understand that in similar polygons, corresponding angles are congruent and the ratios of corresponding sides are equal; understand the concepts of similar figures and scale factor.

m33


H H H


G.TR.07.04 Solve problems about similar figures and scale drawings.

m33


H H H


G.TR.07.05 Show that two triangles are similar using the criteria: corresponding angles are congruent (AAA similarity); the ratios of two pairs of corresponding sides are equal and the included angles are congruent (SAS similarity); ratios of all pairs of corresponding sides are equal (SSS similarity); use these criteria to solve problems and to justify arguments.


H H H


m33



G.TR.07.06 Understand and use the fact that when two triangles are similar with scale factor of r, their areas are related by a factor of r2. m33


H H H


DATA & PROBABILITYCopyright © 2008 International Center for Leadership in Education Math –


Grade 7


RankMEAP Curriculum

Survey Priority

Represent data and interpretD.RE.07.01 Represent and interpret data using circle graphs, stem and leaf plots, histograms, and box-and-whisker plots, and select appropriate representation to address specific questions.

m5


D.AN.07.02 Create and interpret scatter plots and find line of best fit and use an estimated line of best fit to answer questions about the data.


H H H

Compute statistics about datasetsD.AN.07.03 Calculate and interpret relative frequencies and cumulative frequencies for given data sets.


L H M

D.AN.07.04 Find and interpret the median, quartiles, and interquartile range of a given set of data.

m15 Understand the characteristics of measures of central tendency (i.e., mean, median, and mode).

H H Hm42

Understand the concepts and applications of quartiles (i.e., distributing groups into four equal frequencies) and percentiles (i.e., distributing individuals into one-hundred groups of equal frequency).



Grade 8


RankMEAP Curriculum

Survey Priority

NUMBER & OPERATIONSUnderstand real number conceptsN.ME.08.01 Understand the meaning of a square root of a number and its connection to the square whose area is the number; understand the meaning of a cube root and its connection to the volume of a cube.

m19


M H H

N.ME.08.02 Understand meanings for zero and negative integer exponents. m19 Understand the definitions and properties of rational and

irrational numbers. M H Hm24 Understand the basic properties and laws of exponents and

scientific notation. N.ME.08.03 Understand that in decimal form, rational numbers either terminate or eventually repeat, and that calculators truncate or round repeating decimals; locate rational numbers on the number line; know fraction forms of common repeating decimals, e.g., 0.1= 1/9; 0.3= 1/3 .

m19


M H H

N.ME.08.04 Understand that irrational numbers are those that cannot be expressed as the quotient of two integers, and cannot be represented by terminating or repeating decimals; approximate the position of familiar irrational numbers, (e.g.,√2 , √3 , π) on the number line.

m19


M H H

N.FL.08.05 Estimate and solve problems with square roots and cube roots using calculators.

m44

Perform operations with radicals such as addition, subtraction, multiplication, and division of two or more irrational numbers and express as the square root of a positive integer or as the product of a rational number and the square root of a positive integer.

M M M

m68 Apply arithmetic methods for obtaining a rational approximation of an irrational number (e.g., radical).

N.FL.08.06 Find square roots of perfect squares and approximate the square roots of nonperfect squares by locating between consecutive integers, e.g., √130 is between 11 and 12.

m44

Perform operations with radicals such as addition, subtraction, multiplication, and division of two or more irrational numbers and express as the square root of a positive integer or as the product of a rational number and the square root of a positive integer.

M M M

m68 Apply arithmetic methods for obtaining a rational approximation of an irrational number (e.g., radical).

Solve problems



Grade 8


RankMEAP Curriculum

Survey Priority

N.MR.08.07 Understand percent increase and percent decrease in both sum and product form, e.g., 3% increase of a quantity x is x + .03x = 1.03x.


M H H

m7


N.MR.08.08 Solve problems involving percent increases and decreases. m1


M H H

m7


N.FL.08.09 Solve problems involving compounded interest or multiple discounts. m1


M H H

m7


N.MR.08.10 Calculate weighted averages such as course grades, consumer price indices, and sports ratings.


M H H

m7


N.MR.08.11 Solve problems involving ratio units such as miles per hour, dollars per pound, or persons per square mile.


M H Hm7


m33




Grade 8


RankMEAP Curriculum

Survey Priority

ALGEBRAUnderstand the concept of non-linear functions using basic examplesA.RP.08.01 Identify and represent linear functions, quadratic functions, and other simple functions including inverse functions (y = k/x), cubics (y = ax3) roots, (y =√x ), and exponentials (y = ax , a > 0), using tables, graphs, and equations.


H M H



m69 Graph the exponential function and understand its characteristics.

m73

Understand inverse functions as the set of ordered pairs obtained by interchanging the first and second elements of each pair belonging to a one-on-one function. Use one-on-one functions to create symmetric figures consisting of the graphs of a function and its inverse function.

A.PA.08.02 For basic functions, e.g., simple quadratics, direct and indirect variation, and population growth, describe how changes in one variable affect the others.


m81 Solve and graphically sketch problems involving two variables that exhibit direct and indirect variation.

A.PA.08.03 Recognize basic functions in problem context, e.g., area of a circle is π r2, volume of a sphere is 4/3 π r3, and represent them using tables, graphs, and formulas.


L H M

m17 Compute the volume of three-dimensional figures (solids).

m30Know how to measure circle quantities (e.g., area, angle formed by two secants, circumference, length of segments, etc.).


A.RP.08.04 Use the vertical line test to determine if a graph represents a function in one variable.

m32 Use direct proof and indirect proof sequencing techniques to reach a conclusion. Direct proof uses the Laws of Reasoning to create an orderly arrangement of steps leading to a conclusion. Indirect proof uses an initial assumption that the conclusion is false, and through a series of logically sound reasoning steps the statement may be proved otherwise.

L H M



Grade 8


RankMEAP Curriculum

Survey Priority


Understand and represent quadratic functionsA.RP.08.05 Relate quadratic functions in factored form and vertex form to their graphs and vice versa; in particular, note that solutions of a quadratic equation are the x-intercepts of the corresponding quadratic function.

m40Understand appropriate terminology used to define relations and functions and their properties (e.g., domain, range, function composition, inverses, etc.). L M M

m53Apply the zero property of multiplication to find the solution of quadratic equations.

A.RP.08.06 Graph factorable quadratic functions, finding where the graph intersects the x axis and the coordinates of the vertex; use words “parabola” and “roots”; include functions in vertex form and those with leading coefficient –1, e.g., y = x2 – 36, y = (x – 2)2 – 9; y = – x2; y = – (x – 3)2.

m86

Know how to graphically sketch basic conic sections (e.g., circles and parabolas) using their equations, and graphically solve systems of equations.

L L L

Recognize, represent, and apply common formulasA.FO.08.07 Recognize and apply the common formulas:(a + b)2 = a2 + 2 ab + b2

(a – b)2 = a2 – 2 ab + b2

(a + b) (a – b) = a2 – b2 ; represent geometrically.

m81

Solve and graphically sketch problems involving two variables that exhibit direct and indirect variation.

L L L

A.FO.08.08 Factor simple quadratic expressions with integer coefficients, e.g., x2

+ 6x + 9, x2 + 2x –

3 and x2 – 4; solve simple quadratic equations, e.g., x2

= 16 or x2 = 5 (by taking square roots); x2

– x – 6 = 0, x2

– 2x = 15 (by factoring); verify solutions by evaluation.

m22

Understand factoring a composite number into its prime factors, and how to find the largest monomial factor of a polynomial to write the polynomial as the product of the monomial and a polynomial. L H M

m53 Apply the zero property of multiplication to find the solution of quadratic equations.

A.FO.08.09 Solve applied problems involving simple quadratic equations. m22

Understand factoring a composite number into its prime factors, and how to find the largest monomial factor of a polynomial to write the polynomial as the product of the monomial and a polynomial. L H M

m53 Apply the zero property of multiplication to find the solution of quadratic equations.

Understand solutions and solve equations, simultaneous equations, and linear inequalities



Grade 8


RankMEAP Curriculum

Survey Priority

A.FO.08.10 Understand that to solve the equation f(x) = g(x) means to find all values of x for which the equation is true, e.g., determine whether a given value, or values from a given set, is a solution of an equation (0 is a solution of 3x2 + 2 = 4x + 2, but 1 is not a solution).

m12

Analyze the truth value of simple sentences by stating whether a simple objective statement (closed sentence) is true or false, or whether a statement containing pronouns or variables (open sentence) becomes true or false upon replacement of those pronouns or variables.

H H Hm40

Understand appropriate terminology used to define relations and functions and their properties (e.g., domain, range, function composition, inverses, etc.).

m71Know how to find the graphic solution of systems of linear equations (e.g., find the point(s) common to a quadratic-linear pair).

A.FO.08.11 Solve simultaneous linear equations in two variables by graphing, by substitution, and by linear combination; estimate solutions using graphs; include examples with no solutions and infinitely many solutions.

m45Know the equation of a line and interpret graphically using the slope-intercept form (i.e., y = mx+b), and the point-slope form (i.e., y-b = m(x-a)). L M M

m71Know how to find the graphic solution of systems of linear equations (e.g., find the point(s) common to a quadratic-linear pair).

A.FO.08.12 Solve linear inequalities in one and two variables, and graph the solution sets. m35

Find the solution of linear equations and inequalities where the variable appears on both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x+2(x-3) = -4x+5 for x). H H H

m82Know how to find the graphic solution of systems of linear inequalities (e.g., graph the solution set or region of the coordinate plane common to both inequalities).

A.FO.08.13 Set up and solve applied problems involving simultaneous linear equations and linear inequalities.

m71Know how to find the graphic solution of systems of linear equations (e.g., find the point(s) common to a quadratic-linear pair). L L L

m82Know how to find the graphic solution of systems of linear inequalities (e.g., graph the solution set or region of the coordinate plane common to both inequalities).

GEOMETRYUnderstand and use the Pythagorean TheoremG.GS.08.01 Understand at least one proof of the Pythagorean Theorem; use the Pythagorean Theorem and its converse to solve applied problems including perimeter, area, and volume problems.


H H Hm17 Compute the volume of three-dimensional figures (solids).

m21 Use the Pythagorean theorem to compute side lengths of right triangles.

G.LO.08.02 Find the distance between two points on the coordinate plane using the distance formula;

m9 Know how to compute the distance between two points (i.e., length of a line segment) on a coordinate plane.

H H H



Grade 8


RankMEAP Curriculum

Survey Priority

recognize that the distance formula is an application of the Pythagorean Theorem. m21 Use the Pythagorean theorem to compute side lengths of

right triangles. Solve problems about geometric figuresG.SR.08.03 Understand the definition of a circle; know and use the formulas for circumference and area of a circle to solve problems.

m10 Understand the properties of circles (e.g., radius, arc, diameter, chord, secant, tangent, etc.).

H H Hm30

Know how to measure circle quantities (e.g., area, angle formed by two secants, circumference, length of segments, etc.).

G.SR.08.04 Find area and perimeter of complex figures by sub-dividing them into basic shapes (quadrilaterals, triangles, circles).


H H H

m17 Compute the volume of three-dimensional figures (solids).

m21 Use the Pythagorean theorem to compute side lengths of right triangles.


G.SR.08.05 Solve applied problems involving areas of triangles, quadrilaterals, and circles. m13 Compute the perimeter and area of two-dimensional

figures.

H H Hm21 Use the Pythagorean theorem to compute side lengths of right triangles.


Understand concepts of volume and surface area, and apply formulas G.SR.08.06 Know the volume formulas for generalized cylinders ((area of base) x height), generalized cones and pyramids ( 1/3(area of base) x height) and spheres ( 4/3 π (radius) 3 ) and apply them to solve problems.

m17

Compute the volume of three-dimensional figures (solids).

H H H

G.SR.08.07 Understand the concept of surface area, and find the surface area of prisms, cones, spheres, pyramids, and cylinders.


H H Hm30

Know how to measure circle quantities (e.g., area, angle formed by two secants, circumference, length of segments, etc.).

Visualize solidsG.SR.08.08 Sketch a variety of two-dimensional representations of three-dimensional solids including orthogonal views (top, front, and side),

m29 Know the classification and properties of solid figures such as prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres.

H H H



Grade 8


RankMEAP Curriculum

Survey Priority

picture views (projective or isometric), and nets, use such two-dimensional representations to help solve problems. m33




m75

Understand the concepts of direct and opposite isometries (i.e., transformations that preserve distance such as reflections, rotations, translations, and glide reflections, and be able to graphically apply them.

Understand and apply concepts of transformation and symmetryG.TR.08.09 Understand the definition of a dilation from a point in the plane, and relate it to the definition of similar polygons.

m49Apply transformation concepts to understand and create congruent and similar figures. H M H

G.TR.08.10 Understand and use reflective and rotational symmetries of two-dimensional shapes, and relate them to transformations to solve problems.


H M H

DATA & PROBABILITYDraw, explain, and justify conclusions based on dataD.AN.08.01 Determine which measure of central tendency (mean, median, mode) best represents a data set, e.g., salaries, home prices for answering certain questions; justify the choice made.

m15

Understand the characteristics of measures of central tendency (i.e., mean, median, and mode). H H H

D.AN.08.02 Recognize practices of collecting and displaying data that may bias the presentation or analysis.


H H H

m15 Understand the characteristics of measures of central tendency (i.e., mean, median, and mode).

m36 Understand the characteristics of measures of dispersion (i.e., range, mean deviation, variance, and standard deviation).



Grade 8


RankMEAP Curriculum

Survey Priority

m42

Understand the concepts and applications of quartiles (i.e., distributing groups into four equal frequencies) and percentiles (i.e., distributing individuals into one-hundred groups of equal frequency).

Understand probability concepts for simple and compound eventsD.PR.08.03 Compute relative frequencies from a table of experimental results for a repeated event, and be able to answer questions about the result, using relationship of probability to relative frequency.


M H H

m20


D.PR.08.04 Apply the Basic Counting Principle to find total number of outcomes possible for independent and dependent events, and calculate the probabilities using organized lists or tree diagrams.

m56

Use the Counting Principle to determine the probability of events occurring jointly (e.g., if one activity can occur in any of m ways and another in any one of n ways, then the total number of ways both activities can occur is mn).

M M M

D.PR.08.05 Understand the relationship of probability to relative frequency.

m20


L H M

D.PR.08.06 Understand the difference between independent and dependent events, and recognize common misconceptions involving probability, e.g., Alice rolls a 6 on a die three times in a row; she is just as likely to roll a 6 on the fourth roll as she was on any previous roll.


M H H

m20


D.AN.08.07 Compute relative frequencies from a table of experimental results for a repeated event; understand the relationship of experimental probability to relative frequency; answer questions regarding the results.


L H M

m20



Michigan MathematicsStrands/High SchoolContent Expectations


RankMME NESS Priority

STRAND 1: QUANTITATIVE LITERACY AND LOGICSTANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONSBased on their knowledge of the properties of arithmetic, students understand and reason about numbers, number systems, and the relationships between them. They represent quantitative relationships using mathematical symbols, and interpret relationships from those representations. L1.1 Number Systems and Number SenseL1.1.1 Know the different properties that hold in different number systems, and recognize that the applicable properties change in the transition from the positive integers, to all integers, to the rational numbers, and to the real numbers.

M2Understand and apply basic algebraic properties (commutative and associative laws of addition and multiplication, distributive law of multiplication over addition, and identities and inverses). H H H

M35Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value.

L1.1.2 Explain why the multiplicative inverse of a number has the same sign as the number, while the additive inverse of a number has the opposite sign.

M35Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value. H M H

L1.1.3 Explain how the properties of associativity, commutativity, and distributivity, as well as identity and inverse elements, are used in arithmetic and algebraic calculations.

M2Understand and apply basic algebraic properties (commutative and associative laws of addition and multiplication, distributive law of multiplication over addition, and identities and inverses).

H H H

M7Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution.

M11

Apply variables in expressions and equations to solve problems (ie., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world).

M35 Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value

L1.1.4 Describe the reasons for the different effects of multiplication by, or exponentiation of, a positive number by a number less than 0, a number between 0 and 1, and a number greater than 1.

M20

Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. H M H

M35Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value





L1.1.5 Justify numerical relationships (e.g., show that the sum of even integers is even; that every integer can be written as 3m+k, where k is 0, 1, or 2, and m is an integer; or that the sum of the first n positive integers is n (n+1)/2).

M10


H H H

L1.1.6 Explain the importance of the irrational numbers and in basic right triangle trigonometry; the importance of π because of its role in circle relationships; and the role of e in applications such as continuously compounded interest.

M18

Understand the properties of curcles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations.

H M HM24

Understand the concepts of recurrence relations and apply them to solve consumer mathematics problems involving such things as percentage rates, personal loans, simple interest, compound interest, installment buying, mortgage rates, etc.

M28Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent).

M48 Express, graph, and interpret exponential and logarithmic functions.

L1.2 Representations and RelationshipsL1.2.1 Use mathematical symbols (e.g., interval notation, set notation, summation notation) to represent quantitative relationships and situations.

M45 Solve linear inequalities and graph the solution set on a number line. H L M

M60 Evaluate and use finite sequence and series as a systematic and useful means of quantifying things.

L1.2.2 Interpret representations that reflect absolute value relationships (e.g. l x - a l ≤ b, or a ± b) in such contexts as error tolerance.

M35Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value.

H M M

L1.2.3 Use vectors to represent quantities that have magnitude and direction; interpret direction and magnitude of a vector numerically, and calculate the sum and difference of two vectors.

M52

Understand the characteristics and uses of vectors (e.g., representations of velocity and force) and perform basic operations on vectors (e.g., vector addition and scalar multiplication). A vector is a physical element possessing magnitude and direction.

M

L1.2.4 Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data; understand and critique data displays in the media.

M14Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population

M H H

M21 Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit).





M31Understand the characteristics of measures of dispersion (i.e., range, mean deviation, variance, and standard deviation).

L1.2.5 Read and interpret representations from various technological sources, such as contour or isobar diagrams. M21

Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit).

L M M

L1.3 Counting and Probabilistic ReasoningL1.3.1 Describe, explain, and apply various counting techniques (e.g., finding the number of different 4-letter passwords; permutations; and combinations); relate combinations to Pascal’s triangle; know when to use each technique.

M32Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly M M M

M51Determine combinations (the various groupings a set may be arranged in without regard to order) and permutations (arrangements of a set where order matters).

L1.3.2 Define and interpret commonly used expressions of probability (e.g., chances of an event, likelihood, odds).

M32Determine the probability of single and compound events using the Counting Principle to determine the probability of independent events occurring jointly.

M M M

L1.3.3 Recognize and explain common probability misconceptions such as “hot streaks” and “being due.” M5

Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. M H H

M32Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly.

STANDARD L2: CALCULATION, ALGORITHMS, AND ESTIMATIONStudents calculate fluently, estimate proficiently, and describe and use algorithms in appropriate situations (e.g., approximating solutions to equations.) They understand the basic ideas of iteration and algorithms.

L2.1 Calculation Using Real and Complex NumbersL2.1.1 Explain the meaning and uses of weighted averages (e.g., GNP, consumer price index, grade point average).

M14 Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population).

M H H





M21


M31 Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation).

L2.1.2 Calculate fluently with numerical expressions involving exponents; use the rules of exponents; evaluate numerical expressions involving rational and negative exponents; transition easily between roots and exponents.

M20

Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents.

H M H

L2.1.3 Explain the exponential relationship between a number and its base 10 logarithm, and use it to relate rules of logarithms to those of exponents in expressions involving numbers.

M20

Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents.

H M HM37

Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function.

M48 Express, graph, and interpret expontntial and logarithmic functions.

L2.1.4 Know that the complex number i is one of two solutions to x2 = -1. M25 Perform operations and solve equations containing

complex numbers H M H

L2.1.5 Add, subtract, and multiply complex numbers; use conjugates to simplify quotients of ) complex numbers.

M25Perform operations and solve equations containing complex numbers H M H

2.1.6 Recognize when exact answers aren’t always possible or practical; use appropriate algorithms to approximate solutions to equations (e.g., to approximate square roots)

M35 Use the properties of ral (real and irrational) numbers and demonstrate understanding of ordering and absolute value.

H M HM38 Create or evaluate algorithms (e.g., computer program codes, flowcharts) designed to accomplish a specific task.

M39Apply techniques to obtain a rational approximation or estimate of a quantity or number (including irrational numbers such as radicals).

L2.1.7 Understand the mathematical bases for the differences among voting procedures. M38 Create or evaluate algorithms (e.g., computer program

codes, flowcharts) designed to accomplish a specific task. L M M

L2.2 Sequences and Iteration





L2.2.1 Find the nth term in arithmetic, geometric, or other simple sequences. M37

Define and applyt he properties of relations and functions (domain, range, function, composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. H M H

M60 Evaluate and use finite sequence and series as systematic and useful means of quanitifying things.

L2.2.2 Compute sums of finite arithmetic and geometric sequences. M60 Evaluate and use finite sequence and series as systematic

and useful means of quanitifying things.H L M

M69Apply summation notation to taking the sum of an expression using limits (e.g., take the sum of 3i+1, from i = 1 to 5).

L2.2.3 Use iterative processes in such examples as computing compound interest or applying approximation procedures. M24

Understand the concepts recurrence relations and how they are applicable to such things as percentage rates, personal loans, simple interest, compound interest, installment buying, mortgage rates, etc. H M H

M39Apply techniques to obtain a rational approximation or estimate of a quantity or number (including irrational numbers such as radicals).

L2.2.4 Compute sums of infinite geometric sequences. M63

Understand the concepts and apply the uses of functions and limits (i.e., conduct limiting processes using functions to investigate infinite series and sequences).

L L L

STANDARD L3: MEASUREMENT AND PRECISIONStudents apply measurement units and calculations, and understand the concept of error.L3.1 Measurement Units, Calculations, and ScalesL3.1.1 Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly.

M8 Solve problems using units of metric measure and convert between metric and English/customary units.

M H HM13

Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real-world situations (e.g., scale drawings).

L3.1.2 Describe and interpret logarithmic relationships in such contexts as the Richter scale, the pH scale, or decibel measurements (e.g., explain why a small change in the scale can represent a large change in intensity); solve applied problems.

M20

Understand and apply the basic properties and laws of exponents and scientific notation to solve problems including those with fractional, negative, and zero exponents.

M M M

L3.2 Understanding Error





L3.2.1 Determine what degree of accuracy is reasonable for measurements in a given situation; express accuracy through use of significant digits, error tolerance, or percent of error; describe how errors in measurements are magnified by computation; recognize accumulated error in applied situations.

M12Understand accuracy and precision of measurement, round off numbers according to the correct number of significant figures, and determine percent error.

M H H

M13

Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real-world situations (e.g., scale drawings).

L3.2.2 Describe and explain round-off error, rounding, and truncating. M6 Demonstrate understanding of, and accurately apply, place

value to round off numbersM H H

M12Understand accuracy and precision of measurement, round off numbers according to the correct number of significant figures, and determine percent error.

L3.2.3 Know the meaning of and interpret statistical significance, margin of error, and confidence level. M21


M M M


STANDARD L4: MATHEMATICAL REASONING, LOGIC, AND PROOFStudents understand mathematical reasoning as being grounded in logic and proof and can distinguish mathematical arguments from other types of arguments. They can interpret arguments made about quantitative situations in the popular media. Students know the language and laws of logic and can apply them in both mathematical and everyday settings. They write proofs using direct and indirect methods and use counterexamples appropriately to show that statements are false.L4.1 Mathematical ReasoningL4.1.1 Distinguish between inductive and deductive reasoning, identifying and providing examples of each. M10


H H H

L4.1.2 Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic.

M10


M H H





M14Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population).


M49 Analyze the truth value of compound sentences by creating truth tables.

L4.1.3 Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the logical structure of mathematics; identify and give examples of each.

M4Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations.

H H H

M10


M23 Use the Pythagorean theorem to compute side lengths of right triangles.

M41

Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship with intuitive concepts (i.e., colliner points, coplanar points, opposite rays, and parallel lines).

L4.2 Language and Laws of LogicL4.2.1 Know and use the terms of basic logic (e.g., proposition, negation, truth and falsity, implication, if and only if, contrapositive, and converse). M10

Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. H H H


L4.2.2 Use the connectives “NOT,” “AND,” “OR,” and “IF…,THEN,” in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives.

M49

Analyze the truth value of compound sentences by creating truth tables.

H L M

L4.2.3 Use the quantifiers “THERE EXISTS” and “ALL” in mathematical and everyday settings and know how to logically negate statements involving them.

M10







L4.2.4 Write the converse, inverse, and contrapositive of an “If…, then…” statement; use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original while the inverse and converse are not.

M49

Analyze the truth value of compound sentences by creating truth tables.

H L M

L4.3 ProofL4.3.1 Know the basic structure for the proof of an “If…, then…” statement (assuming the hypothesis and ending with the conclusion) and know that proving the contrapositive is equivalent.

M10



L4.3.2 Construct proofs by contradiction; use counterexamples, when appropriate, to disprove a statement. M10


H H H

L4.3.3 Explain the difference between a necessary and a sufficient condition within the statement of a theorem; determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied.

M10



STRAND 2: ALGEBRA AND FUNCTIONSSTANDARD A1: EXPRESSIONS, EQUATIONS, AND INEQUALITIESStudents recognize, construct, interpret, and evaluate expressions. They fluently transform symbolic expressions into equivalent forms. They determine appropriate techniques for solving each type of equation, inequality, or system of equations, apply the techniques correctly to solve, justify the steps in the solutions, and draw conclusions from the solutions. They know and apply common formulas.A1.1 Construction, Interpretation, and Manipulation of Expressions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)





A1.1.1 Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables. M11

Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). H H H

M37


A1.1.2 Know the definitions and properties of exponents and roots, transition fluently between them, and apply them in algebraic expressions. M20


H M H

A1.1.3 Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities (e.g., differences of squares and cubes).

M29

Factor a composite number into its prime components and use least common denominators or least common multiples to solve equations. H M H

A1.1.4 Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x – 1) (1 – x2 + 3); simplify 9x - x3x + 3)

M36Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a+2)+(3a-1) = 5a+1).

H M H

A1.1.5 Divide a polynomial by a monomial.

M62

Perform division of a polynomial by a monomial by dividing powers with like bases, using the rules for the division of powers with like bases to simplify fractions with monomial denominators and reducing fractions to lowest terms.

H L M

A1.1.6 Transform exponential and logarithmic expressions into equivalent forms using the properties of exponents and logarithms including the inverse relationship between exponents and logarithms.

M20

Understand and apply the basic properties and laws of exponents and scientific notation to solve problems including those with fractional, negative, and zero exponents. H M H


A1.1.7 Transform trigonometric expressions into equivalent forms using basic identities such as M28

Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). L M M

M55Perform the general solution of triangles by using the law of Sines and Law of Cosines to obtain the angle and side length measurements of any triangle.





A1.2 Solutions of Equations and Inequalities (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)A1.2.1 Write equations and inequalities with one or two variables to represent mathematical or applied situations, and solve. M11

Apply variables in expressions and equations to solve problems (ie., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). H H H

M27


A1.2.2 Associate a given equation with a function whose zeros are the solutions of the equation. M37

Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. H M H

M53 Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.).

A1.2.3 Solve (and justify steps in the solutions) linear and quadratic equations and inequalities, including systems of up to three linear equations with three unknowns; apply the quadratic formula appropriately.

M10


H H HM27

Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x+2(x-3) = -4x+5 for x).

M47 Solve quadratic equations by applying various tools or techniques.

M65Find the graphic solution of systems of linear inequalities (i.e., graph the solution set or region of the coordinate plane common to both inequalities).

A1.2.4 Solve absolute value equations and inequalities, (e.g. solve l x - 3 l ≤ 6), and justify steps in the solution. M10


M27

Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x+2(x-3) = -4x+5 for x).





A1.2.5 Solve polynomial equations and equations involving rational expressions (e.g. solve -2x(x2 + 4x+3) = 0; solve x - 1x + 6 = 3), and justify steps in the solution.

M10



A1.2.6 Solve power equations (e.g., (x + 1)3 = 8) and equations including radical expressions (e.g., 3x - 7 = 7), justify steps in the solution, and explain how extraneous solutions may arise.

M10


M20


A1.2.7 Solve exponential and logarithmic equations (e.g., 3(2x) = 24), 2 ln(x + 1) = 4), and justify steps in the solution.

M7Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. H H H


A1.2.8 Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable, and justify steps in the solution.

M7Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution.

H H HM10


M27


A1.2.9 Know common formulas (e.g., slope, distance between two points, quadratic formula, compound interest, distance = velocity • time), and apply appropriately in contextual situations.

M19Compute the distance between two points on a coordinate plane (i.e., length of a line segment) and determine the midpoint of a line segment between two points.

H M HM24


M46 Know the equation for the slope of a line and compute slope given the coordinates of two points.





A1.2.10 Use special values of the inverse trigonometric functions to solve trigonometric equations over specific intervals (e.g., 2sin x – I = 0 for 0 ≤ x ≤ 2).

M37


H M H

STANDARD A2: FUNCTIONSStudents understand functions, their representations, and their attributes. They perform transformations, combine and compose functions, and find inverses. Students classify functions and know the characteristics of each family. They work with functions with real coefficients fluently.A2.1 Definitions, Representations, and Attributes of FunctionsA2.1.1 Recognize whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function; and identify its domain and range. M21

Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). H M H

M37


A2.1.2 Read, interpret, and use function notation, and evaluate a function at a value in its domain. M37


M56Know how to express a linear function using the functional notation f(x) = mx+b, and determine the ordered pairs.

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words, and translate among representations. M21

Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). H M H

M37






A2.1.4 Recognize that functions may be defined by different expressions over different intervals of their domains; such functions are piecewise-defined (e.g., absolute value and greatest integer functions).

M37


H M H

A2.1.5 Recognize that functions may be defined recursively, and compute values of and graph simple recursively defined functions (e.g., f(0) = 5, and f(n) = f(n-1) + 2 ).

M24

Understand the concepts of recurrence relations and apply them to solve consumer mathematics problems involving such things as percentage rates, personal loans, simple interest, compound interest, installment buying, mortgage rates, etc. H M H

M37


A2.1.6 Identify the zeros of a function and the intervals where the values of a function are positive or negative, and describe the behavior of a function, as x approaches postive or negative infinity, given the symbolic and graphical representations.

M37


H M HM47 Solve quadratic equations by applying various tools or techniques.

M63Understand the concepts and apply the uses of functions and limits (i.e., conduct limiting processes using functions to investigate infinite series and sequences).

A2.1.7 Identify and interpret the key features of a function from its graph or its formula(e), (e.g. slope, intercept(s), asymptote(s), maximum and minimum value(s), symmetry, average rate of change over an interval, and periodicity).

M37


H M HM44Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b), and the point-slope form (y-b = m(x-a)).


M59Understand the characteristics of maxima and minima and be able to mathematically determine maximum and minimum points on a graph or curve.

A2.2 Operations and TransformationsA2.2.1 Combine functions by addition, subtraction, multiplication, and division. M37


H M H





A2.2.2 Apply given transformations (e.g., vertical or horizontal shifts, stretching or shrinking, or reflections about the x- and y-axes) to basic functions, and represent symbolically.

M54

Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry.

H L M

A2.2.3 Recognize whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs (e.g., f(x) = x3 and g(x) = x1/3).

M37


H M H

A2.2.4 If a function has an inverse, find the expression(s) for the inverse. M37


L M M

A2.2.5 Write an expression for the composition of one function with another; recognize component functions when a function is a composition of other functions.

M37


L M M

A2.2.6 Know and interpret the function notation for inverses and verify that two functions are inverses using composition. M37


L M M

A2.3 Families of Functions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)A2.3.1 Identify a function as a member of a family of functions based on its symbolic, or graphical representation; recognize that different families of functions have different asymptotic behavior at infinity, and describe these behaviors.

M37



A2.3.2 Describe the tabular pattern associated with functions having constant rate of change (linear); or variable rates of change.

M44Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b), and the point-slope form (y-b = m(x-a)).

H L MM53 Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.).

M61Use the process of differentiation (i.e., derivatives) to determine tangents, maxima and minima, velocity, and acceleration.





A2.3.3 Write the general symbolic forms that characterize each family of functions. (e.g., f(x) = A0ax; f(x) = AsinBx ) M37


H M H


M56Know how to express a linear function using the functional notation f(x) = mx+b, and determine the ordered pairs.

M64

Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y=sin x, y= cos x, and y=tan x, where the measure of the angle x is expressed in radians).

A2.4 Lines and Linear FunctionsA2.4.1 Write the symbolic forms of linear functions (standard [i.e., Ax + By = C, where B ≠ 0], point-slope, and slope-intercept) given appropriate information, and convert between forms.


H L MM46 Know the equation for the slope of a line and compute slope given the coordinates of two points.

M56Know how to express a linear function (e.g., y = 1/3x+5) using the functional notation f(x) = 1/3x+5, and determine the ordered pairs.

A2.4.2 Graph lines (including those of the form x = h and y = k) given appropriate information. M44

Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b), and the point-slope form (y-b = m(x-a)). H L M


A2.4.3 Relate the coefficients in a linear function to the slope and x- and y-intercepts of its graph. M44

Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b), and the point-slope form (y-b = m(x-a)).

H L MM46 Know the equation for the slope of a line and compute slope given the coordinates of two points.

M56Know how to express a linear function (e.g., y = 1/3x+5) using the functional notation f(x) = 1/3x+5, and determine the ordered pairs.

A2.4.4 Find an equation of the line parallel or perpendicular to given line, through a given point; understand and use the facts that non-vertical

M4 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations.

L H M





parallel lines have equal slopes, and that non-vertical perpendicular lines have slopes that multiply to give -1.



A2.5 Exponential and Logarithmic FunctionsA2.5.1 Write the symbolic form and sketch the graph of an exponential function given appropriate information. (e.g., given an initial value of 4 and a rate of growth of 1.5, write f(x) = 4 (1.5)x).

M48

Express, graph, and interpret exponential and logarithmic functions. H L M

A2.5.2 Interpret the symbolic forms and recognize the graphs of exponential and logarithmic functions (e.g., f(x) = 10 x, f(x) = log x, f(x) = ex, f(x) = ln x); recognize the logarithmic function as the inverse of the exponential function.

M48 Express, graph, and interpret exponential and logarithmic functions. H L H

A2.5.3 Apply properties of exponential and logarithmic functions (e.g., ax+y = axa y; log(ab)= log a + log b).

M48 Express, graph, and interpret exponential and logarithmic functions H L M

A2.5.4 Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and understand how the base affects the rate of growth or decay.

M48

Express, graph, and interpret exponential and logarithmic functions H L M

A2.5.5 Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time. M48 Express, graph, and interpret exponential and logarithmic

functions H L M

A2.6 Quadratic FunctionsA2.6.1 Write the symbolic form and sketch the graph of a quadratic function given appropriate information (e.g., vertex, intercepts, etc.).

M53 Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). H L M

A2.6.2 Identify the elements of a parabola (vertex, axis of symmetry, direction of opening) given its symbolic form or its graph, and relate these elements to the coefficient(s) of the symbolic form of the function.


A2.6.3 Convert quadratic functions from standard to vertex form by completing the square. M37


H M H







A2.6.4 Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function.

M47 Solve quadratic equations by applying various tools or techniques. H L M


A2.6.5 Express quadratic functions in vertex form to identify their maxima or minima, and in factored form to identify their zeros.


H L MM53 Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.).

M59Understand the characteristics of maxima and minima and be able to mathematically determine maximum and minimum points on a graph or curve.

A2.7 Power Functions (including roots, cubics, quartics, etc.)A2.7.1 Write the symbolic form and sketch the graph of power functions. M20

Understand and apply the basic properties and laws of exponents and scientific notation to solve problems including those with fractional, negative, and zero exponents. H M H

M37


A2.7.2 Express direct and inverse relationships as functions (e.g., y = kxn and y = kx-n, n > 0) and recognize their characteristics (e.g., in y = x3, note that doubling x results in multiplying y by a factor of 8).

M37


H M H

A2.7.3 Analyze the graphs of power functions, noting reflectional or rotational symmetry. M57 Solve and graphically sketch problems involving two

variables that exhibit direct and indirect variation. H M H

A2.8 Polynomial FunctionsA2.8.1 Write the symbolic form and sketch the graph of simple polynomial functions. M53 Express, graph, and interpret polynomial functions (linear,

quadratic, cubic, etc.). H L M

A2.8.2 Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2.


A2.8.3 Determine the maximum possible number of zeros of a polynomial function, and understand the relationship between the x-intercepts of the graph and the factored form of the function.

M53

Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). H L M





A2.9 Rational FunctionsA2.9.1 Write the symbolic form and sketch the graph of simple rational functions. M53 Express, graph, and interpret polynomial functions (linear,

quadratic, cubic, etc.).H L M


A2.9.2 Analyze graphs of simple rational functions (e.g., 2x + 1x - 1xx2 - 4;f(x) = g(x) = ) and understand the relationship between the zeros of the numerator and denominator and the function’s intercepts, asymptotes, and domain.

M37


H M HM47 Solve quadratic equations by applying various tools or techniques.



A2.10 Trigonometric FunctionsA2.10.1 Use the unit circle to define sine and cosine; approximate values of sine and cosine (e.g., sin 3, or cos 0.5); use sine and cosine to define the remaining trigonometric functions; explain why the trigonometric functions are periodic.

M10


M64

Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (i.e., y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians).

A2.10.2 Use the relationship between degree and radian measures to solve problems. M64


H L M

A2.10.3 Use the unit circle to determine the exact values of sine and cosine, for integer multiples of ⁄6 and ⁄4. M64


H L M

A2.10.4 Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, andlocation of maxima and minima.

M37 Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function.

H M H





M59Understand the characteristics of maxima and minima and be able to determine maximum and minimum points mathematically on a graph or curve.

M64


A2.10.5 Graph transformations of basic trigonometric functions (involving changes in period, amplitude, and midline) and understand the relationship between constants in the formula and the transformed graph.

M54

Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. H L M

M64


STANDARD A3: MATHEMATICAL MODELINGStudents construct or select a function to model a real-world situation in order to solve applied problems. They draw on their knowledge of families of functions to do so. A3.1 Models of Real-world Situations Using Families of Functions.A3.1.1 Identify the family of function best suited for modeling a given real-world situation (e.g., quadratic functions for motion of an object under the force of gravity; exponential functions for compound interest; trigonometric functions for periodic phenomena. In the example above, recognize that the appropriate general function is exponential (P = P0at)

M24


M M MM48 Express, graph, and interpret exponential and logarithmic

functions


M64






A3.1.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. In the example above, substitute the given values P0 = 300 and a = 1.02 to obtain P = 300(1.02)t.

M11

Apply variables in expressions and equations to solve problems (ie., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world).

M H HM48 Express, graph, and interpret exponential and logarithmic functions

M64


A3.1.3 Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled. In the example above, the exact solution is 365.698, but for this problem an appropriate approximation is 365.

M10


M M M

A3.1.4 Use methods of linear programming to represent and solve simple real-life problems. M38 Create or evaluate algorithms (e.g., computer program

codes, flowcharts) designed to accomplish a specific task. M M M

STRAND 3: GEOMETRY AND TRIGONOMETRYSTANDARD G1: FIGURES AND THEIR PROPERTIESStudents represent basic geometric figures, polygons, and conic sections and apply their definitions and properties in solving problems and justifying arguments, including constructions and representations in the coordinate plane. Students represent three-dimensional figures, understand the concepts of volume and surface area, and use them to solve problems. They know and apply properties of common three-dimensional figures.G1.1 Lines and Angles; Basic Euclidean and Coordinate GeometryG1.1.1 Solve multi-step problems and construct proofs involving vertical angles, linear pairs of angles supplementary angles, complementary angles, and right angles.

M10


M15Classify angles by measure (acute, right, obtuse, and straight) and understand angle relationships (supplementary, complementary, and vertical).





G1.1.2 Solve multi-step problems and construct proofs involving corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles.

M4 Understand the properties of and parallel, perpendicular, and intersecting lines in problem-solving situations.

H H HM10


M15Classify angles by measure (acute, right, obtuse, and straight) and understand angle relationships (supplementary, complementary, and vertical).

G1.1.3 Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass. M10


M42Use geometric methods, such as using an unmarked straightedge and compass, to complete basic geometric constructions.

G1.1.4 Given a line and a point, construct a line through the point that is parallel to the original line using straightedge and compass; given a line and a point, construct a line through the point that is perpendicular to the original line; justify the steps of the constructions.


H H HM10



G1.1.5 Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint.

M19Compute the distance between two points on a coordinate plane (length of a line segment) and determine the midpoint of a lien segment between two points.

H M H

G1.1.6 Recognize Euclidean Geometry as an axiom system; know the key axioms and understand the meaning of and distinguish between undefined terms (e.g., point, line, plane), axioms, definitions, and theorems.


H H HM10


M41

Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship withintuitive concepts (e.e., collinear points, coplanar points, opposite rays, and parallel lines).





G1.2 Triangles and Their PropertiesG1.2.1 Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle is the sum of the two remote interior angles. M10


M34

Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations.

G1.2.2 Construct and justify arguments and solve multi-step problems involving angle measure, side length, perimeter, and area of all types of triangles.


H H HM10


G1.2.3 Know a proof of the Pythagorean Theorem and use the Pythagorean Theorem and its converse to solve multi-step problems. M10


H H H

M23 Apply the Pythagorean Theorem to right trianglesG1.2.4 Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º triangles and 45º- 45º- 90º triangles. M10


M34


G1.2.5 Solve multi-step problems and construct proofs about the properties of medians, altitudes, and perpendicular bisectors to the sides of a triangle, and the angle bisectors of a triangle; using a straightedge and compass, construct these lines.

M10



G1.3 Triangles and TrigonometryG1.3.1 Define the sine, cosine, and tangent of acute angles in a right triangle as ratios of sides; solve problems about angles, side lengths, or areas using trigonometric ratios in right triangles.


H M H





G1.3.2 Know and use the Law of Sines and the Law of Cosines and use them to solve problems; find the area of a triangle with sides a and b and included angleusing the formula Area = (1/2) a b sin.

M55

Perform the general solution of triangles by using the law of sines and law of cosines to obtain the angle and side length measurements of any triangle. H L M

G1.3.3 Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples, and apply in various contexts.


H M H

G1.4 Quadrilaterals and Their PropertiesG1.4.1 Solve multi-step problems and construct proofs involving angle measure, side length, diagonal length, perimeter, and area of squares, rectangles, parallelograms, kites, and trapezoids.


H H HM10


M34


G1.4.2 Solve multi-step problems and construct proofs involving quadrilaterals (e.g., prove that the diagonals of a rhombus are perpendicular) using Euclidean methods or coordinate geometry.

M10


M34


G1.4.3 Describe and justify hierarchical relationships among quadrilaterals, (e.g. every rectangle is a parallelogram). M10


M34


G1.4.4 Prove theorems about the interior and exterior angle sums of a quadrilateral.

M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems.

H H H





M34


G1.4.5 Understand the definition of a cyclic quadrilateral, and know and use the basic properties of cyclic quadrilaterals.

M34Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations.

L M M

G1.5 Other Polygons and Their PropertiesG1.5.1 Know and use subdivision or circumscription methods to find areas of polygons (e.g., regular octagon, non-regular pentagon).


H H HM18

Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle qualtities (lengths of line segments, angle measure within a circle, circumference, and ares) in problem-solving situations.

M34


G1.5.2 Know, justify, and use formulas for the perimeter and area of a regular n-gon and formulas to find interior and exterior angles of a regular n-gon and their sums.


H H HM10


M34


G1.6 Circles and Their PropertiesG1.6.1 Solve multi-step problems involving circumference and area of circles. M18


H M H

G1.6.2 Solve problems and justify arguments about chords (e.g., if a line through the center of a circle is perpendicular to a chord, it bisects the chord) and lines tangent to circles (e.g., a line tangent to a


H H H





circle is perpendicular to the radius drawn to the point of tangency). M18


G1.6.3 Solve problems and justify arguments about central angles, inscribed angles and triangles in circles. M10


M18


G1.6.4 Know and use properties of arcs and sectors, and find lengths of arcs and areas of sectors. M18


H M H

G1.7 Conic Sections and Their PropertiesG.1.7.1 Find an equation of a circle given its center and radius; given the equation of a circle, find its center and radius. M18

Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle qualtities (lengths of line segments, angle measure within a circle, circumference, and ares) in problem-solving situations. H M H

M66Know how to sketch basic conic sections (e.g., circles and parabolas) using their equations, and graphically solve systems of equations.

G1.7.2 Identify and distinguish among geometric representations of parabolas, circles, ellipses, and hyperbolas; describe their symmetries, and explain how they are related to cones.

M66

Know how to sketch basic conic sections (e.g., circles and parabolas) using their equations, and graphically solve systems of equations. H L M

G1.7.3 Graph ellipses and hyperbolas with axes parallel to the x- and y-axes, given equations.

M30

Know and apply the components and properties of the rectangular coordinate system, (i.e., x - y axis, origin, quadrants, abscissa (x-coordinate) and ordinate (y-coordinate), and the general representation of a point (x,y)). H M H

M66Know how to sketch basic conic sections (e.g., circles and parabolas) using their equations, and graphically solve systems of equations.





G1.7.4 Know and use the relationship between the vertices and foci in an ellipse, the vertices and foci in a hyperbola, and the directrix and focus in a parabola; interpret these relationships in applied contexts.

M66

Know how to sketch basic conic sections (e.g., circles and parabolas) using their equations, and graphically solve systems of equations. L L L

G1.8 Three- Dimensional FiguresG1.8.1 Solve multi-step problems involving surface area and volume of pyramids, prisms, cones, cylinders, hemispheres, and spheres.


H H HM26

Know the classification and properties of three dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids.

G1.8.2 Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres, and spheres. M26

Know the classification and properties of three dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. H M H

M54

Apply transformations (reflection, rotation, translation, and dilation)of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry.

STANDARD G2: RELATIONSHIPS BETWEEN FIGURESStudents use and justify relationships between lines, angles, area and volume formulas, and 2- and 3-dimensional representations. They solve problems and provide proofs about congruence and similarity.G2.1 Relationships Between Area and Volume FormulasG2.1.1 Know and demonstrate the relationships between the area formula of a triangle, the area formula of a parallelogram, and the area formula of a trapezoid.


H H HM34


G2.1.2 Know and demonstrate the relationships between the area formulas of various quadrilaterals


H H H





(e.g., explain how to find the area of a trapezoid based on the areas of parallelograms and triangles). M34


G2.1.3 Know and use the relationship between the volumes of pyramids and prisms (of equal base and height) and cones and cylinders (of equal base and height).

M26


H M H

G2.2 Relationships Between Two-dimensional and Three-dimensional RepresentationsG2.2.1 Identify or sketch a possible 3-dimensional figure, given 2-dimensional views (e.g., nets, multiple views); create a 2-dimensional representation of a 3-dimensional figure.

M34


H M HM26


M18


G2.2.2 Identify or sketch cross-sections of 3-dimensional figures; identify or sketch solids formed by revolving 2-dimensional figures around lines.

M26Know the classification and properties of three dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids.

H M H

G2.3 Congruence and SimilarityG2.3.1 Prove that triangles are congruent using the SSS, SAS, ASA, and AAS criteria, and for right triangles, the hypotenuse-leg criterion. M10


M54


G2.3.2 Use theorems about congruent triangles to prove additional theorems and solve problems, with and without use of coordinates.


H H H





M30

Know and apply the components and properties of the rectangular coordinate system, (i.e., x - y axis, origin, quadrants, abscissa (x-coordinate) and ordinate (y-coordinate), and the general representation of a point (x,y)).

M54


G2.3.3 Prove that triangles are similar by using SSS, SAS, and AA conditions for similarity. M10

Apply transformations (reflection, rotation, translation, and dilation)of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. H H H

M54


G2.3.4 Use theorems about similar triangles to solve problems with and without use of coordinates. M10


H H HM30

Know and apply the components and properties of the rectangular coordinate system, (i.e., x - y axis, origin, quadrants, abscissa (x-coordinate) and ordinate (y-coordinate), and the general representation of a point (x,y)).

M54


G2.3.5 Know and apply the theorem stating that the effect of a scale factor of k relating one two dimensional figure to another or one three dimensional figure to another, on the length, area, and volume of the figures is to multiply each by k, k2, and k3, respectively.

M3 Use proportional reasoning to solve real-world problems. H H H

M9 Compute the perimeter and area of common two dimensional figures.

M13 Use the technique of dimensional analysis to convert units of measure (e.g., convert km/hr to m/min) including drawing to scale and applying ratio to real- world situations.





M26


STANDARD G3: TRANSFORMATIONS OF FIGURES IN THE PLANEStudents will solve problems about distance-preserving transformations and shape-preserving transformations. The transformations will be described synthetically and, in simple cases, by analytic expressions in coordinates.G3.1 Distance-preserving Transformations: IsometriesG3.1.1 Define reflection, rotation, translation, and glide reflection and find the image of a figure under a given isometry. M54

Apply transformations (reflection, rotation, translation, and dilation)of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. H L M

M68Apply transformation concepts to examine two-dimensional aspects or representations of three-dimensional figures (e.g., isometries and nets).

G3.1.2 Given two figures that are images of each other under an isometry, find the isometry and describe it completely. M54



G3.1.3 Find the image of a figure under the composition of two or more isometries, and determine whether the resulting figure is a reflection, rotation, translation, or glide reflection image of the original figure.

M54



G3.2 Shape-preserving Transformations: Dilations and IsometriesG3.2.1 Know the definition of dilation, and find the image of a figure under a given dilation. M54


H L M





G3.2.2 Given two figures that are images of each other under some dilation, identify the center and magnitude of the dilation. M54


H L M

G3.2.3 Find the image of a figure under the composition of a dilation and an isometry. M54

Apply transformations (reflection, rotation, translation, and dilation)of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. L L L


STRAND 4: STATISTICS AND PROBABILITYSTANDARD S1: UNIVARIATE DATA- EXAMINING DISTRIBUTIONSStudents plot and analyze univariate data by considering the shape of distributions and analyzing outliers; they find and interpret commonly-used measures of center and variation; and they explain and use properties of the normal distribution.S1.1 Producing and Interpreting PlotsS1.1.1 Construct and interpret dot plots, histograms, relative frequency histograms, bar graphs, basic control charts, and box plots with appropriate labels and scales; determine which kinds of plots are appropriate for different types of data; compare data sets and interpret differences based on graphs and summary statistics.


H H HM21

Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions,critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit).

M31 Understand and apply measures of dispersion (i.e., range, mean deviation, variance, and standard deviation).

S1.1.2 Given a distribution of a variable in a data set, describe its shape, including symmetry or skewness, and state how the shape is related to measures of center (mean and median) and measures of variation (range and standard deviation) with particular attention to the effects of outliers on these measures.


H H H

M21 Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions,critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit).






S1.2 Measures of Center and VariationS1.2.1 Calculate and interpret measures of center including: mean, median, and mode; explain uses, advantages and disadvantages of each measure given a particular set of data and its context.


H H H

M21


S1.2.2 Estimate the position of the mean, median, and mode in both symmetrical and skewed distributions, and from a frequency distribution or histogram.


H H H

M21


S1.2.3 Compute and interpret measures of variation, including percentiles, quartiles, interquartile range, variance, and standard deviation.


H M HM43

Understand and apply the concepts and applications of quartiles (distributing groups into four equal sizes), percentiles (distributing individuals into 100 groups of equal size), and random distribution to understand and interpret data.

S1.3 The Normal DistributionS1.3.1 Explain the concept of distribution and the relationship between summary statistics for a data set and parameters of a distribution.


M M MM43

Understand and apply the concepts and applications of quartiles (distributing groups into four equal sizes), percentiles (distributing individuals into 100 groups of equal size), and random distribution to understand and interpret data.

S1.3.2 Describe characteristics of the normal distribution, including its shape and the relationships among its mean, median, and mode.

M14 Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population).

H H H





M21


S1.3.3 Know and use the fact that about 68%, 95%, and 99.7% of the data lie within one, two, and three standard deviations of the mean, respectively in a normal distribution.

M31

Understand and apply measures of dispersion (i.e., range, mean deviation, variance, and standard deviation). H M M

S1.3.4 Calculate z-scores, use z-scores to recognize outliers, and use z-scores to make informed decisions.

M31Understand and apply measures of dispersion (i.e., range, mean deviation, variance, and standard deviation). H M H

STANDARD S2: BIVARIATE DATA – EXAMINING RELATIONSHIPSStudents plot and interpret bivariate data by constructing scatterplots, recognizing linear and nonlinear patterns, and interpreting correlation coefficients; they fit and interpret regression models, using technology as appropriate.S2.1 Scatterplots and CorrelationS2.1.1 Construct a scatterplot for a bivariate data set with appropriate labels and scales.

M21


H M H

S2.1.2 Given a scatterplot, identify patterns, clusters, and outliers; recognize no correlation, weak correlation, and strong correlation. M21


H M H

S2.1.3 Estimate and interpret Pearson’s correlation coefficient for a scatterplot of a bivariate data set; recognize that correlation measures the strength of linear association.

M21


H M H

S2.1.4 Differentiate between correlation and causation; know that a strong correlation does not imply a cause-and-effect relationship; recognize the role of lurking variables in correlation.

M21


H M H





S2.2 Linear RegressionS2.2.1 For bivariate data which appear to form a linear pattern, find the least squares regression line by estimating visually and by calculating the equation of the regression line; interpret the slope of the equation for a regression line.

M21


H M H


S2.2.2 Use the equation of the least squares regression line to make appropriate predictions.

M21


H

STANDARD S3: SAMPLES, SURVEYS, AND EXPERIMENTSStudents understand and apply sampling and various sampling methods, examine surveys and experiments, identify bias in methods of conducting surveys, and learn strategies to minimize bias. They understand basic principles of good experimental design.S3.1 Data Collection and AnalysisS3.1.1 Know the meanings of a sample from a population and a census of a population, and distinguish between sample statistics and population parameters.

M21


M M M

S3.1.2 Identify possible sources of bias in data collection and sampling methods and simple experiments; describe how such bias can be reduced and controlled by random sampling; explain the impact of such bias on conclusions made from analysis of the data; and know the effect of replication on the precision of estimates.

M21


M M M

S3.1.3 Distinguish between an observational study and an experimental study, and identify, in context, the conclusions that can be drawn from each. M21


M M M





S3.1.4 Design simple experiments or investigations to collect data to answer questions of interest; interpret and present results. M21


L M M

S3.1.5 Understand methods of sampling, including random sampling, stratified sampling, and convenience samples, and be able to determine, in context, the advantages and disadvantages of each.

M21


L M M

S3.1.6 Explain the importance of randomization, double-blind protocols, replication, and the placebo effect in designing experiments and interpreting the results of studies.

M17 Understand the importance of random sampling and sample size in generating representative data.

L M MM21


S3.2.1 Explain the basic ideas of statistical process control, including recording data from a process over time. M21


L M M

S3.2.2 Read and interpret basic control charts; detect patterns and departures from patterns.

M21


H

STANDARD S4: PROBABILITY MODELS AND PROBABILITY CALCULATIONStudents understand probability and find probabilities in various situations, including those involving compound events, using diagrams, tables, geometric models and counting strategies; they apply the concepts of probability to make decisions.S4.1 ProbabilityS4.1.1 Understand and construct sample spaces in simple situations (e.g., tossing two coins, rolling two number cubes and summing the results).

M5 Examine problem-solving situatins involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events.

M H H






S4.1.2 Define mutually exclusive events, independent events, dependent events, compound events, complementary events and conditional probabilities; and use the definitions to compute probabilities.

M5

Examine problem-solving situatins involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. M H H


S4.1.3 Design and carry out an appropriate simulation using random digits to estimate answers to questions about probability; estimate probabilities using results of a simulation; compare results of simulations to theoretical probabilities.


L H M


S4.2 Application and RepresentationS4.2.1 Compute probabilities of events using tree diagrams, formulas for combinations and permutations, Venn diagrams, or other counting techniques.

M5

Examine problem-solving situatins involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events.

M H HM32Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly.

M51Determine combinations (the various groupings a set may be arranged in without regard t order) and permutations (arrangements of a set where order matters).

S4.2.2 Apply probability concepts to practical situations, in such settings as finance, health, ecology, or epidemiology, to make informed decisions.

M5

Examine problem-solving situatins involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. M H H



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