# GTPS Curriculum 6 th Grade Math. Topic: Topic 1- Numeration

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1 9 days / September Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. Topic: Topic 1- Numeration /Enduring What are whole number and decimal place values? How can whole numbers and decimals be written, compared, and ordered? Our number system is based on groups of ten. Whenever we get 10 in one place value, we move to the next greater place value. Decimal place values are just an extension of whole number place values to numbers less than one. Numbers can be used to tell how many. Place values can be represented using exponents. Place value can be used to compare and order numbers. The value of an equation containing unknowns can be true or false depending on the replacement values of the unknowns. Patterns can be used to determine the locations of the decimal point in the product when multiplying a decimal by 10, 100, or Place Value 1.2 Comparing and Ordering Whole Numbers 1.3 Exponents and Place Value 1.4 Decimal Place Value 1.5 Multiplying and Dividing by 10, 100, and 1, Comparing and Ordering Decimals 1.7 Make an Organized List Topic Readiness Test Teacher Observation Daily Quick check Masters Topic Tests Performance Tasks

2 Suggested Blocks of Instruction: 10 days / September / October Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve onevariable equations and inequalities. 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.2.a Write expressions that record operations with numbers and with letters standing for numbers 6.EE.3 Apply the properties of operations to generate equivalent expressions 6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Topic: Topic 2-Variables, Expressions, and Properties /Enduring What are algebraic expressions and how can they be written and evaluated? What arithmetic number relationships, called properties, are always true? Some mathematical phrases can be represented using a variable in an algebraic expression. The value of an algebraic expression can be found by replacing the variable(s) with given number(s) and doing the calculation that results. You can add (or multiply) two numbers in any order. Three numbers can be grouped and added (or multiplied) in any order. There is an agreed upon order in which operations are carried out in a numerical expression. The Distributive Property of Multiplication over Addition lets you multiply a sum by multiplying each addend separately and then finding the sum of the products. There is more than one way to do a mental calculation. Mental calculation techniques change numbers so the calculation is easy to do mentally. Properties of operations make some calculations easy to do mentally. Some quantities have a mathematical relationship; the value of one quantity can be found if you know the value of the other quantity. Patterns can sometimes be used to identify a relationship between two quantities. Some problems can be solved by recording and organizing data in a table and by finding and using numerical patterns in the table. 2.1 Using Variables to Write Expressions 2.2 Properties of Operations 2.3 Order of Operations 2.4 The Distributive Property 2.5 Mental Math 2.6 Evaluating Expressions 2.7 Using Expressions to Describe Patterns 2.8 Make a Table Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

3 12 days /October Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.EE.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order. 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Topic: Topic 3-Operations with Decimals /Enduring How are sums, differences, products, and quotients involving decimals estimated and found? Rounding is a process for finding the multiple of 10, 100, etc. or of 0.1, 0.01, etc. closest to a given number. Standard addition and subtraction algorithms break calculations into simpler calculations using place value. The standard multiplication algorithm involving decimals is an extension of the standard algorithm for multiplying whole numbers. The sharing interpretation of division can be used to model the standard division algorithm. The standard division algorithm involving decimals is an extension of the standard algorithm for dividing whole numbers. There is an agreed upon order for which operations in a numerical expression are performed. The value of an equation containing unknowns can be true or false depending on the replacement values of the unknowns. Variables are used to represent numbers in mathematical expressions, equations, or inequalities. A number divided by a decimal can be represented as an equivalent calculation using place value to change the divisor to a whole number. 3.1 Estimating Sums and Differences 3.2 Adding and Subtracting 3.3 Estimating Products and Quotients 3.4 Multiplying Decimals 3.5 Dividing Whole Numbers 3.6 Dividing by a Whole Number 3.7 Dividing Decimals 3.8 Evaluating Expressions 3.9 Solutions for Equations and Inequalities 3.10 Multiple-Step Problems Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

4 Suggested Blocks of Instruction: 7 days Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve onevariable equations and inequalities. 6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers Topic: Topic 4-Solving Equations /Enduring What procedures can be used to solve equations? The same number can be added or subtracted from both sides of an equation and not change the equality. Multiplying or dividing both sides of an equation by the same nonzero number does not change the equality. Solving an equation involves finding the value of the unknown that makes the equation true. There is more than one way to solve an equation. Information in a problem can often be shown using a picture and used to understand and solve the problem. Some problems can be solved by writing and completing a number sentence or equation. 4.1 Properties of Equality 4.2 Solving Addition and Subtraction Equations 4.3 Draw a Picture and Write an Equation 4.4 Solving Multiplication and Division Equations 4.5 Draw a Picture and Write and Equation Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

5 Suggested Blocks of Topic: Topic 5-Number and Fraction Concepts Instruction: 9 days / Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor. How can numbers be broken apart into factors? How can fractions be represented and simplified? Every counting number can be divided by 1 and itself. Every positive integer (except the number 1) can be represented as a product of one or more prime numbers in exactly one way disregarding the order of the factors. There is always a greatest number that divides evenly each of two whole numbers. A fraction describes the division of a whole into equal parts. The same fractional amount can be represented by an infinite set of different but equivalent fractions. A fraction can be expressed in its simplest form by dividing the numerator and denominator by common factors, including the greatest common factor, until there are no common factors other than 1. Some real-world quantities have a mathematical relationship; the value of one quantity can be found if you know the value of the other quantity. Commonalities in attributes of objects or situations can be found and used to make generalizations about relationships. 5.1 Factors, Multiples, and Divisibility 5.2 Prime Factorization 5.3 Greatest Common Factor 5.4 Understanding Fractions 5.5 Equivalent Fractions 5.6 Fractions in Simplest Form 5.7 Make and Test Conjectures Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

6 Suggested Blocks of Instruction: 7 days Topic: Topic 6-Decimals, Fractions, and Mixed Numbers / Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. How are decimals and fractions related? A fraction describes the division of a whole into equal parts, and it can be interpreted in more than one way depending on the whole to be divided. A decimal is another name for a fraction. Mixed numbers can also be represented using decimals. Fractional amounts greater than 1 can be represented in different ways. Whole number amounts can be represented as fractions. When the numerator and denominator are equal, the fraction is equal to 1. Information in a problem can often be shown by drawing a picture or a diagram that can be used to understand and solve the problem. 6.1 Fractions and Division 6.2 Fractions and Decimals 6.3 Improper Fractions and Mixed Numbers 6.4 Decimal Forms of Fractions and Mixed Numbers 6.5 Draw a Picture Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

8 Suggested Blocks of Instruction: 7 days Topic: Topic 8-Multiplying Fractions and Mixed Numbers /Enduring These lessons reinforce concepts and skills required for Topic 9. What are standard procedures for estimating and finding products of fractions and mixed numbers? The product of a fraction, x/y and a whole number can be interpreted as dividing the whole into y parts and then taking x of those parts. Products involving fractions can be interpreted as finding the area of a rectangle. Rounding and compatible numbers can be used to estimate the product of fractions or mixed numbers. The product of a fraction and a whole number can be fund by dividing the whole number by the denominator and then multiplying by the numerator. The product of two fractions can be found by writing the product of the numerators over the product of the denominators. The product of two mixed numbers can be found by first changing the mixed numbers to improper fractions, and then multiplying the improper fractions. Some problems can be solved by first finding and solving a sub-problem(s) and then using that answer(s) to solve the original problem. 8.1 Multiplying a Fraction and a Whole Number 8.2 Estimating Products 8.3 Multiplying Fractions 8.4 Multiplying Mixed Numbers 8.5 Multiple-Step Problems Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

9 Suggested Blocks of Topic: Topic 9- Dividing Fractions and Mixed Numbers Instruction: 9 days Objectives/CPI s/ Standards / Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Apply and extend previous understanding of numbers to the system of rational numbers. 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.. 6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. What are standard procedures from estimating and finding quotients of fractions and mixed numbers? Rounding and compatible numbers can be used to estimate the quotient of mixed numbers. When dividing by a fraction that is less than 1, the quotient is greater than the dividend. A division expression with a fraction divisor can be changed to an equivalent multiplication expression. The quotient of two mixed numbers can be found by changing the mixed numbers to improper fractions, then changing the division expression to an equivalent multiplication expression. Equations with fractions and mixed numbers can be solved using properties of equality and inverse operations. Some problems can be solved by identifying elements that repeat in a predictable way. 9.1 Understanding Division of Fractions 9.2 Dividing a Whole Number by a Fraction 9.3 Dividing Fractions 9.4 Estimating Quotients 9.5 Dividing Mixed Numbers 9.6 Solving Equations 9.7 Look for a Pattern Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

10 Suggested Blocks of Instruction: 8 days Topic: Topic 10-Integers Objectives/CPI s/standard / Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values 6.NS.6.c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 6.NS.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram 6.NS.7.d Distinguish comparisons of absolute value from statements about order 6.G.3 Draw polygons in the coordinate plane use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate How are integers related to whole numbers? How can integers be added, subtracted, multiplied, and divided? Numbers that are the same distance from 0 on the number line are opposites. Integers are the counting numbers, their opposites, and zero. Numbers to the right of 0 are positive and numbers to the left of 0 are negative. A number to the right of another on the number is the greater number. Each rational number can be associated with a unique point on the number line. Absolute value is used to define the distance from a number to zero, regardless of whether the number is positive or negative. The Cartesian Coordinate System is a scheme that uses two perpendicular number lines intersecting at zero to tell the location of points in the plane. Some problems can be solved by reasoning about the conditions in the problem Understanding Integers 10.2 Comparing and Ordering Integers 10.3 Rational Numbers on a Number Line 10.8 Absolute Value 10.9 Graphing Points on a Coordinate Plane Use Reasoning Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

11 7 days Lessons in this topic reinforce concepts and skills required for Topic EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another Topic: Topic 11-Properties of Two-Dimensional Figures /Enduring How can angles be measured, drawn, and classified? What are special shapes and how can they be described and compared? Point, line, line segment, ray, and plane are the core attributes of space objects. Lines can be classified by their relationship to other lines. An angle is formed by two intersecting lines or by two different rays with a common endpoint. Angles can be classified by their size. Polygons can be described and classified by their sides and angles. The angles of a triangle can form a straight line so the sum of the measures of the angles is 180 degrees. Polygons can be described and classified by their sides and angles. A quadrilateral can be broken into two triangles. So the sum of the measures of the angles of a quadrilateral is 360 degrees. Some problems can be solved by recording and organizing data in a table and by finding and using numerical patterns in the table Basic Geometric Ideas 11.2 Measuring and Drawing Angles 11.4 Triangles 11.5 Quadrilaterals 11.9 Make a Table and Look for a Pattern Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

12 8 days Understand ratio concepts and use ration reasoning to solve problems. 6.RP.1 Understanding the concept of a ratio and use ratio language to describe a ratio relationship between two quantities 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship 6.RP.3 Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.RP.3.b Solve unit rate problems including those involving unit pricing and constant speed 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation Topic: Topic 12-Ratios, Rates, and Proportions /Enduring What are ratios and rates and how are they used in solving problems? A ratio is a special relationship between two quantities where for every x units of one quantity there are y units of another quantity. The quantities being compared in a ratio are called terms. In a proportional relationship there are an infinite number of ratios equal to the lowest terms or constant ratio. Equal ratios can be found by multiplying both terms by the same non-zero number. A rate is a special ratio that compares two quantities with different units of measure. A unit rate is a rate that compares a quantity to one unit of another quantity. A formula is a common relationship between quantities expressed as a n equation. A special proportional relationship involves distance (d), rate (r), and time (t). The formula showing this relationship is d = r X t. Rates are easily compared when each is expressed as a unit rate. Information in a problem can often be shown using a picture to understand and solve the problem Understanding Ratios 12.2 Equal Ratios and Proportions 12.3 Understanding Rates and Unit Rates 12.4 Comparing Rates 12.5 Distance, Rate, and Time 12.6 Draw a Picture Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

13 7 days Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.RP.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. 6.RP.3.b Solve unit rate problems including those involving unit pricing and constant speed Topic: Topic 13-Solving Proportions /Enduring What procedures can be used to solve proportions? Some procedures problems can be solved by generating equal ratios using multiplication or division. Some proportions can be solved by finding and using the common factor that relates the terms. Some proportions can be solved by finding and using the unit amount. Tape diagrams and double number line diagrams can show ratio relationships and be used to reason about solutions to problems. Equivalent ratios can be represented in a table, and the pairs of values can be plotted on the coordinate plane. Mathematical explanations can be given using words, pictures, numbers, or symbols. A good explanation should be correct, simple, complete, and easy to understand Using Ratio Tables 13.2 Using Unit Rates 13.3 Applying Ratios 13.4 Writing to Explain 13.5 Ratios and Graphs Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

14 9 days Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.RP.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. Topic: Topic 14- Understanding Percent /Enduring What is the meaning of percent? How can percent be estimated and found? A percent is a special kind of ratio in which a part is compared to a whole with 100 parts. The whole is 100%. Percent is relative to the size of the whole. Finding a percent of a whole is like finding a fractional part of a whole. You can find the percent of a number by changing the percent to a decimal and multiplying or using a proportion. The whole can be found when you are given a percent and a part. A number line and a proportion can be used to help solve for the missing whole. A part of a whole or a part of a set can be represented by a fraction, a decimal, and a percent. Some percents can be approximated by simple fractions and used to estimate the percent of a number. Answers to problems should always be checked for reasonableness, and this can be done in different ways Understanding Percent 14.2 Fractions, Decimals, and Percents 14.3 Percents Greater Than 100 and Less Than Estimating Percent 14.5 Finding the Percent of a Number 14.6 Applying Percents: Finding the Whole 14.7 Reasonableness Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

15 6 days Reason about and solve one-variable equations and inequalities. 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equations Topic: Topic 15 Equations and Graphs /Enduring How can equations be graphed? What patterns can be found in the graphs of equations? A solution to an inequality is a value that makes the inequality true. Patterns can sometimes help identify the relationship between quantities, and an equation can be written describing the relationship. Graphs of relationships in the form y = ax and y = x + a (a is a real number) are straight lines. The graph of y = ax passes through the origin. The graph of y = x + a does not pass through the origin, unless a equals zero. Some problems can be solved by using objects to act out the actions in the problems. Some problems can be solved by reasoning about the conditions in the problems Patterns and Equations 15.4 Graphing Equations 15.6 Understanding Inequalities 15.7 Act It Out and Use Reasoning Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

16 8 days Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Topic: Topic 16-Measurement /Enduring How can customary and metric measurements be converted to other units? How are customary and Metric units related? Measurements can be represented in equivalent ways using different units. Relationships exist that enable you to convert between units by multiplying or dividing. The smaller the units on the scale of a measuring instrument, the more precise the measurement is. Time can be expressed using different units that are related to each other. The time between the start and end of an event is called elapsed time and can be calculated. Some problems can be solved by reasoning about conditions in the problem Converting Customary Measures 16.2 Converting Metric Measures 16.3 Units of Measure and Precision 16.4 Relating Customary and Metric Measures 16.5 Elapsed Time 16.6 Use Reasoning Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

17 6 days Solve real-world and mathematical problems involving area, surface area, and volume. 6.EE.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Topic: Topic 17-Perimeter and Area /Enduring What are the meanings of perimeter and area? How can the perimeter and area of certain shapes be found? The distance around a figure is its perimeter. Formulas exist for finding the perimeter of some polygons and some formulas may be represented in more than one way. The measure of a region inside a shape is its area, and area can be found using square units. The area of some irregular shapes can be found by decomposing the shape into polygons for which formulas exist for finding area. The formula for area of a parallelogram is derived from the formula for area of a rectangle. The formula for area of a triangle is derived from the formula for area of a parallelogram. Some problems can be solved by using objects to act out the actions in the problem Perimeter 17.2 Area of Rectangles and Irregular Figures 17.3 Area of Parallelograms and Triangles 17.6 Use Objects Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

18 7 days Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V= l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Topic: Topic 18- Volume and Surface Area /Enduring What is the meaning of volume and how can volume be found? What is the meaning of surface area and how can surface area be found? How can the volume of certain figures by found? A polyhedron is a three-dimensional figure made of flat surfaces. The shapes of these flat surfaces and the way they are connected at edges and vertices determine the characteristics of the polyhedron. Formulas for find the area of polygons can be used to find the surface area of some solids. Volume is a measure of the amount of space inside a solid figure. Volume can be measured by counting the number of cubic units needed to fill a threedimensional object. The volume of rectangular prisms with fractional edge lengths can be determined in the same way as volume of rectangular prisms with wholenumber edge lengths. Some problems can be solved by using objects to act out the actions in the problem. Some problems can be solved by reasoning about the conditions in the problems Solid Figures 18.2 Surface Area 18.3 Volume of Rectangular Prisms 18.4 Volume with Fractional Edge Lengths 18.5 Use Objects and Reasoning Topic Readiness Test Teacher Observation Daily Quick Check Masters Topic Tests Performance Tasks

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