Middle School Math Series: Course 2cdn.carnegielearning.com/assets/worktext-files/Carnegie... · · 2014-11-19Middle School Math Series: ... Middle School Math Series: Course 2

Middle School Math Series: Course 2 1

1 Ratios and Rates

Chapter Lesson Title Key Math Objective CCSS Key Terms

1.1

Show Someone You Care — Send Flowers!

Introduction to Ratios and Rates

• Identify ratios, rates, and unit rates.• Use ratios, rates, and unit rates to analyze problems.

6.RP.17.RP.1

• Ratio• Rate• Proportion• Equivalent ratios• Unit rate• Scaling up• Scaling down

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1.2

Making Punch

Ratios, Rates, and Mixture Problems

• Use ratios to make comparisons.• Use rates and proportions to solve mixture problems. 7.RP.1 N/A �

1.3For the Birds

Rates and Proportions

• Write ratios and rates.• Write proportions.• Scale up and scale down proportions.

7.RP.1 • Convert �

1.4Tutor Time!

Using Tables to Solve Problems

• Use tables to represent equivalent ratios.• Solve proportions using unit rates.

7.RP.17.RP.2.b

N/A

Middle School Math Series: Course 2

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1.5

Looks Can Be Deceiving!

Using Proportions to Solve Problems

• Solve proportions using the scaling method.• Solve proportions using the unit rate method.• Solve proportions using the means and extremes method.

7.RP.17.RP.2.c7.RP.3

• Variable• Means and extremes• Solve a proportion• Inverse operations

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1.6

The Price Is … Close

Using Unit Rates in Real World Applications

• Estimate and calculate values using rates.• Use unit rates to determine the best buy.

7.RP.17.RP.2.b7.RP.3

N/A � �



2 Direct Variation and Constant of Proportionality


2.1What Makes You Tap Your Feet?

Introduction to Direct Variation

• Determine how quantities in different situations vary.• Use multiple representations to explore the types of variation.

7.RP.17.RP.2.a7.RP.2.d

• Direct variation (direct proportion)• Origin �

2.2

Building Bird Feeders Is for the Birds!

Determining Equivalent Ratios

• Determine if the points on a graph are equivalent ratios. 7.RP.2.a N/A � �

2.3

Kids Just Wanna Have Fun!

Determining and Applying the Constant of Proportionality

• Determine the constant of proportionality.• Solve problems using the proportional relationship between two variables.

7.RP.2.b7.RP.2.c • Constant of proportionality �

2.4

Stop That Speeding Snail?

Using the Constant of Proportionality to Solve Proportions

• Determine if there is a constant of proportionality between two variables.

7.RP.2.a7.RP.2.b7.RP.2.c7.RP.3

N/A

This chapter develops the representations of proportional relationships between quantities using words, tables, equations, and graphs. The constant of proportionality is introduced and used to solve various real-world problems.

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2.5

The Man Who Ran from Marathon to Athens

Graphing Direct Proportions

• Graph relationships that are directly proportional.• Interpret the graphs of relationships that are directly proportional.

7.RP.2.a7.RP.2.b7.RP.2.c7.RP.2.d

N/A

2.6Racing to the Finish Line!

Using Direct Proportions

• Determine if two variables are directly proportional or vary directly.• Interpret relationships that are direct proportions.• Solve direct variation problems using the equation y = kx.

7.RP.2.a7.RP.2.b7.RP.2.c

N/A �

2.7

Connecting Representations of Proportional Relationships

Interpreting Multiple Representations of Direct Proportions

• Determine if relationships represented in words, tables, equations, or graphs are directly proportional.• Interpret the meaning of direct proportions represented in words, tables, equations, and graphs.• Determine and interpret the constant of proportionality for variables that are directly proportional and represented in words, tables, equations, and graphs.

7.RP.2.a7.RP.2.b7.RP.2.c7.RP.2.d

N/A �



3 Rates and Percents


3.1

Give Me a Ballpark Figure of the Cost

Estimating and Calculating with Percents and Rates

• Estimate and calculate values using rates.• Estimate and calculate the values of percents.

7.RP.17.RP.3 N/A �

3.2One Size Fits All?

Solving Percent Problems• Solve problems involving percents.

7.RP.17.RP.2.c7.RP.3

N/A � �

3.3

Mathematics and Nutrition

Using Proportions and Percent Equations

• Solve proportions.• Solve percent equations.

7.RP.2.c7.RP.3

• Percent equation �

3.4Be Mindful of the Fees!

Using Percents

• Calculate simple interest.• Calculate the percent of increase.• Calculate the percent of decrease.• Calculate discount of base price.• Calculate tax on a purchase.• Calculate depreciate.

7.RP.2.a7.RP.2.c7.RP.3

• Interest• Principal• Simple interest• Percent increase• Percent decrease• Depreciate

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3.5

Shoe Super Store

Solving Percent Problems Involving Proportions

• Solve percent problems using direct variation.• Write equations to show the constant of proportionality.

7.RP.2.a7.RP.2.b7.RP.2.c7.RP.3

• Commission �

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This chapter extends the use of proportional reasoning to solve percent problems.

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4 Addition and Subtraction with Rational Numbers


4.1

Math Football

Using Models to Understand Integers

• Represent numbers as positive and negative integers.• Use a model to represent the sum of a positive and a negative integer.

7.NS.1.a7.NS.1.b N/A � �

4.2Walk the Line

Adding Integers, Part 1

• Model the addition of integers on a number line.• Develop a rule for adding integers. 7.NS.1.b N/A � �

4.3Two-Color Counters

Adding Integers, Part 2

• Model the addition of integers using two-color counters.• Develop a rule for adding integers.

7.NS.1.a7.NS.1.b7.NS.1.c

• Additive inverses � � �

4.4What's the Difference?

Subtracting Integers

• Model subtraction of integers using two-color counters.• Model subtraction of integers on a number line.• Develop a rule for subtracting integers.

7.NS.1.a7.NS.1.b7.NS.1.c7.NS.1.d

• Zero pair � � � �

This chapter uses models to develop a conceptual understanding of addition and subtraction with respect to the set of integers. These strategies are formalized through questioning, and then extended to operations with respect to the set of rational numbers.

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4.5

What Do We Do Now?

Adding and Subtracting Rational Numbers

• Add and subtract rational numbers.7.NS.1.b7.NS.1.c7.NS.1.d

N/A �



5 Multiplication and Division with Rational Numbers


5.1Equal Groups

Multiplying and Dividing Integers

• Multiply integers.• Divide integers.

7.NS.2.a7.NS.2.b7.NS.2.c

N/A � �

5.2

What's My Product or Quotient?

Multiplying and Dividing Rational Numbers

• Multiply rational numbers.• Divide rational numbers.

7.NS.2.a7.NS.2.b7.NS.2.c

N/A �

5.3

Properties Schmoperties

Simplifying Arithmetic Expressions with Rational Numbers

• Simplify arithmetic expressions using the number properties and the order of operations.7.NS.1.d7.NS.2.a7.NS.2.c

N/A

5.4

Building a Wright Brothers' Flyer

Evaluating Expressions with Rational Numbers

• Model a situation with an expression using rational numbers.• Evaluate rational expressions.

7.NS.1.d7.NS.2.a7.NS.2.b7.NS.2.c7.NS.3

N/A

5.5

Repeat or Not? That Is the Question!

Exact Decimal Representations of Fractions

• Use decimals and fractions to evaluate arithmetic expressions.• Convert fractions to decimals.• Represent fractions as repeating decimals.

7.NS.1.d7.NS.2.d

• Terminating decimals• Non-terminating decimals• Repeating decimals• Non-repeating decimals• Bar notation

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This chapter use models to develop a conceptual understanding of multiplication and division with respect to the set of integers. These strategies are formalized, and then extended to operations with respect to the set of rational numbers.

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6 Numerical and Algebraic Expressions and Equations


6.1What's It Really Saying?

Evaluating Algebraic Expressions• Evaluate algebraic expressions. 6.EE.6

7.NS.3

• Variable• Algebraic expression• Evaluate an algebraic expression

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6.2

Express Math

Simplifying Expressions Using Distributive Properties

• Write and use the distributive properties.• Use distributive properties to simplify expressions.

7.NS.37.EE.17.EE.2

• Distributive Property of Multiplication over Addition• Distributive Property of Multiplication over Subtraction• Distributive Property of Division over Addition• Distributive Property of Division over Subtraction

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6.3Reverse Distribution

Factoring Algebraic Expressions

• Use the distributive properties to factor expressions.• Combine like terms to simplify expressions.

7.EE.17.EE.2

• Factor• Common factor• Greatest common factor (GCF)• Coefficient• Like terms• Combining like terms

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6.4

Are They the Same or Different?

Verifying That Expressions Are Equivalent

• Simplify algebraic expressions.• Verify that algebraic expressions are equivalent by graphing, simplifying, and evaluating expressions.

7.EE.17.EE.2 N/A � �

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This chapter focuses on the use of properties to interpret, simplify, add, subtract, and factor linear expressions.

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6.5

It Is Time to Justify!

Simplifying Algebraic Expressions Using Operations and Their Properties

• Simplify algebraic expressions using operations and their properties. 7.EE.17.EE.2 N/A �



7 Solving Equations and Inequalities


7.1Picture This

Picture Algebra

• Use drawings to represent daily life situations.• Develop different methods to represent daily life situations mathematically.

7.EE.27.EE.3

7.EE.4.a• Equation � �

7.2Maintaining a Balance

Solving Equations

• Develop an understanding of equality.• Use properties of equality to solve equations represented with algebra tiles.• Solve one-step equations.

7.EE.37.EE.4.a

• Properties of Equality � �

7.3Planning a Graduation Party

Solving Two-Step Equations

• Write two-step equations.• Solve two-step equations.• Check solutions algebraically.• Verify solutions within a problem context.

7.EE.37.EE.4.a

• Two-step equation �

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7.4Solving in Big-Time Style

Using Two-Step Equations

• Write two-step equations.• Solve two-step equations.• Verify solutions within a problem context.

7.EE.17.EE.27.EE.3

7.EE.4.a

N/A

7.5

We're Shipping Out!

Solving and Graphing Inequalities in One Variable

• Write simple inequalities.• Graph one-variable inequalities.• Solve one-variable inequalities.

7.EE.37.EE.4.a7.EE.4.b

• Solve an inequality � �



8 Solving Problems with Equations and Inequalities


8.1

Some Places Are Expensive; Some Places Are More Affordable

Multiple Representations of Problem Situations

• Use different methods to represent a problem situation.• Identify advantages and disadvantages of using a particular representation.• Solve two-step equations.

7.EE.37.EE.4.a N/A �

8.2Plastic Containers

Using Two-Step Equations

• Write and use two-step equations.• Compare two problem situations. 7.EE.4.a N/A �

8.3

Just Another Saturday

Solving More Complicated Equations

• Solve equations containing fractions.• Solve equations with variables on both sides.• Verify solutions within a problem context.

7.EE.37.EE.4.a

• Multiplicative inverse• Multiplying by the reciprocal �

8.4

Climbing El Capitan

Making Sense of Negative Solutions

• Use multiple representations to analyze problem situations.• Interpret negative solutions to problem situations.• Identify independent and dependent variables.• Evaluate algebraic expressions.• Solve algebraic equations.

7.EE.37.EE.4.a N/A

This chapter focuses on the use of algebraic reasoning to solve real-world problems posed with positive and negative rational numbers. Tables, graphs, and equations are used throughout.

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8.5Flying in the Ocean

Rate of Change

• Calculate the unit rate of change.• Interpret the unit rate of change in a problem situation.

7.EE.37.EE.4.a7.EE.4.b

• Unit rate of change �

8.6

Emptying a Tank

Using Multiple Representations to Solve Problems

• Use multiple representations to analyze problem situations. 7.EE.37.EE.4.a N/A



9 Using Geometry


9.1

Earth Measure

Introduction to Geometry and Geometric Constructions

• Sketch and draw figures.• Use a compass to construct circles.• Use geometric tools to duplicate line segments.

7.G.2

• Geometry• Protractor• Compass• Straightedge• Sketch• Draw• Construct• Geometric construction• Point• Line• Plane• Coplanar lines• Skew lines• Line segment• Endpoints• Arc• Congruent line segments• Congruent• Intersection

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9.2

Angles and More Angles

Measuring and Constructing Angles

• Measure and construct angles.• Duplicate angles.• Bisect angles.• Review congruent gures.

7.G.2

• Ray• Angle• Sides of an angle• Vertex• Degrees (°)• Acute angle• Right angle• Obtuse angle• Straight angle• Congruent angles• Bisect• Angle bisector

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This chapter introduces points, lines, and planes as the essential building blocks of geometry. Key terms and basic constructions are introduced that will allow students to construct and compare triangles in the Chapter 10.

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9.3

Special Angles

Complements, Supplements, Midpoints, Perpendiculars, and Perpendicular Bisectors

• Calculate the supplement of an angle.• Calculate the complement of an angle.• Construct a perpendicular line.• Construct a perpendicular bisector.• Construct the midpoint of a segment.• Classify adjacent angles, linear pairs, and vertical angles.

7.G.27.G.5

• Supplementary angles• Complementary angles• Perpendicular• Midpoint of a segment• Segment bisector• Perpendicular bisector• Adjacent angles• Linear pair• Vertical angles

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10 Triangles


10.1

Pulling a one-eighty!

Triangle Sum, Exterior Angle, and Exterior Angle Inequality Theorems

• Prove the Triangle Sum Theorem.• Explore the relationship between the interior angle measures and the side lengths of a triangle.• Identify the remote interior angles of a triangle.• Identify the exterior angle of a triangle.• Explore the relationship between the exterior angle measures and two remote interior angles of a triangle.• Prove the Exterior Angle Theorem.• Prove the Exterior Angle Inequality Theorem.

7.G.2

• Triangle Sum Theorem• Remote interior angles of a triangle• Exterior Angle Theorem• Exterior Angle Inequality Theorem

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10.2Triangle Construction 1

Constructing Triangles• Construct triangles to determine uniqueness. 7.G.2 N/A � �

10.3

Triangle Construction 2

Congruent Figures and Constructing Congruent Triangles

• Define congruent figures.• Determine whether figures are congruent.• Determine whether triangles are congruent.• Construct congruent triangles.

7.G.2

• Geometric figures• Congruent geometric figures• Corresponding sides• Corresponding angles• Included angle• Included side

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This chapter investigates the conditions that determine unique triangles.

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10.4Pasta Anyone?

Triangle Inequality Theorem

• Explore the relationships between the side lengths of a triangle and the measures of its interior angles.• Prove the Triangle Inequality Theorem.

7.G.2 • Triangle Inequality Theorem � �



11 Scale Drawings and Scale Factor


11.1

Bigger and Smaller

Scale Drawings, Scale Models, and Scale Factors

• Use scale models to calculate measurements.• Use scale factors to enlarge and shrink models. 7.G.1 • Scale factor �

11.2Say Cheese!

Applications of Ratio• Work with applications of similarity and scale factor. 7.G.1 • Aspect ratio

11.3No GPS? Better Get the Map Out!

Exploring Scale Drawings

• Work with applications of similarity and scale factor.• Use scale drawings and maps. 7.G.1 • Scale drawings � �

11.4Houses for Our Feathered Friends

Creating Blueprints

• Use scale drawings to create three-dimensional models.• Use three-dimensional models to create blueprints. 7.G.1 N/A �

This chapter explores scale drawings, scale models, and scale factors through real-world problems.

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12 Circles


12.1Introduction to Circles

Circles, Radius, and Diameter

• Define circle.• Identify the center, radius, and diameter of a circle. 7.G.4

• Circle• Center of a circle• Radius of a circle• Diameter of a circle

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12.2But Most of All, I Like Pi!

Circumference of a Circle

• Measure the circumference of a circle.• Explore the relationship between the diameter and the circumference of a circle.• Write a formula for the circumference of a circle.• Use a formula to determine the circumference of a circle.

7.G.4 • Pi �

12.3One Million Sides!

Area of a Circle

• Explore the circumference and area of circles inscribed in regular polygons.• Write a formula for the area of a polygon in terms of the perimeter.• Explore the relationship between the circumference of a circle and the area of a circle.

7.G.4 • Inscribed circle � �

12.4It's About Circles!

Unknown Measurements

• Use the area and circumference formulas to solve for unknown measurements.• Use composite figures to solve for unknown measurements. 7.G.4 N/A �

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This chapter develops the formula for the area of a circle. Problems are solved involving circles and circle formulas.

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13 Slicing Three-Dimensional Figures


13.1Slicing and Dicing

Slicing Through a Cube• Sketch, model, and describe cross-sections formed by a plane passing through a cube. 7.G.3 • Cross-section � �

13.2

The Right Stuff

Slicing Through Right Rectangular Prisms

• Sketch, model, and describe cross-sections formed by a plane passing through a right rectangular prism. 7.G.3 N/A � �

13.3

And Now On to Pyramids

Slicing Through Right Rectangular Pyramids

• Sketch, model, and describe cross-sections formed by a plane passing through a right rectangular pyramid. 7.G.3

• Pyramid• Base of a pyramid• Lateral faces of a pyramid• Lateral edges of a pyramid• Vertex of a pyramid• Height of a pyramid• Regular pyramid• Slant height of a pyramid

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13.4

Backyard Barbecue

Introduction to Volume and Surface Area

• Explore the volume of a solid.• Explore the surface area of a solid.• Create a net for a concave polyhedron.

7.G.6 • Volume• Surface area � �

13.5

Famous Pyramids

Applying Volume and Surface Area Formulas

• Explore the volume of a pyramid.• Explore the surface area of a pyramid. 7.G.6 N/A �

This chapter explores planes intersected with three-dimensional geometric figures. Real-world problems are solved involving area, volume, and surface area of three-dimensional figures.

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14 Data Collection


14.1

We Want to Hear from You!

Formulating Questions and Collecting Data

• Collect data from a survey and analyze the results.• Differentiate between a census and a sample.• Differentiate between a parameter and a statistic.

7.SP.17.SP.2

• Survey• Data• Population• Census• Sample• Parameter• Statistic

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14.2

Dealing with Data: Selecting a Sample

Collecting Data through Random Sampling

• Differentiate between a random sample and a sample that is not chosen randomly.• Use several methods to select a random sample.

7.SP.17.SP.2

• Random sample• Random number generator• Random number table

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14.3Floor Plans and Tiles

Random Sampling

• Investigate how results from a random sample are more reliable in representing the population than results from a sample that is not random.• Use sampling and proportional reasoning to predict the value of a population parameter.

7.SP.17.SP.2 • Dot plot (line plot) � � � �

14.4

What Does the Data Mean?

Using Samples, Centers, and Spreads to Describe Data

• Analyze measures of center for samples from a population.• Analyze measures of variation for samples from a population.

7.SP.17.SP.27.SP.4

• Variability• Spread• Range• Mean absolute deviation• Deviation from the mean

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14.5Taking a Survey

Using Sample Size

• Investigate the sample size of a survey.• Investigate the concept that as the sample size increases, the statistic obtained gets closer to the actual population parameter.

7.SP.17.SP.2

• Sample size

This chapter emphasizes data collection, sampling, random samples, and frequencies.

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15 Comparing Populations


15.1

Checking the Papers and the Blogs

Comparing Measures of Center of Two Populations

• Calculate the measures of center and measures of variability for two populations.• Compare the difference of the measures of center for two populations to their measures of variability.

7.SP.37.SP.4 N/A �

15.2

Can Podcasts Affect Ratings?

Comparing Measures of Center of Two Populations

• Compare the measures of center for random samples from two populations.• Use measures of center to draw conclusions about two populations.

7.SP.27.SP.37.SP.4

N/A � �

15.3

Finding Your Spot to Live

Drawing Conclusions About Two Populations

• Compare the measures of variability for random samples from two populations.• Use measures of variability to draw conclusions about two populations.

7.SP.27.SP.37.SP.4

N/A � � �

This chapter uses measures of variablily to compare inferences across two populations.

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16 Introduction to Probability


16.1

Rolling, Rolling, Rolling ...

Defining and Representing Probability

• Differentiate between an outcome and an event for an experiment.• List the sample space for an experiment.• Determine the probability for an event.• Understand that the probability for an event is between 0 and 1 that can be expressed as a fraction, decimal, or percent.• Determine that the sum of the probabilities of the outcomes of an experiment is always 1.

7.SP.5

• Outcome• Experiment• Sample space• Event• Simple event• Probability• Equally likely

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16.2

Toss the Cup

Determining Experimental Probability

• Conduct trials of an experiment.• Predict the experimental probability of an event using the results from the trials of an experiment.• Use proportional reasoning to predict the probability of random events.

7.SP.57.SP.6

7.SP.7.a7.SP.7.b

• Experimental probability � � �

16.3

Double Your Fun

Determining Theoretical Probability

• Calculate the experimental and theoretical probability of an experiment.• Determine the difference between experimental and theoretical probability.• Use proportional reasoning to predict probability of random events.

7.SP.57.SP.6

7.SP.7.a7.SP.7.b

• Theoretical probability � � �

16.4A Toss of a Coin

Simulating Experiements

• Conduct trials using a simulation to determine probability.• Conduct a large number of trials to demonstrate that experimental probability approaches theoretical probability.

7.SP.57.SP.6

7.SP.7.a7.SP.7.b

• Simulation• Trial � �

16.5Roll the Cubes Again

Using Technology for Simulations• Use technology to simulate a large number of trials of an experiment.

7.SP.67.SP.7.a7.SP.7.b

• Spreadsheet

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17 Probability of Compound Events


17.1Is It Better to Guess?

Using Models for Probability

• Determine the probability model for an experiment.• Construct and interpret a uniform probability model.• Construct and interpret a non-uniform probability model.

7.SP.7.a7.SP.7.b

• Probability model• Uniform probability model• Non-uniform probability model

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17.2

Three Girls and No Boys?

Creating and Using Probability Models

• Determine the probability model for an experiment.• Construct and interpret a non-uniform probability model.

7.SP.7.a7.SP.7.b7.SP.8.a7.SP.8.b

• Tree diagram• Complementary events � � � �

17.3

Pet Shop Probability

Determining Compound Probability

• Use probability models to calculate compound probabilities.

7.SP.7.a7.SP.7.b7.SP.8.a7.SP.8.b

• Compound event � � �

17.4

What Type of Blood Do You Have?

Simulating Probability of Compound Events

• Use simulations to estimate compound probabilities. 7.SP.8.c N/A � �

This chapter focuses on probabilities of compound events. Simulations are used to generate frequencies for compound events.

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Middle School Math Series: Course 2cdn.carnegielearning.com/assets/worktext-files/Carnegie... · · 2014-11-19Middle School Math Series: ... Middle School Math Series: Course 2