Pg. 1 10/26/2015. Grade 6 Math Connects Course 1. Start Smart Review of Grade 5. Smart Start pages MA Common Core Standards: Time Frame:

Transcription

1 Grade 6 Math Connects Course 1 Pg. 1 10/26/2015 Start Smart Review of Grade 5 16 days This section may also be done in pieces as an introduction to later chapters. Smart Start pages What are the four steps to problem-solving? The four steps to problem-solving are Understand, Plan, Solve and Check. 2. What are some strategies for problem-solving? Typical strategies for problem-solving are: guess, check and revise; look for a pattern; make an organized list; draw a diagram; act it out; solve a simpler problem; work backwards; eliminate possibilities; use logical reasoning; make a model 3. How do you add and subtract decimals? To add or subtract decimals, the decimal points must be aligned to align digits of the same place-value. 4. What is the Greatest Common Factor and when is it used? The largest number that divides without a remainder into two or more given numbers is the greatest common factor (GCF). It is used to simplify fractions and in problem solving. 5. What is the Least Common Multiple and when is it used? The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a product of a whole number and the numbers. It is used in problem-solving and when adding and subtracting fractions. 6. How do you add and subtract fractions? Fractions are added and subtracted when the denominators are common. Unlike fractions need to be renamed so they have a common denominator and then numerators are added or subtracted. 7. How do you convert measurements in customary units? Customary units, (inches, feet, yards and miles) have conversion factors that are multiplied or divided. Going from larger units to smaller units, requires multiplication, and going from smaller units to larger units require division. 8. How do you convert measurements in metric units? Metric units are related by multiples of ten. Going from larger units to smaller units, requires multiplication by a power of 10, and going from smaller units to larger units require division by a power of When is a bar graph used? A bar graph is used to compare categories of data. 10. When is a line graph used? A line graph is used to show how a set of data changes over a period of time. 11. What are the key components to any graph? All graphs should have a title, labels on the horizontal and vertical axes, a consistent scale along each axis (they do not need to be the same on each axis). 1

2 Vocabulary: continuous, prime number, prime factorization, Pg. 2 10/26/2015 Targeted Skill(s): 1. When given a problem to solve, students will read it for understanding, plan a strategy to solve the problem, use the plan to solve the problem and if it doesn t work, to revise the plan and start again, and finally to check the accuracy of the solution. 2. Students will try one or more strategies to solve a problem. These include: guess, check and revise; look for a pattern; make an organized list; draw a diagram; act it out; solve a simpler problem; work backwards; eliminate possibilities; use logical reasoning; and make a model 3. Students can add and subtract decimals with accuracy including those with up to three decimal places. Assessment Practices: When these topics are included within other chapters, the assessments for the chapters will included these topics. 2

3 Pg. 3 10/26/ 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 1 Write and evaluate numerical expressions involving whole-number exponents. 6 EE 2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); 16 days 3 weeks Chapter 1 Multiply and Divide Decimals -inch graph paper for aligning student work, base ten blocks, 1. How do you estimate the product of decimals? round each number, then multiply. 2. How do you multiply decimals by whole numbers? use base 10 blocks to model problems as necessary recognize multiplication as repeated addition and put the decimal point in the product the same number of decimal places from the right as it was in the original decimal factor. 2. How do you multiply a decimal by a decimal? use the standard algorithm (counting places to the right of the decimal place) to multiply decimals and decimals 3. What happens in the standard algorithm when there aren t enough places in the answer to put the decimal point where it belongs? annex a zero 4. How do you estimate the quotient of decimals? round either the divisor or the dividend to a compatible whole number and divide 5. How do you divide decimals by whole numbers? divide as with a whole number but put the decimal point in the quotient directly above where it starts 6. How do you divide decimals by decimals? use the standard algorithm (multiply the divisor and dividend by a power of 10 so the divisor is a whole number) 7. How do you represent numbers with exponents? the exponent on the base is the number of times the base is used as a factor 8. How do you mentally multiply decimals by powers of 10? multiplying by a power of 10 greater than 1 moves the decimal point to the right the same number of places as the number of zeros in the power of 10, multiplying by a power of 10 less than 1 moves the decimal point to the left the same number of places as the number of zeros in the power of 10, and annex zeros as necessary 9. How do you mentally divide decimals by powers of 10? dividing by a power of 10 greater than 1 moves the decimal point to the left the same number of places as the number of zeros in the power of 10 dividing by a power of 10 less than 1 moves the decimal point to the right the same number of places as the number of zeros in the power of 10 Vocabulary: annex, compatible numbers, base, exponent, product, factor, quotient, coefficient, power 3

4 Targeted Skill(s): 1. Students will estimate the product of decimals and whole numbers by rounding, and then judge the reasonableness of their answers. 1. Students will use the standard algorithm (counting places to the right of the decimal place) to multiply whole numbers and decimals. 2. Students will estimate the product of decimals and decimals and judge the reasonableness of their answers. 2. Students will use the standard algorithm (counting places to the right of the decimal place) to multiply decimals and decimals, including those up to three decimal places. 3. Students will annex additional zeros as necessary to put the decimal point in the correct place. 4. Students will round the divisor or the dividend (or both) to compatible whole numbers to estimate the quotient of decimals. 5. Students will use the standard algorithm to divide decimals by whole numbers. 6. Students will use the standard algorithm to divide decimals by decimals. 7. Students will express products as the multiplication of like factors or in exponential form. 9. Students will multiply decimals by powers of ten by moving the decimal place and annexing zeros as necessary. 10. Students will divide decimals by powers of ten by moving the decimal place and annexing zeros as necessary. Pg. 4 10/26/2015 Assessment Practices: 4

5 Pg. 5 10/26/ 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. 15 days 3 weeks Chapter 2 Multiply and Divide Fractions 1. What are some ways to model a part of a whole number? Typical models of fractions are parts of rectangles or circles, bar diagrams, pattern blocks and number lines 2. What strategies can be used to estimate the product of fractions? Rounding off and/or using compatible numbers will give an estimate of the product of fractions. 3. What models can be used for multiplying a fraction and a whole number? Repeated addition models (both pictorial and arithmetic), and shading rectangles are ways to model multiplication. 4. What is the common algorithm for multiplying fractions and whole numbers? The whole number can be changed to a fraction and then the numerators are multiplied and the denominators are multiplied. Final answers must be in simplest form. 5. How can multiplication of fractions be modeled? Each dimension of a rectangle can be divided into the number of parts indicated by each of the denominators and that creates a new number of parts in the whole. Shading the value of each fraction on a dimension, describes the new number of shaded units in the new number of units in the rectangle as a whole. 6. What are the standard algorithms for multiplying fractions. Multiplying the numerators and then the denominators of fractions will give an answer that may then need to be simplified. Simplifying by dividing common factors from a numerator and denominator before multiplying is another standard algorithm. 7. What model can be used when multiplying whole numbers and mixed numbers? Using a rectangle with dimensions the length of the whole number and mixed number can be shaded to show multiplication, and then the partial blocks can be combined for a final answer. 8. What algorithms can be used to multiply mixed numbers and fractions or mixed numbers and mixed numbers? One way to multiply mixed numbers is to change them to improper fractions and then to multiply the fractions. It is also possible to multiply each component of each mixed number together and add the results. 9. What is a model for dividing whole numbers by fractions? Diving a rectangle into the whole number of parts, and then each section into the denominator number of parts will set up a model that can then be sectioned off by the numerator number of sections. The final answer is the number of those sections. 10. What is the standard algorithm for dividing by fractions? To divide by a fraction the reciprocal of the divisor is used as a multiplier. The final answer must be in simplest form. 11. What is the standard algorithm for dividing mixed numbers by mixed numbers? The standard algorithm includes changing the mixed numbers to improper fractions and then dividing the fractions and simplifying the answer. Vocabulary: fraction, Greatest Common Factor, Improper fraction, mixed number, reciprocal, simplest form, least common multiple 5

6 Pg. 6 10/26/2015 Targeted Skill(s): 1. Students can model fractional parts using varied drawings or manipulatives. 2. Students can determine if the product of fractions is a reasonable answer by estimating. 3, 5 and 7. Students can show understanding of fraction multiplication by drawing a model or showing repeated addition. 4. Students will multiply fractions and whole numbers correctly. 6. Students will multiply fractions and simplify their answers or by simplifying the problem and then multiplying. 8. Students will be able to change mixed numbers to improper fractions and then multiply correctly, or use the distributive property to multiply. 9. Students can show understanding of fraction division by modeling. 10. Students will be able to find the reciprocal of the divisor and multiply to divide fractions. 11. Students will be able to change mixed numbers to improper fractions and then divide. Assessment Practices: 6

7 Pg. 7 10/26/ RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. 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. a. 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. b. Solve unit rate problems, including those involving unit pricing and constant speed. For example, 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? 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. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. MA.3.e. Solve problems that relate the mass of an object to its volume. 11 days (Hanson 12 days, Whitman 8 days) 2 weeks Chapter 3 Ratios and Rates 1. What are the three ways to express a ratio? 1. Ratios, or comparisons of two quantities can be expressed as: x to y, x : y or 2. What is the process for finding equivalent ratios? 2. Students can use the fraction form and multiply or divide the numerator and the denominator by the same common factor. 3. How does a rate differ from a unit rate? 3. A rate is a comparison of two quantities, in a unit rate, the rate has a denominator of 1 unit. 4. What is a rate of change? 4. The rate of change is usually a unit rate where one quantity changes in relation to another, ex. distance/time 5. What is a ratio table? How can a ratio table be completed? 5. A ratio table is filled with pairs of numbers that have the same ratio. A ratio table can be completed by looking for patterns in the table. 6. What is the process of scaling? How are scale drawings related to ratios? 6. Multiplying or dividing two related quantities by the same number is called scaling. Scaling forward makes the quantities bigger and scaling back makes the quantities smaller. Scale drawings use a scale which is a ratio to represent objects that are too large or too small to be drawn in actual size. 7. What processes can determine if two ratios are equivalent? 7. Two ratios are equivalent if both can be reduced to the same unit rate. Cross multiplying (cross products) will determine if two ratios are equivalent. Multiplying or dividing numerators and denominators horizontally by the same number will also determine equivalent ratios. Vocabulary: equivalent ratios, rate, ratio, ratio table, scaling, unit rate 7

8 Targeted Skill(s): 1. Students can make ratios in three forms from visual models or verbal descriptions. 2. Students will be able to generate equivalent ratios, and/or recognize equivalent ratios. 3. Students will be able to determine a unit rate from a given rate, or from a given verbal description. 4. Students will be able to use information from a graph to determine the rate of change. 5. Students will be able to create a ratio table to solve problems involving equivalent ratios. 6. Students will be able to determine actual distances using a scale drawing and scale. 7. Students will determine if ratios are equivalent by changing both to their unit rates and comparing. 7. Students will determine if ratios are equivalent by finding cross products. 7. Students will determine if ratios are equivalent by horizontally multiplying or dividing numerators and denominators by the same value. 8. Students will solve problems using ratios and rates. Pg. 8 10/26/2015 Assessment Practices: 8

9 Fractions, Decimals and Percents Pg. 9 10/26/ RP.3c 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. 15 days 3 weeks Chapter 4 Fractions, Decimals and Percents 1. How can decimals be changed into fractions or mixed numbers? When read correctly,.4 is 4 tenths and that can be written as a fraction and then simplified. A mixed number will have a whole number component with the fraction. 2. How can fractions and mixed numbers be changed into decimals? Some fractions can be made into equivalent fractions that have denominators which are powers of 10 and when correctly can be easily converted to decimals. Other fractions must have the numerator divided by the denominator. Mixed numbers will have a whole number component in addition to the decimal. 3. How is a repeating decimal written? A repeating decimal has a horizontal line over the portion of the decimal that repeats. 4. How is a percent written as a fraction. A percent is a ratio that compares a number to 100, ex. 45% is 45 out of 100 or which should then be simplified. 5. How can a fraction be written as a percent? A fraction can be written as an equivalent fraction with a denominator of 100, then the numerator is the percent. 6. How can a percent be written as a decimal? A percent can be written as a fraction with 100 as a denominator, and that fraction can be changed to a decimal, or the percent can be divided by 100 and the % sign removed. Division by 100 moves the decimal point 2 places to the left. 7. How can a decimal be written as a percent? A decimal can be written as a percent by multiplying the number by 100 and using a % sign. Multiplying a number by 100 moves the decimal point 2 places to the right. 8. How are percents greater than 100% or less than 1% written as fractions and decimals? Percents greater than 100% will form improper fractions with a denominator of 100. That expression gets changed to a mixed number, and only the fraction part needs to be made into a decimal. Percents less than 1% can be divided by 100 and the % sign removed. Division by 100 moves the decimal point 2 places to the left. The decimal can be read correctly to make a fraction, and simplified as necessary. 9. What models can be used to compare fractions? Fractions can be compared using fraction tiles, or number lines. 10. What is the arithmetic process for comparing fractions? Fractions are easiest to compare when they have common denominators. Find the LCD (or any common denominator) and write equivalent fractions using the LCD, then compare the numerators. 9

10 Pg /26/ What is the process for comparing fractions, decimals and percents? Comparing fractions, decimals and percents is easiest when they are all in the same format, and then be put on a number line, or put in inequality or equation. 12. How can the percent of a number be calculated? Percents can be changed to fractions and decimals and multiplied by the number to find the percent of a number. Targeted Skill(s): Vocabulary: compatible numbers, equivalent ratios, least common denominator (LCD), percent, rational number, terminating decimal, repeating decimal Students will be able to convert between decimals and fractions or mixed numbers. Students will be able to find repeating decimals and write them with the appropriate symbolism. Students will be able to convert between fractions and percents or decimals and percents including percents >100% and < 1% Students will use models to compare fractions as necessary. Students will solve problems using percents, decimals and fractions. Assessment Practices: 10

11 Pg /26/ EE.1 Write and evaluate numerical expressions involving whole-number exponents. 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation Subtract y from 5 as 5 y. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld 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). For example, use the formulas V = s 3 and A = 6s 2 to find the volume and surface area of a cube with sides of length s = ½. 6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, 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. 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). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. 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. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; 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. b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 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.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. 19 days (Whitman 14 days, Hanson 14 days) 3 weeks Chapter 5 Algebraic Expressions 1. What is the order of operations and why do we need it? The order of operations is the set of rules for the steps in computation so that everyone finds the same value for a numerical expression. 2. What is meant by evaluate an algebraic expression? Variables will be replaced by numbers and the order of operations will be followed to find a value for the expression. 3. How can a verbal description be translated into an algebraic expression? 11

12 After rewording the verbal description into only the most important words, the first step in translating a verbal description into an algebraic expression is to define the variable. Then that variable can be used to write the algebraic expression. 4. What are properties, and what properties are used to generate equivalent expressions? Properties are statements that are true for any number. The Commutative Property, Associative Property, Identity Property, Distributive Property, the Multiplicative Property of Zero and the Identity Property of Addition, are used to make equivalent expressions. 5. How can the Distributive Property be modeled? The Distributive Property can be modeled by the area model, repeated addition and with algebra tiles. Pg /26/2015 Targeted Skill(s): Vocabulary: algebraic expression, equivalent expression, evaluate, numerical expression, order of operations, variable, algebra, term, constant, properties, Commutative Property, Associative Property, Identity Property, Distributive Property Students will use the Order of Operations to find the value of expressions that include grouping symbols, exponents, and operations. Students will evaluate algebraic expressions with multiple terms. Students will translate verbal descriptions into simple algebraic expressions. Students will model the Distributive Property and use it to solve mental math problems and in simplifying algebraic expressions. Students will recognize the use of the Multiplicative Property of Zero, the Commutative, Associative, Identity, and Distributive Properties by name, and employ them in computation as necessary. Assessment Practices: 12

13 Pg /26/ NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.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. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; 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. b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 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 Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative positions of two numbers on a number line diagram. For example, interpret 3 > 7 as a statement that 3 is located to the right of 7 on a number line oriented from left to right. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write 3 o C > 7 o C to express the fact that 3 o C is warmer than 7 o C. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars. d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars. 6.EE.2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. 10 days 3 weeks Chapter 6 Equations 1. What equations can be solved by using mental math and/or the guess, check and revise strategies? Single step equations with obvious answers or a limited number of possible answers can be solved by these methods. 2. What does an addition equation look like and how can it be solved? In an addition equation there is a number added to the variable on one side of the equation which is equal to a number. Applying the Subtraction Property of Equality means using the inverse operation, subtraction, to subtract the number added to the variable from both sides of the equation leaving the solution. 3. What does a subtraction equation look like and how can it be solved? In an subtraction equation there is a number subtracted from the variable on one side of the equation which is equal to a number. Applying the Addition Property of Equality means using the inverse operation, addition, to add the number subtracted from the 13

14 Pg /26/2015 variable to both sides of the equation leaving the solution. 4. What does a multiplication equation look like and how can it be solved? In a multiplication equation the variable is multiplied by a number, the coefficient, on one side of the equation. Applying the Division Property of Equality means using the inverse operation, division, to divide both sides of the equation by the coefficient, leaving the solution. 5. What does a division equation look like and how can it be solved? In a division equation the variable is divided by a number on one side of the equation. Applying the Multiplication Property of Equality means using the inverse operation, multiplication, to multiply both sides of the equation by that number, leaving the solution. 6. How can a single step equation be modeled? Math manipulatives where variables are different than numbers, balances and bar diagrams can be used to model single step Equations, and algebra tiles can be used to model two step equations. Vocabulary: coefficient, equation, equal sign, solution, solve, inverse operations, Subtraction Property of Equality, Addition Property of Equality, Multiplication Property of Equality, Division Property of Equality, Targeted Skill(s): 1. Students will solve single step equations by showing the inverse operation step, by doing mental math or by the guess, check and revise strategy. 2. Students will recognize an Addition Equation and use subtraction to solve it. 3. Students will recognize a Subtraction Equation and use addition to solve it. 4. Students will recognize a Multiplication Equation and use division to solve it. 5. Students will recognize a Division Equation and use multiplication to solve it. 6. Students will be able to model or interpret a model of a one-step equation with math manipulatives, balances, bar diagrams and/or algebra tiles. Assessment Practices: DDM for chapter 6 14

15 Pg /26/ 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 equation. 15 days 3 weeks Chapter 7 Functions, Inequalities and Integers 1. What is a relation and how can it be displayed? A relation is a set of ordered pairs. It can be displayed as a set of ordered pairs, a table of values or as points plotted on a graph. 2. What is the difference between a function and a relation? A function is a relation that assigns exactly one output value to one input value. 3. How can the input (or x value) and the function rule, or the output and the function rule, or the input and output values be used to complete a function table? The input value is an x value and the output value is the y value. Substituting the x value into the function rule will give the y value, and visa versa. Given both x and y values the function explains how they relate. 4. How can an arithmetic sequence be used as a specific example of a function? Consecutive numbers in an arithmetic sequence differ by the same amount. 5. How can a geometric sequence be used as a specific example of a function? Consecutive numbers in a geometric sequence differ because they are multiplied by a common number. 6. What are the four different representations of functions? Functions can be represented in verbal, tabular, graphical and algebraic form. 7. How is a linear function a special kind of function or equation? Points on a linear function make a line. 8. What methods can be used to solve inequalities? Solving inequalities can be done by mental math and guess/check and revise methods. 9. What is an inequality and how is it graphed? An inequality is a math sentence containing a <, >, sign. An inequality has either an open or closed circle on one end of a segment, and an arrow on the other. The parts of the number line under covered by the arrow, segment or colored in circle are the solution to the inequality. 10. What numbers are included in the set of numbers called integers? 15

16 Pg /26/2015 The integers are the set of positive and negative whole numbers and zero. The negative integers are to the left of zero, and the positive integers are to the right of zero. Zero is neither positive nor negative. 11. What does absolute value mean, and what is the symbol for it? The absolute value of a number is its distance from zero on the number line. Absolute value is indicated by vertical absolute value bars, for example: 12. What is a coordinate plane, and how is it used? The coordinate plane is the space defined by the intersection of the x-axis and y-axis. These axes intersect at the origin, (0,0) breaking the plane into quadrants I IV indicated by Roman Numerals. The positive values are to the right of zero on the x- axis, and above zero on the y-axis. Ordered pairs, (x,y) can be plotted as points on the grid created by the intersection of these axes. Vocabulary: absolute value, coordinate plane, function, function rule,function table, graph, inequality, integer, linear function, ordered pair, origin, quadrants, relation, sequence, term, x-axis, x-coordinate, y-axis, y-coordinate. Targeted Skill(s): 1. Students will use ordered pairs to graph relations and lists of ordered pairs or a table to describe a relation. 2. Students can determine if something is a relation or a function. 3. Students can determine the missing value when given two of the three parts to a function table: input, output, function rule 4. Students will extend and describe arithmetic sequences using algebraic expressions. 5. Students will extend and describe geometric sequences using algebraic expressions. 6. Students will be able to represent a function in verbal, graphical, tabular and algebraic forms. 7. Students will recognize a linear function as a collection of points that make a line. 8. Students will solve simple inequalities by mental math, and guess/check and revise methods. 9. Students will be able to write and graph inequalities. 10. Students will be able to recognize, read and write integers. 11. Students will be able to find the absolute value of integers. 12. Students will be able to locate and plot ordered pairs on any quadrant in the coordinate plane. Key parts of the coordinate plane will be able to be labeled. Assessment Practices: 16

17 Pg /26/ G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. 3-4 days 1 week Chapter 8 Properties of Triangles and Quadrilaterals section 1A-1C 1. What are some basic geometric figures and what parts can be named on them? Basic geometric figures are: point, line, ray, line segment, plane, intersecting lines, parallel lines and perpendicular lines. Points are labeled with capital letters, lines are labeled by two capital letters and a line above them, a ray is labeled by two capital letters and a ray above them, a line segment is labeled with the capital letters of the endpoints of the segment and a segment above them, a plane is labeled by three capital letters of points on the plane. 2. In what ways can pairs of lines relate to one another? Pairs of lines can be parallel (non-intersecting and staying the same distance apart), intersecting (they cross once), or perpendicular (they intersect to form right angles. 3. What is an angle and how is it named? An angle is formed by two rays that meet at a common endpoint called a vertex. Angles are named with three capital letters, one from a side, the middle one from the vertex of the angle and the last from the other side. 4. How can angles be measured? Angles are measured with a protractor that aligns the zero line on one side of the angle, and the vertex on the center of the protractor. There are two scales on the protractor, and the one to use is the one that matches where the angle lines up with 0 degrees. Angles less than 90 degrees are smaller than a right angle. 5. How can angles be classified? Angles are classified according to their measures. Angles measuring exactly 90 are right angles, angles measuring less than 90 are acute angles, angles measuring more than 90 are obtuse angles, and angles measuring exactly 180 are straight angles. Vocabulary: point, line, ray, line segment, plane, congruent, intersecting lines, perpendicular lines, parallel lines, angle, vertex, degree, right angle, acute angle, obtuse angle, straight angle. Targeted Skill(s): 1. Students will identify by name the basic geometric figures and parts on them, and label them with the correct letters and symbols. 2. Students will be able to distinguish between parallel, perpendicular and intersecting lines. 3. Students will be able to classify angles as obtuse, acute, right or straight according to the angle measures. Assessment Practices: 17

18 Pg /26/ G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. MA.1.a. Use the relationships among radius, diameter, and center of a circle to find its circumference and area. MA.1.b.Solve real-world and mathematical problems involving the measurements of circles. 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 = lwh and V = bh 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 areas of these figures. Apply these techniques in the context of solving real-world and mathematical problems. 20 days 4 weeks Chapter 9 Perimeter, Area and Volume 1. How do you find the area of a parallelogram, or the missing dimension of a parallelogram, given its area and one dimension? The formula for the area of a parallelogram is. Given any two values you can substitute those values into the formula and solve for the value of the remaining variable. 2. How do you find the area of a triangle, or the missing dimension of a triangle, given its area and one dimension? The formula for the area of a triangle is or 3. How do you find the circumference of a circle? There are two common formulas for the circumference of a circle, (circumference = π times the diameter) and (circumference = 2 times π times the radius) 4. How do you find the area of a circle? The area of a circle formula,, can be used to find the area. 5. How do you find the area of composite figures? The area of a composite figure is the sum of the areas of the regions that make it. 6. How do you find the volume of rectangular prisms and cubes? The Volume of a rectangular prism or a cube can be found with the formula. A specific formula for the volume of a cube addresses the fact that a cube has equal measures for the length, width and height. 7. How do you find the surface area of rectangular prisms and cubes? The surface area of a rectangular prism or cube is the sum of the surface area of each face. The general formula for the surface area of a rectangular prism is, while the specific formula for a cube is because there are 6 faces with areas of Vocabulary: base, composite, circle, volume, circumference, area, height, diameter, radius, pi, perimeter, surface area, prism, volume, cubic units, square units 18

19 Pg /26/2015 Targeted Skill(s): 1. Students will find the areas and missing dimensions of parallelograms 2. Students will find the areas and missing dimensions of triangles 3. Students will describe the relationship between the diameter and radius of a circle 4. Students will find the circumference of circles 5. Students will find the area of circles 6. Students will find the area of composite figures 7. Students will find the volume of rectangular prisms and cubes 8. Students will find the surface area of rectangular prisms and cubes Assessment Practices: 19

20 Pg /26/ SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. MA.4.a. Read and interpret circle graphs. 6.SP.5 Summarize numerical data sets in relation to their context, such as by: 6.SP.5a Reporting the number of observations. 6.SP.5b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.5c Summarize numerical data sets in relation to their context, such as by: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 14 days 3 weeks Chapter 11 Analyze Data and Graphs (and Additional Lesson #10 pg. 815) 1. What are the measures of central tendency and how are they found? The measures of central tendency are mean, median and mode. The mean is the average, the median is the middle number and the mode is the number that occurs most often 2. What is the range and what does it say about the data? The range is the difference between greatest and least values. It describes the spread of the data. 3. How can data be presented in a box and whisker plot? To make a box and whisker plot the median is plotted, dividing the rest of the data into two halves. Then the median of each half is plotted and a rectangle is drawn using these two medians as the ends of the rectangle. This gives 50% of all the data within the rectangle. Whiskers are the line segments connecting these medians to the minimum value or the maximum value. If there is an outlier, the outlier is plotted as a single point, but not used in any of the calculations for median, or connected with the whiskers. 4. Which measure of central tendency is most appropriate for a given situation? The mean is most appropriate when there is no extreme data. The median is used when the data has extreme values, and the mode is used when there are many repeated numbers and the question asks for the most likely number. 5. How is a data point determined to be an outlier? What impact does an outlier have? Students can determine when an outlier exists and when it affects the mean, median or mode 6. How is a stem and leaf plot made? The data in a stem-and-leaf plot is organized by place value. The stems are the digits in the greatest common place value, the leaves are the digits in the remaining place value. A back-to-back stem- 20

21 Pg /26/2015 and-leaf plot has leaves for one set of data on one side of the stem, and leaves for another set of data are on the other side of the stem. It is used to compare data sets. 7. What information can be determined from a histogram? Students can determine which range of values is most/least common and comparative clumps of data 8. How can circle graphs be used to analyze data? Circle graphs are used when the total amount of data can be divided into fractional parts of a whole or percentage parts of 100%. 9. How can scale be used to create a misleading graph? A scale with shorter range will make differences in the data appear larger, just as a scale with a greater range will make differences in the data appear lesser. A broken scale may be used effectively or to make a misleading representation. 10. What is the mean absolute deviation and how do you find it? The mean absolute deviation is the average distance between each data value and the mean and it is found by finding the absolute value of the difference between each piece of data and the mean added together to get a sum which is then divided by the n number of observations (pieces of data). It is similar to the standard deviation but it doesn t require using the square root. Targeted Skill(s): Vocabulary: histogram, mean, median, mode, histogram, statistics, data, average, range, box and whisker plot, outlier, frequency, stem and leaf plot, circle graphs, broken scale, mean absolute deviation 1. Students can find mean, median, and mode without calculator 2. Students can calculate the range 3. Students can use medians and highest and lowest point to create a box and whisker plot 4. Students can determine which measure of central tendency is most appropriate for a set of data 5. Students can determine and identify outlier values within a set 6. Students can create a stem and leaf plot from a set of data 7. Students can interpret ranges and comparative clumps of data read from a histogram 8. Students can reason from a circle graph. 9. Students can determine whether or not a graph or pictorial representation is misleading due to the scale factor. 10. Students can find the mean absolute deviation and use it to make conclusions about data. Assessment Practices: 21

22 Pg /26/2015 Topics not on MCAS/PARCC Assessments 20 days Vocabulary: Targeted Skill(s): Assessment Practices: 22