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Collegium Charter School Grade 6 Math PA Core Standards Scope & Sequence Global Vision We Use Math in Our Everyday Lives Grade 6 PA Common Core Scope and Sequence 2014-2015 Standards of Mathematical Practice (Habits of Mind) 6th grade: 1. Make sense of problems and persevere in solving them. Solve problems involving ratios and rates and discuss how they solved them. Solve real world problems through the application of algebraic and geometric concepts. o Seek the meaning of a problem and look for efficient ways to represent and solve it. o Check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?” 5. o o 2. Reason abstractly and quantitatively. o Represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. o Contextualize to understand the meaning of the number or variable as related to the problem. o Decontextualize to manipulate symbolic representations by applying properties of operations. 3. Construct viable arguments and critique the reasoning of others. o Construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). o Refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. o Pose questions like “How did you get that?”, “Why is that true?”, and “Does that always work?” o Explain their thinking to others and respond to others’ thinking. 4. Model with mathematics. o Model problem situations symbolically, graphically, tabularly, and contextually. o Form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. o Begin to explore covariance and represent two quantities simultaneously. o Use number lines to compare numbers and represent inequalities. o Use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. o Connect and explain the connections between the different representations. o Use all representations as appropriate to a problem context. Use appropriate tools strategically. Consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. o Decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data. o Use physical objects or applets to construct nets and calculate the surface area of three dimensional figures. o 6. 7. Attend to precision. o Continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. o Use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities. Look for and make use of structure. Routinely seek patterns or structures to model and solve problems. Recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. o Apply properties to generate equivalent expressions (i.e. 6 + 2x = 2 (3 + x) by distributive property). o Solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality, c=6 by division property of equality). o Compose and decompose two‐ and three‐dimensional figures to solve real world problems involving area and volume. o o 8. Look for and express regularity in repeated reasoning. Use repeated reasoning to understand algorithms and make generalizations about patterns. o Solve and model problems. They may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization. o Connect place value and their prior work with operations to understand algorithms to fluently divide multi‐digit numbers and perform all operations with multi‐digit decimals. o Informally begin to make connections between covariance, rates, and representations showing the relationships between quantities o 2 Grade 6 PA Common Core Scope and Sequence 2014-2015 Investigation, Task and Problem-Solving Rubric 18 points + 3 advanced Below Basic - 1 Basic - 2 Proficient - 3 Advanced - 4 Understands the Problem Doesn't understand enough to get started or make progress Understands enough to solve part of the problem or to get part of the solution Understands the problem Uses Information Appropriately Uses inappropriate information Uses some appropriate information correctly Uses all appropriate information correctly Applies Appropriate Procedures Applies inappropriate procedures Applies some appropriate procedures Applies completely appropriate procedures Uses Representations Uses a representation that gives little or no significant information about the problem Uses a representation that gives some Uses a representation that clearly important information about the depicts the problem problem Uses a representation that is unusual in its mathematical precision Answers the Problem No answer or wrong answer based upon an inappropriate plan Copying error, computational error, partial answer for problem with multiple answers, no answer statement, answer labeled incorrectly Correct solution of problem and made a general rule about the solution or extended the solution to a more complicated solution Communicates Effectively No explanation, or Partial explanation explanation doesn’t make Some steps described with partial sense. connection between representations (work) and the math. Grade 6 critical areas: 1. Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems. 2. Completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers. 3. Writing, interpreting, and using expressions and equations. 4. Developing understanding of statistical thinking. Correct solution Identifies special factors that influence the approach before starting the problem Explains why procedures are appropriate to the problem. Essential mathematics in the solution is connected to any work or representations used in solving. PA Core Standard Areas for Math Grade 6 M.06 M.06 M.06 M.06 M.06 A-N = The Number System A-R = Ratios and Proportional Relationships B-E = Expressions and Equations C-G = Geometry D-S = Statistics and Probability 3 Grade 6 PA Common Core Scope and Sequence 2014-2015 CMQ1 How do we use numbers and operations in our everyday lives? Big Ideas Essential Questions (CMQs) OPERATION MEANINGS & RELATIONSHIPS: The same number How are mathematical operations (addition, subtraction, multiplication, sentence (e.g. 12-4 = 8) can be associated with different division) related to each other? concrete or real-world situations, AND different number How can mathematical operations be related to real-world situations? sentences can be associated with the same concrete or realHow can I use common factors or common multiples in solving problems? world situation. How can I use the distributive property express addition? How can the distributive property be modeled using manipulatives? EQUIVALENCE: Any number, numerical expression, algebraic How does multiplying a decimal divisor and dividend by an appropriate expression, or equation can be represented in an infinite power of ten change it into an equivalent calculation? number of ways that have the same value. Concepts & Competencies Compute fluently with multi-digit numbers and find common factors and multiples Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation . Assessment Anchor Descriptors MO6 A-N.2.1 Compute with multi-digit numbers using the four arithmetic operations with or without a calculator. MO6 A-N.2.2 Apply number theory concepts (specifically, factors and multiples). 4 Grade 6 PA Common Core Scope and Sequence 2014-2015 CM1 Numbers & Operations FK Assessments CM1 Vocabulary FK: Teacher-created assessment. Each word is worth one point. Teacher may differentiate method of assessment. May be more than one test. Grade 6 Vocabulary List in Progress as of June 30, 2014 Array Common multiples Factor Multiple Product Squares Common factors Distributive property Inverse operations Operation Quotient terms CMQ1 Numbers & Operations Student completes task or investigation at any time during the CM unit that addresses Big Ideas of this Unit. Standards of Mathematical Practice in this task are evaluated by the problem solving rubric. Tasks may be differentiated according to student interest, ability level, etc. CM1FK1 Numbers & Operations Solve problems involving operations (+, –, ×, ÷), straight computation, or word problems with whole numbers & decimals (thru thousandths). Find the greatest common factor (GCF) of two whole numbers less than or equal to 100 Find and the least common multiple (LCM) of two whole numbers less than or equal to 12. Apply 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. Example: Express 36 + 8 as: 4 (9 + 2). Limit numbers 1 through 100. 5 Grade 6 PA Common Core Scope and Sequence CMQ2How do we use ratios in our everyday lives? Big Ideas PROPORTIONALITY: If two quantities vary proportionally, that relationship can be represented as a linear function on a graph or in a table. In a proportional relationship there are an infinite number of ratios equal to the lowest terms or constant ratio. 2014-2015 Essential Questions (CMQs) How can graphs and tables help us see how numbers are related? Where can examples of ratios and rates be found? How are cross products and unit rates helpful in determining whether two ratios are equivalent? Why is a percent ratio based on 100? How can a proportion be used to solve a problem involving percent? How is dividing whole numbers similar to dividing fractions? How can dividing fractions be modeled using area, sets, or a number line? OPERATIONS MEANINGS & RELATIONSHIPS: Dividing a fraction by a fraction divisor can be thought of as finding the number of fractional pieces in the dividend. Concepts & Competencies Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates. Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems Assessment Anchor Descriptors M06.A-R.1.1 Represent and/or solve real-world and mathematical problems using rates, ratios, and/or percents. M06.A-N.1.1 Solve real-world and mathematical problems involving division of fractions. 6 Grade 6 PA Common Core Scope and Sequence 2014-2015 CM2 Ratios FK Assessments CM2 Vocabulary: Teacher created assessment. Each word is worth one point. Teacher may differentiate the method of assessment. May be more than one test. Grade 6 Vocabulary List in Progress as of June 30, 2014 constant speed denominator numerator percent proportion ratio unit price coordinate plane equivalent ratio ordered pair product quotient ratio notation unit rate CMQ2 Ratios Student completes task or investigation at any time during the CM unit that addresses Big Ideas of this Unit. Standards of Mathematical Practice in this task are evaluated by the problem solving rubric. Tasks may be differentiated according to student interest, ability level, etc. CM2FK1 Ratios Use ratio language and notation (such as 3 to 4, 3:4, 3/4) to describe a ratio relationship between two quantities. Ex 1: “The ratio of girls to boys in a math class is 2:3, because for every 2 girls there are 3 boys.” Ex 2: “For every five votes candidate A received, candidate B received four votes.” Find the unit rate a/b associated with a ratio a: b (with b≠0), and use rate language in the context of a ratio relationship. Ex 1: “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” Ex 2: “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” Construct tables of equivalent ratios relating quantities with whole-number measurements. Find missing values in the tables, and/or plot the pairs of values on the coordinate plane. Use tables to compare ratios. CM2FK2 Unit rate and percent Solve unit rate problems including those involving unit pricing and constant speed. Ex: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? 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. CM2FK3 Fraction Quotients Interpret and compute quotients of fractions (including mixed numbers), and solve word problems involving division of fractions by fractions. Solving division expressions by understanding using multiplicative inverse (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc Ex 1: Given a story context for (2/3) ÷ (3/4), explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. Ex 2: How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Ex 3: How many 2 1/4-foot pieces can be cut from a 15 1/2-foot board? 7 Grade 6 PA Common Core Scope and Sequence CMQ3 How do we use integers in our everyday lives? Big Ideas NUMBERS — The set of real numbers is infinite, and each real number can be associated with a unique point on the number line. ORIENTATION & LOCATION: Objects in space can be oriented in an infinite number of ways, and an object’s location in space can be described quantitatively. 2014-2015 Essential Question (CMQs) What are integers? What are opposite numbers? What is the greatest integer on the number line? What is the least? How are negative and positive numbers used in the real world? How do we name points on the number line that are not named by integers? What is the meaning of absolute value on the number line? How do we use coordinates to determine the distance between two points on the coordinate plane? Concepts & Competencies Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. Assessment Anchor Descriptors M06.A-N.3.1 Demonstrate how positive and negative numbers are used together to describe quantities having opposite directions or values and locations on the number line and coordinate plane. M06.A-N.3.2 Understand ordering and absolute value of rational numbers. 8 Grade 6 PA Common Core Scope and Sequence 2014-2015 CM3 Integers FK Assessments CM3 Vocabulary FK: Teacher-created assessment. Each word is worth one point. Teacher may differentiate method of assessment. May be more than one test. Grade 6 Vocabulary List in Progress as of June 30, 2014 absolute value debit integer number line origin rational number x-coordinate coordinate elevation magnitude opposite integer positive numbers vertical y-axis coordinate grid horizontal negative numbers ordered pair quadrants x-axis y-coordinate credit CMQ3 Integers Student completes task or investigation at any time during the CM unit that addresses Big Ideas of this Unit. Standards of Mathematical Practice in this task are evaluated by the problem solving rubric. Tasks may be differentiated according to student interest, ability level, etc. CM3FK1 Integers Represent quantities in real-world contexts using positive and negative numbers, explaining the meaning of zero in each situation (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). Determine the opposite of a number and recognize that the opposite of the opposite of a number is the number itself. (e.g., – (–3) = +3, and that 0 is its own opposite) Locate and plot integers and other rational numbers on a horizontal or vertical number line. Locate and plot pairs of integers and other rational numbers on a coordinate plane. Write, interpret, and explain statements of order for rational numbers in real-world contexts. Example: Write –3°C > –7°C to express the fact that –3°C is warmer than –7°C. Interpret the absolute value of a rational number as its distance from 0 on the number line and as a magnitude for a positive or negative quantity in a real-world situation. Example: For an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars, and recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars. Solve real-world and mathematical problems by plotting points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 9 Grade 6 PA Common Core Scope and Sequence 2014-2015 CMQ4 How do we use algebra in our everyday lives? Big Ideas (CM) VARIABLE: Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations. EQUIVALENCE: Numbers & numerical expressions can be named in an infinite number of different, but equivalent ways e.g. 24 x 6 = (20 + 4) x 6 Essential Questions (CMQs) What is the correct order for performing mathematical operations? How does changing the order of operations affect the outcome when simplifying an expression? What are variables, and what can they represent? How can we solve for the unknown? How can verbal statements be translated into algebraic expressions? How does the result change when the value of the variable is changed? How can graphs and tables be used to represent relationships between variables? What is an inequality and what types of situations can it represent? EQUATIONS & INEQUALITIES: Rules of arithmetic and algebra can be used together with notions of equivalence to transform equations and inequalities so solutions can be found. Concepts & Competencies: Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities. Assessment Anchor Descriptors M06.B-E.1.1 Identify, writes, and evaluates numerical and algebraic expressions. M06.B-E.2.1 Create, solve, and interpret one-variable equations or inequalities in real-world and mathematical problems. M06.B-E.3.1 Use variables to represent two quantities in a real-world problem that change in relationship to one another. 10 Grade 6 PA Common Core Scope and Sequence 2014-2015 CM 4 Algebra FK Assessments CM 4 Vocabulary FK: Teacher-created assessment. Each word is worth one point. Teacher may differentiate method of assessment. May be more than one test. Grade 6 Vocabulary List in Progress as of June 30, 2014 algebraic expressions coefficient coordinate grid/plane dependent variable difference equation equivalent evaluate factor formula function table graphs independent variable inequality inverse operation numerical expression operation order of operations ordered pair origin point product quantity quotient simplify solution substitution sum term variable x- coordinate y-coordinate CMQ4 Algebra Student completes task or investigation at any time during the CM unit that addresses Big Ideas of this Unit. Standards of Mathematical Practice in this task are evaluated by the problem solving rubric. Tasks may be differentiated according to student interest, ability level, etc. CM4FK1 Expressions Write and evaluate numerical expressions involving whole-number exponents. Write algebraic expressions from verbal descriptions. Example: Express the description “five less than twice a number” as 2y – 5. Identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient, quantity). Example: Describe the expression 2(8 + 7) as a product of two factors. Evaluate expressions at specific values of their variables, including expressions that arise from formulas used in real-world problems. Example: Evaluate the expression b2 – 5 when b = 4. Apply the properties of operations to generate equivalent expressions. Apply the distributive property to the expression: 3 (2 + x) to produce the equivalent expression 6 + 3x. Apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y). Apply properties of operations to y + y + y to produce the equivalent expression 3y. CM4FK2 Equations & Inequalities Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Write algebraic expressions to represent real-world or mathematical problems. 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. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem and/or represent solutions of such inequalities on number lines. CM4FK3: Variable Relationships Write an equation to express the relationship between the dependent and independent variables. Example: In a problem involving motion at a constant speed of 65 units, write the equation d = 65t to represent the relationship between distance and time. Analyze the relationship between the dependent and independent variables using graphs and tables, and/or relate these to an equation. 11 Grade 6 PA Common Core Scope and Sequence 2014-2015 CMQ5 How do we use geometry in our everyday lives? Big Ideas (CM) SHAPES & SOLIDS: Two- and three-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes. ORIENTATION & LOCATION: Objects in space can be oriented in an infinite number of ways, and an object’s location in space can be described quantitatively. Essential Questions (CMQs) How do we use geometry to make sense of the real world? How are geometric shapes constructed? What is the relationship between the shape of an object and its use? What is the relationship between plane figures and solid figures? Can any geometric shape be transformed into any other shape? Why or why not? Objects can be analyzed, sorted, and compared by attributes. How can we use the coordinates of vertices of polygons to find side lengths and area? Concepts & Competencies Solve real-world and mathematical problems involving area, surface area, and volume. Students in Grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They prepare for work on scale drawings and constructions in Grade 7 by drawing polygons in the coordinate plane. Assessment Anchor & Descriptors M06.C-G.1 Understand how to solve real-world and mathematical problems involving area, surface area, and volume. M06.C-G.1.1 Find area, surface area, and volume by applying formulas and using various strategies. 12 Grade 6 PA Common Core Scope and Sequence 2014-2015 CM 5 Geometry FK Assessments CM5 Vocabulary FK: Teacher-created assessment. Each word is worth one point. Teacher may differentiate method of assessment. May be more than one test. Grade 6 Vocabulary List in Progress as of June 30, 2014 Area Base Compound Polygon Congruent Edges Face formula Irregular polygon Net Ordered Pair Parallel Parallelogram Perpendicular Points Polygon Quadrant Quadrilateral Rectangle Rectangular Prism Rectangular prism Regular Polygon Rhombus Solid Square Square units Surface area Trapezoid Triangle Vertices (vertex) Volume CMQ5 Geometry Student completes task or investigation at any time during the CM unit that addresses Big Ideas of this Unit. Standards of Mathematical Practice in this task are evaluated by the problem solving rubric. Tasks may be differentiated according to student interest, ability level, etc. CM5FK1 Geometry Determine the area of triangles and special quadrilaterals (i.e., square, rectangle, parallelogram, rhombus, and trapezoid). Formulas will be provided. Determine the area of irregular or compound polygons. Example: Find the area of a room in the shape of an irregular polygon by composing and/or decomposing the original shape. Determine the volume of right rectangular prisms with fractional edge lengths. Formulas will be provided. Given coordinates for the vertices of a polygon in the coordinate plane, use the coordinates to find side lengths and area of the polygon (limited to triangles and special quadrilaterals). Formulas will be provided. 13 Grade 6 PA Common Core Scope and Sequence 2014-2015 CMQ6 How do we use data in our everyday lives? Big Ideas (CM) DATA COLLECTION: Some questions can be answered by collecting and analyzing data, and the question to be answered determines the data that needs to be collected and how best to collect it. DATA REPRESENTATION: Data can be represented visually using tables, charts, and graphs. The type of data determines the best choice of visual representation. DATA DISTRIBUTION: There are special numerical measures that describe the center and spread of numerical data sets. Essential Questions (CMQs) What kinds of problems can be solved with data analysis? When is it helpful to collect more than one set of data? How does the size of the sample determine how closely the data represents the population? Why and how can we represent the same data in a variety of ways? How does scale influence how we read data in graphic representations? How does the question to be answered, determine the best measure of central tendency to use? (i.e., mean, median, mode) How do outliers affect the mean, median, and mode? Concepts & Competencies Develop understanding of statistical variability & learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected. Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. Students recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Assessment Anchor & Descriptors M06D-S.1.1 Display, analyzes, and summarizes numerical data sets in relation to their context. 14 Grade 6 PA Common Core Scope and Sequence 2014-2015 CM 6 Data FK Assessments CM6 Vocabulary FK: Teacher-created assessment. Each word is worth one point. Teacher may differentiate method of assessment. May be more than one test. Grade 6 Vocabulary List in Progress as of June 30, 2014 Absolute Deviation Box-and-whisker plots Data Distribution Dot Plot Histogram Interquartile range Interquartile variability Interval Line graph Line plot Mean Median Mode Outlier Picture graph (pictograph) Population Range Sample space Scale Statistical variability Tally CMQ6 Data Student completes task or investigation at any time during the CM unit that addresses Big Ideas of this Unit. Standards of Mathematical Practice in this task are evaluated by the problem solving rubric. Tasks may be differentiated according to student interest, ability level, etc. CM6FK1 Data Display numerical data in plots on a number line, including dot plots, histograms, and box-and-whisker plots. Determine quantitative measures of center (e.g., median, mean, and/or mode) and variability (e.g., range, interquartile range, and/or mean absolute deviation). Describe any overall pattern and any deviations from the overall pattern with reference to the context in which the data were gathered. Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 15