Math Curriculum

6^th Grade Math ENVISION MATH PROGRAM OVERVIEW

v Topic1-NUMBERATION

Ø TOPICS COVERED

§ PlaceValue

§ Comparingand Ordering Whole Numbers

§ Exponentsand Place Value

§ DecimalPlace Value

§ Multiplyingand Dividing by 10, 100, and 1,000

§ Comparingand Ordering Decimals

§ PROBLEMSOLVING-Make an Organized List

Ø COMMON CORE STANDARDS

§ 6.NS.2 The Number System -Fluently divide multidigit numbers using the standard algorithm

§ 6.NS.3 The Number System -Fluently add, subtract,multiply, and divide multi-digit decimals using the standard algorithm for eachoperation

§ 6.EE.1 Expressions and Equations- Writeand evaluate numerical expressions involving whole-number exponents.

v Topic2-VARIABLES, EXPRESSIONS, AND PROPERTIES

Ø TOPICS COVERED

§ UsingVariables to Write Expressions

§ Propertiesof Operations

§ Orderof Operations

§ TheDistributive Property

§ MentalMath

§ EvaluatingExpressions

§ UsingExponents to Describe Patterns

§ PROBLEMSOLVING-Make a Table

Ø COMMON CORE STANDARDS

§ 6.EE.2 Expressions and Equations -Writeand evaluate numerical expressions involving whole-number exponents

§ 6.EE.2a Expressions and Equations-Write expressions that record operations with numbers and with letters standingfor numbers. For Example: express the calculation “Subtract y from 5” as 5-y.

§ 6.EE.2b Expressions and Equations-Identify parts of an expression using mathematical terms; view more one or moreparts if an expression as a single entity. For Example: describe theexpression 2(8+7) as a product of two factors; view (8+7) as both a singleentity and a sum of two terms.

§ 6.EE.2c Expressions and Equations-Evaluate expressions at specific values of their variables. Include expressions that arise from formulasused in real-world problems. Performarithmetic operations, including those involving whole-number exponents, in theconventional order when there are no parentheses to specify a particular order(Order of Operations). For Example: use the formulas V=s^2 and A =6s^2 to find the volume and surface area of a cube with sides of length s=1/2.

§ 6.EE.3 Expressions and Equations- Applyproperties of operations to generate equivalent expressions. For Example: apply the distributive propertyto the expression 3 (2+x) to produce the equivalent expression 6 +3x; apply thedistributive property to the expression 24x+18y to produce the equivalentexpression 6 (4x+3y); apply properties of operations to y+y+y to produce theequivalent expression 3y.

§ 6.NS.2 The Number System- Fluentlydivide multi-digit numbers using the standard algorithm

§ 6.NS.3 The Number System- Fluently add,subtract, multiply, and divide multi-digit decimals using the standardalgorithm for each operation.

v Topic3-OPERATIONS WITH DECIMALS

Ø TOPICS COVERED

§ EstimatingSums and Differences

§ Addingand Subtracting

§ EstimatingProducts and Quotients

§ MultiplyingDecimals

§ DividingWhole Numbers

§ Dividingby a Whole Number

§ DividingDecimals

§ EvaluatingExpressions

§ Solutionsfor Equations and Inequalities

§ PROBLEMSOLVING-Multiple-Step Problems

Ø COMMON CORE STANDARDS

§ 6.NS.2 The Number System- Fluentlydivide multi-digit numbers using the standard algorithm

§ 6.NS.3 The Number System- Fluently add,subtract, multiply, and divide multi-digit decimals using the standardalgorithm for each operation.

§ 6.EE.5 Expressions and Equations-Understand solving an equation or inequality as a process of answering aquestion: which values from a specified set, if any, make the equation orinequality true? Use substitution to determine whether a given number in aspecified set makes an equation or inequality true.

§ 6.EE.6Expressions and Equations- UseVariables to represent numbers and write expressions when solving a real-worldor mathematical problem; understand that a variable can represent an unknownnumber, or, depending on the purpose at hand, any number in a specified set.

v TOPIC4 SOLVING EQUATIONS

Ø TOPICS COVERED

§ Propertiesof Equality

§ SolvingAddition and Subtraction Equations

§ SolvingMultiplication and Division Equations

§ PROBLEMSOLVING- Draw a picture to write an equation

Ø COMMON CORE STANDARDS

§ 6.EE.4 Expressions and Equations-Identify when two expressions are equivalent (i.e, when the two expressionsname the same number regardless of which value is substituted into them). ForExample: the expressions y+y+y and 3y are equivalent because they name the samenumber regardless of which number y stands for.

§ 6.EE.7Expressions and Equations- Solvereal-world and mathematical problems by writing and solving equations of theform x + p= q and px=q for cases in which p, q, and x are all nonnegativerational numbers.

§ 6.NS.2 The Number System- Fluentlydivide multi-digit numbers using the standard algorithm

v TOPIC5 NUMBER AND FRACTION CONCEPTS

Ø TOPICS COVERED

§ Factors,Multiples, and Divisibility

§ PrimeFactorization

§ GreatestCommon Factor

§ UnderstandingFractions

§ EquivalentFractions

§ Fractionsin Simplest Form

§ PROBLEMSOLVING-Make and Test Conjectures

Ø COMMON CORE STANDARDS

§ 6.NS.1 The Number System- Interpret and compute quotients of fractions, andsolve word problems involving division of fractions, by using visual fractionmodels and equations to represent the problem. For Example: create a story context for (2/3) x (3/4) and use a visualfraction model to show the quotient; use the relationship betweenmultiplication and division to explain that (2/3) x (3/4) = 8/9 because ¾ of8/9 is 2/3. (In general, (a/b) x (c/d) = ad / bc.) How much chocolate will eachperson get if 3 people share ½ lb of chocolate equally? How many ¾ cup servingsare in 2/3 of a cup of yogurt? How wide is a rectangular strip of land withlength ¾ mi and area 1/2 square mile?

§ 6.NS.4 The Number System- Find the greatest common factor of two wholenumbers less than or equal to 100 and the least common multiple of two wholenumbers less than or equal to 12. Use the distributive property to express asum of two whole numbers 1-100 with a common factor as a multiple of a sum oftwo whole numbers with no common factor. ForExample: express 36 + 8 as 4(9+2)

v TOPIC6 DECIMALS, FRACTIONS, and MIXED NUMBERS

Ø TOPICS COVERED

§ Fractionsand Division

§ Fractionsand Decimals

§ ImproperFractions and Mixed Numbers

§ DecimalsForms of Fractions and Mixed Numbers

§ PROBLEMSOLVING-Draw a Picture

Ø COMMON CORE STANDARDS

§ 6.NS.3 The Number System- Fluently add,subtract, multiply, and divide multi-digit decimals using the standardalgorithm for each operation.

v TOPIC7 ADDING AND SUBTRACTING FRACTIONS AND MIXED NUMBERS

Ø TOPICS COVERED

§ Addingand Subtracting: Like Denominators

§ LeastCommon Multiple

§ Addingand Subtracting: Unlike Denominators

§ EstimatingSums and Differences of Mixed Numbers

§ AddingMixed Numbers

§ SubtractingMixed Numbers

§ PROBLEMSOLVING-MAKE A TABLE

Ø COMMON CORE STANDARDS

§ 6.RP.1 Ratios and ProportionalRelationships- Understand the concept of a ratio and use ratio language todescribe a ratio relationship between two quantities. For Example: “The ratio of wings to beaks in the bird house at the zoowas 2:1, because for every 2 wings there was 1 beak.” “For every vote candidateA received, candidate C received nearly three votes.”

v TOPIC8 MULTIPLYING FRACTIONS AND MIXED NUMBERS

Ø TOPICS COVERED

§ Multiplyinga fration and a whole number

§ EstimatingProducts

§ MultiplyingFractions

§ MultiplyingMixed Numbers

§ PROBLEMSOLVING Multiple-Step Problems

Ø COMMON CORE STANDARDS

§ 6.G.1 Geometry- Find the area of righttriangles, other triangles, special quadrilaterals, and polygons by composinginto rectangles or decomposing into triangles and other shapes; apply thesetechniques in the context of solving real-world and mathematical problems.

v TOPIC9 DIVIDING FRACTIONS AND MIXED NUMBERS

Ø TOPICS COVERED

§ UnderstandingDivision of Fractions

§ Dividinga Whole Number by a Fraction

§ DividingFractions

§ EstimatingQuotients

§ DividingMixed Numbers

§ SolvingEquations

§ PROBLEMSOLVING-Look for a Pattern

Ø COMMON CORE STANDARDS

§ 6.NS.6 The Number System- Understand a rational number as a point on the number line. Extend number line diagramsand coordinate axes familiar from previous grades to represent points on theline and in plane with negative number coordinates.

v TOPIC10 INTEGERS

Ø TOPICS COVERED

§ UnderstandingIntegers

§ Comparingand Ordering Integers

§ RationalNumbers on a Number Line

§ AddingIntegers

§ SubtractingIntegers

§ MultiplyingIntegers

§ DividingIntegers

§ AbsoluteValue

§ GraphingPoints on a Coordinate Plan

§ PROBLEMSOLVING-Use Reasoning

Ø COMMON CORE STANDARDS

§ 6.NS.5 The Number System- Understanding that positive and negative numbersare used together to describe quantities having opposite directions or values(e.g. temperature above/below zero, elevation above/below sea level,credits/debits, positive/negative electric charge); use positive and negativenumbers to represent quantities in real-world contexts, explaining the meaningof 0 in each situation.

§ 6.NS.6a The Number System- Recognize opposite signs of numbers as indicatinglocations on opposite sides of 0 on the number line; recognize that theopposite of the opposite of a number is the number itself and that 0 is its ownopposite.

§ 6.NS.6b The Number System- Understand signs of numbers in ordered pairs asindicating location in quadrants of the coordinate plane; recognize that whentwo ordered pairs differ only by signs, the location of the points are relatedby reflections across one or both axes.

§ 6.NS.6c The Number System- Find and position integers and other rationalnumbers on a horizontal or vertical number line diagram; find and positionpairs of integers and other rational numbers on a coordinate plane.

§ 6.NS.7a The Number System- Interpret statements of inequalities asstatements about the relative position of two numbers on a number line diagram.For Example: interpret -3 > -7 as astatement that -3 is located to the right of -7 on a number line oriented fromleft to right.

§ 6.NS.7b The Number System-Write, interpret, and explain statements of orderfor rational numbers in real-world contexts. For Example: write -3 degrees C > -7 degrees C to express the factthat -3 degrees C is warmer that – 7 degrees C.

§ 6.NS.7c The Number System-Understand the absolute value of a rational numberas its distance from 0 on the number line; interpret absolute value as magnitudefor a positive or negative quantity in a real-world situation. ForExample: for an account balance of -30 dollars, right |-30|=30 to describe thesize of the debt in dollars.

§ 6.NS.7d The Number System- Distinguish comparisons of absolute value fromstatements about order. For Example:recognize that an account balance less than -30 dollars represents a debtgreater than 30 dollars.

§ 6.NS.8 The Number System- Solve real-world and mathematical problems bygraphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute valueto find distance between points with the same first coordinate or the samesecond coordinate.

§ 6.G.3 Geometry- Draw polygons in thecoordinate plane given coordinates for the vertices; use coordinates to findthe length of a side joining points with the same first coordinate of the samesecond coordinate. Apply thesetechniques in the context of solving real-world and mathematical problems.

v Topic11 PROPERTIES OF TWO-DIMENSIONAL FIGURES

Ø TOPICS COVERED

§ BasicGeometric Ideas

§ Measuringand Drawing Angles

§ AnglePairs

§ Triangles

§ Quadrilaterals

§ Circles

§ Transformationsand Congruence

§ Symmetry

§ PROBLEMSOLVING-Make a Table and Look for a Pattern

Ø COMMON CORE STANDARDS

§ 6.G.3 Geometry- Draw polygons in thecoordinate plane given coordinates for the vertices; use coordinates to findthe length of a side joining points with the same first coordinate of the samesecond coordinate. Apply these techniquesin the context of solving real-world and mathematical problems.

§ 6.EE.9 Expressions and Equations- Usevariables to represent two quantities in a real-world problem that change inrelationship to one another; write an equation to express one quantity ,thought of as the dependent variable, in terms of the other quantity, thoughtof as the independent variable. Analyzethe relationship between the dependent and independent variables using graphsand tables, and relate these to the equation. For Example: in a problem involving motion at constant speed, list andgraph ordered pairs of distances and times, and write the equation d=65t torepresent the relationship between distance and time.

v Topic12- RATIOS, RATES, AND PROPORTIONS

Ø TOPICS COVERED

§ UnderstandingRatios

§ EqualRatios and Proportions

§ UnderstandingRates and Unit Rates

§ ComparingRates

§ Distance,Rate, and Time

§ PROBLEMSOLVING-Draw a picture

Ø COMMON CORE STANDARDS

§ 6.RP.2 Ratios and ProportionalRelationships- Understand the concept of a rate a/b associated with aration a:b with b not equal to 0 and use rate language in the context of aration relationship. For Example: “This recipe has a ratio of 3cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup ofsugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

§ 6.RP.3 Ratios and ProportionalRelationships- Use ratio and ratereasoning to solve real-world and mathematical problems, e.g., by reasoningabout tables of equivalent ratios, tape diagrams, double number line diagrams,or equations.

§ 6.RP.3b Ratios and ProportionalRelationships- Solve unit rate problems including those involving unitpricing and constant speed.

v Topic13-SOLVING PROPORTIONS

Ø TOPICS COVERED

§ UsingRatio Tables

§ UsingUnit Rates

§ ApplyingRatios

§ PROBLEMSOLVING-Writing to Explain

§ Ratiosand Graphs

§ Mapsand Scale Drawings

Ø COMMON CORE STANDARDS

§ 6.RP.3a Ratios and ProportionalRelationship- Make tables of equivalent ratios relating quantities withwhole-numbers measurements, find missing values in the tables, and plot thepairs of values in the tables, and plot the pairs or values on the coordinateplane. Use tables to compare ratios.

§ 6.RP.3b Ratios and ProportionalRelationships- Solve unit rate problems including those involving unitpricing and constant speed. For Example: if it took 7 hours to mow 4lawns, then at that rate, how many lawns could be mowed in 35 hours? At whatrate were lawns being mowed?

v Topic14-UNDERSTANDING PERCENT

Ø TOPICS COVERED

§ UnderstandingPercent

§ Fractions,Decimal, and Percents

§ PercentsGreater than 100 and less than 1

§ EstimatingPercent

§ Findingthe Percent of a Number

§ ApplyingPercents: Finding the whole

§ PROBLEMSOLVING-Reasonableness

Ø COMMON CORE STANDARDS

§ 6.RP.3c Ratios and ProportionalRelationships-Find a percent of a quantity as a rate per 100 (e.g. 30% ofquantity means 30/100 times the quantity); solve problems involving finding thewhole, given a part and the percent.

v Topic15 Equations and Graphs

Ø TOPICS COVERED

§ Equationswith more than one operation

§ Patternsand equations

§ Morepatterns and equations

§ Graphingequations

§ Graphingequations with more than one operation

§ UnderstandingInequalities

§ PROBLEMSOLVING-Act it Out and Use Reasoning

Ø COMMON CORE STANDARDS

§ 6.EE.8 Expressions and Equations-Writean inequality of the form x>c or x<c to represent a constraint orcondition in a real-world or mathematical problem. Recognize that inequalities of the formx>c and x<c have infinitely many solutions; represent solutions of suchinequalities on number line diagrams.

v TOPIC16 MEASUREMENT

Ø TOPICS COVERED

§ ConvertingCustomary Measures

§ ConvertingMetric Measures

§ Unitsof Measure and Precision

§ RelatingCustomary and Metric Measures

§ ElapsedTime

§ PROBLEMSOLVING-Use Reasoning

Ø COMMON CORE STANDARDS

§ 6.RP.3d Ratios and ProportionalRelationships-Use ratio reasoning to convert measurement units; manipulateand transform units appropriately when multiplying or dividing quantities.

v TOPIC17 PERIMETER AND AREA

Ø TOPICS COVERED

§ Perimeter

§ Areaof Rectangles and Irregular Figures

§ Areaof Parallelograms and Triangles

§ Circumference

§ Areaof a Circle

§ PROBLEMSOLVING-Use Objects

Ø COMMON CORE STANDARDS

§ 6.G.4 Geometry- Representthree-dimensional figures using nets made up of rectangles and triangles, anduse the nets to find the surface area of these figures. Apply these techniquesin the context of solving real-world and mathematical problems.

v TOPIC18 VOLUME AND SURFACE AREA

Ø TOPICS COVERED

§ SolidFigures

§ SurfaceArea

§ Volumeof Rectangular Prisms

§ Volumeof Fractional Edge Lengths

§ PROBLEMSOLVING-Use Objects and Reasoning

Ø COMMON CORE STANDARDS

§ 6.G.2 Geometry- Find the volume of aright rectangular prism with fractional edge lengths by packing it with unitcubes of the appropriate unit fraction edge lengths, and show that the volumeis the same as would be found by multiplying the edge lengths of theprism. Apply the formulas V= l w h andV=bh to find volumes of right rectangular prisms with fractional edge lengthsin the context of solving real-world and mathematical problems.

v TOPIC19 DATA AND GRAPHS

Ø TOPICS COVERED

§ StatisticalQuestions

§ Lookingat Data Sets

§ Mean

§ Median,Mode, Range

§ FrequencyTables and Histograms

§ BoxPlots

§ Measuresof Variability

§ AppropriateUse of Statistical Measures

§ SummarizingData Distributions

§ PROBLEMSOLVING-Try, Check, and Revise

Ø COMMON CORE STANDARDS

§ 6.SP.1 Statistics and Probability-Recognize a statistical question as one that anticipates variability in thedate related to the question and accounts for it in the answer. For Example; “How old am I?” is not astatistical question, but “How old are the students in my school?” is astatistical question because one anticipates variability in students’ age.

§ 6.SP.2 Statistics and Probability-Understand that a set of data collected to answer a statistical question has adistribution which can be described by its center, spread, and overall shape.

§ 6.SP.3 Statistics and Probability-Recognize that a measure of center for a numerical data set summarizes all ofits values with a single number, while a measure of variation describes how itsvalues with a single number.

§ 6.SP.4 Statistics and Probability-Display numerical data in plots on a number line, including do plots,histograms, and box plots

§ 6.SP.5 Statistics and Probability- Summarizenumerical data sets in relation to their context.

§ 6.SP. 5a Statistics and Probability-Reporting the number of observations

§ 6.SP.5b Statistics and Probability-Describing the nature of the attribute under investigation, including how itwas measured and its units of measure.

§ 6.SP. 5c Statistics and Probability-Giving quantitative measures of center (media and/or mean) and variability(interquartile range and/or mean absolute deviation), as well as describing anyoverall pattern and any striking deviations from the overall pattern withreference to the context in which the data were gathered.

§ 6.SP.5d Statistics and Probability-Relating the choice of measures of center and variability to the shape of thedata distribution and the context in which the data were gathered.

Last Modified on March 4, 2013