Grade 6 Unit 1 Unit 1: Operations and Statistical Variability Timeframe: September/October Common Core State Standards, 2010 Key Areas of Focus in the CCSS: 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. For example, create a story context for (2/3) (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) (3/4) = 8/9 because ¾ of 8/9 is 2/3. In general, (a/b) (c/d) = (ad/bc) How much chocolate will each person get if 3 people share 1`/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length ¾ mi and area ½ sq. mi.? 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 standards algorithm for each operation. 6.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, 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.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. For example, How old am I? is not a statistical question, but How old are the students in my school? is a statistical question because one anticipates variability in students ages. 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, Grade 6 Unit 1 1
Grade 6 Unit 1 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.5 Summarize numerical data sets in relation to their context, such as by: 6.SP.5c 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 deviation from the overall pattern with reference to the context in which the data were gathered. 6.SP.5d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Additional Standards: Career Ready Practices - 9.1.8.A.6 Explain how income affects spending decisions. Unit 1 Essential Outcomes 1. Compute quotients of fractions. 6.NS.1 2. Construct visual fraction models to represent quotients and explain the relationship between multiplication and division of fractions. 6.NS.1 3. Solve real-world problems involving quotients of fractions and interpret the solutions in the context given. 6.NS.1 4. Fluently add, subtract, multiply and divide multi-digit decimals and whole numbers using standard algorithms. 6.NS.2; 6.NS.3 Grade 6 Unit 1 2
Grade 6 Unit 1 5. Use positive and negative numbers to describe quantities in real-world situations. 6.NS.5 6. Calculate, compare, and interpret measures of center and variability in a data set to answer a statistical question. (Including median, mean, interquartile range, mean absolute deviation and overall pattern). 6.SP.1; 6.SP.2; 6.SP.3; 6.SP.5c,d 7. Solve real -world problems involving positive and negative integers while using a budget. 9.1.8.A.6 Major Content Supporting Content Additional Content(Identified by PARCC Model Content Frameworks). Bold type indicates grade level fluency requirements. (Identified by PARCC Model Content Frameworks). Selected Opportunities for Connection to Mathematical Practices 1. Make sense of problems and persevere in solving them. EO #3: Involve problems that include several givens or those that must be carefully deconstructed before they can be solved. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. EO #2: Visual fraction models are required. 5. Use appropriate tools strategically. EO #2: Tools will include diagrams, words, and equations. 6. Attend to precision. EO #6: The use of precise language is needed when answering statistical questions. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. All content presented at this grade level has connections to the standards for mathematical practices. Grade 6 Unit 1 3
Grade 6 Unit 1 Big Idea/Enduring Understanding(s) 1. All operations are related to one another. There are many different interpretations and solutions when using addition, subtraction, multiplication, and division of rational numbers. 2. Whole numbers, integers, decimals, and fractions are real numbers. Each real number can be associated with a unique point on a number line. Essential Questions for Learners EO #1: Why do you invert and multiply when dividing fractions? EO #2: What can I use as a visual model and how can this help me solve multiplication and division of fractions? EO #3: When will I use the division of fractions in the real world? (ie: recipes) EO #4: When and why are decimals used in the real world? EO #5: What is the significance of positive and negative? numbers? When are positive and negative numbers used? EO #6: How are median, mean, IQR, mean absolute deviation, and overall Guiding Questions for Teachers What methods would you use to find the quotient of two fractions? How can you use patterns and models to solve problems? What is necessary to interpret solutions in a given context? What is the purpose of the decimal when adding, subtracting, multiplying, and dividing decimals and whole numbers? How can the use of adding and subtracting integers be modeled on a number line? Can patterns be used to show why adding and subtracting rules make sense? How do you calculate the measure of center and IQR? Grade 6 Unit 1 4
Grade 6 Unit 1 pattern relevant in life? Unit 1: Operations and Statistical Variability Academic Technical Vocabulary Vocabulary: quotient visual fraction model interpret fluently standard algorithm multi-digit positive negative value opposites statistical variability data spread measure of center measure of variability quantitative mean Prerequisite Knowledge Students in Fifth Grade were expected to: add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition, subtraction,multiplication, and division fluently multiply multi-digit whole numbers using the standard algorithm divide a unit fraction by a non-zero whole number and interpret by creating a story context or visual fraction model divide a whole number by a unit fraction and interpret by creating a story context or visual fraction model Grade 6 Unit 1 5
Grade 6 Unit 1 median interquartile range (IQR) mean absolute deviation data distribution overall pattern reciprocal inverse operations solve real world problems involving division of unit fractions by whole numbers or whole numbers by unit fractions interpret a fraction as a division of the numerator by the denominator; solve word problems where division of whole numbers leads to fractional or mixed number answers Summative Performance Task: This would be project based, teacher developed, and common. Grade 6 Unit 1 6
Grade 6 Unit 1 Grade 6 Unit 1 7
Grade 6 Unit 2 Unit 2: Expressions Timeframe: November/December Common Core State Standards, 2010 Key Areas of Focus in the CCSS: 6.EE.1 Write and evaluate numerical expressions involving whole number exponents. 6.EE.2 Read, write, and evaluate expressions in which letters stand for numbers. 6.EE.2a 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. 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. 6.EE.2c 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). For example, use the formulas V = s3and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2. 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 1
Grade 6 Unit 2 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.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 1 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). Additional Standards: Unit 2 Essential Outcomes 1. Use mathematical language to identify parts of an expression. 6.EE.2 2. Write and evaluate numerical expressions involving whole number exponents. 6.EE.1 3. Read, write, and evaluate expressions in which letters stand for numbers (Including formulas that arise from real-world contexts). 4. Apply the properties of operations to generate equivalent expressions (Including the distributive property; for example, express 36 + 8 as 4(9 + 2) and y + y + y = 3y. 5. Identify when two expressions are equivalent; for example, are the two expressions equal? 81 + 18 and 9(9 + 2). 6. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two numbers less than or equal to 12. 6.EE.2 6.EE.3; 6.NS.4 6.EE.4 6.NS.4 2
Grade 6 Unit 2 Major Content Supporting Content Additional Content(Identified by PARCC Model Content Frameworks). Bold type indicates grade level fluency requirements. (Identified by PARCC Model Content Frameworks). Selected Opportunities for Connection to Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. E.O. #5 Listen to arguments of others about the equivalence of two expressions and decide if they make sense. Ask useful questions to clarify. 4. Model with mathematics. E.O. # 1, 2, and 3 Use expressions that arise from real-world context. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. All content presented at this grade level has connections to the standards for mathematical practices. Big Idea/Enduring Understanding(s) 1. Expressions often contain a variable and a mathematical operation. 2. Expressions are used to write equations and inequalities. 3. Expressions represent functions as verbal rules, equations, tables, and graphs. Essential Questions for Learners Guiding Questions for Teachers 3
Grade 6 Unit 2 EO #1: What are the parts of an expression? EO #2: How can a numerical expression be written? EO #3: What is the purpose of a variable in an expression? EO #4: How is the distributive property used to solve a written expression? EO #5: How does the value of any expression remain the same for any value of x? EO #6: How can you use factor lists or the prime factorizations to find the GCF or LCM? What is the purpose for identifying the parts of an expression? When would you use a numerical expression? How do you know where to place the variable in an algebraic expression? How can the distributive property be used to simplify algebraic expressions? How can you make an equivalent equation? What is the purpose of a GCF and /or LCM? Unit 2: Expressions Academic Technical Vocabulary Vocabulary: evaluate constant numerical expressions exponents term coefficient Pre-Requisite Knowledge Students in Fifth Grade were expected to: add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition, 4
Grade 6 Unit 2 sum product factor variable Order of Operations (PEMDAS) Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Distributive Property Identity Property of Addition Identity Property of Multiplication equivalent common factor greatest common factor (GCF) least common multiple (LCM) prime number composite number relatively prime subtraction,multiplication, and division. add and subtract fractions (including mixed numbers) with unlike denominators. evaluate numerical expressions with parentheses, brackets or braces. write numerical expressions when given a word problem or a scenario in words and use words to interpret numerical expressions. Summative Performance Task: This would be project based, teacher developed, and common. 5
Grade 6 Unit 2 6
Grade 6 Unit 3 Unit 3: Equations and Inequalities Timeframe: January/February Common Core State Standards, 2010 Key Areas of Focus in the CCSS: 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.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. 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. 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.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. 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 = Grade 6 Unit 3 1
Grade 6 Unit 3 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. Additional Standards: Unit 3 Essential Outcomes 1. Use variables to represent numbers and write expressions when solving real world or mathematical problems. 6.EE.6 2. Solve an equation or inequality to answer the question: which values from a specified set, if any, make the equation or inequality true? and check the solution using substitution to determine whether a given number in a specified set makes an equation or inequality true. (including formulas V=lwh and V=bh). 6.EE.5 3. Write and solve one step equations that represent real world or mathematical problems. 6.EE.7 4. 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 represent them on a number line diagram. 5. Find the area of right triangles, other triangles, special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and other shapes to solve real world or mathematical problems. 6. Represent three dimensional figures using nets made of rectangles and triangles, and use the nets to find the surface area of the figures in the context of solving real world and mathematical problems. 7. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes. Show that the volume is the same as it would be if found by multiplying the edge lengths. 6.EE.8 6.G.1 6.G.4 6.G.2 Grade 6 Unit 3 2
Grade 6 Unit 3 Major Content Supporting Content Additional Content(Identified by PARCC Model Content Frameworks). Bold type indicates grade level fluency requirements. (Identified by PARCC Model Content Frameworks). Selected Opportunities for Connection to Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. E.O. #3 Order and justify steps to reach a solution to a one step equation. 4. Model with mathematics. E.O. #6 The use of 2-D nets to solve surface area problems. 5. Use appropriate tools strategically. 6. Attend to precision. E.O. #2 Real-world context involving careful attention to units of measure. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. E.O. #1 The use of variables to represent real-world context over time. All content presented at this grade level has connections to the standards for mathematical practices. Big Idea/Enduring Understanding(s) 1. Mathematicians describe simple relationships by creating and analyzing equations, expressions, and tables. 2. Mathematicians write, interpret, and use expression, equations, and formulas. Grade 6 Unit 3 3
Grade 6 Unit 3 Essential Questions for Learners EO #1 What is the purpose of a variable? EO #2 What does inequality mean? EO #3 When would a bar model be used? EO #4 What is a constraint? EO #5 Why do I need to know how to find the area of a triangle? EO #6 What are the characteristics of a 3D figure? EO #7 What is the formula for finding the area of a rectangular prism? Guiding Questions for Teachers Why is a variable represented by a letter? How can you check to see if your solution is accurate? What is needed to create a bar model? How can you represent an inequality on a number line? How are triangles and rectangles related? How can you find the area of a 3D figure using a net? How can you determine the volume of a rectangular prism? Unit 3: Equations and Inequalities Academic Technical Vocabulary Vocabulary: Prerequisite Knowledge Students in Fifth Grade were expected to: Grade 6 Unit 3 4
Grade 6 Unit 3 expression inequality equation value substitution value formula constraint specified set nonnegative rational quadrilateral polygon isosceles right triangle composing decomposing scalene equilateral obtuse acute three dimensional nets surface area volume unit cubes right rectangular prism identify attributes of a twodimensional shape based on attributes of the groups and categories in which the shape belongs. classify two- dimensional figures in a hierarchy based on properties. find the area of a rectangle with fractional side lengths by tiling unit squares and multiplying side lengths. measure volume by counting the total number of same size cubic units required to fill a figure without gaps or overlaps. choose an appropriate cubic unit based on the attributes of the 3- dimensional figure you are measuring. apply formulas to solve real world and mathematical problems involving volumes of right rectangular prisms and composites of same. evaluate numerical expressions with parentheses, brackets or braces. write numerical expressions when given a word problem or a scenario Grade 6 Unit 3 5
Grade 6 Unit 3 in words and use words to interpret numerical expressions. Summative Performance Task: This would be project based, teacher developed, and common. Grade 6 Unit 3 6
Grade 6 Unit 3 Grade 6 Unit 3 7
Grade 6 Unit 4 Unit 4: Review Timeframe: March (4 weeks) Common Core State Standards Key Areas of Focus in the CCSS: 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. For example, create a story context for (2/3) (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) (3/4) = 8/9 because ¾ of 8/9 is 2/3. In general, (a/b) (c/d) = (ad/bc) How much chocolate will each person get if 3 people share 1`/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length ¾ mi and area ½ sq. mi.? 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standards algorithm for each operation. 6.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, 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.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. For example, How old am I? is not a statistical question, but How old are the students in my school? is a statistical question because one anticipates variability in students ages. 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 Grade 6 Unit 4 1
Grade 6 Unit 4 variation describes how its values vary with a single number. 6.SP.5c 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 deviation from the overall pattern with reference to the context in which the data were gathered. 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. 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. 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 1 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). 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. 6.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. 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. Grade 6 Unit 4 2
Grade 6 Unit 4 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. 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.NS.7 Understand ordering and absolute value of rational numbers. 6.NS.7c 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. 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 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.3b 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? Grade 6 Unit 4 3
Grade 6 Unit 4 6.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. 6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Additional Standards: Career Ready Practices - 9.1.8.B.2 Construct a simple personal savings and spending plan based on various sources of income. Career Ready Practices - 9.1.8.A.6 Explain how income affects spending decisions. Unit 4 Essential Outcomes 1. Solve real-world problems involving quotients of fractions and interpret the solutions in the context given. 6.NS.1 2. Fluently add, subtract, multiply and divide multi-digit decimals and whole numbers using standard algorithms. 6.NS.2; 6.NS.3 3. Calculate, compare, and interpret measures of center and variability in a data set to answer a statistical question. (Including median, mean, interquartile range, mean absolute deviation and overall pattern). 6.SP.1; 6.SP.2; 6.SP.3; 6.SP.5c,d 4. Use mathematical language to identify parts of an expression. 6.EE.2 5. Write and evaluate numerical expressions involving whole number exponents. 6.EE.1 6. Read, write, and evaluate expressions in which letters stand for numbers (Including formulas that arise from real-world contexts). 6.EE.2 Grade 6 Unit 4 4
Grade 6 Unit 4 6. Apply the properties of operations to generate equivalent expressions (Including the distributive property; for example, express 36 + 8 as 4(9 + 2) and y + y + y = 3y. 6.EE.3; 6.NS.4 7. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two numbers less than or equal to 12. 6.NS.4 8. Use variables to represent numbers and write expressions when solving real world or mathematical problems. 6.EE.6 9. Solve an equation or inequality to answer the question: which values from a specified set, if any, make the equation or inequality true? and check the solution using substitution to determine whether a given number in a specified set makes an equation or inequality true. (including formulas V=lwh and V=bh). 10. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes. Show that the volume is the same as it would be if found by multiplying the edge lengths. 6.EE.5 6.G.2 11. 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 represent them on a number line diagram. 6.EE.8 12. Find the area of right triangles, other triangles, special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and other shapes to solve real world or mathematical problems. 6.G.1 Grade 6 Unit 4 5
Grade 6 Unit 4 13. Represent three dimensional figures using nets made of rectangles and triangles, and use the nets to find the surface area of the figures in the context of solving real world and mathematical problems. 6.G.4 14. Locate positive and negative rational numbers on the number line and explain the meaning of absolute value of a rational number as indicating locations on opposite sides of zero on the number line. 6.NS.6; 6.NS.7 15. Write and compare rational numbers using inequality signs. 6.NS.7 16. Plot ordered pairs in all four quadrants on the coordinate plane and describe their reflections. 6.NS.6 17. Draw polygons in the coordinate plane given the coordinates of the vertices and use the coordinates to solve real world distance, perimeter, and area problems. 6.G.3 18. Solve real world problems mathematically by graphing points in all four quadrants of the coordinate plane. Use the absolute value of the differences of their coordinates to find distances between points with the same first coordinate or same second coordinate. 19. Display numerical data in plots on the number line (including dot plots, histograms, and box plots) and summarize in relation to their context. 6.NS.8 6.SP.4; 6.SP.5a,b Grade 6 Unit 4 6
Grade 6 Unit 4 20. Explain the relationship of two quantities or measures of a given ratio and use ratio language to describe the relationship between the two quantities. For example, The ratio of wings to beaks in the birdhouse at the zoo was 2:1, because for every 2 wings there was 1 beak. For every vote candidate A received, candidate C received nearly three votes. 21. Use rate language in the context of a ratio relationship to describe a unit rate a/b associated with a ratio a:b with b 0. For example, 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. We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger. 22. Use ratio and rate reasoning to solve real world and mathematical problems which include making tables of equivalent ratios, solving unit rate problems, finding percent of a quantity as a rate per 100. 6.RP.1 6.RP.2 6.RP.3 23. Use ratio and rate reasoning to convert measurement units (manipulate and transform units appropriately when multiplying or dividing quantities). 6.RP.3 24. Analyze the relationship between the dependent and independent variables in an equation using graphs and tables. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. 6.EE.9 Selected Opportunities for Connection to Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. Grade 6 Unit 4 7
Grade 6 Unit 4 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Big Idea/Enduring Understanding(s) 1. All operations are related to one another. There are many different interpretations and solutions when using addition, subtraction, multiplication, and division of rational numbers. 2. Whole numbers, integers, decimals, and fractions are real numbers. Each real number can be associated with a unique point on a number line. 3. Expressions often contain a variable and a mathematical operation. 4. Expressions are used to write equations and inequalities. 5. Expressions represent functions as verbal rules, equations, tables, and graphs. 6. Mathematicians describe simple relationships by creating and analyzing equations, expressions, and tables. 7. Mathematicians write, interpret, and use expression, equations, and formulas. 8. A ratio relates two or more quantities, and when two ratios are equivalent, the data are said to be proportional. Proportions allow us to scale up or down data while keeping the ratio intact. 9. Rational numbers can be used, expressed, and manipulated in a variety of ways. Students will be applying their knowledge of positive and negative numbers by viewing them in everyday contexts (temperatures, recipes, taxes, money, etc). Essential Questions for Learners Guiding Questions for Teachers Grade 6 Unit 4 8
Grade 6 Unit 4 EO #1: When will I use the division of fractions in the real world? (ie: recipes) EO #2: When and why are decimals used in the real world? EO #3: What is the significance of positive and negative? numbers? When are positive and negative numbers used? EO #4: How are median, mean, IQR, mean absolute deviation, and overall pattern relevant in life? EO #5: What are the parts of an expression? EO #6: How is the distributive property used to solve a written expression? EO #7: How can you use factor lists or the prime factorizations to find the GCF or LCM? EO #8: How can a numerical expression be written? EO #9: What does inequality mean? EO #10: Why do I need to know how to find the area of a triangle? EO #11: What are the characteristics of a 3D figure? What is necessary to interpret solutions in a given context? What is the purpose of the decimal when adding, subtracting, multiplying, and dividing decimals and whole numbers? How can the use of adding and subtracting integers be modeled on a number line? Can patterns be used to show why adding and subtracting rules make sense? How do you calculate the measure of center and IQR? What is the purpose for identifying the parts of an expression? How can the distributive property be used to simplify algebraic expressions? What is the purpose of a GCF and /or LCM? When would you use a numerical expression? How can you represent an inequality on a number line? How are triangles and rectangles related? How can you find the area of a 3D Grade 6 Unit 4 9
Grade 6 Unit 4 EO #12: What is the formula for finding the area of a rectangular prism? EO#13: How can you make equal ratios? EO#14: What in everyday life can be described in unit rates? EO#15: What are some customary units of length and what do you measure with them? figure using a net? How can you determine the volume of a rectangular prism? How do you think a unit rate can be used to solve a complex problem? How can you use a unit rate to solve for an unknown measure or quantity? How can you change between units of measurement? Unit 4: Review Academic Technical Vocabulary statistical measure of center measure of variability visual fraction model quantitative mean median interquartile range (IQR) mean absolute deviation Prerequisite Knowledge Students in (previous) Grade were expected to: Students were taught the skills within this unit throughout the year. This unit is designed to review those skills, and the students are asked to apply those skills and learned strategies to various types of problems throughout this unit. Grade 6 Unit 4 10
Grade 6 Unit 4 data distribution overall pattern reciprocal inverse operations numerical expressions exponents Distributive Property greatest common factor (GCF) least common multiple (LCM) prime number composite number relatively prime expression inequality equation three dimensional nets surface area volume ratio unit rate coordinate plane dependent variable independent variable ordered pairs absolute value rational Grade 6 Unit 4 11
Grade 6 Unit 4 Grade 6 Unit 4 12
Grade 6 Unit 4 Grade 6 Unit 4 13
Grade 6 Unit 5 Unit 5: Rational Numbers Timeframe: April Common Core State Standards, 2010 Key Areas of Focus in the CCSS: 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.NS.6a 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. 6.NS.6b 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. 6.NS.6c 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. 6.NS.7a Interpret statements of inequality as statements about the relative position 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. 6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write 3 ⁰C > 7 ⁰C to express the fact that 3⁰C is warmer than 7 ⁰C. 6.NS.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as Grade 6 Unit 5 1
Grade 6 Unit 5 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. 6.NS.7d 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.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Additional Standards: Unit 5 Essential Outcomes 1. Locate positive and negative rational numbers on the number line and explain the meaning of absolute value of a rational number as indicating locations on opposite sides of zero on the number line. 6.NS.6; 6.NS.7 Grade 6 Unit 5 2
Grade 6 Unit 5 2. Write and compare rational numbers using inequality signs. 6.NS.7 3. Plot ordered pairs in all four quadrants on the coordinate plane and describe their reflections. 6.NS.6 4. Interpret and explain absolute value as magnitude for a positive or negative quantity in a real-world situation. 6.NS.7 5. Solve real world problems mathematically by graphing points in all four quadrants of the coordinate plane. Use the absolute value of the differences of their coordinates to find distances between points with the same first coordinate or same second coordinate. 6. Draw polygons in the coordinate plane given the coordinates of the vertices and use the coordinates to solve real world distance, perimeter, and area problems. 7. Display numerical data in plots on the number line (including dot plots, histograms, and box plots) and summarize in relation to their context. 6.NS.8 6.G.3 6.SP.4; 6.SP.5a,b Major Content Supporting Content Additional Content(Identified by PARCC Model Content Frameworks). Bold type indicates grade level fluency requirements. (Identified by PARCC Model Content Frameworks). Selected Opportunities for Connection to Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. E.O. #2 Use inequality symbols to make comparisons. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. E.O. #6 Represent polygons on a coordinate plane. 5. Use appropriate tools strategically. E.O. #7 Use spreadsheets when working with data sets with a large quantity of data points. Grade 6 Unit 5 3
Grade 6 Unit 5 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. All content presented at this grade level has connections to the standards for mathematical practices. Big Idea/Enduring Understanding(s) 1. Rational numbers can be used, expressed, and manipulated in a variety of ways. Students will be applying their knowledge of positive and negative numbers by viewing them in everyday contexts (temperatures, recipes, taxes, money, etc). Unit 5: Rational Numbers Academic Technical Vocabulary Vocabulary: positive negative rational absolute value inequality plot Prerequisite Knowledge Students in Fifth Grade were expected to: Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the Grade 6 Unit 5 4
Grade 6 Unit 5 coordinates coordinate plane reflection interpret magnitude quadrants coordinate (first - fourth) axes ordered pairs integers opposite reflection inequality relative ordering horizontal vertical vertices polygon perimeter area summarize plane located by using an ordered pair of numbers, called its coordinates. represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. classify two- dimensional figures in a hierarchy based on properties. make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Summative Performance Task: This would be project based, teacher developed, and common. Grade 6 Unit 5 5
Grade 6 Unit 5 Grade 6 Unit 5 6
Grade 6 Unit 6 Unit 6: Ratio and Proportion Timeframe: May Common Core State Standards, 2010 Key Areas of Focus in the CCSS: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak. For every vote candidate A received, candidate C received nearly three votes. 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. For example, 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. We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger. 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.3b 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? 6.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. 6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Grade 6 Unit 6 1
Grade 6 Unit 6 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. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Additional Standards: Career Ready Practices - 9.1.8.B.2 Construct a simple personal savings and spending plan based on various sources of income. Unit 6 Essential Outcomes 1. Explain the relationship of two quantities or measures of a given ratio and use ratio language to describe the relationship between the two quantities. For example, The ratio of wings to beaks in the birdhouse at the zoo was 2:1, because for every 2 wings there was 1 beak. For every vote candidate A received, candidate C received nearly three votes. 2. Use rate language in the context of a ratio relationship to describe a unit rate a/b associated with a ratio a:b with b 0. For example, 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. We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger. 3. Use ratio and rate reasoning to solve real world and mathematical problems which include making tables of equivalent ratios, solving unit rate problems, finding percent of a quantity as a rate per 100. 4. Use ratio and rate reasoning to convert measurement units (manipulate and transform units appropriately when multiplying or dividing quantities). 6.RP.1 6.RP.2 6.RP.3 6.RP.3 5. Use variables to represent two quantities that change in relationship to one another in a real world problem and 6.EE.9 Grade 6 Unit 6 2
Grade 6 Unit 6 write an equation to express one quantity, thought of as the dependent variable, in terms of another quantity, thought of as the independent variable. 6. Analyze the relationship between the dependent and independent variables in an equation using graphs and tables. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. 6.EE.9 7. Create a savings and spending plan based on various sources of income. 9.1.8.B.2 Major Content Supporting Content Additional Content(Identified by PARCC Model Content Frameworks). Bold type indicates grade level fluency requirements. (Identified by PARCC Model Content Frameworks). Selected Opportunities for Connection to Mathematical Practices 1. Make sense of problems and persevere in solving them. E.O. #5 Relating variables to real world context. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. E.O. #6 Use graphs and tables to represent a dependent/independent relationship. 5. Use appropriate tools strategically. 6. Attend to precision. E.O. #4 Converting measures. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. All content presented at this grade level has connections to the standards for mathematical practices. Big Idea/Enduring Understanding(s) Grade 6 Unit 6 3