BOSTON PUBLIC SCHOOLS Course 437: Middle School Pre-Algebra Scope and Sequence Guide 2012-2013 Background The 2012-2013 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., in particular the learning trajectories for grades 5 through 8. The new state frameworks incorporate the Common Core State Standards as adopted by the Massachusetts Department of Elementary and Secondary Education (ESE). 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.

Accordingly, new curricular resources have been acquired by Boston Public Schools to support instruction in these areas, as well as transition to the 2011 Massachusetts Frameworks:

• 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 2012-2013 academic year, professional development opportunities will be offered to continue the study of our transition to the 2011 Massachusetts Frameworks. Through this process, schools are urged to regularly consult the Middle School Math page of MyBPS and MyLearningPlan for curriculum and instruction and professional development announcements. A note on terminology: we often hear the “Common Core State Standards” referenced in the media and in Federal statements on education policy. Massachusetts adopted the Common Core State Standards, and have incorporated them into the Massachusetts Frameworks. Accordingly, the new standards are correctly referred to as the “2011 Massachusetts Frameworks.”

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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. 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.

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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. In addition, any highlighted standards within this document are listed on the MCAS Assessable Standards lists for spring 2013 and 2014. See the DESE website for more details: http://www.doe.mass.edu/mcas/transition/2013-14g10math.html?section=list. Essential Question(s) for the course: What does it mean for two quantities to be related proportionally? How are properties of numbers and operations applied to different classes of numbers (whole numbers, integers, rational numbers)? How are expressions and linear equations used to communicate mathematically? How are area, surface area, and volume used to solve problems with 2-dimensional and 3-dimensional shapes? How are probability models used to investigate chance processes? Why is the ability to draw inferences about populations based on representative sampling important? What will students have learned by the end of this course? In Grade 7, instruction focuses on four critical areas: (1) developing understanding of an applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

• Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratio and proportion to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.

• Students develop a unified understanding of number, recognizing fractions, decimals, and percents as different representations of rational numbers. Students extend operations to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction and multiplication and division. Applying these properties and viewing negative numbers in terms of everyday contexts, students explain and interpret rules for operations with negative numbers. They use the

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arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

• Students extend their work from grade 6, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. To prepare for Grade 8 work on congruence and similarity, they reason about relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects.

• Students build on previous work with single data distributions to compare two data distributions and address questions about differences between populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences.

According to the MA Curriculum Frameworks 2011, “in grade 7, instructional time should focus on 1.) developing understanding of and applying proportional relationships; 2.) developing understanding of operations with rational numbers and working with expressions and linear equations; 3.) solving problems involving scale drawings, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and 4.) drawing inferences about populations based on samples.” The transition from additive to proportional thinking is a key stage in a student’s mathematical development. Real-world applications allow students to identify changing attributes in a context, to use ratios as multiplicative comparisons of two quantities, and to reason about the effect of change in one or both quantities. Sense making is key to the development of symbolic representation and algorithms for solving problems with ratios and proportions. Mathematical communication is based on the use of numbers and symbols to represent and to act on quantities. Knowledgeable use of operations (addition, subtraction, multiplication, and division) on all rational numbers in both real-world and mathematical problems is an expected goal. Formalizing the use of variables to represent unknown or changing quantities in expressions and equations further extends mathematical communication and reasoning. Students recognize constant rate of change problems and represent them with tables, graphs, and equations. Students use properties of numbers and operations to solve equations. The course emphasis on proportional relationships, operations with rational numbers, equivalent expressions, and solving equations continues in the study of geometry, probability, and statistics. Explicit connections to these big ideas will strengthen student learning and emphasize discipline coherence. Students explore the relationship between scale, angle measure, area, surface area, and volume with 2-dimensional and 3-dimensional shapes. They investigate chance processes and develop, use, and evaluate probability models. They use random sampling to draw inferences about a population, and develop strategies to compare populations.

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What must students have learned by the middle of the academic year? By mid-year, students should have worked extensively with real-world and mathematical problems to reason proportionally and solve problems. They should compute fluently with rational numbers. They should recognize change in a linear pattern, identify the variables, and represent the relationship with tables, graphs, and equations. They should be able to recognize and form equivalent algebraic expressions and equations, and to solve equations.

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Unit Title: Similarity & Congruence Primary Curricular Resource: Stretching & Shrinking Estimated Instructional Time: 16 days Overarching Questions: 1. What is the same and what is different about two similar figures? 2. What determines whether two shapes are similar? 3. When figures are similar, how are the lengths, areas, perimeters, angles, and scale factors related? 4. How can I use information about similar figures to solve problems in real life?

Instructional Notes: • Fluency marker: Draw, construct and describe geometrical figures and describe the relationships between them. • The 2011 Massachusetts Frameworks establish a strong focus on similarity. • Instruction in this unit should maintain a strong emphasis on similarity, congruence, and solving problems involving scale drawings. • Since Variables & Patterns is now being taught in grade 6, basic coordinate geometry (Quadrant I) needs to be reviewed.

Concepts developed in this unit • Identify similar figures by comparing corresponding parts. • Use scale factors and ratios to describe and predict relationships among the side lengths, angle measures, perimeters and areas of similar

figures. • Reproducing a scale drawing at a different scale • Solve problems involving scale drawings of geometric figures

Standards for Mathematical Practice #3 Construct viable arguments and critique the reasoning of others. #4 Model with mathematics. #7 Look for and make use of structure.

Prior knowledge expected • Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. • Interpret multiplication as scaling. • Write, read and evaluate expressions in which letters stand for numbers. • Find the area and perimeter of triangles and quadrilaterals. • Draw polygons in the coordinate plane.

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Terms developed in this unit coordinate pair, corresponding, equivalent ratios, image, nested triangles, ordered pair, proportion, proportional relationship, relationship, scale factor, scaling, similar, table, x-coordinate, y-coordinate

Terms from prior units or experience angle, area, base, coordinate graph, degree, dimensions, equivalent fractions, factor, height, length, perimeter, quadrilateral, ratio, width, x-axis, y-axis

Learning Outcomes

7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For example, if a person walks ! mile in each " hour, compute the unit rate as the complex fraction !/" miles per hour, equivalently 2 miles per hour.

7.G.1 Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an unknown angle in a figure.

TERM

MA 2011 Framework

Citation

After completing each section, students will be able to:

Primary Curricular

Resource

Days

1

7.G.1

° Use algebraic rules to produce similar figures on a coordinate grid. ° Focus student attention on both lengths and angles as criteria for similarity. ° Contrast similar figures with non-similar figures. ° Understand the role multiplication plays in similarity relationships. ° Understand the effect on the image if a number is added to the x and y

coordinates. ° Develop more formal ideas of the meaning of similarity, including the

vocabulary of scale factor. ° Understand the relationships of angles, side lengths, perimeters, and areas of

similar polygons.

STRETCHING & SHRINKING Inv. 2 Similar Figures 2.1 Making Similar Figures 2.3 Scale Factors

° 4 days

1

7.G.1 7.G.5

° Construct similar quadrilateral and triangles from smaller, congruent figures. ° Connect the ratio of areas of 2 similar figures to scale factors. ° Generalize relationship between scale factor and area, and to scale factors

less than 1. ° Subdivide a figure into smaller, similar figures. ° Use scale factors to make similar shapes. ° Find missing measures in similar figures using scale factor.

STRETCHING & SHRINKING Inv. 3 Similar Polygons 3.3 Scale Factors and Similar Shapes

1 day

8

1

7.RP.1 7.G.1

° Use ratios of corresponding sides within a figure to determine whether two figures are similar.

° Use ratios to identify similar triangles. ° Use ratios of corresponding sides or scale factors to find missing lengths in

similar figures.

STRETCHING & SHRINKING Inv. 4 Similarity and Ratios 4.1 Ratios within Similar Parallelograms 4.2 Ratios within Similar Triangles 4.3 Using Similarity to Find Measurements

°

4 days

1

7.RP.1 7.G.1

° Apply knowledge of similar triangles and similar quadrilaterals. ° Develop a technique for indirect measurement. ° Practice measuring lengths to solve problems.

STRETCHING & SHRINKING Inv. 5 Using Similar Triangles and Rectangles 5.1: Using Shadows to Find Heights 5.3: Finding Lengths within Similar Triangles

3 days

End Unit Similarity & Congruence 12 Days of Core Instruction

4 Discretionary/Assessment Days Unit Title: Ratios & Proportional Relationships Primary Curricular Resources: Bits & Pieces III; Comparing & Scaling Instructional Time: 21 days Overarching Questions: 1. How can a comparison be expressed in different but useful ways? 2. What type of comparison will give the most useful information? 3. How can given comparison data be used to make predictions about unknown quantities?

Instructional Notes: • Fluency marker: Analyze proportional relationships and use them to solve real-world and mathematical problems. • Fluency marker: Solve real-life and mathematical problems using numerical and algebraic expressions and equations. • The 2011 Massachusetts Frameworks establish a strong focus on ratios and proportional relationships. • Instruction in this unit should maintain a strong emphasis on proportional reasoning. • Investigation 4 from Bits & Pieces III is an essential review in order to address the details of standards 7.EE.2 and 7.RP.3.

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Concepts developed in this unit • Analyze proportional relationships to solve real-world and mathematical problems

Standards for Mathematical Practice Focus #1 Make sense of problems and persevere in solving them. #3 Construct viable arguments and critique the reasoning of others. #8 Look for and express regularity in repeated reasoning.

Prior knowledge expected • Understand the concept of ratio and the concept of unit rate as it relates to ratios. • Use ratio and rate reasoning to solve real-world problems • Write, read and evaluate expressions in which letters stand for numbers • Know that Pi is the ratio of diameter to circumference of a circle.

Academic language developed in this unit coefficient, percent decrease, percent increase, mark ups, mark downs, rate, rate table, scaling (as a method of comparison), scale drawing, tax, unit rate

Academic language from prior units or experience data, decimal, denominator, equivalent fractions, equivalent ratios, fraction, improper fraction, mixed number, numerator, percent, proportion, proportional relationship, ratio, relationship, scale, scale factor, scaling, similar, survey, table

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Learning Outcomes

7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For example, if a person walks ! mile in each " hour, compute the unit rate as the complex fraction !/" miles per hour, equivalently 2 miles per hour.

7.RP.2 Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table, or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r ) where r is the unit rate.

7.RP.3 Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns,

gratuities and commissions, fees, percent increase and decrease, percent error. 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it

are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

7.G.1 Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

TERM

MA 2011 Framework

Citation

After completing each section, students will be able to:

Primary Curricular

Resource

Days

1 7.EE.2

7.RP.3

° Using percent to solve problems involving tax, discount, and tip. BITS & PIECES III Inv. 4 Using Percents 4.1 Determining Tax

4.3 Using Discounts

2 days

11

1

7.RP.2 7.RP.3

° Choose an appropriate method to make comparisons among quantities using ratios, percents, fractions, rates, or differences.

COMPARING & SCALING Inv. 1 Making Comparisons

1.1 1.1 Exploring Ratios and Rates 1.2 Analyzing Comparison Statements

1.3 Writing Comparison Statements

3 days

1

7.RP.2

7.RP.3

° Find equivalent forms of given ratios and rates to scale comparisons up and down.

COMPARING & SCALING Inv. 2 Comparing Ratios, Fractions, Percents 2.1 Developing Comparison Strategies 2.2 More Comparison Strategies 2.3 Scaling Ratios

4 days

1

7.RP.1

7.RP.2

7.RP.2.b 7.RP.2.c 7.RP.2.d 7.RP.3

° Find and interpret unit rates, and use them to make comparisons ° Use unit rates to write an equation to represent a pattern in a table of data ° Introduce “steady” or “average” rate of progress. ° Understand that a rate is a comparison of two quantities and requires

appropriate labeling.

COMPARING & SCALING Inv. 3 Comparing and Scaling Rates 3.1 Making and Using a Rate Table 3.2 Finding Rates 3.3 Unit Rates and Equations 3.4 Two Different Rates

5 days

1

7.RP.2

7.RP.2.a

7.RP.2.b

7.RP.3

7.G.1

° Apply proportional reasoning to solve problems ° Set up and solve proportions

COMPARING & SCALING Inv. 4 Making Sense of Proportions 4.1 Setting Up and Solving Proportions 4.3 Developing Strategies for Solving Proportions

3 days

12

End Comparing and Scaling 17 Days of Core Instruction

4 days discretionary/assessment

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Unit Title: Operations with Rational Numbers Primary Curricular Resource: Accentuate the Negative Instructional Time: 17 days Overarching Questions:

1. How do you decide if sums, differences, products, and quotients of two numbers are positive, negative, or zero? 2. Which operations on rational numbers are commutative? 3. What is the order of operations and why is it important? 4. What does it mean to say that multiplication distributes over addition?

Instructional Notes: • Fluency Marker: Solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and

decimals). • Fluency Marker: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational

numbers. • Fluency Marker: Use properties of operations to generate equivalent fractions. • The 2011 Massachusetts Frameworks establish a strong focus on operations with positive and negative numbers. • Instruction in this unit should maintain a strong emphasis on rational numbers in any form (whole numbers, fractions, and decimals). • Investigation 1 of Accentuate the Negative is taught in grade 6.

Concepts developed in this unit • Extend rules for addition, subtraction, multiplication, and division to all rational numbers. • Understand negative numbers in everyday contexts. • Explain and interpret rules for adding, subtracting, multiplying, and dividing with negative numbers.

Standards for Mathematical Practice Focus #4 Model with mathematics. #8 Look for an express regularity in repeated reasoning.

Prior knowledge expected • Fluently add, subtract, multiply and divide multi-digit decimals using standard algorithms for each operation. • Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. • Understand a rational number as a point on a number line. • Understand ordering and absolute value of rational numbers. • Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.

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Academic language developed in this unit absolute value, additive inverse, algorithm, associative property, commutative property, distributive property, estimate, evaluate, expression, integers, inverse operations, negative number, non-zero divisor, number line, number sentence, opposites, order of operations, positive number, rational numbers, repeating decimal, signed number, substitution, terminating decimal

Academic language from prior units or experiences algorithm, decimal, exponent, fact family, mixed number, rational number, reciprocal, repeating decimal, terminating decimal, x-axis, y-axis

Learning Outcomes

7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two

constituents are oppositely charged. b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive

or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the

properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or

eventually repeats.

7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers,

fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9# inches long in the center of a door that is 27! inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

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TERM

MA 2011 Framework

CItation

After completing each section, students will be able to:

Primary Curricular

Resource

Days

1

7.NS.1

7.NS.1.a

7.NS.1.b

7.NS.1.c

7.NS.1.d

7.NS.3 7.EE.3

° Model addition and subtraction of integers using distance/directions on a number line and a chip model.

° Understand absolute value as distance from zero on a number line. ° Develop and use algorithms for adding and subtracting integers. ° Demonstrate that the Commutative Property holds for addition but not

subtraction of rational numbers. ° Solve real-world and mathematical problems involving addition and

subtraction of integers. ° Solve equations with missing values by using related fact families. ° Extend graphing with positive and negative coordinates to all four quadrants.

ACCENTUATE THE NEGATIVE Inv. 2 Adding and Subtracting Integers 2.1 Introducing Addition of Integers 2.2 Introducing Subtraction of Integers 2.3 Addition and Subtraction Relationships 2.4 Fact Families 2.5 Coordinate Graphing

6 days

1

7.NS.2

7.NS.2.a

7.NS.2.b

7.NS.2.c

7.NS.2.d

7.NS.3

7.EE.3

° Model the relationship between repeated addition and multiplication of integers using a number line.

° Develop and use algorithms for multiplying and dividing integers. ° Explore division of integers using fact families. ° Solve real-world and mathematical problems involving multiplication and

division of integers.

ACCENTUATE THE NEGATIVE Inv. 3 Multiplying and Dividing Integers 3.1 Introducing Multiplication of Integers 3.3 Introducing Division of Integers 3.4 Multiplying Integers with the Product Game

3 days

2

7.NS.1

7.NS.1.d

7.NS.2

7.NS.2.c

7.EE.3

° Explore the use of the order of operations for computation. ° Model the Distributive Property with areas of rectangles ° Develop and use the Distributive Property of multiplication over addition and

subtraction. ° Apply the Distributive Property to solve real-world and mathematical

problems.

ACCENTUATE THE NEGATIVE Inv. 4 Properties of Operations 4.1 Order of Operations 4.2 Distributing Operations 4.3 Distributive Property and Subtraction

5 days

End Unit Operations with Rational Numbers 14 Days of Core Instruction

3 days discretionary/assessment days

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Unit Title: Proportional Reasoning represented in Linear Relationships Primary Curricular Resources: Moving Straight Ahead, Common Core Investigations Instructional Time: 35 days Overarching Questions:

1. What patterns in the problem suggest a linear relationship? 2. How can a linear relationship be represented in a problem, with a table, in a graph, or with an equation? 3. How do changes in one variable affect changes in a related variable; and how can these changes be captured in a table, graph or

equation? 4. How are tables, graphs and equations interpreted to answer questions about linear relationships? 5. How can you find the slope of a line using a table, graph or equation? 6. How does finding the slope compare to finding the rate of change between two variables in a linear relationship?

Instructional Notes: • Fluency marker: Analyze proportional relationships and use them to solve real-world and mathematical problems. • Fluency marker: Solve real-life and mathematical problems using numerical and algebraic expressions and equations. • The 2011 Massachusetts Frameworks establish a strong focus on ratio and proportional reasoning through linear relationships

represented by the use of expressions, equations, tables, and graphs. This unit on linear relationships explicitly connects the constant of proportionality (unit rate) to the different representations.

• Instruction in this unit should maintain a strong emphasis on linear relationships. • A review of graphing relationships is essential to this unit. • Note that while slope is now encompassed within the 8th grade standards, the material in this unit provides a deeper understanding of

direct proportionality, which is a 7th grade standard. The unit rate is shown on the graph of a proportional relationship by the change between points (0,0) and (1, r), where r is the unit rate.

• Replace Moving Straight Ahead Inv. 3.5 with Variables & Patterns Inv. 2 ACE problem 6, page 38, to address the objective of applying the point of intersection of two equations in a real-world context.

• Common Core Investigation 1 will be taught after Moving Straight Ahead. Teachers need to make explicit connections between proportionality and writing equations.

Concepts developed in this unit • Recognize situations in which two or more variables have a linear relationship • Construct tables, graphs and equations that express linear relationships • Understand the connections between linear equations and the patterns in the tables and graphs of those equations; rate of change; slope;

and y-intercept • Solve linear equations • Use tables, graphs, and equations to answer questions about linear relationships

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Standards for Mathematical Practice Focus #1 Make sense of problems and persevere in solving them. #4 Model with mathematics. #5 Use appropriate tools strategically. #7 Look for and make use of structure.

Prior knowledge expected • Write, read, and evaluate expressions in which letters stand for numbers. • Evaluate expressions at specific values of their variables. • Apply properties of operations to generate equivalent expressions. • Identify when two expressions are equivalent. • Use substitution to determine whether a given number in a specified set makes an equation or inequality true. • Use variables to represent numbers and write expressions when solving real-world or mathematical problems.

Academic language developed in this unit constant term, dependant variable, equation, independent variable, linear functions, linear relationship, origin, pattern, point of intersection, quadrants, rise, rule, run, scale, slope, solution, substitution, x-intercept, y-intercept

Academic language from prior units or experience coordinate pair, coordinate graph, dependent variable, equation, graph, independent variable, origin, pattern of change, rate, ratio, rule, scale, table, variable

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Learning Outcomes 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different

units. For example, if a person walks ! mile in each " hour, compute the unit rate as the complex fraction !/" miles per hour, equivalently 2 miles per hour.

7.RP.2 Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table, or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r ) where r is the unit rate.

7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers,

fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9# inches long in the center of a door that is 27! inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve

problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x ÷ q) = r, where p, q, and r are specific rational numbers. Solve

equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

TERM

MA 2011

Framework Citation

After completing each section, students will be able to:

Primary Curricular

Resource

Days

19

2

7.EE.3

7.EE.4

7.EE.4.a

7.RP.1

° Recognize problem situations in which two or more variables have a linear relationship to each other

° Construct tables, graphs, and symbolic equations that express linear relationships

° Translate information about linear relations given in a table, a graph, or an equation to one of the other forms

MOVING STRAIGHT AHEAD Inv. 1 Walking Rates 1.1 Finding and Using Rates 1.2 Linear Relationships in Tables, Graphs, and Equations 1.3 Using Linear Relationships

1.4 Recognizing Linear Relationships

5 days

2

7.EE.3

7.EE.4

7.EE.4.a

7.EE.4.b

7.RP.1

7.RP.2.a 7.RP.2.b 7.RP.2.c 7.RP.2.d

° Understand the connections between linear equations and the patterns in the tables and graphs of those equations: rate of change, slope, and y-intercept

MOVING STRAIGHT AHEAD Inv. 2 Exploring Linear Relationships with Graphs and Tables 2.1 Finding the Point of Intersection 2.2 Using Tables, Graphs, and Equations 2.3 Comparing Equations

2.4 Comparing Tables, Graphs, and Equations

5 days

2

7.EE.1

7.EE.3

7.EE.4

7.EE.4.a

° Apply the point of intersection of two equations in a real-world context ° Solve linear equations ° Solve problems and make decisions about linear relationships using

information given in tables, graphs, and symbolic expressions ° Use tables, graphs, and equations of linear relations to answer questions

MOVING STRAIGHT AHEAD Inv. 3 Solving Equations 3.1 Solving Equations with Tables and Graphs 3.2 Exploring Equality 3.3 Writing Equations 3.4 Solving Linear Equations

VARIABLES & PATTERNS

Inv. 2 ACE Problem 6, page 38

5 days

20

2

7.EE.1

7.EE.3

7.EE.4

7.EE.4.a

° Understand the slope as the ratio of vertical change to horizontal change ° Use information in tables, graphs, and equations to find the slope ° Recognize when the rate of change is equivalent from information in tables,

graphs, and equations

MOVING STRAIGHT AHEAD Inv. 4 Exploring Slope 4.1 Using Rise and Run 4.2 Finding the Slope of a Line 4.3 Exploring Patterns with Lines

4 days

2

7.RP.1

7.RP.2.a

7.RP.2.d

° Compute unit rates associated with ratios of fractions ° Decide whether two quantities are in a proportional relationship ° Explain what any point (x,y), including (0,0) and (1,r) where r is a unit rate, on

a graph of a proportional relationship means in terms of the situation

COMMON CORE INVESTIGATIONS Inv. 1 Graphing Proportions Problem 1.1 Problem 1.2 Problem 1.3

4 days

2

7.EE.1

7.EE.2

° Expand algebraic expressions by applying the properties of operations ° Use equivalent expressions in a problem context to show how quantities in

the problem are related

COMMON CORE INVESTIGATIONS Inv. 2 Equivalent Expressions Problem 2.1 Problem 2.2

3 days

2

7.EE.4.b ° Solve real world problems leading to one- and two-step inequalities ° Graph the solutions to one- and two-step inequalities and interpret the

solution set in the context of the problem

COMMON CORE INVESTIGATIONS Inv. 3 Inequalities Problem 3.1 Problem 3.2 Problem 3.3

4 days

End Unit Linear Relationships 30 days of Core Instruction

5 Discretionary/Assessment Days

21

Unit Title: Two-dimensional and Three-dimensional Geometry Primary Curricular Resources: Filling & Wrapping, Common Core Investigations Instructional Time: 32 days Overarching Questions: 1. What is the difference between surface area and volume? 2. Is it possible for 3-dimensional shapes to have the same volume, but different surface area? 3. How are the volumes of cylinders, cones, and spheres related? 4. What are some strategies in finding surface area and volume?

Instructional Notes: • Fluency marker: Solve real-life and mathematical problems involving angle measure, area, surface area and volume. • The 2011 Massachusetts Frameworks establish a strong focus on two- and three- dimensional shapes to solve problems on surface area

and volume. • Instruction in this unit should maintain a strong emphasis on surface area and volume. • During Inv. 5, teachers should make explicit connections between proportionality and scaling of boxes. • Surface area and volume of cylinders are needed for complete understanding of Investigation 3; however, surface area and volume of

cylinders are not specifically noted in CCSS for grade 7. This is a grade 8 standard 8.G.9.

Concepts developed in this unit • Relate 2-dimensional figures to 3-dimensional figures by examining cross-sections. • Solve real-world and mathematical problems involving area, surface area and volume of two and three-dimensional figures.

Standards for Mathematical Practice Focus #1 Make sense of problems and persevere in solving them. #4 Model with mathematics. #5 Use appropriate tools strategically. #6 Attend to precision.

Prior knowledge expected • Find the area of triangles, quadrilaterals, and other polygons by composing into rectangles or decomposing into triangles and other

shapes. • Understand radius, diameter and center of a circle and use them to find the circumference and area. • Find the volume of a right rectangular prism, including those with fractional edge lengths. • Represent 3-dimensional figures using nets made up of rectangles and triangles; and use the nets to find the surface area of these figures.

22

Academic language developed in this unit adjacent angle, area, base, complementary angles, cone, cube, cylinder, edge, formula, face, geometric figure, height, intersecting lines, net, polygon, prism, quadrilateral, rectangular prism, right prism, right triangle, sphere, supplementary angles, surface area, triangle, unit cube, vertical angle, volume

Academic language from prior units or experience acute angle, angle, circle, circumference, diagonal, diameter, dimensions, exterior angle, interior angle, obtuse angle, parallel lines, parallelogram, perpendicular lines, pi, radius, right angle, transversal, vertex

Learning Outcomes

7.G.1 Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and

right rectangular pyramids. 7.G.4 Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between

the circumference and area of a circle. 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve

simple equations for an unknown angle in a figure. 7.G.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of

triangles, quadrilaterals, polygons, cubes, and right prisms. 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different

units. For example, if a person walks ! mile in each " hour, compute the unit rate as the complex fraction !/" miles per hour, equivalently 2 miles per hour.

TERM

MA 2011

Frameworks

After completing each section, students will be able to:

Primary Curricular

Resource

Days

23

3

7.G.6 ° Explain how to find the total area of all the faces of a rectangular box. ° Explain how to find the number of identical cubes it will take to fill a

rectangular box. ° Compare several different nets for a given box.

FILLING & WRAPPING Inv. 1 Building Boxes 1.1 Making Cubic Boxes

1.2 Making Rectangular Boxes 1.3Rectangular Prisms

1.4 Surface Area of Rectangular Prisms

4 days

3

7.G.6 ° Find volume and surface area of rectangular prisms ° Determine what type of arrangement will produce the least or greatest surface

area given a fixed volume.

FILLING & WRAPPING Inv. 2 Designing Rectangular Boxes 2.1 Finding Surface Area 2.2 Finding the Least Surface Area

2.3 Filling Rectangular Boxes

5 days

3

7.G.4 ° Know the formulas for area and circumference of circles ° Describe the relationship between the circumference and area of a circle

COMMON CORE INVESTIGATIONS Inv. 4 Geometry Topics

Problem 4.2

1 day

3

7.G.4

7.G.6

° Find volume and surface area of right prisms and cylinders. (see instructional note re: cylinders)

FILLING & WRAPPING Inv. 3 Prisms and Cylinders 3.1 Finding the Volumes of Other Prisms 3.2 Finding the Volumes of Cylinders

3.3 Finding the Surface Area of Cylinders and

Prisms

4 days

3

7.G.3 ° Describe 2-dimensional cross sections of 3-dimensional figures COMMON CORE INVESTIGATIONS Inv. 4 Geometry Topics

Problem 4.1

2 days

24

3

7.G.1

7.G.6

7.RP.1

° Determine how the volume and surface area of a rectangular prism changes as any of its dimensions is scaled up or down.

FILLING & WRAPPING Inv. 5 Scaling Boxes 5.1 Doubling the Volume of a Rectangular Prism 5.2 Applying Scale Factors to Rectangular Prisms

5.3 Similarity and Scale Factors

5 days

3

7.G.5 ° Use facts about supplementary, complementary, vertical, and adjacent angles to solve single and multi-step problems

° Solve simple equations for unknown angles in a figure

COMMON CORE INVESTIGATIONS Inv. 4 Geometry Topics

Problem 4.4

2 days

End Unit 2-dimensional and 3-dimensional Geometry 23 days of Core Instruction,

9 Discretionary/Assessment Days

25

Unit Title: Probability Primary Curricular Resource: What Do You Expect? Instructional Time: 16 days Overarching Questions:

1. What are some strategies you can use to find experimental probabilities? 2. What are some strategies you can use to find theoretical probabilities? 3. How do you use probabilities, either theoretical or experimental, to decide if a particular game of chance is fair? 4. How can you determine the probability of compound events? 5. How would you calculate the expected value for a probability situation? 6. How can you use expected value to help make decisions?

Instructional Notes: • Fluency Marker: Use ratios and proportional reasoning to evaluate probability models. • The 2011 Massachusetts Frameworks establish a strong focus on investigating chance processes and developing, using

and evaluating probability models. • Instruction in this unit should maintain a strong emphasis on probability models and probability of compound events.

Concepts developed in this unit • Investigate chance processes and develop, use, and evaluate probability models.

Standards for Mathematical Practice Focus #2 Reason abstractly and quantitatively. #3 Construct viable arguments and critique the reasoning of others. #4 Model with mathematics.

Prior knowledge expected • Compare two fractions with different numerators and different denominators. • Multiply fractions. • Understand the place value system as it relates to decimals. • Find percent of a quantity as a rate per 100. • Translate among fractions, decimals and percents.

26

Academic language developed in this unit expected value, prediction, sample space, tree diagram, unlikely event

Academic language from prior units or experience certain outcome, chance, equally likely events, event experimental probability, fair game, favorable outcome, impossible outcome, outcome, possible, probability, probable, theoretical probability, trial

Learning Outcomes 7.SP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers

indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ! indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency,

and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is

not good, explain possible sources of the discrepancy. a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For

example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the

approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

TERM

MA 2011

Frameworks Citation

After completing each section, students will be able to:

Primary Curricular Resource

Days

27

3

7.SP.5 ° Review basic probability concepts, such as fair game, experimental probability, theoretical probability, and fraction notation for expressing probabilities

WHAT DO YOU EXPECT? Inv. 1 Evaluating Games of Chance 1.1 Experimental and Theoretical Probability 1.2 Using and Organized List

1.3 Determining Whether a Game is Fair

3 days

3

7.SP.8 7.SP.8.a 7.SP.8.b 7.SP.8.c

° Use an area model to analyze the theoretical probabilities for two-stage outcomes ° Simulate and analyze probability situations involving two-stage outcomes ° Distinguish between equally likely and non–equally likely outcomes by collecting data

and analyzing experimental probabilities

WHAT DO YOU EXPECT? Inv. 2 Analyzing Situations Using an Area Model 2.1 Area Models and Probability 2.2 Using an Area Model

2.3 Finding the Best Arrangement

4 days

3

7.SP.6 7.SP.7 7.SP.7.a 7.SP.7.b 7.SP.8 7.SP.8.a 7.SP.8.b 7.SP.8.c

° Understand the difference between the probability of an outcome and the long-term average of many trials in a situation with a payoff

° Determine the expected value in a probability situation ° Use probability to make decisions

WHAT DO YOU EXPECT? Inv. 3 Expected Value 3.1 Simulating a Probability Situation 3.2 Finding Expected Value

3.3 Comparing Expected Values

4 days

End Unit Probability 11 days of Core Instruction

5 Discretionary/Assessment Days

28

Unit Title: Statistics Primary Curricular Resource: Data Distributions; Samples & Populations Instructional Time: 43 days Overarching Questions: 1. Meaningful Statistics:

What question was asked that resulted in these data being collected? How do you think the data were collected? Why are these data represented using this kind of presentation/display? What are the ways to describe the data distribution? What questions do the data raise? What is your response?

2. What makes a sample representative? 3. What does it mean to infer? How does random sampling allow me to draw inferences about a population? 4. What strategies can I use to compare two different data sets? Where do the data cluster in the distributions? How are the mean, median, range, and variability used to understand and describe data distributions? 5. Can I use my results to make predictions or generalizations about the populations?

29

Instructional Notes: o Fluency Marker: Use percentages and proportional reasoning to draw inferences about a population from a sample. o The 2011 Massachusetts Frameworks emphasize drawing inferences about populations based on relevant samples. o Instruction in this unit should maintain a focus on valid generalizations about a population derived from statistical analysis of

representative samples. o Use of relevant data promotes student engagement and learning. Consider the following data sets and resources for

classroom use. o Statistics Unit – Suggested Links for Data

http://www.cityofboston.gov/doit/databoston/ http://www.bostonindicators.org/Indicators2008/Default.aspx http://www.cityofboston.gov/dnd/U_Neighborhood_Profiles.asp http://www.bostonredevelopmentauthority.org/Research/ResearchSubject.asp?SubjectID=25 http://metrobostondatacommon.org/ http://www.hsph.harvard.edu/hyvpc/research/ http://quickfacts.census.gov/qfd/states/25/2507000.html http://www.census.gov/

o Frameworks 6.SP.1 through 6.SP5 will need to be included/reviewed in 7th grade in 2012-2013 only. Data Distributions Inv 1 and Samples and Populations Inv 1 will move to 6th grade in 2013-2014.

o The 7th grade statistics standards require use of both Data Distributions and Samples and Populations. o Manipulation of data sets with technology will help students understand how changes in data affect median and mean Data

Distributions Problem 2.4. o Another real-world problem with local relevance could be substituted here in place of Problem 3.2 application practice.

Concepts developed in this unit • Use random sampling to draw inferences about a population. • Note connection to major work of grade: Opportunity to apply percentages and proportional reasoning. • Draw informal comparative inferences about two populations.

Standards for Mathematical Practice Focus #2 Reason abstractly and quantitatively. #3 Construct viable arguments and critique the reasoning of others. #4 Model with mathematics. #5 Use appropriate tools strategically.

30

Prior knowledge expected • Recognize a statistical question as one that anticipates variability. • Understand that a data distribution can have different measures of center and can be described by identifying clusters, peaks,

gaps, and symmetry. • Display numerical data in line plots, histograms, circle graphs (MA.4.a), and box plots. • Summarize numerical data sets by quantitative measures of center (median and/or mean), and variability (interquartile range

and/or absolute deviation).

Academic language developed in this unit Data Distributions attribute, counts, distribution measures of center, ordered value bar graph, range, value of an attribute, value bar graph, variability of a set of numerical data

Samples and Populations box-and-whisker plot, convenience sample, five-number summary, histogram, population, random sample, sample, systematic sample, voluntary response sample, quartile

Academic language from prior units or experience categorical data, line plot, mean, median, mode, numerical data, outlier, scatter plot, stem-and-leaf plot, bar graph, coordinate graph, data, range, survey, table, x-axis, y-axis

31

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.

MA.4.a. Read and interpret circle graphs. 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. c. 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. d. 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.

7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a

population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or

simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers

by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

32

TERM

MA2011

Frameworks Citation

After completing each section, students will be able to:

Primary Curricular Resource

Days

4

6.SP.1 6.SP.2 6.SP.4 6.SP.5

° Recognize that variability occurs when data are collected. ° Use properties of distributions to describe variability. ° (Statistics such as measures of central tendency (mean, median, ° mode), Variability (outliers, range), and Characteristics of shape ° (clumps, gaps, skewed distributions)) ° Use counts or percents to report frequencies of data. ° Read a data table and use it to make a bar graph. ° Use line plots and value bar graphs to display data distributions. ° Decide if a difference among data values matters in the context of the problem. ° Identify sources of variability.

DATA DISTRIBUTION Inv. 1 Making Sense of Variability 1.1 Variability in Categorical Data 1.2 Variability in Numerical Counts 1.3 Variability in Numerical Measurements 1.4 Two Kinds of Variability

5 days

4

6.SP.3

7.SP.4

° Recognize that variability occurs when data are collected. ° Use properties of distributions to describe variability. ° Decide if a difference among data values matters in the context of the problem. ° Read and understand bar graphs and line plots used to display data distributions ° Understand and use different models to make sense of the mean ° Decide when to use the mean, median, or mode to describe a distribution ° Develop and use strategies for comparing equal-size and unequal-size data sets ° Understand when and how changes in data values affect the median or mean ° Relate the shape of a distribution to the location of its mean and median

DATA DISTRIBUTION Inv. 2 Making Sense of Measures of Center 2.1 The Mean as an Equal Share 2.2 The Mean as a Balance Point in a Distribution 2.3 Repeated Values in a Distribution 2.4 Measures of Center and

Shapes of Distributions

5 days

4

7.SP.1

7.SP.4

° Decide when to use the mean, median, or mode to describe a distribution ° Recognize the importance of having the same scales on graphs that are used to

compare data distributions ° Develop and use strategies for comparing equal-size data sets to solve problems ° Understand and use three components of graph comprehension: ° Reading the data: use data in the graph to answer explicit questions ° Reading between the data: interpret and integrate information presented in a graph ° Reading beyond the data: extend, predict, or infer from data to

answer implicit questions

DATA DISTRIBUTION Inv. 3 Comparing Distributions: Equal Numbers of Data Values 3.1 Measuring and Describing Reaction Times 3.2 Comparing Reaction Times 3.3 Comparing Many Data Values

3.4 Comparing Larger Distributions

5 days

33

4

7.SP.1

7.SP.4

° Develop and use strategies for comparing unequal-size data sets to solve problems ° Use shape of a distribution to estimate locations of the mean and median ° Understand and use three components of graph comprehension: ° Reading the data: use data in the graph to answer explicit questions ° Reading between the data: interpret and integrate information

presented in a graph ° Reading beyond the data: extend, predict, or infer from data to

answer implicit questions

DATA DISTRIBUTION Inv. 4 Comparing Distributions: Unequal Numbers of Data Values 4.1 Representing Survey Data

4.2 Comparing Speed

3 days

4

6.SP.4 ° Learn how to make and use histograms, five-number summaries, and box-and-whisker plots

° Compare sample distributions using measures of center (mean, median), measures of variability (minimum and maximum values, range), and data displays that group data (histograms and box-and-whisker plots)

SAMPLES AND POPULATIONS Inv. 1 Comparing Data Sets 1.1 From Line Plots to Histograms 1.2 Using Histograms 1.3 Box-and-Whisker Plot

1.4 Analyzing Data

5 days

4

7.SP.1 7.SP.2 7.SP.4

° Distinguish between a sample and a population ° Use data from samples to make predictions about a population ° Consider various ways of developing a sample plan ° Select a random sample from a population ° Use sampling distributions, measures of center, and measures of variability to

describe and compare samples ° Note the effect of sample size on distribution and summary statistics

SAMPLES AND POPULATIONS Inv. 2 Choosing a Sample from a Population 2.1 Using a Sample to Make Predictions 2.2 Selecting a Sample 2.3 Choosing Random Samples 2.4 Choosing a Sample Size

5 days

4

7.SP.1 7.SP.3 7.SP.4

° Distinguish between samples and populations ° Use information from samples to draw conclusions about populations

SAMPLES AND POPULATIONS Inv. 3 Solving Real-World Problems

3.1 Comparing and Analyzing Data

2 days

End Unit Statistics 30 days of Core Instruction

13 Discretionary/Assessment Days