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BOSTON PUBLIC SCHOOLS Course 436 – Mathematics 6 Scope and Sequence Guide 2013-2014 Background The 2013-2014 Scope and Sequence documents are the product of many hours of analysis, planning, and professional development among representatives of all corners of Boston’s mathematics community: teachers, administrators, and math office staff. This domain-by-domain, standard-by-standard work has been driven by the ongoing goal of providing students with wider access to Algebra 1 in grade 8 as well as to higher level math courses, including advanced placement. Our work has been informed by 2011 Massachusetts Frameworks, as well as the PARCC Model Content Frameworks for Mathematics, with particular focus on the learning trajectories for grades 5 through 8. The title, “Common Core State Standards” is often referenced in the media and in Federal statements on education policy. Massachusetts adopted the Common Core State Standards, and have incorporated them into the Commonwealth’s curriculum frameworks. Accordingly, the new 2011 standards are correctly referred to as the “Massachusetts Curriculum Framework for Mathematics.” The scope and sequence documents reflect the Common Core Instructional Shifts for Mathematics: Focus, Coherence, and Rigor. 1. Focus strongly where the standards focus. 2. Coherence: Think across grades, and link to major topics within grades. 3. Rigor: Require fluency, application and deep understanding. New curricular resources have been acquired by Boston Public Schools, and distributed to schools during the 2012-2013 school year to support instruction in these areas, as well as transition to the 2011 Massachusetts Frameworks. BPS will continue to use these supporting resources in forming mathematics instruction:

• Grade 6: CMP2 Common Core Investigations • Grade 7: CMP2 Common Core Investigations and Data Distributions • Grade 8: CMP2 Common Core Investigations and Kaliedoscopes, Hubcaps, and Mirrors.

Throughout the academic year, professional development opportunities will be offered to support our implementation Massachusetts Frameworks. Through this process, schools leaders and teachers are urged to regularly consult the Middle School Math page of MyBPS and MyLearningPlan for curriculum and instruction and professional development announcements. Additionally, the BPS


Curriculum and Instruction Weebly page has been established as a resource, and can be accessed at http://bpscurriculumandinstruction.weebly.com/

The Standards for Mathematical Practice The standards for mathematical practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices are based on two sets of processes and proficiencies which retain longstanding importance in mathematics education: the NCTM Process Standards (Problem Solving, Reasoning and Proof, Communication, Representation, Connections) and the strands of mathematical proficiency established by the National Research Council report, “Adding it Up.” These proficiencies include: Adaptive Reasoning, Strategic Competence, Conceptual Understanding (the comprehension of mathematical concept, operations, and relations), Procedural Fluency (the skill that enables students to apply procedures flexibly, accurately, efficiently, and appropriately), and Productive Disposition (the habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). The Standards for Mathematical Practice describe the way in which developing mathematics students engage with the subject matter throughout learning trajectories. Accordingly, the Boston Public Schools Scope and Sequences documents for Secondary Mathematics Courses are designed to address the need to connect the mathematical practices to mathematical content throughout mathematics instruction. Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. In grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”. Standard for Mathematical Practice 2: Reason abstractly and quantitatively In grade 6, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations. Standard for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others In grade 6, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always work?” They explain their thinking to others and respond to others’ thinking.


Standard for Mathematical Practice 4: Model with Mathematics In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students begin to explore covariance and represent two quantities simultaneously. Students use number lines to compare numbers and represent inequalities. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context. Standard for Mathematical Practice 5: Use appropriate tools strategically Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 6 may decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data. Additionally, students might use physical objects or applets to construct nets and calculate the surface area of three-dimensional figures. Standard for Mathematical Practice 6: Attend to precision In grade 6, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities. Standard for Mathematical Practice 7: Look for and make use of structure Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality), c=6 by division property of equality). Students compose and decompose two- and three- dimensional figures to solve real world problems involving area and volume. Standard for Mathematical Practice 8: Look for and express regularity in repeated reasoning In grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization. Students connect place value and their prior work with operations to understand algorithms to fluently divide multi-digit numbers and perform all operations with multi-digit decimals. Students informally begin to make connections between covariance, rates, and representations showing the relationships between quantities. The Need for Efficiency and Collaboration


It is our collective responsibility to work together to continually seek and develop strategies that add power and efficiency to the instruction process. It is also our responsibility to ensure that we prepare our students adequately for each subsequent year of learning, realizing that no matter the grade at which we teach, we are preparing young minds for success in high school, college, and career. The Boston Public Schools, as well as the developers of Connected Mathematics 2, have identified a workshop model of instruction as the most efficient model currently available to optimize student gains in mathematics. Workshop is an inquiry-based model. It is the expectation of the district that the workshop model of instruction will be the primary and core instructional model used in all math classes in the district. In broad terms, the intent of the model is to provide an instructional framework that supports a blend of individual and group strategies that vary from explicitly teacher led direct instruction experiences to student led collaborative experiences. It is a model that emphasizes the recognition of students’ prior learning and allows students to increase their confidence by using their prior knowledge along side of their teachers in a variety of individual and group experiences. The savings in time that come from continual student engagement with their teacher and the confidence that comes from skillfully using prior knowledge to develop new knowledge, creates the energy and momentum among all members of the classroom community that is a necessary prerequisite for the successful completion of this scope and sequence. In short, classes who master our curricula collaborate to apply efficient strategies that combine the strengths of the teacher with the strengths of the students to achieve our learning goals. They are classes in which everyone understands the mission of the day, week, month, and year and everyone works collaboratively to achieve it. They are classes that are grounded in the common belief that we will be successful in our academic and personal goals. Instructional Shifts and Focus for Grade 6 In grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division, and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. (1) Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates.


(2) Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems. Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. (3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities. (4) Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. Students recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected. Students in grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They prepare for work on scale drawings and constructions in grade 7 by drawing polygons in the coordinate plane. Essential Question(s) for the course:

1. How do we use reasoning about multiplication and division to solve ratio and rate problems about quantities?

2. What is the meaning of fractions in relation to the meanings of multiplication and division?


3. What is the purpose of variables in mathematical expressions?

4. How do we build upon our understanding of number to develop the ability to think statistically?

What must students have learned by the middle of the academic year? By mid-year, students should have worked extensively with rational numbers. Students should compute fluently with multi-digit whole numbers and decimals using the standard algorithm, and apply number theory concepts (including prime factorization) to the solution of problems. Students should also interpret and compute sums, differences, products, and quotients of fractions. Students should also understand the concept of a ratio, a unit rate, and use ratio and rate reasoning to solve real-world and mathematical problems. What will students have learned by the end of this course? In grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division, and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. In this first transition year to the full implementation of the 2011 Frameworks, the Fraction Unit remains a large part of the instructional year to ensure that there is a strong foundation before beginning ratio and proportion work. As the fraction work shifts in focus in upcoming years to grades 3, 4, and 5, the Data Unit would move earlier into the year and both the Ratio & Proportion and Expressions & Equations Units would expand to full rigor. In addition to the learning in the first half of the year, students will apply and extend previous understandings of arithmetic to algebraic expressions. This will include numerical expressions involving whole-number exponents, using letters as variables to describe an unknown or constant, and apply the properties of operations to generate equivalent expressions. Students will reason about and solve one-variable equations and inequalities as well as represent and analyze quantitative relationships between dependent and independent variables. In Geometry, students will solve real-world and mathematical problems involving area, surface area, and volume, including working with circles and three-dimensional figures. Students will be able to work with negative numbers within a real-world context including locating numbers along a number line and a four-quadrant coordinate plane. Students will also order and find the absolute value of rational numbers. Working with data, students will develop an understanding of statistical variability and be able to summarize and describe data distributions.


Unit #1: Whole Numbers and Prime Time Primary Curricular Resource: Prime Time Estimated Instructional Time: 25 days Overarching Questions:

• What strategies do we have to find factors of a number and what value do factors and common factors have?

• How do we find multiples of a number and what value do multiples and common multiples have?

• What makes a number prime or composite, even or odd, square or not square?

• How do we break down a number into a unique string of prime factors?

Standards for Mathematical Practice Focus • Attend to precision (SMP 6) in use of vocabulary around factors,

multiples, and divisibility including when solving real world least common multiple and greatest common factor problems, factors and multiples will need to be labeled clearly.

• Model with mathematics (SMP 4) and apply skills in number sense and number theory to solve a variety of real world problems using prime factorization trees and Venn diagrams for LCMs and GCFs.

Instructional Notes:

• Teachers should give a whole number operation pre-assessment to gear teaching of multiplication and division referencing the US algorithm. Teachers should have a series of “Do Now’s” that will foster fluency of multiplication and division of real world problems as well as pure computation. The overview of the number system in which our standards are used to guide our teaching is to compute fluently with multi-digit numbers and find common factors and multiples. In our third unit Bits 1,2, 3 our teaching drives us to use multiplication and division to apply and extend previous understandings of multiplication and division to divide fractions by fractions.

• The Prime Time Partner Quiz is a good assessment of key concepts. • For ideas on how to teach the US algorithm of division to achieve fluency, please see the TERC Investigation Family Handbook notes on the Middle

School Mathematics page on MyBPS under Grade 6 Concepts developed in this unit

• Increase knowledge of and flexibility with multiplication and division relationships

• Become familiar with factors and multiples and relationships between them

• Determine whether one number is a factor or multiple of another • Classify numbers as prime and composite • Fluently find common factors and multiples (specifically LCM and

GCF) • Relate dividing and finding factors of a number • Determine prime factorization of a number

Prior knowledge expected (add standards) • Fluently add, subtract multi-digit whole numbers using standard

algorithm • Multiply fluently by multiples of ten • Use arrays to represent multiplication

Terms developed in this unit: multiplication combinations, multiples of ten, prime number, square number, composite number, factor, product, multiple, dividend, divisor, quotient, divisible, inverse relationship, rectangular array, unmarked array, dimension,

Terms from prior units or experiences: odd number, even number, whole number, counting number, natural number, place value, digit, ones, tens, hundreds, thousands, number line, multiplication, congruent, representation, doubling, halving, equal, less than (use symbols),


exponent greater than (use symbols), estimation Learning Outcomes

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. 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.NS.MA.4.a Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems. • 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. T E R M

MA 2011 Framework

Citation After completing each investigation, students will be able to: Days

Primary Curriculum Resource

Discretionary Days √ Beginning of year classroom routines √ Notebook Set Up √ Classroom Expectations

3

6.NS.2 √ Demonstrate knowledge of whole number computations √ Use the US Standard algorithm for multiplying and dividing multi-digit

numbers. http://betterlesson.com/course/5570/6th-grade-mathematics lesson 1 Number Sense Long Division

5

Teacher Created Whole Number Review (Pre-assessment and work on the standard

algorithm)

6.NS.4 √ Recognize that factors come in pairs and that these pairs can be visualized as dimensions on a rectangle

√ Determine whether numbers are prime or composite; even or odd; and square or not square based on factor pairs

1 Prime Time 2.1

6.NS.4 √ Classify numbers by their properties using Venn diagrams √ Develop understanding of factors and multiples and common factors and

multiples.

1 Prime Time 2.3

6.NS.4 √ Recognize common multiples √ Find patterns in common multiples to reason, predict to solve problems

2 Prime Time 3.1

6.NS.4 √ Recognize and find multiples of numbers √ Develop strategies for finding least common multiple

2 Prime Time 3.2

6.NS.4 √ Recognize and find common factors or numbers √ Develop strategies for finding common factors √ Find patterns of common factors of numbers to reason and predict to solve

problems

2 Prime Time 3.3

6.NS.4 √ Recognize and develop strategies for finding Greatest Common Factor (GCF)

√ Understand when Greatest Common Factor and Least Common Multiple

2 Prime Time 3.4


are helpful for solving problems √ Classify numbers by their properties using Venn diagrams √ Develop understanding of factors and multiples and common factors and

multiples. 6.NS.4

6.NS.MA.4a 6.EE.1

√ Find factorizations of numbers √ Find prime factorization (and use exponential notation) √ http://betterlesson.com/course/5570/6th-grade-mathematics √ Lesson 1 and 2 Expressions and Equations Order of Operations (perform

Order of Operations including exponents)

2 Prime Time 4.2

6.NS.4 6.NS.MA.4a 6.NS.MA.4a

√ Find Common factors, Greatest Common Factors, Common multiples, Least Common Multiples using prime factorization

√ Understand prime numbers are building blocks for whole numbers

2 Prime Time 4.3

6.NS.4 √ Use primes, composites, factors, multiples, and square numbers to reason mathematically and solve an interesting real world problem.

2 Prime Time 5.1

√ 1 Number Systems Assessment

End of Unit 1: October 8,2013 Unit #2: Fractions, Decimals, and Percentages Primary Curricular Resource: Bits & Pieces I, Bits & Pieces II, and Bits & Pieces III Estimated Instructional Time: 30 days Overarching Questions:

• How do we represent situations as fractions, decimals, and percents? (Bits I)

• What is the most efficient/effective way to represent a quantity in a given situation? (Bits I)

• Why is it important to understand and use equivalent fractions to reason about situations? (Bits I)

• How do you add, subtract, multiply, and divide fractions and decimal fractions? (Bits II)

• How to recognize when addition, subtraction, multiplication, or division is the appropriate operation to solve a problem? (Bits II)

• How is the inverse relationship between addition and subtraction, and between multiplication and division used in solving problems involving fractions and decimal fractions? (Bits II)

Standards for Mathematical Practice Focus • Reason abstractly and quantitatively (SMP 2). Students will need to

create a coherent representation of the problem at hand; consider the units involved; attend to the meanings of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects.

• Model with mathematics (SMP 4). Students will need to apply proportional reasoning to real world events or problems, as well as interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

• Attend to precision (SMP 6). Students will need to communicate precisely to others using clear definitions in discussion with others and in their own reasoning. Students will need to state the meaning of the symbols they use, including using the equal sign consistently and


• When is it appropriate to use decimal fractions instead of common fractions? (Bits III)

• Why is knowledge of equivalent fractions essential in working with decimals? (Bits III)

• Why is knowledge of place value (base-ten system) essential in working with decimals? (Bits III)

•

appropriately. Students will need to calculate accurately and efficiently and make the bridge between the elementary expectation of giving carefully formulated explanations to each other to the entry high school expectation of examining claims and making explicit use of definitions.

Instructional Notes: • Fraction and decimal work has largely shifted to 5th grade under the 2011 Massachusetts Frameworks including:

o using equivalent fractions as a strategy to add and subtract fractions, o applying and extending previous understandings of multiplication and division to multiply and divide fractions, o performing operations (+, -, x, ÷) with multi-digit whole numbers and with decimals to hundredths.

• As a major cluster in grade 6, students will be expected to apply and extend previous understandings of multiplication and division to divide fractions by fractions. This completes the extension of operations to fractions. (6.NS.1)

• Fraction work is pivotal to key advances between grades 6 and 7 and students’ understanding of proportional relationships (ratios including percents and fractions) needs to be solid.

• Standards 6.NS.1 and 6.NS.3 highlight the fluency expectations for students working with word problems involving division of fractions by fractions and using the standard algorithm for adding, subtracting, multiplying, and dividing decimals.

Concepts developed in this unit Bits & Pieces II

• Use benchmarks and other strategies to estimate the reasonableness of results of operations with fractions

• Develop ways to model products and quotients with areas, strips, and number lines

• Use estimate and exact solutions to make decisions • Look for and generalize patterns in numbers • Use knowledge of fractions and equivalence of fractions to develop

algorithms for dividing fractions • Recognize when addition, subtraction, multiplication, or division is the

appropriate operation to solve a problem • Write fact families to show the inverse relationship between addition

and subtraction, and between multiplication and division • Solve problems using arithmetic operations on fractions

Bits & Pieces III

• Build on knowledge about operations with fractions and whole numbers

• Develop and use benchmarks and other strategies to estimate the answers to computations with decimals

• Develop meaning of and algorithms for operations with decimals • Use the relationship between decimals and fractions to develop and

understand why decimal algorithms work

Prior knowledge expected (add standards) Bits & Pieces II

• Interpreting fractions as part-whole relationships; combining and comparing fractions, partitioning and repartitioning fractions, finding equivalent fractions; factorization of numerators or denominators (Elementary Units and Prime Time)

• Estimating to check reasonableness of answers • Inverse operations in whole number settings (Elementary Units: Fact

Families/Related Operations) • Interpreting fractions as part-whole relationships; combining and

comparing fractions, partitioning and repartitioning fractions, finding equivalent fractions (Bits & Pieces I and Elementary Units)

Bits & Pieces III • Interpreting decimals as fractions; understanding place value of

decimals, combining and comparing decimals, performing mathematical operations with fractions (Elementary Unit and Bits & Pieces II)

• Connecting fractions, decimals, and percents to check the reasonableness of answers, estimating to check reasonableness of answers; developing and applying algorithms for performing fraction calculations; developing algorithms for finding the area and perimeter of 2 dimensional shapes (Elementary Unit & Shapes & Designs)

• Inverse operations in whole number settings; inverse operations in fraction settings; finding an unknown dimension given area


• Use the place value interpretation of decimals to make sense of shortcut algorithms for operations

• Generalize number patterns to help make sense of decimal operations • Choose between addition, subtraction, multiplication, or division as an

appropriate operation to use to solve a problem • Solve problems using operations with decimals • Use understanding of operations and the meaning of percents to solve

percent problems of the form a% of b equals c for any one of the variables a, b, or c

• Create and interpret circle graphs

(Elementary Units) • Defining, comparing, and applying percents (Bits & Pieces II)

Terms developed in this unit: Bits & Pieces I base ten number system, benchmark, decimal, denominator, equivalent fraction, fraction, improper fraction, mixed number, numerator, percent, place value, place value-chart, power of ten, ratio, rational number, unit fraction, tenths grid, hundredths grid, thousandths grid Bits & Pieces II algorithm, fact family, mathematical sentence, overestimate, reciprocal, underestimate Bits & Pieces III repeating decimal, terminating decimal, power of ten, difference, dividend, divisor, factor, mean, product, quotient, sum

Terms from prior units or experiences: Bits & Pieces I factor, greatest common factor, least common multiple, key Bits & Pieces II benchmark, decimal, factor, fraction, percent, unit fraction, denominator, equivalent fraction, improper fraction, least common multiple (LCM), mixed number, multiple, numerator, number sentence Bits & Pieces III algorithm, base ten, benchmark, decimal, denominator, estimate, fact family, mathematical sentence, number sentence, numerator, percent, place value

Learning Outcomes 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 3/4 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 3/4 mi and area 1/2 square mi?

6.NS.6c 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 Find and position pairs of integers and other rational numbers on a coordinate plane. 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.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. T E R M

MA 2011 Framework



6.NS.6c

Pre-assessment Quiz Use benchmarks to estimate the size of fractions and compare fractions

√ Develop strategies for finding equivalent fractions √ Develop strategies for finding fractions in between fractions

5

Bits & Pieces I Investigation 2.3/2.4

6.RP.1 6.RP.3c

√ Introduce percents as a part-whole relationship where the whole is not out of 100, but scaled to be “out of 100.”

√ Use fraction partitioning and fraction benchmarks to make sense of percents.

1 Bits & Pieces I Investigation 4.1

6.RP.1 6.RP.3c

√ Develop strategies, including percents, to use in comparisons where the whole is less than 100.

√ Understand that comparing situations with different numbers of trials is difficult unless we use percents or some other form of equivalent representation.


6.RP.1 6.RP.3c

√ Work with situations where the whole is sometimes greater than 100 and sometimes less than 100.

√ Develop connections between representations (fractions, decimals, and percents).

√ Develop strategies for expressing data in percent form.


6.RP.1 6.RP.3c

√ Relate and develop connections between fractions, decimals, and percents.

√ Move from percents to other representations and from other representations to percents.


6.NS.1 √ Use models to represent a fraction divided by a fraction. √ Develop and use strategies for dividing a fraction by a fraction. √ Understand when division is an appropriate operation.

1

Bits & Pieces II Investigation 4.3

6.NS.1 Note: Day 6 of Investigation 4 focus on standard 6.NS.1 to create a story/real-world context based on expressions in the 4.4 groups (fraction ÷whole #),

2

Bits & Pieces II Investigation 4.4


(whole # ÷fraction), (fraction ÷fraction), and (mixed # division). √ Develop an efficient algorithm to solve any fraction division problem. √ Explore the inverse operation of multiplication and division.

6.NS.3 Note: Bits & Pieces III focuses on fluency and use of the algorithm for

adding, subtracting, multiplying, and dividing decimals. √ Use benchmarks and decimal-fraction relationships to develop estimation

strategies for finding decimal sums. √ Develop place-value understanding of decimal addition and subtraction. √ Develop strategies for adding and subtracting decimal numbers.

1 Bits & Pieces III Investigation 1.1 and 1.2

6.NS.3 √ Connect strategies for addition and subtraction of decimals to addition and subtraction of fractions with powers of ten in the denominator.

√ Relate renaming fractions to have common denominators to the notion of adding values with the same place value.

√ Develop and use efficient algorithms for adding and subtracting decimals. √ Explore the inverse relationship between addition and subtraction in

decimal settings.

1 Bits & Pieces III Investigation 1.3 and 1.4

6.NS.3 √ Estimate the relative size of a decimal product prior to finding an exact answer.

√ Develop place value understanding of decimal multiplication. √ Solve problems that require decimal multiplication √ Consider how finding a decimal part of and a fraction part of a quantity

affects the relative size of a produce.

1 Bits & Pieces III Investigation 2.1

6.NS.3 Note: Investigation 2.2 treat as a “missing product” problem (e.g. Fact Families x 6 = 0.36 as a pre-algebra concept)

√ Use place value to reason about decimal multiplication. √ Explore the relationship between factors and products in decimal

multiplication.


6.NS.3 √ Develop estimation strategies for finding decimal products. √ Use estimation as a strategy for finding exact decimal products.


6.NS.3 √ Generalize an approach to placing the decimal point into a product that involves counting and adding decimal places.

√ Consider when various strategies are useful for finding decimal products. √ Understand what happens to place value and the position of the decimal

when you multiply by powers of 10. √ Develop and use an efficient algorithm for multiplying decimals.

1

Bits & Pieces III Investigation 2.4

6.NS.3 √ Choose between division, multiplication, addition, or subtraction as an appropriate operation to use to solve a problem.



√ Use models and the context to find solutions to division problems. √ Estimate to find approximate solutions.

6.NS.3 √ Use the relationship between decimals and fractions to develop and

understand decimal division. √ Use the common denominator approach to fraction division as a strategy

to help understand and develop an algorithm for dividing decimals. √ Relate the division algorithm to place value understanding.


6.NS.3 √ Use knowledge about computation with fractions to understand algorithms for division with decimals.

√ Use place value to develop an algorithm for division with decimals. √ Develop and use efficient algorithms for dividing decimals. √ Explore the inverse relationship between multiplication and division in fact

families.

1

Bits & Pieces III Investigation 3.3

6.NS.3 √ Understand and predict the decimal representation of a fraction (terminating or repeating).


.6.RP.3c √ Understand that a percent is a decimal fraction with a denominator of 100. √ Represent $1.00 as 100 pennies, and relate this to partitioning a number

line into 100 parts. √ Represent percents as decimals and use decimal computation to compute

percents.


6.RP3c √ Represent percents as decimals and use decimal computation to compute percents.

√ Explore the relationship between 1% and 10% and use these to compute 5%, 15%, and 20% tips.

√ Work backwards to find the amount of the bill if you know the tip and the percent of tip for the bill.


6.RP.3c √ Use percents in estimating or computing taxes, tips, and discounts. √ Find what percent one number is of another number. √ Solve problems using percents.


6.RP.3c √ Develop a strategy for finding the percent of discount an amount taken off a price represents.

√ Use percents in estimating taxes, tips, and discounts.


6.RP.3c √ Use percents in estimating or computing taxes, tips, and discounts. √ Solve problems using percents.



6.SP.MA.4a √ Create and interpret circle graphs to represent data. √ Solve problems using percents.


6.NS.3, 6.RP.3c 6SP.MA.4a

• Review Material and Assessment •

2 Bits & Pieces III Assessment

End of Unit 2: November 21, 2013 Unit #3: Ratio and Proportions Primary Curricular Resource: Common Core Investigations and Online Resources Estimated Instructional Time: 28 days Overarching Questions:

• How can two quantities be related? • How can this relationship be used to predict

increased/decreased quantities? • What does a ratio look like in a graphical or tabular

representation? • How can a unit rate be used to compare different situations?

Standards for Mathematical Practice Focus • Make sense of problems and persevere in solving them (SMP 1) • Reason abstractly and quantitatively (SMP 2) • Attend to precision (SMP 6) • Look for and make use of structure, Look for an express

regularity in repeated reasoning (SMP 7)

Instructional Notes: • Common Core Investigations need to be supplemented; a suggested online unit is included. • Graphing is introduced in Investigation 1.3—students have not yet been exposed to this • An online investigation is included as a real-world link and reinforcement of unit rates • READ THE TEACHER NOTES for the Common Core Investigations (wording, launch ideas are very helpful) • Many of the numbers in the Common Core book are “unfriendly.” Please consider doing the problems first yourself!! • A few daily lessons are built around a handful of the Common Core exercise problems. Supplement as needed. • Calculators should be used as a tool to move past arithmetic to allow time for deep understanding of mathematical concepts that are the

focus of the lesson. • Grade 7 begins with Similarity and Congruence from CMP2, so student understanding of RP standards needs to be strong

Concepts developed in this unit

• Describe a relationship between two quantities as a ratio • Represent ratios as fractions, decimals, and percents • Recognize that a/b is a different form of a:b

Prior knowledge expected (add standards) • Create equivalent fractions • Find the GCF • Understand fractions as representations of division


• Understand a rate is a ratio comparing two quantities measured in different units

• Understand that a unit rate is a rate for which one of the numbers being compared is 1 unit

• Represent ratio and rates in tables, graphs, tape diagrams, and equations

• Plot values in a table as points on a coordinate plane (Common Core Investigation 3)

• Use unit rates to calculate unit pricing and constant speed • Understand that percents of a quantity are rates out of 100 • Convert measurement units using ratios • Relate the mass of objects and volume using ratios

• Fluently use the standard algorithm for division

Learning Outcomes 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 ¾ 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.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.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

T E R M

MA 2011 Framework



6.RP.1 6RP3

NOTE: In problems 1.1 A and B, students should find minutes per poster (not poster per minute, as written)

• Create unit rates using a ratio • Write different forms of ratios

Better Lessons: Lesson 1 Intro to Ratio http://betterlesson.com/lesson/305439/lesson-01-intro-to-ratios http://betterlesson.com/lesson/305440/lesson-02-ratios-and-vocabulary

3 Common Core Investigation 1.1


6RP2, 6RP3 • Compare ratios

• Create ratio tables • Plot pairs of values on the coordinate plane


6RP2, 6RP3 • Calculate and compare unit rates

2 Common Core Investigation 1 Question 23

(also #16-18, 24-29, 30)

6RP2, 6RP3b Note: Consider Scaffolded Instruction/Modeling and Group Poster Presentations • Use unit rate to solve real life problems

2 Common Core Investigation 1

Question 31 (also #12, 13)

6RP2 Note: Consider Scaffolded Instruction/Modeling and Group Poster Presentations • Calculate unit rates • Compare ratios


6RP3d • Convert units on a map 1 Common Core Investigation 1 Question 22

6RP3d • Convert units of measurement 1 Common Core Investigation 1.4

6RP3d Review/Discretionary Days Massachusetts DESE-created unit: “Ratios and Rates” http://www.doe.mass.edu/candi/model/sample.html Better Lessons: Lesson 3 Intro to Tape Diagrams http://betterlesson.com/lesson/305441/lesson-03-intro-to-tape-diagrams • Better Lessons: Lesson 4 Tape Diagrams for Part Whole

diagrams • http://betterlesson.com/lesson/305442/lesson-04-tape-diagrams-for-part-

whole-problems •

12 Teacher-Created or Use Additional

Resources listed in the center.


• Better Lessons: Lesson 5 Ratio Tables [Insert link here!] • http://betterlesson.com/lesson/305443/lesson-05-ratio-tables • : • Better Lessons: Lesson 6 Use Tape Diagrams and Ratio Tables

to solve Problems • http://betterlesson.com/lesson/305444/lesson-06-use-tape-diagrams-and-ratio-

tables-to-solve-problems • • http://betterlesson.com/lesson/305445/lesson-07-intro-to-rate-and-unit-rate : • Better Lessons: Lesson 8 Real World Rate and Unit Rate

Problems • http://betterlesson.com/lesson/305446/lesson-08-real-world-rate-and-unit-rate-

problems • Better Lessons: Lesson 9 Ratio tables- graphing • http://betterlesson.com/lesson/305447/lesson-09-ratio-tables-graphing • Better Lessons: Lesson 10 Percent as a ratio using double

number-lines • http://betterlesson.com/lesson/305448/lesson-10-percent-as-a-ratio-using-

double-number-lines • Better Lessons: Lesson 11 comparing ratio problem solving

tools http://betterlesson.com/lesson/305449/lesson-11-comparing-ratio-problem-solving-

tools Better Lessons: Lesson 12 Ratio and Rate Review http://betterlesson.com/lesson/305450/lesson-12-rate-ratio-review End of Unit Assessment

End of Unit 3: January 14,2014 1 Unit #4: Negative Numbers and Four Quadrant Graphing Primary Curricular Resource: Accentuate the Negative and Common Core Investigations


Estimated Instructional Time: 18 days Overarching Questions:

• How are positive and negative numbers used to describe quantities? • What is the absolute value of a number and how is it used

mathematically? • How are coordinate grids and the measure of distance related as a

representation of change?

Standards for Mathematical Practice Focus • Make sense of problems and persevere in solving them (SMP 1).

When calculating the sum, difference, product, or quotient, as well as distance and reflection of points on a coordinate grid, students will have to determine if the positive/negative solution makes sense given the context of the problem.

• Construct viable arguments and critique the reasoning of others (SMP 3). Students will be asked to make justify their solutions using inequalities, equations, and graphing.

• Model with mathematics (SMP 4). Students will represent all rational numbers on number lines, a four quadrant coordinate grid, and with inequalities.

• Use appropriate tools strategically (SMP 5). Students will need to identify and use the appropriate available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator.


• The major standards for negative numbers and graphing in four quadrants are taught in 2012-2013 (6.NS.5-8). • As the instructional time for fractions decreases when there is an increased instructional time for fractions in grades 3, 4, and 5, this unit will expand in

scope and depth to include the majority of Accentuate the Negative to address 6.NS. 5-8 in more depth and to include standards 6.EE.4 and 6.G.3 Concepts developed in this unit

• Reason about the order and absolute value of rational numbers • Use appropriate notation to indicate positive and negative numbers • Locate rational numbers (positive and negative fractions, and

decimals and zero) on a number line. • Compare and order rational numbers • Understand the relationship between a positive or negative number

and its opposite (additive inverse) • Develop algorithms for operations with positive and negative numbers • Write mathematical sentences (including inequalities) to show

relationships. • Write and use related fact families for addition/subtraction and

multiplication/division to solve simple equations with missing facts. • Use parenthesis and order of operations to make computational

sentences clear. • Understand and use the commutative property for addition and

multiplication of positive and negative numbers

Prior knowledge expected (add standards) • Fluency with positive integers and rational numbers on a number line,

including performing addition and subtraction. • Knowledge of “fact families”


• Apply the distributive property with positive and negative numbers to simplify expressions and solve problems

• Use positive and negative numbers to graph in four quadrants and to model and answer questions about applied settings.

Academic language developed in this unit:

opposite, integer, rational number, absolute value, coordinate plane, ordered pairs, origin, inequality, Commutative property, Distributive property, additive inverse, Associative property, Quadrant I, II, III, IV

Academic language from prior units or experiences: positive, negative, inverse, algorithm, coordinate grid, fact family, mathematical sentence, inequality, number line, number sentence, operations, ordered pair

Learning Outcomes 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 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. 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 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.

6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3oC > –7oC to express the fact that –3oC is warmer than –7oC.

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.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 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.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. •


T E R M

MA 2011 Framework



6.NS.5 6.NS.6

6.NS.6a 6.NS.7d

Note: Common Core Investigation 3.1 can be used, if needed, to additionally support student understanding of the concept of opposites on a number line. Locate positive and negative numbers on a number line and compare and order them. Understand relationship between a positive or negative number and its opposite.

2 Accentuate the Negative 1.1

6.NS.5 6.NS.6

6.NS.6a 6.NS.7a, b, c

Write number sentences to reflect the actions and results of changes in situations and find missing values. Understand that an integer and its additive inverse are called opposites. http://betterlesson.com/lesson/363045/lesson-04-integers

3 Accentuate the Negative

1.2

6.NS.7c Better Lessons Absolute Value http://betterlesson.com/lesson/305456/lesson-03- absolute-value 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. Distinguish comparisons of absolute value from statements about order.

3 Better Lessons Absolute Value

6.NS.6c 6.NS.7c 6.NS.7d 6.NS.8 6.EE.5 6.EE.8

Interpret real-world problems and represent them as an inequality and graph the inequality on a number line. Provide multiple solutions to inequalities involving negative numbers.

2 Common Core Investigation

3.6

6.NS.6 6.NS.6c

Graph positive and negative coordinates in all 4 quadrants. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

2 Accentuate the Negative 2.5

6.NS.6b 6.NS.6c 6.NS.7c

Graph points on a four-quadrant coordinate plane with identical x-values or identical y-values. Measure length along a line segment on a four-quadrant coordinate plane given two points.


3.4

6.NS.6b 6.NS.6c

6.G.3

Graph points on a four-quadrant coordinate plane and connect line segments to create a polygon, given the coordinates are vertices of the polygon.


3.5 6.NS.5

6.NS.6a-c 6.NS.7a-d

Practice using a vertical number line using Additional Practice Skill Pages 31-32 (in the teachers’ guide of the Common Core Investigation book)

1 Common Core Investigation 3

(Additional 6.NS.5

6.NS.6a-c 6.NS.7a-d

! Additional Practice to be determined by teacher based on student performance to increase fluency working with rational numbers on both a number line and a four quadrant coordinate grid

1 Common Core Investigation/

Accentuate the


6.NS.8 Negative 6.NS.5

6.NS.6a-c 6.NS.7a-d

6.NS.8

! 1 Number Sense: Negative Numbers &

Four Quadrant Graphing

Assessment Review and Assessment End of Unit 4: February 10,20 Unit #5: Expressions and Equations Primary Curricular Resource: Common Core Investigations and Variables and Patterns Estimated Instructional Time: 24 days Overarching Questions:

• What are variables and how do we represent them in algebra? • What models can you use to represent a relationship between

variables? • What is the connection between a linear relationship and a ratio? • How do we represent a general rule to show the relationship between

two variables? • How can we use the general rule to find values of the dependent

variable? • Why and how do we solve equations? • What are some of the major properties of arithmetic operations? How

do we use these in algebra?

Standards for Mathematical Practice Focus • In this unit, students will be called upon to think reason abstractly and

quantitatively (SMP #2) using letters to represent variables, a much higher level of abstraction than was expected in earlier grades.

• Students will rely heavily on the use of models (tables, graphs, and equations) to solve real world mathematical problems (SMP #4). Their facility with utilizing these models will be critical to their success in later pre-algebra, algebra, and higher-level math classes. Students should be able to see the correspondence between these three models and use them to solve problems (SMP #1).

• Looking for and making use of structure (SMP #7) is one of the most heavily utilized standards of mathematical practice in algebra. To create tables, students will need to identify patterns, extend them, and create a symbolic rule for all possibilities.

• When graphing and making tables, students should be able to see and express regularity in repeated reasoning (SMP #8), making the same jumps on the graph and table when modeling linear relationships and describing why this happens. They should also be able to see that not all graphs create straight lines, using appropriate scaling on their axes.


• There is a heavy focus in the 6th grade standards on algebraic thinking using expressions and equations, significantly more than in the 2004 Frameworks. Many of the current 6th grade standards were formerly 7th grade standards.

• Standard 6.EE.9 is a very large standard covering a number of different topics. It is flushed out in the first 2 investigations and investigation 3.1 of Variables and Patterns.


• Students should be able to fluently convert between different forms of a linear relationship- table to graph, table to equation, graph to table, graph to equation, equation to table, and equation to graph.

• In 2013, standards 6.EE.3, 6.EE.4, 6.EE.5, and 6.EE.8 aren’t assessed for MCAS but are essential to success in the 7th and 8th grades. • Students should be shown the connection between unit rates and linear relationships. • While students may model situations with a y-intercept not at 0, they are not expected to be fluent in writing equations that represent non-proportional

linear relationships. • Standard 6.EE.8 in which students are expected to write, represent, and interpret inequalities of x < c or x > c are addressed in the unit on extending the

number system using number lines. • Standard 6.EE.2 in which students are write, read, and evaluate expressions in which letters stand for numbers, will be extended in the next unit on

geometry. In that unit, students will use formulas to calculate perimeter, area, circumference, surface area, and volume of two and three-dimensional figures.

• Standard 6.EE.1 in which students are expected to write and evaluate numerical expressions involving whole number exponents, was addressed in the first unit on whole number unit at the beginning of the year.

• For the Common Core Investigations, teachers should explicitly model evaluating expressions for given values of a variables and solving one-step equations. End of investigation exercises and additional practice problems are good sources for practice.


• Identify quantitative variables in situations and distinguish between dependent and independent variables.

• Recognize situations where changes in variables are related to useful patterns

• Describe patterns of change shown in words, tables, and graphs of data.

• Construct tables and graphs to display relations among variables. • Observe relationships between two quantitative variables as shown in

a table, graph, or equation and describe how the relationship can be seen in each of the other forms of representation.

• Use algebraic symbols to write rules and equations relating variables. • Use tables, graphs, and equations to solve problems. • Write algebraic expressions based on real world problems and

evaluate them given values • Connect the concept of rate and ratio to tables, graphs, and equations • Create and solve simple equations (in the forms of x ± p = q or px = q)

using given output values. • Identify and generate equivalent expressions using properties of

operations, including the distribute property.

Prior knowledge expected (add standards) • Identifying and extending patterns in number and geometry • Gathering, organizing, displaying, and interpreting data in one- and

two-dimensional graphs and tables in order to look for patterns and relationships

• Developing operations algorithms for fractions, decimals, and percents.

• Compute unit rates in the form a/b where b ≠ 0. • Make tables of equivalent ratios and plot these points on a coordinate

grid. • Read coordinate graphs to find sets of values (coordinate points). • Evaluate arithmetic expressions using the order of operations.

Terms developed in this chapter: associative property, change, commutative property, coordinate graph, coordinate pair, dependent variable, distance/time/rate of speed, distributive property, equations/formula, expression, income/cost/profit, independent variable, inverse operations, plot, point, range of values, relationship, rule, scale, table, variable, x-axis, x-coordinate, y-axis, y-coordinate

Terms from prior chapters or experiences: Data, factors, line plot, order of operations, pattern, ratio, rate, rate table, unit rate


Learning Outcomes • T E R M

MA 2011 Framework



6.EE.6 6.EE.2a, 6.EE.2b

6.EE.2c

√ Use variables to represent numbers and write expressions solving a real world mathematical problem

√ Evaluate expressions at specific values of their variables √ https://betterlesson.com/lesson/285012/equations-expressions-lesson-03


Better Lessons

Expressions and Equations Lesson 03:

Writing Algebraic Expressions

Better Lessons


Writing Algebraic Expressions Lesson 2

6.EE.2,

6.EE.2a 6.EE.2c, 6.EE.4



6.EE.6 6.EE.2a, 6.EE.2b

6.EE.2c


√ Evaluate expressions at specific values of their variables


6.EE.6 6.EE.2a, 6.EE.2b

6.EE.2c




6.EE.6 6.EE.2a, 6.EE.2b

6.EE.2c




6.EE.6 6.EE.7 6.EE.2 6.EE.2c


√ Solve real-world and mathematical problems by writing and solving equations is the form x + p = q and px = q


Better Lessons


Evaluating Expressions


6.EE.7 √ Solve real-world and mathematical problems by writing and solving equations is the form x + p = q and px = q

1 Common Core End of Investigation 2 Exercises # 27-34, 38, 67-68, 71 and

other supplemental materials at teacher’s choosing

6.EE.3 6.EE.4

√ Apply the properties of operations to produce equivalent expressions (distributive property and combining like terms)

√ Identify when two expressions are equivalent


6.EE.9 √ Create a data table given a real world situation √ Analyze the relationship between two variables in a table

1 Variables & Patterns 1.1

6.EE.9 √ Create a coordinate graph given a data table √ Analyze the relationship between two variables in a graph


6.EE.9 √ Create a coordinate graph given a data table √ Analyze the relationship between two variables in a table and graph √ Describe possible patterns of change between points on a coordinate

graph of a non-linear relationship


6.EE.9 √ Create a data table given a coordinate graph √ Analyze the relationship between two variables in a table and graph


6.EE.9 √ Analyze the relationship between two variables in a table and graph √ Compare two data sets given in different forms (one graph and one table) √ Make predictions about points not seen on tables and graphs given the

pattern


6.EE.9 √ Use variables to represent two quantities in a real world problem that change

√ Create a coordinate graph given a data table √ Make predictions about points not seen on tables and graphs given the

pattern


6.EE.9 √ Identify the dependent and independent variables in a real world problem √ Represent the relationship between two variables in a real world problem

on a sketch of a coordinate graph √ Describe patterns seen in a graph and relate them to a real world story


6.EE.9 6.EE.6

√ Write an equation to represent one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the dependent variable.


√ Analyze the relationship between two variables in a table and graph and relate these to the equation


End of Unit 5: March 26, 2014 Unit #6: Geometry Primary Curricular Resource: Better Lessons


Estimated Instructional Time: 25 days Overarching Questions:

• Solve real world and mathematical problems involving area, surface

area and volume • How do you determine whether area or perimeter of a figure is

involved? • What attributes of a shape are important to measure? • What are you finding when measuring area and when measuring

perimeter? • How can you decompose and compose to find area?

Standards for Mathematical Practice Focus • In Covering and Surrounding Inv.1 Designing Bumper Cars finding

perimeter and area and relating it to cost of designs • Investigations 3,4 and 5 the concepts of measuring rectangles,

parallelograms and triangles and circles. • Using decomposing shapes and composing to other shapes to apply

formulas or strategies for measuring area • Plotting points /vertices of polygons and measuring horizontal and

vertical sides

Instructional Notes: • Teachers need to assess students prior knowledge of polygons. • Investigation 1 Covering and Surrounding CC 3,MD5-8*need to more than tiles

6.G.1 is covered in Inv 3 Measuring Triangles and Inv 4 Measuring Parallelograms Inv 5 Measuring irregular shapes and circles • 6.G.2 covered in CC Investigation 4 Measurement covers nets and prisms surface area and volume • 6.G.3 covered in CC Investigations 3 Integers and the Coordinate Plane note *Shapes and Designs ACE 39 • 6.G.4 CC Inv4 3D shapes using nets


• Use area and relate area to covering a figure • Use perimeter and relate perimeter to surrounding a figure • Analyze what is means to measure area and perimeter • Develop strategies for finding area of rectangular shapes and non-

rectangular shapes • Discover relationships between perimeter and area including each can

vary while the other stays fixed • Analyze how the area of a triangle and the area of a parallelogram are

related to the area of a rectangle • Develop formulas and procedures stated in words and symbols for

finding areas and perimeter of rectangles, parallelograms, triangles, and circles

• Develop techniques for estimating the area and perimeter of irregular shapes

• Recognize situations in which measuring area or perimeter will help answer practical questions

• Draw polygons in the coordinate plane when given coordinates for vertices

• Find volume of a right triangular prism by packing it with unit cubes • Apply formulas V=lwh V=bh to solve real world problems • Represent three dimensional figures using nets made of rectangle and

Prior knowledge expected (add standards) • Recognition of polygons angles sum of angle measures of polygons,

side lengths • Understand effects of side lengths on shapes of polygons • Properties and attributes of polygons • Effects of side lengths on shape of polygon


triangles • Use nets of three dimensional figures to solve real –world problems

involving surface area Terms developed in this chapter, area, base, circumference, diameter, height, length, perimeter, pi, radius, width , opposite, integer, coordinate plane, quadrants, ordered pairs, origin, net, prism, surface area, volume

Terms from prior chapters or experiences: diagonal, isosceles, parallelogram, perpendicular, rectangle, right angle, right triangle, scalene, trapezoid, triangle

Learning Outcomes 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.MA.1a Use the relationships among radius, diameter, and center of a circle to find its circumference and area. 6.G.MA.1b 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.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.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. MA.6.RP.3e Solve problems that relate the mass of an object to its volume T E R M

MA 2011 Framework



6.G.1 • An introduction to area and perimeter and the study of the relationship between them through problems involving bumper car designs

1 Covering and Surrounding

1.1 6.G.1 • Demonstrate shapes with the same area need not have the same

perimeter 1 Covering and

Surrounding 1.2

6.G.1 • Develop formulas for area and perimeter of rectangles


1.3 6.G.1 • Investigation of how perimeter can vary 1 Covering and

Surrounding 2.1

6.G.1 • Stretching the perimeter when a constant area Develop a conceptual understanding of area and perimeter Apply an understanding of area and perimeter of nonrectangular figures composing and decomposing http://betterlesson.com/lesson/236787/lesson-02-area-of-a-right-triangle


2.2 Better Lessons Lesson 2 Area of a Right Triangle


http://betterlesson.com/lesson/236788/lesson-03-area-of-triangles

Better Lessons Lesson 3 Area of a triangle

6.G.1 • Identify the maximum and minimum possible areas for a rectangle with a constant perimeter


2.3 6.G.1 • Estimate the area and perimeter of triangles on a grid 1 Covering and

Surrounding 3.1

6.G.1

• Develop the understanding of base and height of a triangle • Change orientation of triangles to better understand base and height and

how they are related to area •

1

Covering and Surrounding

3.2

6.G.1

• Study triangles with common base to develop formula for area of a triangle


3.3 6.G.1 • Draw triangles satisfying given constraints 1 Covering and

Surrounding 3.4

6.G.1

• Estimate area and perimeter of parallelograms on grid paper 1 Covering and Surrounding

4.1 6.G.1 • Relating parallelograms to triangles students develop a rule for finding

area of a parallelogram by answering the question how is the area of a triangle related to the area of a parallelogram?

http://betterlesson.com/lesson/236786/lesson-01-area-of-parallelogram

•


4.2 Better Lessons Lesson 1 Area of a Parallelogram

6.G.1

• Designing parallelograms under constraints • Parks Hotels and Quilts apply techniques for finding perimeter and area of

parallelograms to real world problems • http://betterlesson.com/lesson/236789/lesson-05-practice • http://betterlesson.com/lesson/236598/lesson-06-quiz-more-practice • Develop techniques for estimating areas and perimeter of irregular figures

2 Covering and Surrounding 4.3 and 4.4

Better Lessons Lesson 4 Area of a Trapezoid Better Lessons Lesson 5 Practice Area Better Lessons Lesson 6 Quiz and More Practice 5.1

6.G.1.a • Discover the relation ship between a diameter and the circumference discuss terms diameter, radius area and circumference



6.G.1.a • Discover the relation ship between a diameter and the circumference discuss terms diameter, radius area and circumference http://betterlesson.com/lesson/236605/lesson-07-circumference-1-of-2


5.2 Better Lesson 7

Circumference 1-of-2 6.G.1.a • Develop techniques for estimating the area of a circle use ideas about

area and perimeter to solve real world problems http://betterlesson.com/lesson/236614/lesson-08-circumference-2-of-2

•


5.3 Better Lesson 8

Circumference 2-of-2

6.G.1.a • Discover that it takes slightly more than three radius square to equal the area of a circle and use the discovery to develop a formula for the area of a circle

http://betterlesson.com/lesson/236618/lesson-09-circle-area-1-of-2 http://betterlesson.com/lesson/236636/lesson-10-circle-area-2-of-2


5.4 Better Lesson 9 Circle

area 1-of-2 Better Lesson 10 circle

area 2-of-2 6.G.4 • Represent three dimensional figures using nets made up of rectangles and

triangles to find the surface area and apply the concepts to real world problems (ACE exercises 1-15 also should be used for real world

• problems) • Develop a definition of surface area Additional practice 1- Skill 1-4 Check

Up 1-2 • http://betterlesson.com/lesson/236646/lesson-11-surface-area-1-of-2

http://betterlesson.com/lesson/236658/lesson-12-surface-area-2-of-2 •


Better Lesson 11Surface

Area 1-of-2 Better Lesson Surface 2-

of 2

6.G.2 • Apply the volume formulas V=lwh V=bh to solve real world problems Checkup 3-4Skill 7-10 Additional practice 2 ACE 15-22

• http://betterlesson.com/lesson/305451/lesson-14-volume http://betterlesson.com/lesson/236663/lesson-13-review-for-test

•

2 Common Core Investigation 4.2 Better Lesson 14

Volume Better Lesson Review

for Test 6.G.3 http://betterlesson.com/lesson/305452/lesson-15-graphing-polygons

1 Common Core Investigation 3.5 Better Lesson 15

Graphing Polygons

MA.6.RP.3d Solve problems that relate the mass of an object to its volume Scope of standard: • Connect the relationship between volume and mass using the formula mass = volume × density.

TBD

End of Unit Assessment 1 End of Unit 6: May 7, 2014


Statistics Unit #7: Statistics Primary Curricular Resource: Data About Us, Common Core Investigations, and Better Lessons Estimated Instructional Time: 20 days Overarching Questions:

• What are the different ways we can represent a set of data? • What is the difference between categorical and numerical data? • What are the ways to summarize a set of numerical data? How do we

find the “typical number” in a data set? • How is the data spread out or shaped, and how can we see that in

different data representations? • How are the measures of center affected by the distribution of the

data? • What does the shape or distribution of the data tell us about the thing

being measured?

Standards for Mathematical Practice Focus • When calculating measures of center and variability, calculation errors

are common. Students need to be taught to ask the question, “Does my answer make sense?” given the data set (SMP #1)

• Students will be asked to make conjectures about real world phenomenon and make connections given patterns in a data set. (SMP #3)

• Students will be given multiple models to model data sets, including line plots, stem and leaf plots, histograms, and circle graphs. Students will need to choose which model best fits the data they are given (SMP #4)

• In this unit, students will need to attend to precision (SMP #6) when they are calculating measures of center and variability. Labels are critical as students talk about different types of data and different measures.


• Standards 6.SP.1, 6.SP.2, and 6.SP.3 are recurring themes throughout the entire unit. Teacher should emphasize what types of questions are statistical questions, the ideas of center, spread, and overall shape, and that measures of center summarize a data set with a single number and measures of variability describe how values vary with a single number.

• Standard 6.SP.5.b is woven throughout this unit. Every time a display of data is interpreted and analyzed, this standard is addressed. • When teaching the Common Core Investigations, be sure to use the exercises at the end of the unit for additional practice. • Circle graphs are taught in Bits and Pieces III Investigation 5.


• Understand and use the process of data investigation; posing questions, collecting and analyzing data distributions, and making interpretations to answer questions.

• Represent distributions of data using line plots, bar graphs, stem-and-leaf plots, and coordinate graphs.

Prior knowledge expected (add standards) • Making and using line plots to display a data set (in fractions of a unit)

(Grade 5) • Analyzing and classifying counting numbers • Representing the number of proper factors of a counting number • Graphing rectangle lengths and widths with constant perimeter or


• Compute the mean, median, mode, and range of the data. • Distinguish between categorical data and numerical data and identify

which graphs and statistics may be used to represent each kind of data.

• Make informed decisions about which graphs and which measures of center (mean, median, or mode) and range may be used to describe a distribution of data.

• Develop strategies for comparing distributions of data.

constant area • Ordering numbers from least to greatest, counting • Comparing, counting, and ordering numbers, spreads of measures • Using arithmetic operations; learning the meaning of rational numbers.

Learning Outcomes Learning Outcomes: 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 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.

6.SP.MA.4a 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 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.

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.

• T E R M

MA 2011 Framework



6.SP.1 6.SP.4

• Display numerical data in plots (line plots, bar graphs, and histograms) 1 Data About Us 1.1

6.SP.1 6.SP.5.c 6.SP.3

• Give measures of center (mode, median) and variability (range) • Create data sets given measures of center and variability

1 Data About Us 1.2

6.SP.1 • Give measures of center (median) of a numerical data set 1 Data About Us


6.SP.5.c • Create data sets to change the median 1.3 6.SP.1

6.SP.5.c • Give measures of center (mode, median) and variability (range) • Distinguish between numerical and categorical data

1 Data About Us 1.4

6.SP.1 6.SP.4

6.SP.5.c

• Display numerical data in line plots (stem and leaf plots) • Give measures of center (median, mode) and variability (range)

1 Data About Us 2.1

6.SP.1 6.SP.4

6.SP.5.c 6.SP.5.d

• Compare data sets using measures of center • Relate choice of measure of center and variability to the shape of the data

distribution • Identify outliers in a data set and their effect on measures of center

1 Data About Us 2.2

6.SP.1 6.SP.4

• Display numerical data in line plots (dot plots) • Interpret data and find patterns in dot plots

1 Data About Us 2.3

6.SP.1 6.SP.4

• Display numerical data in line plots (dot plots) • Interpret data and find patterns in dot plots

1 Data About Us 2.4

6.SP.1 6.SP.5b 6.SP.5.c

• Give measures of center (mean) of a numerical data set 1 Data About Us 3.1

6.SP.5b 6.SP.5.c 6.SP.4

• Give measures of center (mean) of a numerical data set • Create multiple data sets with the same mean • Display numerical data in line plots

1 Data About Us 3.2

6.SP.1 6.SP.2 6.SP.4

6.SP.5.a 6.SP.5b 6.SP.5.c

• Give measures of center (mean, median, mode) of a numerical data set • Determine the effect that new numbers have on measures of center and

variability in a data set • Identify outliers in a data set and their effect on measures of center

2 Data About Us 3.3

6.SP.1 6.SP.4

6.SP.5b 6.SP.5.c 6.SP.5.d 6.SP.3

• Display numerical data in plots (box and whisker plots) • Give measures of center (mean, median, mode) and variability (range, 1st

quartile, 3rd quartile and interquartile range) of a numerical data set • Identify outliers in a data set and their effect on measures of variability • Determine the lower and upper quartiles for a set of data, and us those to

calculate the interquartile range • Use interquartile range as a measure of variability that is less impacted by

outliers • Understand how to use quartiles to describe the data set in fractions of the

set, including ¼, ½, and ¾ .


Better Lessons 21st Century Lessons

“Interquartile Range

6.SP.1 6.SP.4

6.SP.5b 6.SP.5.c

• Display numerical data in plots (box and whisker plots) • Give measures of center and variability given a box and whisker plot • Better Lessons: Statistics Lesson on Box Plots [Insert link here!]


6.SP.1 6.SP.4

6.SP.5b

• Display numerical data in plots (histogram) • Create frequency tables to match histograms

1

Common Core Investigation 5.2


6.SP.5c 6.SP.1

6.SP.3 6.SP.5c 6.SP.5.c

• Give measures of variability (mean absolute deviation) of a numerical data set

• Recognize that a measure of variation describes how a data set’s values vary with a single number


End of Unit 7: June 9, 2014 1

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