Sample | Curriculum for the Digital SAT

Curriculum Digital SAT for the

®

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Curriculum for the Digital SAT® How Curriculum for the Digital SAT® Works: How Mastery for the Digital SAT Works Welcome to Mastery for the Digital SAT, a new take on improving SAT scores.

Students have three paths to boosting their scores: • time management Students have three paths to boosting their scores:

• Time management • Test-taking skills • Content

• test-taking skills • content

Traditional test preparation provides a review of content. It discusses test-taking skills. It reminds students about time management. Students who are already strong test-takers, who already perform well academically, benefit.

Most students, however, forget the content, don’t change their test-taking behaviors, and continue to struggle against the clock.

Our approach is different. We call it the Mastery Pyramid. It works even for students who need the most help in these areas.

• Content Mastery: Engaging direct instruction based on modern practices helpstest-taking students bridge the skills. gap Traditional test preparation provides a review of content.

It best discusses It reminds between classroom learning and the SAT’s assessed standard. By isolating skills, we make each topic easier

to learn. Later, mixed practice with authentic SAT question emulations prepares students for a real testing students about time management.

Students who are already strong test-takers, who already perform environment.

• Test Mastery: Test-taking is a skill that can be learned like any other.content, Our lessons simplify key test-taking well academically, benefit.

Most students, however, forget the don’t change their test-taking strategies, provide many opportunities to practice them, and reinforce the fundamentals again and again until

test-taking behavior actually improves. behaviors, and continue to struggle against the clock.

• Time Mastery: It’s one thing to memorize the time limits. It takes practice to improve pacing and decisionmaking skills. In each lesson, students practice with timed test sections and mini-tests until they’re genuinely able to beat the clock.

Our approach is different. We call it the Mastery Pyramid. It works even for students who need the most help in these areas.

Three Paths to Boosting Scores:

Content Mastery:

HOW MASTERY FOR THE DIGITAL SAT WORKS

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Engaging direct instruction based on modern best practices helps students bridge the gap between classroom learning and the SAT’s assessed standard. By isolating skills, we make each topic easier to learn. Later, mixed practice with authentic SAT question emulations prepares students for a real testing environment.

Test Mastery: Test-taking is a skill that can be learned like any other. Our lessons simplify key test-taking strategies, provide many opportunities to practice them, and reinforce the fundamentals again and again until testtaking behavior actually improves.

Time Mastery: It’s one thing to memorize the time limits. It takes practice to improve pacing and decision-making skills.

In each lesson, students practice with timed test sections and mini-tests until they’re genuinely able to beat the clock.

Curriculum for the Digital SAT®

The Mastery for the Digital SAT Difference

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Authentic Practice

Learning by Doing

Each math book features hundreds of SAT Math questions that are extremely similar to what students will face on the SAT. If they can answer the questions correctly in this book, they’ll get them correct on the test. Other test prep programs don’t come close to the amount of authentic practice we provide.

Unlike traditional test prep programs that resort to “sage on the stage” demonstrations, Mastery for the Digital SAT students learn from what they do in the lessons, not from what you read them from a slide deck. Every instructional unit is centered around student activity and engagement.

Lots of Authentic Practice

Scaffolded Instruction

Did we mention Mastery for the Digital SAT provides students a lot of authentic “at-bats”? For each content dimension, we include literally dozens of practice questions to ensure your students have enough runway to master the skills. With traditional test prep programs, the average number of practice questions provided per skill is 3 (compared to more than 40 in Mastery for the Digital SAT).

SAT items are highly rigorous. Traditional test prep aims directly at rehearsing the items, which can leave many students behind. Mastery for the Digital SAT is designed to start at a more remedial level and rapidly scaffold students to higher levels of attainment. We don’t assume students know the basics, and the result is greater student engagement and achievement.

Isolation

Detailed Test-Taking Approaches and Guessing Strategies

We break the test down one content dimension at a time. Students are given the opportunity to master each skill, and given plenty of practice with each skill in isolation, before applying what they have learned to mixed practice. The result is that students better consolidate what they learn and don’t get overwhelmed. Traditional test prep programs fail to provide sufficient practice in isolation.

Interleaving Mastery for the Digital SAT’s mini-tests bridge the gap between isolated practice and the real SAT experience by interspersing questions related to the topic at hand with a wide range of question types.

This helps students improve their problem-solving and time management skills. Mastery for the Digital SAT Math has 23 mini-tests, nearly five times more than traditional test prep programs.

Scripted Lessons Mastery for the Digital SAT is the only SAT prep program that provides comprehensive, scripted lesson plans.

The result is a consistent, score-raising experience for students and minimal planning time for teachers.

Pre-Built Hooks Traditional test prep programs are dry and follow the old “drill and kill” philosophy. If this is your first time running a test prep program, it can be challenging to introduce the content while bringing it to life. We’ve designed the program with hooks and activities built in, making each lesson interesting and engaging.

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Traditional test prep programs are built for highperforming students who are already solid testtakers. Students who struggle with tests also struggle in these programs. Mastery for the Digital SAT radically simplifies key strategies and walks students through each concept step-by-step.

Integrated Time Management Coaching By embedding time management coaching into every section, Mastery for the Digital SAT is by far the most reliable program for improving your students’ time management skills.

Slide Decks and Teacher Resources Unlike traditional test prep designed for tutors to use for weekend programs, we provide everything you need to run an Mastery for the Digital SAT class during the school day, right out of the box, including lesson plans, slide decks, and entrance and exit tickets.

Complete Mastery for the Digital SAT provides the most comprehensive coverage of SAT content dimensions for students at all score ranges.

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Curriculum for the Digital SAT®

What Mastery for the Digital SAT Includes Student Workbooks (4)

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Slide Decks (4 sets)

• Mastery for ther Digital SAT Writing

• Mastery for ther Digital SAT Writing

• Mastery for ther Digital SAT Reading

• Mastery for ther Digital SAT Reading

• Mastery for ther Digital SAT Math: Algebra and Geometry

• Mastery for ther Digital SAT Math: Algebra and Geometry

• Mastery for ther Digital SAT Math: Advanced Math and Data Analysis

• Mastery for ther Digital SAT Math: Advanced Math and Data Analysis

Students work directly in their workbooks. The workbooks allow students to follow along with the classroom activities and include all of the practice materials needed for success on the SAT. No printing required!

Every Mastery for ther Digital SAT chapter is accompanied by a slide deck to provide a visual focus for the class. Every exercise and question that appears in the student workbook also appears in the accompanying slide deck, along with its solution.

Teacher Manuals (4) • Mastery for ther Digital SAT Writing • Mastery for ther Digital SAT Reading • Mastery for ther Digital SAT Math: Algebra and Geometry • Mastery for ther Digital SAT Math: Advanced Math and Data Analysis Teacher manuals (like this one) provide everything you need to facilitate over 40 Mastery for ther Digital SAT lessons, including answer explanations, scripted lesson plans, and definitions of key terms.

Missing a resource?

Contact your school support manager directly by emailing support@masteryprep.com

WHAT MASTERY FOR THE DIGITAL SAT INCLUDES

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Curriculum for the Digital SAT®

Mastery for the Digital SAT Page by Page

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When you teach a Mastery for ther Digital SAT lesson, you’ll work from the teacher manual. Your students will be looking at their workbooks. On the projector will be the slide deck. There are a lot of moving parts. That’s why Mastery for ther Digital SAT uses a system to ensure that you and your class are always, literally, on the same page.

For every page in the student workbook (for example, page 218 on the left in the figure below), there are two pages in the teacher manual (for example, pages 218a and 218b on the right). This is always an exact match.

In the teacher manual, the “a” pages are always on the left, and the “b” pages are always on the right.

The entire student workbook is reprinted in the teacher manuals, so you don’t need to have the student workbook open and in front of you to track what your students are looking at.

STUDENT WORKBOOK MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Approach ENTRANCE Mean, median, and mode questions can be identified by these terms:

• mean • median • mode Review question 4 with your teacher. Circle the part of the question that identifies it as a mean, median, and mode question. Do not answer the question.

ENTRANCE

SOLVE, STRATEGIZE, OR SKIP

EXIT

4

Pets Survey Number of Pets

Town A

Town B

0

10

20

1

140

130

2

110

70

3

100

120

4

60

60

A prospective pet shop owner chose 400 families at random from each of two towns and asked each family how many pets they own. The results are shown in the table. There is a total of 2,800 families in Town A and 4,800 families in Town B. What is the median number of pets for all the families surveyed?

A) 0 B) 1 C) 2

TEACHER MANUAL

D) 3

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

MEAN, MEDiAN, AND MODE

Approach (12 minutes) MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

In this section, you will illustrate how the Entrance step can help students identify mean, median, and mode questions on the SAT.

Approach

SLIDE 218.1

ENTRANCE

I think you’re going to find that mean, median, and mode questions are some of the easiest to identify on the SAT. These questions aren’t even trying to disguise themselves. All we have to look for is at least one of these terms in the question (drumroll, please): mean, median, or mode.

Mean, median, and mode questions can be identified by these terms:

• mean • median • mode Review question 4 with your teacher. Circle the part of the question that identifies it as a mean, median, and mode question. Do not answer the question.

ENTRANCE

SOLVE, STRATEGIZE, OR SKIP

EXIT

SLIDE 218.2

Do you notice anything that identifies question 4 as a mean, median, and mode question?

The question says median.

Circle that identifier with your pencil.

218

Warm-Up

Foundation

Approach

Strategy

Mini-Test

4

Wrap-Up

Review Pets Survey Number of Pets

Town A

0 1

10 140

Town B 20 130

2

110

70

3

100

120

4

60

60

It’s important to take this step to make sure that you understand what kind of question you are being asked. Take 30 seconds to circle the identifying language in questions 5 and 6.

A prospective pet shop owner chose 400 families at random from each of two towns and asked each family how many pets they own. The results are shown in the table. There is a total of 2,800 families in Town A and 4,800 families in Town B. What is the median number of pets for all the families surveyed?

A) 0 B) 1 C) 2 D) 3

218

Warm-Up

Foundation

Approach

Strategy

218a

Mini-Test

Wrap-Up

Review

218b

MASTERY FOR THE DIGITAL SAT PAGE BY PAGE

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Curriculum for the Digital SAT®

Mastery for the Digital SAT Page by Page Each page in the student workbook (page 219, for instance) can have one or more slides that go with it. Each slide number starts with the matching page number in the student workbook and teacher manual (for example, 219.1, 219.2, and 219.3).

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STUDENT WORKBOOK, PAGE 219 MEAN, MEDiAN, AND MODE

Approach Review questions 5 and 6 on your own. Circle the part of the question that identifies it as a mean, median, and mode question. Do not solve the questions.

6

Questions 5 and 6 are examples of studentproduced response questions.

Amount of Hourly Pay Earned by 10 Employees

5 A focus group gives ratings of between 0 and 40, inclusive, for a new product. In the first 10 ratings, the average (arithmetic mean) of the ratings was 29. What is the least value the focus group can give for the 11th rating and still be able to have an average of at least 32 for the first 15 ratings?

Employee

Hourly Pay (Dollars)

Employee

Hourly Pay (Dollars)

Employee 1

34

Employee 6

43

Employee 2

43

Employee 7

46

35

Employee 8

49

35

Employee 9

39

36

Employee 10

38

Employee 3 Employee 4 Employee 5

The table above lists the hourly pay of various employees. According to the table, what was the mean hourly pay, in dollars, of these employees? (Round your answer to the nearest tenth.)

219

Warm-Up

Foundation

SLiDE 1 FOR PAGE 219

Strategy

Mini-Test

SLiDE 2 FOR PAGE 219

SLiDE 3 FOR PAGE 219

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Approach

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Wrap-Up

Review

219

Curriculum for the Digital SAT®

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If you want to start a lesson on pages 212a and 212b in your teacher manual, have your students turn to page 212 in their workbooks, and pull up slide 212.1 on the projector.

TEACHER MANUAL, PAGE 212 MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

MEAN, MEDiAN, AND MODE

Foundation (17 minutes) MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

In this section, students will find the mean and median for a few sets.

Foundation Work on your own to plot each set of values on a number line and determine the mean and median for questions 1–3.

1.

1, 2, 3, 4, 5 Mean: _______________ Median: _____________

2

3

In question 1, the median is 3. How are the other values spaced around that point?

They’re all evenly spaced.

Since they’re evenly spaced, that means the median and mean are the same value. How about question 2?

There are two values at 1, so there is a gap between 1 and 3.

I like to pronounce “median” as “mid-dian” to remind myself that median is looking for the middle term in the set. Work on your own to answer questions 1 through 3. You have 2 minutes.

Number Line:

1

Let’s review the difference between mean and median. To find the mean of a set of data, add all of the values in the set together, then divide by the number of values in the set. To find the median, list the numbers from least to greatest and locate the value in the middle of the set. If there are an even number of values in the set, take the mean of the middle two numbers to determine the median.

4

5

6

7

8

9 10

The middle number, the median, is still the same at 3. With the uneven distribution to the left, though, the mean will get pulled to the left. That makes it smaller than the median. How are the points spread in question 3?

There is a big gap between 4 and 10, but the rest are spread evenly.

SLIDE 212.1

Review the answers when time is up.

2.

1, 1, 3, 4, 5

The value at 10 pulls the mean higher than the median, but the median still remains at 3. The point at 10 is called an outlier. It is farther away from the rest of the points and pulls the mean away from the median. It also stretches the range, or the difference between the highest and lowest number in the data set.

ANSWER KEY

Mean: _______________

SLIDE 212.2

Median: _____________

1. Mean:

Number Line:

1+ 2 + 3 + 4 + 5 =3 5

Median: 3

Question 2 also had a mean different from the median, but would you say it also had an outlier?

No.

Number Line: 1

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9 10

8

9 10

Even though we do see an effect on the mean in question 2, there isn’t an outlier. An outlier is much larger or much smaller than the other values in the data set.

SLIDE 212.3

3.

1+ 1+ 3 + 4 + 5 = 2.8 2. Mean: 5 Median: 3

1, 2, 3, 4, 10

Number Line:

Mean: _______________ Median: _____________ Number Line:

1

2

3

4

5

6

7

SLIDE 212.4

1

2

3

4

5

6

7

8

1 + 2 + 3 + 4 + 10 == 4 3. Mean: 5 Median: 3

9 10

Number Line: 212

Warm-Up

Foundation

Approach

Strategy

Mini-Test

Wrap-Up

Review

1

212a

2

3

4

5

6

7

8

9 10

212b

SLiDE 1 FOR PAGE 212

212

MASTERY FOR THE DIGITAL SAT PAGE BY PAGE

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Curriculum for the Digital SAT®

Mastery for the Digital SAT Page by Page

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As you go through the dialogue on page 212b, helpful slide call-outs tell you when the next slide is relevant (for example, 212.2 and 212.3).

Once you get to the bottom of page 212b, you’ll turn to the next pages in your teacher manual, 212a and 213b, while having your students look at page 213 in their workbooks. You’ll also click the slides to 213.1.

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SLiDE 1 FOR PAGE 212

SLiDE 2 FOR PAGE 212

SLiDE 3 FOR PAGE 212

SLiDE 4 FOR PAGE 212

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Curriculum for the Digital SAT®

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TEACHER MANUAL MEAN, MEDiAN, AND MODE

Foundation (17 minutes) In this section, students will find the mean and median for a few sets.

Let’s review the difference between mean and median. To find the mean of a set of data, add all of the values in the set together, then divide by the number of values in the set. To find the median, list the numbers from least to greatest and locate the value in the middle of the set. If there are an even number of values in the set, take the mean of the middle two numbers to determine the median.

In question 1, the median is 3. How are the other values spaced around that point?

They’re all evenly spaced.

Since they’re evenly spaced, that means the median and mean are the same value. How about question 2?

There are two values at 1, so there is a gap between 1 and 3.

I like to pronounce “median” as “mid-dian” to remind myself that median is looking for the middle term in the set. Work on your own to answer questions 1 through 3. You have 2 minutes.

The middle number, the median, is still the same at 3. With the uneven distribution to the left, though, the mean will get pulled to the left. That makes it smaller than the median. How are the points spread in question 3?

There is a big gap between 4 and 10, but the rest are spread evenly.

SLIDE 212.1

Review the answers when time is up.

The value at 10 pulls the mean higher than the median, but the median still remains at 3. The point at 10 is called an outlier. It is farther away from the rest of the points and pulls the mean away from the median. It also stretches the range, or the difference between the highest and lowest number in the data set.

ANSWER KEY SLIDE 212.2

1+ 2 + 3 + 4 + 5 =3 5 Median: 3

1. Mean:

Question 2 also had a mean different from the median, but would you say it also had an outlier?

No.

Number Line: 1

2

3

4

5

6

7

8

9 10

8

9 10

Even though we do see an effect on the mean in question 2, there isn’t an outlier. An outlier is much larger or much smaller than the other values in the data set.

SLIDE 212.3

1+ 1+ 3 + 4 + 5 = 2.8 5 Median: 3

2. Mean:

Number Line:

1

2

3

4

5

6

7

SLIDE 212.4

1 + 2 + 3 + 4 + 10 == 4 5 Median: 3

3. Mean:

Number Line: 1

2

3

4

5

6

7

8

9 10

212b

MASTERY FOR THE DIGITAL SAT PAGE BY PAGE

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Curriculum for the Digital SAT®

Structure of a Mastery for the Digital SAT Lesson

Every Mastery for ther Digital SAT lesson has the same structure. This sequence of content is designed to help students master SAT content dimensions. It also enables them to gradually build test-taking and time management skills until they’re second nature.

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At the bottom of many student workbook pages is this tracker that shows where you are in the lesson. Every lesson has the same tracker.

1

2

4

5

6

7

Warm-Up

Foundation

Approach

Strategy

Mini-Test

Wrap-Up

8

Review

3

1. Warm-Up: Three timed practice questions introduce your students to the topic and help you differentiate instruction.

2. Foundation: A scaffolded lesson reviews the basics of the content dimension and provides students with skill-building activities that improve their grasp of content.

3. Practice Sets: Three sets of authentic test items are interspersed throughout the lesson, providing isolated practice and a way to hone newly acquired skills.

6. Mini-Test: Students apply what they have learned to a timed emulation of a real SAT testing experience.

Questions from the lesson are mixed with other SAT items.

7. Wrap-Up: Three more timed questions, at the same difficulty level as the warm-up items, help you evaluate students’ progress.

8. Review: A one-page cheat sheet provides a handy recap for students to study before test day.

4. Approach: Students learn how to identify question types, eliminate incorrect answers, and double-check their work.

5. Strategy: There is more than one way to earn points. Students learn test-taking strategies that help them arrive at the right answer even if they’re stuck.

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MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Curriculum for the Digital SAT®

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Every section of an Mastery for ther Digital SAT lesson is designed to stand on its own.

individual sections to suit your students’ needs.

For example, if you feel your students have mastered the instructional content but need additional practice, you could have them focus only on the Warm-Up, Practice Sets, Mini-Test, and Wrap-Up.

You can use the lessons one after another for a onesemester or full-year course. However, you’re also free to pick and choose resources from lessons and

ExTRA PRACTICE

Foundation

Warm-Up

Approach

Strategy

Mini-Test

Wrap-Up

Review

Or if you want to give your students a one-week cram just before test day, you could stick to Warm-Up, Strategy, and Wrap-Up for a lightning-fast review of tips and tricks.

ONE-WEEK CRAM

Warm-Up

Foundation

Approach

Strategy

Mini-Test

Wrap-Up

Review

Teachers who want to avoid test-taking tips and focus on content might prefer the Foundation and Approach sections.

CONTENT FIRSTMASTERY OF THE DIGITAL SAT FEATURES

Warm-Up

Foundation

Approach

xIII

Strategy

Mini-Test

Wrap-Up

Review

For more ideas about how to make the most of Mastery for ther Digital SAT’s modular lesson design, see “How to Use Mastery for ther Digital SAT” later in this guide.

STRUCTURE OF A MASTERY FOR THE DIGITAL SAT LESSON

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Curriculum for the Digital SAT®

MATH

Advanced Math and Data Analysis

Table of Contents How Mastery for the Digital SAT Works.....................................................................................................................................i Chapter 1: Math Introduction ....................................................................................................................................................... 1 Chapter 2: Ratios and Proportions ...........................................................................................................................................13 Chapter 3: Rates...............................................................................................................................................................................41 Chapter 4: Unit Conversion ........................................................................................................................................................ 67 Chapter 5: Interpreting Slope .....................................................................................................................................................91 Chapter 6: Scatterplots and Line Graphs ........................................................................................................................... 123 Chapter 7: Statistics...................................................................................................................................................................... 151 Chapter 8: Percentages ...............................................................................................................................................................179 Chapter 9: Mean, Median, and Mode ..................................................................................................................................209 Chapter 10: Isolating Expressions ..........................................................................................................................................237 Chapter 11: Equivalent Expressions.......................................................................................................................................265 Chapter 12: Polynomial Functions .........................................................................................................................................297 Chapter 13: Nonlinear Equations ...........................................................................................................................................325 Chapter 14: Applying Exponents .......................................................................................................................................... 355 Contributors ................................................................................................................................................................................... 383

Curriculum for the Digital SAT®

RATIOS AND PROPORTIONS CONTENT DiMENSiONS

CHAPTER 2

Rates, proportions, and units Apply proportional relationships, ratios, rates, and units in a wide variety of contexts.

Examples include, but are not limited to, scale drawings and problems in the natural and social sciences.

Ratios and Proportions This chapter will teach students to solve problems

LEARNiNG TARGETS

using ratios and proportions. Specifically, students

1. Use proportions and ratios to solve word problems.

will learn how to apply proportional relationships and ratios to numerical and real-world problems.

2. Implement the basic approach for ratios and proportions questions.

3. Apply the Estimate and Factor Elimination strategies to ratios and proportions questions.

2 CHAPTER

Ratios and Proportions

SCORE DiFFERENTiATiON Skip the Foundation segment if ...

SAT

PSAT

LEARNING TARGETS 1. Use proportions and ratios to solve word problems.

2. Implement the basic approach for ratios and proportions questions.

3. Apply the Estimate and Factor Elimination strategies to ratios and proportions questions.

13

13

Math above 500 Problem Solving and Data Analysis above 8 Math above 480 Problem Solving and Data Analysis above 8

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Warm-Up

Complete the warm-up. When you have finished, double-check your work.

1

2 Propagation Period versus Height Expectancy

The cost of a pair of shoes at Store B is 7 approximately of the cost at Store A. The cost 8 19 of a pair of shoes at Store C is approximately 8

Height expectancy (feet)

12

of the cost at Store A. If a pair of shoes costs $80 at Store A, approximately how much more does it cost at Store C than at Store B ?

A)

$70

B)

$91

D

10

C) $120

8

B

6

A

C

4 2 0

10 15 20 25 30 35 40 45 50 55 60 65 70 Propagation period (days)

A botanist at a plant society created the scatterplot above to understand the relationship between the propagation period and height expectancy of 10 species of plants.

D) $190

Which labeled point represents the plant that has the greatest ratio of height expectancy to propagation period?

A) A B) B C) C D) D

3 A baker takes 3 cupcakes out of every 500 cupcakes home for his family. At this rate, how many cupcakes will be taken home if the baker makes 25,000 cupcakes?

A) 100 B) 150 C) 200 D) 250

14

Warm-Up

Foundation

Approach

Strategy

14a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Warm-Up

(12 minutes)

SLIDE 14.1

Give the class 6 minutes to complete the warm-up. Review the answers briefly before beginning the lesson.

SLIDE 14.2

SLIDE 14.4

1. The correct answer is C. Because the cost of 7 a pair of shoes at Store B is approximately 8 7 of the cost at Store A, multiply by 80 to find 8

3. The correct answer is B. The ratio of cupcakes 3 that the baker takes home is . The ratio 500 of cupcakes that the baker takes home out of 25,000 cupcakes, x, can be represented by x .

25,000 Set the ratios equal to each other. x 6 = 500 25,000

that the cost of the shoes at Store B is $70.

Because the cost of a pair of shoes at Store C is approximately multiply

19 of the cost at Store A, 8

19 by 80 to find that the cost of shoes 8

Multiply both sides by 25,000.

3 x= · 25,000 500

at Store C is $190.

The question asks for the difference between the costs of the pairs of shoes at Store C and Store B. To find the answer, subtract the values as follows: $190 − $70 = $120.

Simplify. x = 150

SLIDE 14.3

2. The correct answer is A. Height expectancy is the y-axis, and propagation period is the x-axis.

The ratio of height expectancy to propagation period for each point is the ratio of the y to x values. Calculating this for all points shows 6 1 that point A has the highest ratio: = .

18 3

14b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

RATiOS AND PROPORTiONS

Foundation Complete the class activity following your teacher’s instructions.

1.

Length of volunteer’s foot: _______________

2.

Length of statue’s foot: __________________

3.

Height of volunteer: _____________________

4.

Height of statue: ________________________

Work with your teacher to identify the given ratio in the question. Do not solve.

5.

At a ballpoint pen factory, an inspector randomly selects 12 pens for inspection from a batch of 500 pens. At this rate, how many pens will the inspector select for inspection from a day’s production of 15,000 pens?

Given ratio: _____________________________

Work on your own to identify the given ratio in the question. Do not solve.

6.

For two similar right triangles, the hypotenuse of the smaller triangle is 5 inches, and the hypotenuse of the larger triangle is 10 inches. If the shorter side of the smaller triangle is 3 inches, what is the length of the shorter side of the larger triangle?

Given ratio: _____________________________

Warm-Up

Foundation

Approach

Strategy

15a

Mini-Test

Wrap-Up

Review

15

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Foundation (17 minutes) In this section, you will introduce students to the basics of ratios and proportions. Students will see how proportions can model word problems and how they can be solved by cross multiplication or scaling.

How tall would the action figure be?

65 ÷ 10 = 6.5 inches

SLIDE 15.1

We are going to build a statue in front of the school. One person in this class will be the inspiration. Hundreds of years from now, people will look at this heroic statue in awe and wonder.

Whom should we honor with the statue?

These examples show how we can use ratios to scale the size of something up, like we did for the statue, or scale down, like we did for the action figure. Many SAT questions ask you to use this idea.

Wait for a volunteer and have them come to the front of the class. We assume the student is “Michael” in this dialogue.

SLIDE 15.2

The keys to solving these problems are: 1) identifying a ratio from given information, 2) setting up a proportion, and 3) solving that proportion. A proportion is an equation stating that two ratios are equal. We’ll practice working through this process step by step.

We want this statue to be majestic: 20 times as tall as Michael. However, we want it to be as accurate as possible in terms of Michael’s proportions. For example, we don’t want the feet to be too long or too short; that would look weird.

How can we figure out how long the statue’s feet should be?

Measure the length of Michael’s foot, then make the statue’s foot 20 times as long.

Look at question 5 in your workbook. There are three quantities given. A ratio relates two quantities at a time, so let’s relate 12 pens selected to 500 pens in the whole batch.

Writing this ratio as a fraction, we get

Measure the length of the volunteer’s foot.

If a ruler or measuring tape is not available, estimate.

selected 12 : . total 500 Notice how we labeled the numerator (the top) and the denominator (the bottom).

How long should the statue’s feet be?

20 · (length of Michael’s foot) Have the volunteer return to their seat.

SLIDE 15.3

Take 30 seconds to identify a ratio in question 6.

Always label the numerator and denominator.

The statue will represent Michael’s features without distorting him, but it will make him 20 times larger. This is because we used a ratio to solve this question. A ratio is a numerical way to describe the size relationship between two quantities. We often see ratios written as fractions.

What did your ratio look like?

3 small 5 shorter : or : large 10 hypotenuse 5

There’s more than one way to correctly relate quantities with a ratio. Relating 5 inches to 10 inches is correct since those are both lengths of a hypotenuse. Relating 3 inches to 5 inches is also correct since those are side lengths of the same triangle.

Next, we decide to make a small action figure of Michael instead of a huge statue.

Ask the volunteer for their height and convert it to inches. Assume for this example that the volunteer is 65 inches tall.

Always relate “like with like.” Writing the ratio 3 for this problem would be incorrect, since 3 10 inches is the short-side length of the smaller

Let’s design the action figure so the ratio of Michael’s foot length to the foot length of the 10 action figure is 10 to 1, or . What do you think 1 we can figure out from this information?

triangle, while 10 inches is the hypotenuse length of the larger triangle. There’s nothing alike about them.

The height of the action figure.

15b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Foundation Work with your teacher to write a proportion for the question. Do not solve.

5.

At a ballpoint pen factory, an inspector randomly selects 12 pens for inspection from a batch of 500 pens. At this rate, how many pens will the inspector select for inspection from a day’s production of 15,000 pens?

Proportion: _______________________

Work on your own to write a proportion for the question. Do not solve.

6.

For two similar right triangles, the hypotenuse of the smaller triangle is 5 inches, and the hypotenuse of the larger triangle is 10 inches. If the shorter side of the smaller triangle is 3 inches, what is the length of the shorter side of the larger triangle?

Proportion: _______________________

16

Warm-Up

Foundation

Approach

Strategy

16a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Foundation (17 minutes) SLIDE 16.1

SLIDE 16.2

Turn to the next page in your workbook. Question

Take 1 minute to set up a proportion for question 6.

5 is there again, but now we’re going to set up a

When time is up, review by calling on individual students.

proportion. We already have the ratio selected 12 : for the problem. Now we need total 500 to set up another ratio that relates the other

Let’s say your given ratio for question 6 was small 5 : . How would your proportion look? large 10 small 5 3 : = large 10 x

known quantity, 15,000, and the unknown quantity, x pens selected for inspection. How

How would your proportion look if you used shorter 3 : ? hypotenuse 5

should this ratio look if we’re staying consistent with the first ratio?

x shorter 3 : = 10 hypotenuse 5

x selected : 15,000 total

Both are correct, as long as your numerators and denominators mean the same thing.

It’s important to relate quantities the same way in both ratios. This lets us write a proportion, which is an equation between ratios: x selected 12 : = total 500 15,000

Why would

15,000 12 = be incorrect?

500 x

The numerators and denominators describe different kinds of quantities.

16b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Foundation Work with your teacher to write a proportion for the question. Do not solve.

5.

At a ballpoint pen factory, an inspector randomly selects 12 pens for inspection from a batch of 500 pens. At this rate, how many pens will the inspector select for inspection from a day’s production of 15,000 pens?

Proportion: _______________________

Work on your own to write a proportion for the question. Do not solve.

6.

For two similar right triangles, the hypotenuse of the smaller triangle is 5 inches, and the hypotenuse of the larger triangle is 10 inches. If the shorter side of the smaller triangle is 3 inches, what is the length of the shorter side of the larger triangle?

Proportion: _______________________

16

Warm-Up

Foundation

Approach

Strategy

16a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Practice Set 1 (12 minutes) SLIDE 19.1

Give the class 6 minutes to complete the practice set. Review the answers before moving to the next section.

SLIDE 19.2

SLIDE 19.3

5 to its 9 approximate decimal form, 4.56. The proportion 70 of vinyl records to inches is , where b is the 4.56 number of vinyl recordsin the stack, and h is

1. The correct answer is A. The number of students that were both in 8th grade and played golf is 5. The total number of students 5 1 = is 50, so the proportion is .

50 10

2. The correct answer is B. Convert 4

the height of the stack. Set h = 9 and solve for b. b 70 = 9 4.56

Cross multiply.

4.56b = 70 · 9 Simplify the right side of the equation.

4.56b = 630 Divide both sides by 4.56 and simplify. b ≈ 138.16 or about 140.

19b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Approach ENTRANCE Ratios and proportions questions can be identified by these terms:

• proportion • rate • ratio

... or by this feature:

• two related quantities being compared to each other Review question 4 with your teacher. Circle the part of the question that identifies it as a ratios and proportions question. Do not solve the question.

ENTRANCE

SOLVE, STRATEGIZE, OR SKIP

EXIT

4 A wildlife biologist at a fishery checks 9 fish at random for disease out of every 200 fish caught.

At this rate, how many fish will be checked if 10,000 fish are caught at the fishery?

A) 350 B) 400 C) 450 D) 500

22

Warm-Up

Foundation

Approach

Strategy

22a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Approach (12 minutes) In this section, you will illustrate how the Entrance step can help students identify ratios and proportions questions on the SAT. SLIDE 22.1

Now that you know the basics of ratios and proportions, it’s time to see how these questions are asked on the SAT. The first step to answering any question is Entrance. During Entrance, you identify what kind of question you are being asked. No matter how good you are at answering ratios and proportions questions, you must be able to identify which questions they are on the SAT, so you know when to put your new knowledge and skills into action.

SLIDE 22.2

Look at question 4 for identifying language such as proportion or ratio or features such as related quantities being compared to each other. Do you notice any of those identifiers?

The question has the word rate and compares a subset of the population of fish with a larger population of fish.

Circle those identifiers with your pencil.

These questions often specifically ask for a ratio or proportion, but if they don’t, they will still compare related quantities. When you see these identifiers in a question, there’s a very good chance you need to set up a proportion.

Ratios and proportions questions sometimes look a lot like rate questions. They often will literally say the word rate, after all! It turns out that proportions and rates are mostly interchangeable, so you wonʼt be in a bad place if your entrance step sends you either way. Today weʼre going to focus on the ratio and proportion way of solving things. The rate way of doing it is covered in the Rates chapter.

Circle the identifying language in questions 5 and 6.

22b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Approach SOLVE, STRATEGIZE, OR SKIP Step 1: Can I solve this quickly and easily?

1. Relate relevant quantities as ratios.

2. If the question’s answer choices have fractions, choose the one relating the specified quantities. The quantity coming after to or of in the question should go in the denominator.

3. If the question’s answer choices are whole numbers, set the unknown’s ratio equal to another ratio and solve the proportion.

4. If the question asks you to compare ratios, convert them to decimals.

Step 2: If not, can I apply a strategy?

Estimate: Look for opportunities to estimate, use number sense, and rule out answer choices that are unreasonable based on the numbers in the problem.

Factor Elimination: Find where the row and column of the data table mentioned in the question intersect, and eliminate choices whose numerators don’t divide that number.

Step 3: Otherwise, I should skip.

If the question is too tough to solve or strategize, skip it. Mark your best guess and move on.

Solve question 4 with your teacher.

ENTRANCE

SOLVE, STRATEGIZE, OR SKIP

EXIT

4 A wildlife biologist at a fishery checks 9 fish at random for disease out of every 200 fish caught.

At this rate, how many fish will be checked if 10,000 fish are caught at the fishery?

A) 350 B) 400 C) 450 D) 500

24

Warm-Up

Foundation

Approach

Strategy

24a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Approach (12 minutes) In this section, you will illustrate the steps to Solve, Strategize, or Skip for ratios and proportions questions using the three questions from the Entrance step.

With question 4, though, we skip part 2 of the solve step and go directly to part 3. Since the choices are whole numbers, what do we need to do?

SLIDE 24.1

Once you confirm you have a ratios and proportions question, it’s time to decide how to answer it. If you can solve the question quickly and easily, you should do so. If you’re stumped, you have other options: You will either need to strategize, using techniques such as Estimate or Factor Elimination, or skip the question. For now, we will focus on the solve part of the approach.

We’ll discuss strategize later.

Set the unknown’s ratio equal to another ratio and solve the proportion.

We know 10,000 fish are caught in the second case; let’s say x fish were checked out of that 10,000. How should this ratio look? x 10,000

Note: The questions discussed in this chapter are similar to those covered in the Rates chapter. You might find that the approach we use here can also be helpful for those questions, and vice versa. If you have already covered rates with your students, it’s a good idea to mention this.

That gives us a proportion of

9 x = .

200 10,000

How would it look if you solved by cross multiplication?

9 · 10,000 = 200x

SLIDE 24.2

Divide both sides by 200.

Look at question 4. The first thing to do is relate relevant quantities as ratios. Before we do that, let’s review what each of the given values specifically refers to.

x = 90,000 ÷ 200 x = 450 fish You could also solve this with a scale factor.

Since 200 fish are caught the first time and 10,000 fish are caught the second time, 10,000 ÷ 200 = 50 times as many fish are caught the second time. This means 9 · 50 = 450 fish would be checked the second time.

For example, the 9 means 9 fish checked for disease. How would you describe the other two given values?

200 fish caught and 10,000 fish caught.

The 9 fish checked were out of the 200 fish caught, so it makes sense to relate them. How would this ratio look?

We didn’t get to part 4 with this problem. What is its advice?

If the question asks you to compare ratios, convert them to decimals.

9 200

That makes it easier to compare the fractions.

This will help you with question 6 on the next page.

Are the answer choices fractions?

No, they’re whole numbers.

Now that we’ve reviewed the solve step for question 4, use it on questions 5 and 6 on your own.

If they were fractions, you would just need to find the right fraction, and your work would be done.

Flip back in your book to Practice Set 1, question 1. This is an example of a question that you answer using part 2 of the solve step.

24b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

RATiOS AND PROPORTiONS

Approach Solve questions 5 and 6 on your own.

5

6 In a group of 225 college students, 5 are majoring in education. At this rate, how many of 8,100 college students are majoring in education?

Average Home Prices for Different Cities from 2014 to 2017

A) 130

Year

City

2014

2015

2016

2017

Dough Hills

2,357,268

2,628,841

2,476,957

3,275,766

C) 230

Cashtown

3,522,549

4,070,228

4,585,950

5,879,718

D) 280

Bill City

375,497

360,236

487,876

490,185

B) 180

Moneyville

12,456,280 10,900,057

9,022,378 12,776,495

Dollar Springs

297,854

450,321

402,397

524,832

Los Pesos

1,364,475

1,547,665

1,430,444

1,648,254

The table above lists the average home prices for each of six different cities from 2014 to 2017.

Of the following, which city’s ratio of its 2014 home price to its 2017 home price is closest to Dollar Springs's ratio of its 2014 home price to its 2017 home price?

A) Dough Hills B) Cashtown C) Moneyville D) Los Pesos

Warm-Up

Foundation

Approach

Strategy

25a

Mini-Test

Wrap-Up

Review

25

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Approach (12 minutes) SLIDE 25.1

Give the class 4 minutes to answer questions 5 and 6. Review the answers afterward.

SLIDE 25.2

SLIDE 25.3

5. The correct answer is B. Relate the given quantities using a ratio.

6. The correct answer is B. Relate the

The ratio is number of college students majoring in education total number of college students 5 or 225 .

given quantities using ratios. The ratio is cost of a home in 2014 . cost of a home in 2017 Dollar Spring’s ratio of 2014 home price to 297, 854 ≈ 0.57.

2017 home price is 524, 832

Write a ratio using the given values. Let x represent the unknown value.

5 x = 225 8,100

Find the ratio of the 2014 to 2017 home price for the cities in the answer choices and convert to decimals.

Cross multiply to solve for x.

Dough Hills:

8,100 · 5 = 225x Simplify.

Cashtown:

3,522,549 ≈ 0.60 5, 879, 718

Moneyville:

12, 456, 280 ≈ 0.97 12, 776, 495

40,500 = 225x Divide both sides by 225. x = 180 students majoring in education

2, 357, 268 ≈ 0.72 3, 275, 766

1,364, 475 Los Pesos: 1,648,254 ≈ 0.83

The closest number to 0.57 is 0.60, Cashtown.

25b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Approach EXIT If you solved the question …

Double-check your work to catch any mistakes. Did you …

• use values that aren’t relevant to the question?

• set up a proportion with mismatched numerators and denominators?

If you used a strategy …

Mark the problem so you can come back to it after you answer the remaining questions. On the second pass, you can double-check that you used the strategy correctly or try to solve the question.

If you skipped the question …

Check that you guessed an answer and marked the question.

Work with your teacher to double-check the math on question 7. identify and correct any errors made by the student who attempted to solve it.

SOLVE, STRATEGIZE, OR SKIP

ENTRANCE

EXIT

7 Student reasoning:

Taco Filling and Tortilla Selection

In the chart, there are a total of 28

Filling

Tortilla

Chicken

Veggie

people picking tacos, but from that,

Flour

12

6

6 people chose veggie and flour, and

Corn

8

2

2 people chose veggie and corn.

The table above shows the type of taco filling and tortilla chosen by people at a party. Each person chose one filling and one tortilla. Of the people who chose flour tortillas, what fraction chose veggie as their filling?

9 14 1 B) 3 4 C) 14

A)

D)

26

Therefore, 8 people from the 28 chose veggies. That means that the fraction of people who chose veggie as their filling is

2 8 or .

7 28

2 7

Warm-Up

Foundation

Approach

Strategy

26a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Approach (12 minutes) In this section, you will illustrate the Exit step for ratios and proportions questions by double-checking an SAT item that has been solved incorrectly.

There is one other error to check for with this type of problem. What is it?

SLIDE 26.1

Once you’ve finished your calculations, you might think you’re done and ready for the next question. But don’t forget this important last step to every problem: the Exit step. If you solved the question, double-check your work. If you used a strategy or guessed, mark the question so you can review it on a second pass.

Setting up a proportion with mismatched numerators and denominators.

All of the denominators, the bottom numbers in the fractions, should be about the same thing. All of the top numbers, the numerators, should also be about the same thing. That’s why you should label your fractions. Always double-check this before moving on to the next problem.

SLIDE 26.2

Since we are focusing on the solve part of the Approach, we’re going to practice doublechecking your work. Look at question 7 in your workbook. A student has already attempted to answer this question. Take 2 minutes to doublecheck the work on this problem and identify any errors made by the student.

Let’s apply the ratios and proportions approach to a practice set.

What error did the student make?

The student found the ratio of all veggie-filled tacos to the total number of tacos. all veggie-filled tacos 8 (6 + 2) 2 = = = total number of tacos (12 + 8 + 6 + 2) 28 7

This type of error can happen easily if you use the wrong information from the table. Here, it happened because the student jumped straight to veggie and ignored an important part of the last sentence. What part did they ignore?

Of the people who chose flour tortillas.

The student should only consider people who chose flour tortillas. Any numbers involving corn tortillas are irrelevant to the question. How should they have answered the question?

The student should have set up a ratio of veggiefilled flour tortillas to the total number of flour tortillas.

6 1 veggie-filled flour = = all flour tortilla tacos 18 3

Only use values relevant to the question. If the student had remembered to check this, they would have avoided this error and gotten another question right.

26b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Strategy ESTIMATE You can answer some ratios and proportions questions if you look for opportunities to estimate, use number sense, and rule out answer choices that are unreasonable based on the numbers in the problem.

1. Estimate the scale factor between two known values in the question. For example, is one value roughly two times as large as the other?

2. Scale the other known value up or down by the same approximate scale factor.

3. Choose the answer closest to the result.

Review question 11 with your teacher. Use the Estimate strategy to answer the question as a class.

11 If 20 pancakes were stacked on top of each other in a column, the column would be approximately 5

3 inches tall. At this rate, which of the following 8

is closest to the number of pancakes it would take to make a 10-inch-tall column?

A)

30

B)

40

C)

80

D) 120

28

Warm-Up

Foundation

Approach

Strategy

28a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Strategy (12 minutes) In this section, you will demonstrate the Estimate strategy using three SAT items.

How many pancakes would there be if the number doubled?

SLIDE 28.1

It can be easy to foul up ratios and proportions questions by messing up arithmetic, hitting the wrong button, or being unclear about whether you should multiply or divide.

40 Now we choose the answer closest to the result.

What choice is closest to 40?

Choice B.

If you are unsure how to solve a ratios and proportions problem, strategize instead. One of the best and quickest techniques to use is the Estimate strategy. This strategy will allow you to use an estimated relationship between two values, the ratio, to approximate the new value.

Having an idea of about how big or small the new value should be will help you eliminate wrong answers and can even land you on the correct one.

That’s the correct answer.

You were able to solve this question without using your calculator because you applied your estimation skills.

You can always use this strategy to eliminate answer choices that don’t make sense with the given information.

This strategy is also used as a time saver. Even if you do know how to apply the solve step, when you are crunched for time, it’s often a good idea to use the Estimate strategy instead. Look out for obvious scale factors. That’s the clearest tipoff that Estimate is the way to go.

SLIDE 28.2

What ratio is being asked for in question 11?

The number of pancakes in a stack compared to the stack’s height.

What is the unknown value in this question?

The number of pancakes in a 10-inch stack.

Try this strategy on questions 12 and 13. Fair warning: you can’t follow the exact Estimate procedure for question 12, but you can still use the concept of estimation to arrive at the correct answer.

The first step is to estimate the scale factor between two known values in the question.

What comparison can you make with the given heights?

The known height for 20 pancakes is a little more than 5 inches. This is about half the height of the requested 10-inch stack.

What’s the scale factor?

Itʼs 2, since we have to multiply 5 inches by 2 to get a 10-inch stack.

What’s the next step in the strategy?

Scale the other known value up or down by the same approximate scale factor.

If the height of the stack of pancakes roughly doubles, what do you think that means for the number of pancakes in the stack?

It means the number of pancakes should also roughly double.

28b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

RATiOS AND PROPORTiONS

Strategy Work on your own using the Estimate strategy to answer questions 12 and 13.

12

13 Nikki earned $30 for 4 hours of work at her job today. How many dollars will she earn tomorrow if she works 10 hours at the same hourly rate?

A)

40

The number of cupcakes a baker sells is directly proportional to the number of people that come into the bakery. The baker sells 180 cupcakes when 10 people come into the bakery. How many cupcakes will the baker sell when 40 people come into the bakery?

B)

75

A) 1,800

C) 120

B)

900

D) 150

C)

720

D)

360

Warm-Up

Foundation

Approach

Strategy

29a

Mini-Test

Wrap-Up

Review

29

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Strategy (12 minutes) SLIDE 29.1

Give the class 4 minutes to complete questions 12 and 13. Review the answers when time has elapsed.

SLIDE 29.2

SLIDE 29.3

12. The correct answer is B. The estimated scale 10 factor from 4 hours to 10 hours is = 2.5.

4 Scale Nikki’s earnings for 4 hours up by 2.5 to

13. The correct answer is C. The estimated scale factor from 10 people to 40 people is 40 ÷ 10 = 4. Scale the known number of cupcakes up by multiplying by 4 and estimating.

estimate her earnings for 10 hours of work.

180 · 4 ≈ 200 · 4 ≈ 800

30 · 2.5 = 75

Since 720 is the answer closest to 800, choice C is the correct answer.

Since 75 is an answer, choice B is the correct answer.

29b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Mini-Test

Complete the mini-test. When you have finished, double-check your work.

1

2 Percent of University Students Who Studied Abroad

U.S. Major City Populations, 2010 to 2013 Year

City

University

Percent of students

University A

32.8%

Atlanta

2010

2011

2012

2013

367,423

352,490

477,386

479,645

University B

39.9%

Chicago

2,276,908

2,539,222

2,392,516

3,164,093

University C

33.4%

Dallas

1,489,217

1,689,155

1,561,217

1,798,939

University D

26.5%

Los Angeles

4,067,068

4,115,275

4,636,705

5,944,791

University E

40.5%

Miami

273,109

412,909

368,966

481,228

University F

46.8%

New York

13,589,001 11,891,262

9,842,835

13,938,337

University G

44.1%

The table above lists the population for six different major cities in the United States from 2010 to 2013.

A survey was given to college seniors at 30 different universities around the country asking if they had studied abroad during their sophomore or junior year. The table above shows the results from 7 of the universities. The median percent of students who studied abroad for all 30 universities was 38.92%. What is the difference between the median percent of students who studied abroad for these 7 universities and the median for all 30 universities?

Of the following, which city’s ratio of its 2011 population to its 2012 population is closest to Dallas’s ratio of its 2011 population to its 2012 population?

A) Atlanta B) Chicago C) Los Angeles

A) 0.08%

D) New York

B) 0.98% C) 1.35% D) 2.42%

34

Warm-Up

Foundation

Approach

Strategy

34a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Mini-Test (30 minutes) SLIDE 34.1

Give the class 20 minutes to complete the mini-test. Review the answers at the end and emphasize the ratios and proportions questions.

SLIDE 34.2

SLIDE 34.3

CENTER AND SPREAD

1. The correct answer is B. List the percentages of students given in the table from least to greatest.

RATES, PROPORTIONS, AND UNITS

2. The correct answer is B. Dallas’s population 1,689,155 ≈ 1.08. ratio from 2011 to 2012 is 1,561, 217 Find the ratio of the 2011 and 2012 population

26.5%, 32.8%, 33.4%, 39.9%, 40.5%, 44.1%, 46.8%

from each city in the answer choices and

The median is the number found in the middle of the list. In this case, the median percentage of the 7 universities is 39.9%.

convert to decimals.

Subtract the reported median of all universities from the median percentage of the 7 given universities to find the difference.

Atlanta:

352,490 ≈ 0.74 477,386

Chicago:

39.9 − 38.92 = 0.98

2,539,222 ≈ 1.06 2,392,516

Los Angeles: New York:

4,115,275 ≈ 0.89 4,636,705

11,891,262 ≈ 1.21 9,842,835

The closest to 1.08 is 1.06, Chicago.

34b

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Mini-Test (30 minutes) SLIDE 35.1

SLIDE 35.3

LINEAR FUNCTIONS

5. The correct answer is B. A total of 8 + 7 = 15 customers bought paperback textbooks, and 7 of them bought composition notebooks.

7 The ratio is .

15

3. The correct answer is A. The given function is in the form of w(d) = md + n. This is a linear function in slope-intercept form, and the coefficient of d represents the slope, or the rate of change. Because slope can be interpreted as the amount of change in the y variable for each change in the x variable, m can be interpreted as the amount of change in the ounces the kitten gains on each day during the period.

SLIDE 35.2

SLIDE 35.4

NONLINEAR FUNCTIONS

6. The correct answer is A. Plug in −3 for x.

2

2 � 3 �3 + 6 �3 ) –3(–3) +6 9+9+6 (–3) x2 – � 3x + 6 ( () = = = – �3 x– �3 – �3 �6 –

RATES, PROPORTIONS, AND UNITS

24 = –4 – �6

4. The correct answer is B. Since the measure 4 of of distance on Map X is approximately 15 4 the measure of distance on Map Y, multiply 15 by 150 to find that the measure of distance on Map X is 40.

Since the measure of distance on Map Z is 17 of the measure of distance approximately 15 on Map Y, multiply

RATES, PROPORTIONS, AND UNITS

17 by 150 to find that the 15

measure of distance on Map Z is 170.

Subtract the two values to find how many more units the measure of distance on Map Z is to Map X. 170 – 40 = 130

35b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Complete the wrap-up. When you have finished, double-check your work.

Wrap-Up 4

Question 6 is an example of a student-produced response question.

Diet and Exercise Information for Men and Women Diet only

Diet and exercise

Total

Men

8

14

22

Women

4

17

21

Total

12

31

43

6 Furlongs and miles are units of measure of distance. They are directly proportional such that 8 miles is equal to 64 furlongs. What distance, in furlongs, is equal to 3 miles?

The table above shows the number of men and women trying to lose weight through diet only or through diet and exercise. What fraction of the women are only dieting?

A)

4 43

B)

4 21

C)

1 3

D)

4 7

5 If 75 sheets of paper are stacked on top of each

1 inches 4 tall. At this rate, which of the following is closest other, the stack will be approximately 3

to the number of sheets of paper it would take to make a 12-inch-tall stack of paper?

A) 280 B) 325 C) 575 D) 900

38

Warm-Up

Foundation

Approach

Strategy

38a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

RATiOS AND PROPORTiONS

Wrap-Up (12 minutes) SLIDE 38.1

Give the class 6 minutes to complete the wrap-up. Review the answers afterward.

SLIDE 38.2

SLIDE 38.4

6. The correct answer is 24. Since 8 miles

4. The correct answer is B. The number of women dieting only is 4. The total number of

is equal to 64 furlongs, the proportional

women in the group is 21. The ratio of women only dieting to the total number of women is 4 21 .

relationship between miles to furlongs is M 1 = , where M is the number of miles and F F 8 is the number of furlongs. Set M = 3.

SLIDE 38.3

1 3 = 8 F

5. The correct answer is A. Convert 3

1 to its 4

Cross multiply.

3·8=1·F

decimal form, 3.25. The ratio of sheets of

Simplify.

F = 24

75 . The ratio of paper in the shorter stack is 3.25

sheets of paper in the 12-inch stack, x, can be found by solving the proportion

x 75 = .

3.25 12

Cross multiply.

3.25x = 75(12) Simplify.

3.25x = 900 Divide both sides by 3.25 and simplify. x ≈ 276.92 The closest answer choice is 280.

38b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ADVANCED MATH AND DATA ANALYSIS

MASTERY FOR THE DiGiTAL SAT: ADVANCED MATH AND DATA ANALYSiS

Review ENTRANCE Ratios and proportions questions can be identified by these terms:

• proportion • rate • ratio

... or by this feature:

• two related quantities being compared to each other SOLVE, STRATEGIZE, OR SKIP Step 1: Can I solve this quickly and easily?

Step 2: If not, can I apply a strategy?

1.

Relate relevant quantities as ratios.

2.

If the question’s answer choices have fractions, choose the one relating the specified quantities. The quantity coming after to or of in the question should go in the denominator.

Estimate: Look for opportunities to estimate, use number sense, and rule out answer choices that are unreasonable based on the numbers in the problem.

Factor Elimination: Find where the row and column of the data table mentioned in the question intersect, and eliminate choices whose numerators don’t divide that number.

3.

If the question’s answer choices are whole numbers, set the unknown’s ratio equal to another ratio and solve the proportion.

4.

If the question asks you to compare ratios, convert them to decimals.

Step 3: Otherwise, I should skip.

If the question is too tough to solve or strategize, skip it. Mark your best guess and move on.

EXIT If you solved the question …

If you used a strategy or skipped the question …

Double-check your work to catch any mistakes.

Did you …

Mark the problem so you can come back to it after you answer the remaining questions. On a second pass, double-check that you guessed an answer or try to solve the question.

• use values that aren’t relevant to the question?

• set up a proportion with mismatched numerators and denominators?

40

Warm-Up

Foundation

Approach

Strategy

40a

Mini-Test

Wrap-Up

Review

MATH

Algebra and Geometry

Table of Contents How Mastery for the Digital SAT Works.......................................................................................................................i Chapter 1: Math Introduction ......................................................................................................................................... 1 Chapter 2: Linear Equations..........................................................................................................................................13 Chapter 3: Interpreting Equations .............................................................................................................................39 Chapter 4: Linear Equations in Context ..................................................................................................................63 Chapter 5: Linear Equations in Two Variables........................................................................................................91 Chapter 6: Systems of Equations ...............................................................................................................................117 Chapter 7: Creating Systems of Equations .......................................................................................................... 147 Chapter 8: Linear Functions ........................................................................................................................................175 Chapter 9: Slope ............................................................................................................................................................203 Chapter 10: Solving Inequalities ...............................................................................................................................237 Chapter 11: Creating Inequalities ............................................................................................................................ 263 Contributors ...................................................................................................................................................................... 291

Curriculum for the Digital SAT®

LINEAR EQUATIONS CONTENT DiMENSiONS

CHAPTER 2

Linear equations, one variable Solve a linear equation in one variable, making strategic use of algebraic structure.

Linear Equations This chapter will teach students how to solve a linear

LEARNiNG TARGETS

equation in one variable, both for the variable and

1. Strategically use algebraic structure to

for other linear expressions involving the variable.

solve a linear equation in one variable.

Specifically, students will learn how to simplify a

2. Implement the basic approach for linear equations questions.

linear equation, apply inverse operations to solve

3. Apply the Solve for the Expression and Working Backward strategies to linear equations questions.

for a variable, and find the value of another linear expression from the solution of the equation.

SCORE DiFFERENTiATiON 2

Skip the Foundation segment if ...

CHAPTER

SAT

Linear Equations

PSAT

LEARNING TARGETS 1. Strategically use algebraic structure to solve a linear equation in one variable.

2. Implement the basic approach for linear equations questions.

3. Apply the Solve for the Expression and Working Backward strategies to linear equations questions.

13

13

Math above 480 Heart of Algebra above 8 Math above 460 Heart of Algebra above 8

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Warm-Up

Complete the warm-up. When you have finished, double-check your work.

1

3 If 5q = 45, what is the value of 10q – 4 ?

A)

6

B)

9

If

4 1 8 1 x– x = + , what is the value of x ?

9 27 9 3

C) 86 D) 90

Questions 2 and 3 are examples of student-produced response questions.

2 4(x – 15) – (5 + x) = 10 What value of x satisfies the equation above?

14

Warm-Up

Foundation

Approach

Strategy

14a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Warm-Up (12 minutes) SLIDE 14.1

Give the class 6 minutes to complete the warm-up. Review the answers briefly before beginning the lesson.

SLIDE 14.2

SLIDE 14.4

1. The correct answer is C. Divide both sides of the equation by 5 to find q.

3. The correct answer is 3. Multiply both sides of the equation by 27.

q=9

12x – x = 24 + 9

Plug in 9 for q into 10q – 4.

Combine like terms.

10(9) – 4 = 90 – 4 = 86

11x = 33 Divide both sides by 11.

SLIDE 14.3

x=3

2. The correct answer is 25. Distribute to eliminate the parentheses.

4x – 60 – 5 – x = 10 Combine like terms.

3x – 65 = 10 Add 65 to both sides of the equation.

3x = 75 Divide both sides of the equation by 3. x = 25

14b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

LiNEAR EQUATiONS

Foundation Write an equation modeling the story described in class.

Equation: ______________________________________________

Solve the equation you just wrote for x. x = ____________________________________________________

Review these examples of linear and nonlinear equations with your teacher.

Linear Equations

Nonlinear Equations

2x + 5 = 6

3x2 + 5 = 6

3,000 – 2x = 6 + 4x

Warm-Up

Foundation

x+

Approach

x = 12

Strategy

15a

Mini-Test

Wrap-Up

Review

15

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Foundation (12 minutes) In this section, you will introduce students to the parts of a linear equation.

Great. Let’s represent the story using an equation, but let’s say that we don’t know yet how much money Hannah started with. Call that amount x. Go ahead and write the equation in your workbook. You have 30 seconds.

SLIDE 15.1

Who would like an all-expenses paid trip to Las Vegas?

Pick a volunteer student. They will be referred to as Hannah here.

What did your equation look like?

10x – 500 = 1,000

Sorry, Hannah, you’re not really getting a trip, but we’re going to pretend you are for an example.

That’s just as good, right? Let’s say Hannah goes to Vegas and starts out hot, winning like crazy.

Not just doubling or tripling the money she started with, but some other multiple—by how much do we think Hannah multiplied the money she took to Vegas?

This is an example of a linear equation in one variable. The only variable, x, is to the first power.

It isn’t in a radical, like a square root.

Let’s solve this equation. Following the order of operations, we would multiply x by 10 first and then subtract 500, fitting with the steps of our story.

Wait for responses and decide on a multiple.

We will assume it is 10 here and on the slides.

When we solve such an equation, we work back through the story and undo each step in reverse order, just as we did when we figured out how much money Hannah started with. Take 30 seconds to solve your equation.

So Hannah is winning big and decides to make one more big bet before calling it quits. All good things must come to an end, and that includes Hannah’s luck—she loses. How much money do you think Hannah lost on the last bet?

How did you solve?

Add 500 to both sides of the equation, then divide both sides by 10. Hannah’s starting amount was $150.

Wait for responses and decide on an amount lost. We will assume it is $500 here and on the slides.

Every time you “undid” a step, you applied an inverse operation. Addition and subtraction are inverse operations of each other, just as multiplication and division undo each other.

Applying the correct inverse operations in the correct reverse order (that is, against the order of operations) is the key to solving linear equations.

Hannah isn’t flat broke, though. How much money did she have left?

Wait for responses and decide on an amount left. We will assume it is $1,000 here and on the slides.

To recap, Hannah multiplied her money by 10, then lost $500 on her last bet and was left with a total of $1,000. How can we figure out how much money Hannah started with?

Add $500 back to $1,000 to get $1,500, or how much money she had before losing on her last bet, then divide $1,500 by 10 to get $150.

SLIDE 15.2

There are two examples of linear equations in your workbook. x to the first power is just written as x, without an exponent. To the right are two examples of nonlinear equations. Why is each not linear?

The first equation has a variable raised to the second power. The second equation takes the square root of a variable.

To figure out the original amount of money Hannah had, we needed to work backward through the story and “undo” each step. Does the order in which we undo the steps matter?

Yes. If we divided $1,000 by 10 we would get $100, then adding $500 would give us $600 instead. This would have been incorrect.

15b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Foundation Review the Order of Operations chart with your teacher.

Order of Operations Parentheses

Exponents

Multiply/Divide

Add/Subtract

2(5 + 1)2 – 10 = _______________

Review this example with your teacher.

For each question, write the sequence of operations applied to x in the correct order, then write the sequence of inverse operations that would cancel those operations.

1. 2x + 4

5.

Operations:

x– �2 4

Operations:

Inverse operations:

Inverse operations:

2. 3x – 6

6. 3(x – 4)

Operations:

Operations:

Inverse operations:

Inverse operations:

3. –2 + 10x

7.

Operations:

Operations:

Inverse operations:

Inverse operations:

4.

2 x+5 3

x +5 3

Operations: Inverse operations:

16

Warm-Up

Foundation

Approach

Strategy

16a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Foundation (12 minutes) In this section, you will review identifying operations and their inverses in linear expressions.

Inverse operations also follow PEMDAS, in reverse. SADMEP. What operations would undo the operations in question 1, and in what order?

Subtract 4. Then divide by 2.

SLIDE 16.1

I’ve mentioned “the order of operations” several times. You can see it illustrated in the diagram in your workbook. The arrow shows the general order you should use to apply operations in an expression. For example, if you see a minus sign and a multiplication sign, you should multiply before doing the subtraction, even if the minus sign comes first in the expression.

In the Inverse operations blank, write –4 and then ÷2.

Next look at question 2. What happens to x in that expression, and in what order?

It gets multiplied by 3, then 6 is subtracted.

Addition and subtraction are grouped together in the diagram since they have the same importance: take care of plus and minus signs one by one, working left to right. Multiplication and division are grouped together this way, too. However, exponents come before any of those four operations. If you see something that needs to be squared, handle that before adding, subtracting, multiplying, or dividing.

Next to Operations, write ×3 and –6. What operations would undo those, and in what order?

Add 6. Then divide by 3.

In the Inverse operations blank, write +6 and ÷3.

Take 3 minutes to complete questions 3–7.

When 3 minutes are up, review the answers.

Parentheses are the most important. If you see anything grouped in parentheses, the operations inside come before anything else.

ANSWER KEY SLIDE 16.3

Look at the example. In the expression 2(5 + 1)2 – 10, what is the first calculation?

Parentheses: 5 + 1 = 6

3. Operations: ×10, –2 Inverse operations: +2, ÷10

Next we have 2(6) – 10. What’s the next calculation?

Exponents: 62 = 36

4. Operations: ÷3, +5

Now we have 2(36) – 10. What’s the next calculation?

Multiplication: 2(36) = 72

5. Operations: –2, ÷ 4

2

Inverse operations: –5, ×3 SLIDE 16.4

It might seem that this goes against the order of operations, but the expression is equivalent to (x – 2) ÷ 4. It’s like every fraction has invisible parentheses around its numerator and denominator.

That gives us 72 – 10. What’s the last calculation?

Subtraction: 72 – 10 = 62

Inverse operations: ×4, +2 6. Operations: –4, ×3

SLIDE 16.2

Subtract first because of the parentheses.

Look at question 1. What happens to x in that expression, and in what order?

It gets multiplied by 2, then 4 is added.

Inverse operations: ÷3, +4 7. Operations: ×

Next to Operations, write ×2 and then +4.

2 , +5 3

2 3 or –5, × 3 2 To divide by a fraction, flip it to get its reciprocal, then multiply.

Inverse operations: –5, ÷

16b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

LiNEAR EQUATiONS

Practice Set 1

Complete the practice set. When you have finished, double-check your work.

1 If

Question 3 is an example of a student-produced response question.

5 6 p= , what is the value of p ?

7 5

A)

25 42

B)

30 35

C)

35 30

D)

42 25

3 9(x + 2) – 3(1 + 2x) = 60 What value of x satisfies the equation above?

LiNEAR EQUATiONS

Practice Set 1

Complete the practice set. When you have finished, double-check your work.

1 If A) B) C) D)

Question 3 is an example of a student-produced response question.

5 6 p= , what is the value of p ?

7 5

3

25 42

9(x + 2) – 3(1 + 2x) = 60

30 35

What value of x satisfies the equation above?

35 30 42 25

2 If

w w + 24 w = , what is the value of ?

4 16 2

A)

8

B)

4

C)

2

D)

1 4

Warm-Up

Foundation

Approach

Strategy

Mini-Test

Wrap-Up

Review

19

2 If

w w + 24 w = , what is the value of ?

4 16 2

A)

8

B)

4

C)

2

D)

1 4

Warm-Up

Foundation

Approach

Strategy

19a

Mini-Test

Wrap-Up

Review

19

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Practice Set 1 (12 minutes) SLIDE 19.1

Give the class 6 minutes to complete the practice set. Review the answers before moving to the next section.

SLIDE 19.2

SLIDE 19.4

3. The correct answer is 15. Distribute to eliminate the parentheses.

1. The correct answer is D. Multiply both sides of 7 the equation by .

5

9x + 18 – 3 – 6x = 60

�6 �7 7 �5 � ( )÷ p = � ( ÷ ) � 5 �7 � �5 �5

Combine like terms.

42 p= 25

Subtract 15 from both sides of the equation.

3x + 15 = 60 3x = 45 Divide both sides of the equation by 3.

SLIDE 19.3

x = 15

2. The correct answer is B. Cross-multiply to eliminate the fractions.

4(w + 24) = 16w Distribute.

4w + 96 = 16w Subtract 4w from both sides of the equation.

96 = 12w Divide both sides of the equation by 12. w=8 Plug in 8 for w into the expression. w 8 = =4 2 2

19b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Approach ENTRANCE Linear equations questions can be identified by this feature:

•

one equation with a single variable raised to the first power

... and the question asks for the value of a variable or expression

Review question 4 with your teacher. Circle and underline the parts of the question that identify it as a linear equations question. Do not solve the question.

ENTRANCE

EXIT

SOLVE, STRATEGIZE, OR SKIP

Question 4 is an example of a student-produced response question.

4 2(3x – 30) – (17 + 4x) = 11 What value of x satisfies the equation above?

20

Warm-Up

Foundation

Approach

Strategy

20a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Approach (12 minutes) In this section, you will illustrate how the Entrance step can help students identify linear equations questions on the SAT. SLIDE 20.1

SLIDE 20.3

OK everyone, pencils ready. You have 1 minute to solve this equation:

Why is question 4 a linear equations question?

There is one equation. It has only one variable, x.

The variable does not have an exponent or appear in a radical or fraction denominator.

SLIDE 20.2 3

Underline the equation. Circle the variables.

These are your identifiers. When a variable has no exponent, it’s understood to be to the first power. It has a 1 as its exponent; it’s just not written. If the variable has a 2 as its exponent, it’s to the second power. A 3 means to the third power, and so on. When a variable is in a radical, like a square root, or in the denominator of a fraction, it’s not to the first power.

x4 – � x 2 + 1 10 = 5 x 3x – � 2x3

Just kidding; believe it or not, I’m not some kind of monster. What about this equation makes it intimidating? I know, “everything.” But specifically, what are the parts of it that are troubling?

The radical, the exponents, the fractions.

Quick question before you try this on your own.

Look at question 6. Does it have two equations?

Guess what—when you have a linear equations question, you don’t have to worry about any of these things. They’re some of the nicer, easierto-solve equations that you’ll see. The best part is, if you get stuck on one, you can work backward from the answer choices to see which one is right. We’ll talk about that strategy later; right now we’re going to discuss identifying these questions.

No. It has one equation and one expression.

Find the identifiers for questions 5 and 6.

Underline the equations and circle the variables.

Entrance steps are supposed to be very fast, so I’ll give you 30 seconds.

The first step to answering any linear equations question is Entrance. During Entrance, you identify what kind of question you are being asked. Look for terms, features, or question patterns that are common to linear equations questions.

Like with most question types, the clue is in the name. A linear equations problem presents one equation. That equation uses one variable (but, like in question 4, it might use that same variable more than once). And that equation is linear, meaning the variable is to the first power.

20b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

LiNEAR EQUATiONS

Approach Review questions 5 and 6 on your own. Circle and underline the parts of the question that identify it as a linear equations question. Do not solve the problems.

5

6 If

If 4x + 5 = 8, what is the value of 8x + 3 ?

4 3 y = , what is the value of y ?

5 4

A)

3

3 5

B)

7

B)

15 16

C)

9

D) 16

C)

16 15

D)

5 3

A)

Warm-Up

Foundation

Approach

Strategy

21a

Mini-Test

Wrap-Up

Review

21

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Approach (12 minutes) SLIDE 21.1

Give the class 30 seconds to circle and underline the identifying language in questions 5 and 6. Review the answers afterward.

SLIDE 21.2

SLIDE 21.3

5. You can identify this as a linear equations question because it has one equation with one variable, y, to the first power.

6. You can identify this as a linear equations question because it has one equation with one variable, x, to the first power.

21b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Approach SOLVE, STRATEGIZE, OR SKIP Step 1: Can I solve this quickly and easily?

1. Simplify the equation by distributing across any parentheses and combining any like terms.

2. Use inverse operations to isolate the variable.

3. If the question asks for the value of the variable, find the answer choice matching the value you found. If the question asks for the value of another expression, plug in the value of the variable.

Step 2: If not, can I apply a strategy?

Solve for the Expression: When the question asks for the value of an expression, compare the coefficient of the expression with the coefficient of the equation. Look for multiples, factors, or any other way you can connect the two and use that connection to solve for the expression.

Working Backward: Relate the answers to the question and the rules, plug in each answer, and eliminate conflicts.

Step 3: Otherwise, I should skip.

If the question is too tough to solve or strategize, skip it. Mark your best guess and move on.

Solve question 4 with your teacher.

ENTRANCE

EXIT

SOLVE, STRATEGIZE, OR SKIP

Question 4 is an example of a student-produced response question.

4 2(3x – 30) – (17 + 4x) = 11 What value of x satisfies the equation above?

22

Warm-Up

Foundation

Approach

Strategy

22a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Approach (12 minutes) In this section, you will illustrate the steps to Solve, Strategize, or Skip for linear equations questions using the three questions from the Entrance step.

Some linear equations questions ask you to do more than just isolate the variable. They want you to use that value to solve for a new expression. We can see this in question 6. So after isolating the variable, you need to make sure you answered the question being asked.

What does question 4 ask you to find?

The value of x.

SLIDE 22.1

Once you know you have a linear equations question, the next step is choosing the best way to solve it. I mentioned earlier that we’d talk about working backward from the answer choices, but it’s actually quicker to just solve these questions when you can. Otherwise, you will either need to strategize or skip the question.

We’ll talk later about techniques to use when you’re stumped.

Since the question is asking for the value of x and not some other expression, we’re done.

These same three steps apply any time you need to solve a linear equations question on the SAT.

SLIDE 22.2

Look at question 4. What should we do first?

Simplify the equation. Distribute the 2 and the negative to get rid of the parentheses.

Now that we’ve reviewed this method for question 4, solve questions 5 and 6 on your own.

How does this change the equation?

It is now 6x – 60 – 17 – 4x = 11.

When we distributed the negative, it changed the sign of the terms in the parentheses. What should be the next step?

Combine like terms.

2x – 77 = 11 What’s our next step?

Isolate the variable using inverse operations. The opposite of –77 is +77, so begin by adding 77 to both sides of the equation.

2x = 88 Note: Students may try to divide by 2 first.

Remind them that inverse operations are applied in reverse order. Addition/subtraction is before multiplication/division.

What is the final step?

Divide both sides of the equation by 2. x = 44

22b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

LiNEAR EQUATiONS

Approach Solve questions 5 and 6 on your own.

5

6 If A)

If 4x + 5 = 8, what is the value of 8x + 3 ?

4 3 y = , what is the value of y ?

4 5

A)

3 5

B)

15 16

C)

16 15

D)

5 3

Warm-Up

3

B)

7

C)

9

D) 16

Foundation

Approach

Strategy

23a

Mini-Test

Wrap-Up

Review

23

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Approach (12 minutes) SLIDE 23.1

Give the class 4 minutes to answer questions 5 and 6. Review the answers afterward.

SLIDE 23.2

SLIDE 23.3

5. The correct answer is B. The equation does not need to be simplified. Use inverse operations to isolate the variable.

Multiply both sides of the equation by 5 4 3 �5 � � ( )� ( )÷ � ÷ y= � 4 �4 � �4 �5

y=

6. The correct answer is C. The equation does not need to be simplified. Use inverse operations to isolate the variable.

5 .

4

Subtract 5 from both sides of the equation.

4x = 3 Divide both sides of the equation by 4.

15 16

x=

3 4

Plug in

3 for x in the expression.

4

�3 � 8� ( )÷ + 3 = 6 + 3 = 9 �4 �

23b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Approach EXIT If you solved the question …

Double-check your work to catch any mistakes. Did you …

• subtract from both sides of an equation when you should have added?

• stop at solving for the variable when the question asked for an expression?

If you used a strategy …

Mark the problem so you can come back to it after you answer the remaining questions. On the second pass, double-check that you used the strategy correctly or try to solve the question.

If you skipped the question …

Check that you guessed an answer and marked the question for review.

Work with your teacher to double-check the math on question 7. Identify and correct any errors made by the student who attempted to solve it.

ENTRANCE

EXIT

SOLVE, STRATEGIZE, OR SKIP

Question 7 is an example of a student-produced response question.

Student reasoning:

7

4p = 28

If 4p = 28, what is the value of 5p + 6 ?

28 4p = 4 4 p=7

24

Warm-Up

Foundation

Approach

Strategy

24a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Approach (12 minutes) In this section, you will illustrate the Exit step for linear equations questions by double-checking an SAT item that has been solved incorrectly.

There is another error to look out for. Students often subtract when they need to add, or vice versa. Look at question 10 in practice set 2. You need to add 30 to both sides, but if you’re going too quickly, you might think, “I need to get rid of the –30.” So you get rid of it and then subtract 30 from the right side. Whoops! This error is listed as the first bullet point in the Exit procedure.

SLIDE 24.1

The final step is Exit. At this point, you need to wrap up the question and move on to the next one. If you solved the question, double-check your work. If you used a strategy or guessed at the answer, mark the question so that you can review it on a second pass.

We’ve covered all three parts of the Approach for linear equations questions. Let’s apply what we’ve learned to a practice set.

SLIDE 24.2

Since we are focusing on the solve part of the Approach, we’re going to practice doublechecking your work. Look at question 7. A student has already attempted to answer this question. Take 2 minutes to double-check the work on this problem and identify any errors made by the student.

What error did the student make?

The student selected the value of p, but the problem is asking for the value 5p + 6.

The student probably got excited when they found the value of p and forgot to plug it back in. How should the student have answered this question?

The student should have substituted 7 for p into 5p + 6.

Simplify.

5(7) + 6 = 35 + 6 = 41 Always double-check what the question is asking for instead of assuming it wants the value of the variable. You can bet that the SAT is going to give you questions meant to catch you in this kind of mistake!

24b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

LiNEAR EQUATiONS

Practice Set 2

Complete the practice set. When you have finished, double-check your work.

8

Question 10 is an example of a student-produced response question.

4n + n + n + n – 5 – 6 = 4 + n + n In the equation above, what is the value of n ?

10

A) –4 11 B) – 4 C) 3 D)

If 5x – 30 = 45, what is the value of x – 6 ?

15 4

9 x +6 = 12, what is the value of x ?

–6 x� 66 A) 13 B) 6

If

13 2 78 D) 11

C)

Warm-Up

Foundation

Approach

Strategy

25a

Mini-Test

Wrap-Up

Review

25

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Practice Set 2 (12 minutes) SLIDE 25.1

Give the class 6 minutes to complete the practice set. Review the answers before moving to the next section.

SLIDE 25.2

SLIDE 25.4

8. The correct answer is C. Combine like terms to simplify the equation.

10. The correct answer is 9. Add 30 to both sides of the equation.

7n – 11 = 4 + 2n

5x = 75

Subtract 2n from both sides of the equation.

Divide both sides of the equation by 5.

5n – 11 = 4

x = 15

Add 11 to both sides of the equation.

Plug in 15 for x into the expression.

5n = 15

x – 6 = 15 – 6 = 9

Divide both sides by 5. n=3 SLIDE 25.3

9. The correct answer is D. Multiply both sides of the equation by x – 6. x + 6 = 12(x – 6) Distribute. x + 6 = 12x – 72 Subtract x from both sides of the equation.

6 = 11x – 72 Add 72 to both sides of the equation.

78 = 11x Divide both sides of the equation by 11. x=

78 11

25b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Strategy SOLVE FOR THE EXPRESSION When the SAT asks you to solve for an expression, manipulate the equation instead of solving for a variable to save time. Make one side of the equation match the expression.

1. Compare the coefficients in the equation and the expression.

2. Make the coefficients equal by multiplying or dividing both sides of the equation.

3. If necessary, make the expression appear in the equation by adding or subtracting.

Review question 11 with your teacher. Use the Solve for the Expression strategy to answer the question as a class.

Question 11 is an example of a student-produced response question.

11 If 4q = 18, what is the value of 8q + 2 ?

26

Warm-Up

Foundation

Approach

Strategy

26a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Strategy (12 minutes) In this section, you will demonstrate the Solve for the Expression strategy using three SAT items.

We still don’t know what q is, but we do know that 8q + 2 is 38. That’s all the SAT is asking for, so that’s all the work we’ll do.

SLIDE 26.1

I’m going to give you a strategy that makes your work easier and quicker—who doesn’t like that?

We call this Solve for the Expression.

Questions 12 and 13 are just like this. Use the Solve for the Expression strategy to solve them on your own.

SLIDE 26.2

Note: This strategy may not work as well when the coefficient in the expression is not a factor or multiple of the coefficient in the equation. In this case, it may be faster to use the standard Solve procedure.

With question 11, how would we normally solve this?

Find the value of q, then find the value of 8q + 2.

That works, but it turns out we can skip a step.

When we need to find the value of an expression, we don’t actually have to find the value of the variable. We don’t have to solve for q. Instead, we can just find the value of 8q + 2. In other words, we’ll Solve for the Expression.

How do we do that? We make one side of the equation match the expression.

Usually we want to isolate q, to get it all alone on one side of the equation. Next we’re going to give it some friends. It can get lonely being a variable.

The first step is to make the coefficients match.

A coefficient is the number multiplied by a variable. For example, in question 11, the 8 in the expression is a coefficient. What is the coefficient in the equation?

The coefficient in the equation is 4.

How can we make it match the expression?

Multiply both sides of the equation by 2.

8q = 36 Does one side of the equation match yet?

What do we need to do?

No. Add 2 to both sides.

8q + 2 = 38

26b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

LiNEAR EQUATiONS

Strategy Work on your own using the Solve for the Expression strategy to answer questions 12 and 13.

12 If 3y + 5 = 7, what is the value of 6y + 2 ?

Question 13 is an example of a student-produced response question.

A)

2

13

B)

5

C)

6

If 3x – 12 = 33, what is the value of x – 4 ?

D) 14

Warm-Up

Foundation

Approach

Strategy

27a

Mini-Test

Wrap-Up

Review

27

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Strategy (12 minutes) SLIDE 27.1

Give the class 4 minutes to complete questions 12 and 13. Review the answers when time has elapsed.

SLIDE 27.2

SLIDE 27.3

12. The correct answer is C. Before calculating the answer, compare the expression to the equation. 6y in the expression is 2 times the value of 3y in the equation.

13. The correct answer is 11. Before calculating the answer, compare the expression to the equation. The coefficient in the equation is 3, while the coefficient in the expression is 1.

Make the coefficients equal by dividing both sides of the equation by 3.

Make the coefficients equal by multiplying both sides of the equation by 2.

x – 4 = 11

6y + 10 = 14 The question asks for 6y + 2, not 6y + 10, so the left-hand side must be reduced by 8.

Subtract 8 from both sides of the equation.

6y + 2 = 6

27b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Strategy WORKING BACKWARD When given a multiple-choice linear equations question, check which answer satisfies the equation.

1. Relate the answers to the question and the rules.

2. Plug in each answer.

3. Eliminate conflicts.

Review question 14 with your teacher. Use the Working Backward strategy to answer the question as a class.

14 4p + p + p + 6 + 2 = p + p + p – 7 In the equation above, what is the value of p ?

28

Warm-Up

A)

4

B) –

8 7

C)

–5

D)

15 7

Foundation

Approach

Strategy

28a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Strategy (12 minutes) In this section, you will demonstrate the Working Backward strategy using three SAT items.

Let’s try choice C. What happens when you plug in –5 for p?

SLIDE 28.1

Let’s say you have a multiple-choice linear equations question. It’s asking you for the value of z, and you’re having a brain freeze.

The equation becomes 4(–5) + (–5) + (–5) + 6 + 2 = (–5) + (–5) + (–5) – 7.

Simplify both sides of this equation and see if it’s true.

–22 = –22 is true, so choice C is the correct answer.

Here’s where Working Backward is the way to go.

This strategy is especially good for questions where solving for the variable algebraically could take a long time. Let’s look at question 14.

If choice C didn’t result in a true statement, what could you do next?

You could either continue to test answer choices, or, if you’re out of time, guess from the remaining two options.

SLIDE 28.2

With this strategy, we try out each answer choice until we find one that doesn’t break the equation.

What’s the first step?

Relate the answers to the question and the rules.

The other two choices would be pretty ugly to work with, so you might be better off guessing instead of spending a lot of time messing with fractions. This question is worth the same as any other multiple choice question, so you don’t want to spend too much time on it.

What’s the relationship between the answers and the question?

They are all possible values for p.

There aren't really any rules to worry about for this question, since p could be any kind of number. What's the next step?

Plug in each answer.

In any case, it’s always better to guess from two answer choices instead of four!

Try the Working Backward strategy on your own on the next page. Watch out: question 16 is trickier than the others.

Since the answers are all possible values for p, we will substitute each choice in for p in the equation and see what works. When you use Working Backward, always start with answer choices that look easy to work with. This saves you time. Which choices look easier to you and why?

Choices A and C because they are not fractions.

Let’s test choice A. What happens when you plug in 4 for p?

The equation becomes 4(4) + 4 + 4 + 6 + 2 = 4 + 4 + 4 – 7.

Simplify both sides of this equation and see if it’s true.

32 ≠ 5 This is what we’ll call a conflict in the Working Backward strategy, so that brings us to the third step: eliminate conflicts. We know that choice A leads to a conflict, so get rid of it.

28b

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Strategy (12 minutes) SLIDE 29.1

Give the class 4 minutes to complete questions 15 and 16. Review the answers when time has elapsed.

SLIDE 29.2

SLIDE 29.3

15. The correct answer is D. All of the choices are values for p. Since all the answer choices are fractions, start with a middle value. Based on the result, you will know if you need to test a larger or smaller number, which can save you time. Start with choice B, and plug it into the equation for p.

16. The correct answer is B. All of the choices are x values for . Start with choice C, and assume 6 x is equal to 6. This means x is 36.

6 Plug 36 into the equation for x and simplify.

6 24 = 36 36 + 30 1 24 = 66 6

�3 ��5 � 7 ( )÷� ( )÷ = � �5 ��7 � 3

Since there is one 5 in the numerator and denominator, they cancel.

3 7 = 7 3 Since the side with the variable is too small, you need to pick a larger choice. Try choice D, and plug it in for p.

Cross-multiply.

1 · 66 = 6 · 24 66 = 6 · 24 Since 66 is smaller than 6 · 24, eliminate choice C. Assume choice D is correct, since it’s the only other whole number option.

�3 ��35 � 7 ( ÷� )( ÷ )= � �5 �� 9 � 3

Since

Factor the 35 and cancel the 5s and a pair of 3s.

x is 10, x must be 60.

6

Plug 60 into the equation for x and simplify.

· 7� 7 �3 ��5 × ( )÷� ( )= ÷ � �5 ��3 ·×3 � 3

24 6 = 60 60 + 30

7 7 = 3 3

1 24 = 10 90

This is a true statement, so choice D is the correct answer.

Cross-multiply.

1 · 90 = 24 · 10 90 = 240 This is not a true statement, so eliminate choice D. The correct answer must be either choice A or choice B. In this case, it is choice B. This question highlights the importance of the first step in the Working Backward procedure.

If you don’t relate the answers to the question, then you might make the mistake of plugging in the choices as values for x. The right way to do Working Backward was to figure out what x was based on the answer choices, then plug that into the equation.

29b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Complete the mini-test. When you have finished, double-check your work.

Mini-Test 1

3 If

12 y 6 = , what is the value of ? y + 10 y 2

A) 10 B)

5

C)

2

D)

1 5

If

9 4 x = , what is the value of x ?

5 9

A)

20 81

B)

4 5

C)

5 4

D)

81 20

If

t +2 = 6, what is the value of t ?

–2 t�

A)

10 7

B)

7 3

C)

14 5

D)

4

2 3

Which of the following is equivalent to 25 4 ?

A)

3

25

B)

4

25

C)

4

5

D) 5 5

32

Warm-Up

Foundation

Approach

Strategy

32a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Mini-Test (30 minutes) SLIDE 32.1

Give the class 20 minutes to complete the mini-test. Review the answers at the end and emphasize the linear equations questions.

SLIDE 32.2

SLIDE 32.4

LINEAR EQUATIONS, ONE VARIABLE

1. The correct answer is B. Cross-multiply to eliminate the fractions.

3. The correct answer is A. Multiply both sides of 5 the equation by .

9 5 �9 � �4 �5 ( ÷ )x = � ( )÷ � 9 �5 � �9 �9

6(y + 10) = 12y Distribute.

6y + 60 = 12y Subtract 6y from both sides of the equation.

x=

60 = 6y Divide both sides of the equation by 6.

y 10 = =5 2 2

t + 2 = 6(t – 2)

EQUIVALENT EXPRESSIONS

Distribute.

2. The correct answer is D. Expand the exponent so that the fraction has 1 in the numerator.

This will make it easier to reduce the expression later on.

t + 2 = 6t – 12 Subtract t from both sides of the equation.

2 = 5t – 12 Add 12 to both sides of the equation.

1

(25) 25 4 = ( 25 4 )3

14 = 5t 1 4

1 1 2 2

Divide both sides of the equation by 5.

1 2

Convert the 25 into ( 2525) since 25 can be

t=

reduced to a whole number.

1

1

2 ) 2 )3 ( 2525

1

25 2 is equal to

LINEAR EQUATIONS, ONE VARIABLE

4. The correct answer is C. Because the question asks for the value of t, the goal is to isolate the variable in the equation. Multiply both sides of the equation by t – 2.

Plug in 10 for y in the expression.

1 3

20 81

SLIDE 32.5

y = 10

SLIDE 32.3

LINEAR EQUATIONS, ONE VARIABLE

25 , which is equal to 5.

1

Therefore, 25 2 can be replaced with 5.

3

1 � 3 3 (5 5 2 )� � ÷ = 5( 5 ) � �

· 55 · 5=5 5 5 55

32b

14 5

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

LiNEAR EQUATiONS

4

4

5

7 The amount of money a landscaper earns is directly proportional to the number of lawns the landscaper mows. The landscaper earns $96 on a day when he mows 6 lawns.

If 3x + 1 = 3, what is the value of 15x + 2 ?

A)

2

B)

9

C) 12

How much money will the landscaper earn if he mows 20 lawns in a week?

D) 15

A) $720 B) $480 C) $320

8

D) $200

2 7 x = , what is the value of x ?

11 7

If

6 4 p and p = 25, what is the value of 5 3m + 1 ?

If m =

A) 61 B) 46 C) 20 D) 15

Warm-Up

Foundation

Approach

Strategy

33a

A)

2 11

B)

22 49

C)

49 22

D)

11 2

Mini-Test

Wrap-Up

Review

33

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Mini-Test (30 minutes) SLIDE 33.1

RATES, PROPORTIONS, AND UNITS

SLIDE 33.4

8. The correct answer is C. Multiply both sides of 7 the equation by .

2

5. The correct answer is C. Since the landscaper earns $96 when he mows 6 lawns, he earns $96 = $16 per lawn. If the landscaper mows 6 20 lawns, he will earn 20 · $16 = $320.

SLIDE 33.2

7 �2 � � 7 �7 ( ÷ ) ( )x � ÷x = � 2 �7 � �11�2

x=

SYSTEMS OF EQUATIONS

6. The correct answer is A. Plug in 25 for p. m=

4 (25) = 20 5

Plug in 20 for m into the expression.

3(20) + 1 60 + 1 61 SLIDE 33.3

LINEAR EQUATIONS, ONE VARIABLE

7. The correct answer is C. Subtract 1 from both sides of the equation.

3x = 2 Divide both sides of the equation by 3. x=

2 3

Plug in

LINEAR EQUATIONS, ONE VARIABLE

2 for x into the expression.

3

�2 � ( ÷ ) +2 15 � �3 �

10 + 2 12

33b

49 22

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Wrap-Up

Complete the wrap-up. When you have finished, double-check your work.

4

6 If 6x + 8 = 13, what is the value of 12x + 4 ?

A)

2(p + 2) + 4(p – 2) = 3p

5

What value of p is the solution of the equation above?

B) 11 C) 14 D) 26

Questions 5 and 6 are examples of student-produced response questions.

5 7 3 q= 3 5 What value of q is the solution of the equation above?

36

Warm-Up

Foundation

Approach

Strategy

36a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

LINEAR EQUATIONS

Wrap-Up (12 minutes) SLIDE 36.1

Give the class 6 minutes to complete the wrap-up. Review the answers afterward.

SLIDE 36.2

SLIDE 36.4

6. The correct answer is 4 . Distribute to 3 eliminate the parentheses.

4. The correct answer is C. Subtract 8 from both sides of the equation.

6x = 5

2p + 4 + 4p – 8 = 3p

Divide both sides of the equation by 6. x=

Combine like terms.

5 6

6p – 4 = 3p

5 Plug in for x into the expression.

6 �5 � ( ÷ ) +4 12 � �6 �

Subtract 6p from both sides of the equation.

10 + 4 = 14

p=

–4 = –3p Divide both sides of the equation by –3.

SLIDE 36.3

9 . Multiply 35 both sides of the equation by 3 .

7 3 �7 � �3 �3 (� )q ( ) ÷q = ÷ 7 �3 � � �5 �7

5. The correct answer is .257 or

Simplify. q=

9 35

Either

9 or its decimal equivalent, .257, are 35

acceptable answers to input.

36b

4 3

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: ALGEBRA AND GEOMETRY

MASTERY FOR THE DiGiTAL SAT: ALGEBRA AND GEOMETRY

Review ENTRANCE

Linear equations questions can be identified by this feature:

• one equation with a single variable raised to the first power ... and the question asks for the value of a variable or expression

SOLVE, STRATEGIZE, OR SKIP

Step 1: Can I solve this quickly and easily?

Step 2: If not, can I apply a strategy?

1.

Simplify the equation by distributing across any parentheses and combining any like terms.

2.

Use inverse operations to isolate the variable.

3.

If the question asks for the value of the variable, find the answer choice matching the value you found. If the question asks for the value of another expression, plug in the value of the variable.

Solve for the Expression: When the question asks for the value of an expression, compare the coefficient of the expression with the coefficient of the equation. Look for multiples, factors, or any other way you can connect the two and use that connection to solve for the expression.

Working Backward: Relate the answers to the question and the rules, plug in each answer, and eliminate conflicts.

Step 3: Otherwise, I should skip.

If the question is too tough to solve or strategize, skip it. Mark your best guess and move on.

EXIT

If you solved the question …

If you used a strategy or skipped the question …

Double-check your work to catch any mistakes.

Did you …

Mark the problem so you can come back to it after you answer the remaining questions. On a second pass, double-check that you guessed an answer or try to solve the question.

• •

38

subtract from both sides of an equation when you should have added? stop at solving for the variable when the question asked for an expression?

Warm-Up

Foundation

Approach

Strategy

38a

Mini-Test

Wrap-Up

Review

Curriculum for the Digital SAT®

READING

Table of Contents How Mastery for the Digital SAT Works...................................................................................................................................... i Chapter 1: Reading Introduction ...................................................................................................................................................1 Chapter 2: Reading Closely .......................................................................................................................................................... 13 Chapter 3: Word Choice ............................................................................................................................................................... 39 Chapter 4: Function of a Detail .................................................................................................................................................. 59 Chapter 5: Illustrating Claims...................................................................................................................................................... 85 Chapter 6: Main Idea .....................................................................................................................................................................107 Chapter 7: Purpose ........................................................................................................................................................................133 Chapter 8: Conclusions ................................................................................................................................................................159 Chapter 9: Arguments ...................................................................................................................................................................185 Chapter 10: Quantitative Information .....................................................................................................................................211 Contributors .....................................................................................................................................................................................243

Curriculum for the Digital SAT®

Parenthetical Elements

ILLUSTRATING CLAIMS PARENTHETICAL ELEMENTS READING CLOSELY

CONTENT DIMENSIONS

CHAPTER 2

Determining implicit meaning The student will draw reasonable inferences

Reading Closely

and logical conclusions from text.

This chapter explains how to recognize reading

LEARNING TARGETS

closely questions on the SAT and introduces a brief,

1. Draw reasonable inferences and make

step-by-step process to eliminate incorrect answers.

logical conclusions from information and ideas explicitly stated in the text.

This chapter also demonstrates how to apply the Predict the Future strategy to answer reading closely

2. Implement the basic approach for reading closely questions.

questions.

3. Apply the Predict the Future strategy and utilize pacing key words to improve scanning techniques on reading closely questions.

2 CHAPTER

Reading Closely

SCORE DIFFERENTIATION Skip the Foundation segment if ...

LEARNING TARGETS 1. Draw reasonable inferences and make logical conclusions from information and ideas explicitly stated in the text.

2. Implement the basic approach for reading closely questions.

3. Apply the Predict the Future strategy and utilize pacing key words to improve scanning techniques on reading closely questions.

13

13

SAT

Reading and Writing above 500 No subscore

PSAT

Reading and Writing above 480 No subscore

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Warm-Up

Complete the warm-up. When you have finished, double-check your work.

11

2 The following text is adapted from Frances Hodgson Burnett’s The Secret Garden:

In 1901, Guglielmo Marconi and his team worked tirelessly to achieve the first transatlantic radio transmission. Due to the Earth’s curvature, standard radio signals could not travel long distances without losing strength and clarity.

Marconi’s team had to find a way to overcome this limitation in order to successfully send a wireless message across the Atlantic Ocean.

After much research and experimentation, they discovered that using lower-frequency radio waves, which could be refracted by the Earth’s ionosphere, would allow the signal to travel the required distance. In December 1901, Marconi’s groundbreaking achievement was realized when his team successfully transmitted a radio signal from Cornwall, England, to Newfoundland, Canada, revolutionizing long-distance communication.

She had begun to wonder why she had never seemed to belong to anyone even when her father and mother had been alive. Other children seemed to belong to their fathers and mothers, but she had never seemed to really be anyone’s little girl. She had had servants, and food and clothes, but no one had taken any notice of her. She did not know that this was because she was a disagreeable child; but then, of course, she did not know she was disagreeable. She often thought that other people were, but she did not know that she was so herself.

According to the text, what is true about the main character?

A) She believes she has always belonged to her parents.

According to the text, why was a standard radio signal unable to travel long distances without losing strength and clarity?

B) She acknowledges her own disagreeable nature.

A) Because of the Earth’s curvature and the limitations of standard radio signals

C) She feels she has never truly belonged to anyone.

B) Because Marconi’s team used the wrong type of equipment for their initial experiments

D) She is aware that she is the reason for her loneliness.

C) Because the Earth’s ionosphere was not yet discovered at the time of Marconi’s research D) Because radio signals were too weak to penetrate the Earth’s atmosphere during long-distance transmission

14

Warm-Up

Foundation

Approach

14a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Warm-Up (16 minutes) Give students 9 minutes to complete the warm-up. Review the answers briefly before beginning the lesson.

SLIDE 14.1

SLIDE 14.2

1. The correct answer is C. Choice A is incorrect because the text states that the main character feels she never really belonged to her parents, even when they were alive.

2. The correct answer is A. Choice B is incorrect because the text does not mention any issues with the equipment Marconi’s team used in their initial experiments. The limitation in the standard radio signals was due to the Earth’s curvature, not the equipment used.

Choice B is incorrect because the text specifically states that the main character does not know she is disagreeable.

Choice C is incorrect because the text does not state or imply when the ionosphere was discovered; the text only states that Marconi’s team discovered how low-frequency radio waves could be refracted by the Earth’s ionosphere.

Choice D is incorrect because while the main character is lonely, she does not realize that her own disagreeableness is the cause of her loneliness.

Choice C is correct because the text directly states that the main character feels she has never truly belonged to anyone, not even her own parents.

Choice D is incorrect because the text does not mention the strength of radio signals being unable to penetrate the Earth’s atmosphere.

Instead, it discusses the Earth’s curvature as the main reason for the limitation with standard radio signals.

Choice A is correct because the text clearly states that standard radio signals could not travel long distances without losing strength and clarity due to the Earth’s curvature. This limitation was what Marconi’s team had to overcome in order to successfully send a wireless message across the Atlantic Ocean.

14b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Foundation Fill in the blank with a distinguishing detail for each suspect. Then draw a line to the related piece of evidence from the crime scene.

Suspect:

Professor Reginald Smythe

Distinguishing detail:

Evidence:

A.

After securing the crime scene, a dimly lit gallery in a local art museum, investigators collected a number of intriguing clues.

A distinguished and slightly eccentric professor of archaeology, Professor Smythe is known for his extensive knowledge of ancient civilizations and artifacts.

He is never seen without his tweed jacket.

Miss Penelope “Penny” Featherstone

1. Under a bench they found a half-eaten chocolate bar.

2. On a nail protruding from the wall where the painting had been stolen, they discovered a small scrap of wool woven in a diagonal pattern.

B.

A quiet and unassuming librarian, Penny often goes unnoticed by those around her. Those who do know her admire her compassion for animals—she owns four cats—and are enamored by her gentle, genial spirit.

Madeline “Maddie” LaRue

3. Out of the trash can, they recovered a crumpled exhibition brochure. It included meticulous notes on many of the paintings— but curiously, notes on the stolen piece were conspicuously absent.

C.

4. Near the entrance of the gallery, they found a small gold locket with an engraved letter “M” on the front.

A young and ambitious reporter, Maddie is known for her nervous energy and obsessive note-taking. Her keen intuition and dogged determination often lead her to uncover secrets others would prefer to keep hidden.

16

Warm-Up

5. On the corner of a painting frame, they found a smattering of two- to three-inch hairs.

Foundation

Approach

16a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Foundation (18 minutes) In this section, students will practice identifying the core points and key phrases of a given detail.

Have you ever been in a car that was driving through the fog? How far ahead could you see?

For letter B, write owns four cats. Which piece of evidence could link to the cats?

Two- to three-inch hairs sounds like cat hairs.

Ask 1–2 students to share.

Draw a line from letter B to number 5 in the evidence column. One more distinguishing detail to find. What stands out about “Maddie” LaRue? Her nervous energy and obsessive note-taking.

What about the fog makes it harder to drive?

It’s hard to see the road ahead of you.

SLIDE 16.1

Do either of those line up with something from the evidence column? Which number?

Number 3 talks about a brochure that somebody wrote meticulous notes on.

Answering reading closely questions can be a little bit like driving in the fog. It can be hard to see where you’re going, but keeping an eye out for key details can keep you on the right track.

Let’s practice! Take a look at the first Foundation exercise in your book. What do you think our goal is here?

To look at the evidence. To solve the crime!

Write obsessive note-taking down for letter C and draw a line to number 3. Learning to align details that describe the same thing or communicate the same ideas with different words is really at the heart of reading closely questions. Certain details jump out of both questions and passages like clues. Connect the dots between them, and you’ve often found your answer!

We’ve got a painting heist on our hands, and it would be nice to find the guilty suspect!

But for the purposes of our test, our goal is to practice recognizing how the same ideas can be communicated with different words. In the exercise, you’ll match that detail to another short statement. For starters, what’s the first suspect’s name?

Professor Reginald Smythe.

Read over the professor’s description. What visual detail stands out?

His tweed jacket.

Under Distinguishing detail, next to letter A, write tweed jacket. Move on to the next column, Evidence, and look for something that could incriminate a person who wears a tweed jacket.

What is it?

Number 2. It mentions a small scrap of wool woven in a diagonal pattern.

Draw a line from letter A in the center column to number 2 in the right-hand column. Next, look back to the left-hand column with the suspect’s names. Who’s the second one, and what’s her distinguishing detail?

“Penny” Featherstone. She owns a whole bunch of cats.

16b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

READING CLOSELY

Foundation Match each sentence from the sentence banks to its correct description. Write each sentence’s letter in the space provided.

1. Sentence Bank A. Watching extreme sports can be relaxing for some.

B. Drag racing is the ultimate sleep aid.

C. The sound of roaring engines makes some people more productive.

Original

Some people watch drag racing to fall asleep.

Extreme Reversal

I could never fall asleep watching extreme sports.

Sounds Good Half-Wrong

I watch street luge to fall asleep.

2. Sentence Bank A. Interpretive dance is certainly a creative approach to communicating with squirrels.

B. Ballroom dance has been used to communicate with squirrels effectively.

C. Trained animals have been used in some ballet performances.

D. Interpretive dance is the ultimate squirrel communication technique.

Original

Interpretive dance is an effective way to communicate with squirrels.

Extreme Reversal

Interpretive dance isn’t an effective way to communicate with squirrels at all.

Sounds Good Half-Wrong

Warm-Up

Foundation

Approach

Strategy

17a

Wrap-Up

Review

17

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

READING CLOSELY

Practice Set 1

Complete the practice set. When you have finished, double-check your work.

21

22

In 1928, physicist Paul Dirac developed a groundbreaking equation that combined quantum mechanics and special relativity to describe the behavior of subatomic particles. His equation led to the prediction of the existence of antimatter, specifically the antielectron, or positron. The discovery of the positron in 1932

by Carl David Anderson confirmed Dirac’s prediction and revolutionized the understanding of particle physics, ultimately earning Dirac the Nobel Prize in Physics in 1933.

The following text is adapted from L. Frank Baum’s The Wonderful Wizard of Oz: The sun had baked the plowed land into a gray mass, with little cracks running through it. Even the grass was not green, for the sun had burned the tops of the long blades until they were the same gray color to be seen everywhere. Once the house had been painted, but the sun blistered the paint and the rains washed it away, and now the house was as dull and gray as everything else.

According to the text, why was Paul Dirac’s equation significant to the field of physics?

According to the text, what is the description of the setting?

A) The equation established the fundamental principles of quantum mechanics for the first time.

A) The house appears to be well-maintained.

B) All the surrounding grass has died.

C) The ground is covered in large cracks.

B) The equation combined quantum mechanics and special relativity to predict the existence of antimatter, which was later confirmed.

D) The environment lacks color.

C) The equation provided an explanation for the behavior of subatomic particles at extremely high speeds.

D) The equation demonstrated that antimatter and matter behave in an identical manner.

Warm-Up

Foundation

Approach

Strategy

19a

Wrap-Up

Review

19

Curriculum for the Digital SAT®

READING CLOSELY

Practice Set 1 (16 minutes) Give students 9 minutes to complete the practice set. Review the answers before moving to the next section.

SLIDE 19.1

SLIDE 19.2

1. The correct answer is B. Choice A is incorrect because the text does not state that Dirac’s equation established the fundamental principles of quantum mechanics for the first time.

2. The correct answer is D. Choice A is incorrect because it states that the house appears to be well-maintained, but the text indicates that the paint has been blistered and washed away, making the house dull and gray.

Choice C is incorrect because it does not mention the prediction of antimatter, which was a significant aspect of Dirac’s equation according to the text.

Choice B is incorrect because it claims that all the surrounding grass has died. The text does not say the grass is dead but rather that the sun has burned the tops of the long blades, making them gray in color.

Choice D is incorrect because the text does not mention any comparison between the behavior of antimatter and matter.

Choice C is incorrect because it states that the ground is covered in large cracks. The text mentions there are little cracks running through the gray mass of plowed land but does not say the ground is covered in large cracks.

Choice B is correct because it accurately reflects the text’s description of Dirac’s equation as significant for combining quantum mechanics and special relativity to predict the existence of antimatter, which was later confirmed through the discovery of the positron.

Choice D is correct because it says the environment lacks color. The text describes various aspects of the setting as gray, dull, and colorless, such as the plowed land, the grass, and the house.

19b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Approach ENTRANCE Reading closely questions can be identified by this feature:

• a question that asks for specific information from the passage ... or one of the following phrases:

• according to the text • based on the text Review question 4 with your teacher. Do not answer the question.

ENTRANCE

EVIDENCE AND ELIMINATION

EXIT

4

The following text is adapted from Charlotte Brontë’s 1847 novel Jane Eyre. Jane Eyre is describing her first impression of Mr. Rochester.

[Mr. Rochester] had a dark face, with stern features and a heavy brow; his eyes and gathered eyebrows looked ireful and thwarted just now; he was past youth, but had not reached middle-age; perhaps he might be thirty-five.

I felt no fear of him, and but little shyness. Had he been a handsome, heroic-looking young gentleman, I should not have dared to stand thus questioning him against his will, and offering my services unasked.

According to the text, what is true about Jane’s impression of Mr. Rochester?

A) She is intimidated by his appearance and demeanor.

B) She finds his appearance approachable and feels comfortable around him.

C) She believes he is too old to be an attractive gentleman.

D) She is unsure of his intentions and remains cautious.

22

Warm-Up

Foundation

Approach

22a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Approach (30 minutes) In this section, students will practice identifying a reading closely question using a key word or phrase.

SLIDE 22.1

SLIDE 22.2

The first step to answering any Reading question is Entrance. During Entrance, you identify what kind of question you are being asked.

Let’s practice using question 4. Who can read the question text only for us?

Select a student to read.

Have a student read the identifying characteristics of a reading closely question aloud.

What clue do you see in the question that tells you it’s a reading closely question?

According to the text. It asks for specific details about Jane’s impression.

Imagine we’re on a birdwatching expedition, and we want to identify a red-tailed hawk. What would we need to know?

We’d need to know what that hawk looks like.

Things like size, color, wing shape, unusual markings.

Take a moment now to find the identifying language in questions 5 and 6 on your own. Don’t answer the questions just yet.

You can spot a red-tailed hawk by—get ready for it—their distinctive russet-red tails. But what do we know if we see a bird that has some of the features of a red-tailed hawk, but it has a whitebanded tail instead of a red one?

Then it’s not a red-tailed hawk. It must be some other kind of bird.

We can use the same kind of reasoning to identify a reading closely question. This type of question will always have a defining characteristic. If you see one of the key words from your book in the question, it’s a giveaway, but do you think they’ll always appear?

Probably not.

They won’t always show up, but they often will— and it’s super helpful when they do. It’s important to identify reading closely questions because it changes how you approach them. Unlike other Reading question types, when you come across a reading closely question, you should dive into the passage and dig for evidence as quickly as possible.

22b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Approach EVIDENCE AND ELIMINATION To answer a reading closely question, follow these steps: 1. Identify the key words and phrases in the question.

2. Scan the passage for related key words.

3. Eliminate answer choices not supported by evidence from the passage.

Answer question 4 with your teacher.

ENTRANCE

EVIDENCE AND ELIMINATION

EXIT

4

The following text is adapted from Charlotte Brontë’s 1847 novel Jane Eyre. Jane Eyre is describing her first impression of Mr. Rochester.

[Mr. Rochester] had a dark face, with stern features and a heavy brow; his eyes and gathered eyebrows looked ireful and thwarted just now; he was past youth, but had not reached middle-age; perhaps he might be thirty-five.

I felt no fear of him, and but little shyness. Had he been a handsome, heroic-looking young gentleman, I should not have dared to stand thus questioning him against his will, and offering my services unasked.

According to the text, what is true about Jane’s impression of Mr. Rochester?

A) She is intimidated by his appearance and demeanor.

B) She finds his appearance approachable and feels comfortable around him.

C) She believes he is too old to be an attractive gentleman.

D) She is unsure of his intentions and remains cautious.

24

Warm-Up

Foundation

Approach

24a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Approach (30 minutes) In this section, students will practice applying the steps of Evidence and Elimination to a reading closely question.

If she wasn’t afraid of him, then she wouldn’t be intimidated by him either. We can eliminate answer choice A. What about the but little shyness part? If she wasn’t feeling shy, what does that let us eliminate?

SLIDE 24.1

Once you know you have a reading closely question on your hands, it’s time to start Evidence and Elimination. Our first step says we need to identify key words and phrases. What do you think a key word is?

A key word is a unique word or phrase that helps us determine the main focus or topic of the question.

Choice D. Why can we eliminate choice D? Choice D says that she remains cautious. But the quote says she wasn’t feeling shy, so she probably isn’t being cautious.

Will key words always appear exactly the same in the question and passage?

No, sometimes the SAT will use synonyms.

Choices A and D are out. Each one contradicts at least part of the sentence, I felt no fear of him, and but little shyness. That leaves us with answer choices B and C. Which one isn’t completely supported by the passage?

Choice C.

Where would be the best place to find key words?

In the question.

SLIDE 24.2

Why isn’t choice C supported by evidence?

In the passage, Jane mentions how old she thinks Mr. Rochester is. But she doesn’t specifically say that she thinks his age makes him too old to be attractive.

Do you see anything in question 4 that could be a key word or phrase?

Jane’s impression of Mr. Rochester.

In this case, the characters’ names appear to be the most important. We need to know something about Mr. Rochester. And specifically, we need to find details about Jane’s impression of him. We’re dealing with a short passage, so you won’t need to do any in-depth skimming.

Mr. Rochester’s name shows up right at the beginning. But where is Jane’s name?

It isn’t there. It says I instead.

What about choice B? Any evidence to support the idea that she finds him approachable and feels comfortable around him?

Jane says that she felt no fear of him. She also goes on to say that she was comfortable questioning him against his will.

Both of these details support choice B, so we know it’s our answer. Use this same process of Elimination to help you answer questions 5 and 6. Find the key words in each question. Then use them to make eliminations and identify the correct answer.

We’re getting the description from Jane’s point of view. Since the passage is from Jane’s point of view, we need a phrase where we see I telling us something about her impression of Mr. Rochester. What in the passage could fit that?

I felt no fear of him, and but little shyness.

We’re told that Jane felt no fear when it came to Mr. Rochester. Can you eliminate any answers based on that?

Answer choice A.

24b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

READING CLOSELY

Approach Answer questions 5 and 6 on your own. Focus on eliminating incorrect answers.

5

6

In 1905, physicist Albert Einstein developed the special theory of relativity, a revolutionary concept that transformed the understanding of space and time in the realm of physics.

This innovative theory introduced the idea that time is relative and can vary depending on the observer’s speed, challenging previous notions of absolute time. The famous equation E = mc², derived from the theory, demonstrates the equivalence of mass and energy. Einstein’s work forever changed the field of physics, laying the foundation for advancements in nuclear research and shaping modern scientific thought.

The following text is adapted from Herman Melville’s 1851 novel Moby Dick. The scene takes place on a Saturday night aboard the whaling ship Pequod, where the sailors are engaged in their work as harpooneers under the leadership of the mysterious Captain Ahab.

It was a Saturday night, and such a Sabbath as followed! Ex officio professors of Sabbath breaking are all whalemen. The ivory Pequod was turned into what seemed a shamble; every sailor a butcher. You would have thought we were offering up ten thousand red oxen to the sea gods [...]

And now suspended in stages over the side, Starbuck and Stubb, the mates, armed with their long spades, began cutting a hole in the body for the insertion of the hook just above the nearest of the two side-fins.

According to the text, what made Einstein’s special theory of relativity a pivotal development in the field of physics?

A) The theory provided the first evidence of the existence of black holes.

Based on the text, what can be inferred about the atmosphere on the Pequod on this particular Saturday night?

B) The theory established that time is influenced by the observer’s speed.

C) The theory disproved the fundamental principles of Newtonian physics.

A) The sailors are focused on their work, even though it is the Sabbath and traditionally a time for rest.

D) The theory explained the behavior of subatomic particles in the presence of strong magnetic fields.

B) The sailors are enjoying a festive celebration, even though their profession is typically serious and dangerous.

C) The sailors are participating in a religious ceremony, even though their primary concern is their work on the ship.

D) The sailors are engaged in a competition, even though their main responsibility is to serve as harpooneers.

Warm-Up

Foundation

Approach

Strategy

25a

Wrap-Up

Review

25

Curriculum for the Digital SAT®

READING CLOSELY

Approach (30 minutes) Give the class 6 minutes to answer questions 5 and 6. Review the answers afterward.

SLIDE 25.1

SLIDE 25.2

5. The correct answer is B. Using the Evidence and Elimination process, first identify key words and phrases in the question. Here, the key phrases are Einstein’s special theory of relativity and pivotal development in the field of physics.

6. The correct answer is A. Using the Evidence and Elimination process, first identify key words and phrases in the question. Here, the key phrases are Saturday night and Sabbath.

Next, scan the passage for related key words.

In the passage, we see that it was a Saturday night and that the sailors were working as butchers on the ship, despite it being the Sabbath, a traditional day of rest.

Next, scan the passage for related key words.

The special theory of relativity shows up in the first sentence and is defined in the second sentence: This innovative theory introduced the idea that time is relative and can vary depending on the observer’s speed, challenging previous notions of absolute time.

Then we eliminate answer choices not supported by evidence from the passage.

Choice B is incorrect because a festive celebration does not fit with our key phrases, nor does anything in the passage suggest that the sailors were celebrating.

Then we eliminate answer choices not supported by evidence from the passage.

Choice A is incorrect because the term is not one of our key words, and the concept does not appear in the passage.

Choice C is incorrect because while one of our key phrases, Saturday night, could imply the Sabbath, there is no mention of a religious ceremony taking place.

Choice C is incorrect because although the passage mentions physics and this word aligns with one of our key words, nothing in the passage states that Einstein’s theory disproved the fundamental principles of Newtonian physics.

Choice D is incorrect because a competition is unrelated to our key phrases, and nothing else in the passage fits with that.

Choice A is correct because the sailors are employed on the Sabbath as butchers, confirming their focus on work.

Choice D is incorrect because neither subatomic particles nor magnetic fields relate to our key words, and the concepts do not appear in the passage.

Choice B is correct because the passage states that Einstein’s special theory of relativity—one of our key phrases—introduced the idea that time is relative and can vary depending upon the observer’s speed.

25b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Approach EXIT If you’ve found the answer …

Check to make sure that the answer matches the details of the passage. Mark your answer and continue to the next question.

If you’re stuck between two choices …

Break the remaining answer choices into pieces and compare them to find the essential difference.

Test each piece against the passage for accuracy.

If you have no idea what the answer is …

Go with your gut. Select the choice that is the most consistent with the context and sounds best.

Review questions 7 and 8. Work with your teacher to choose the correct answer from the remaining choices.

ENTRANCE

EVIDENCE AND ELIMINATION

EXIT

7

The following text is adapted from Jonathan Swift’s 1726 satirical novel Gulliver’s Travels. The narrator, Gulliver, has just woken up and discovered himself in a strange land with a giant figure observing him.

In the midst of the discourse, I was awakened by the noise of a great shout, and found it was day. The first object I beheld was the figure of a man having no resemblance in any manner to those I had seen before; he was only twice my length, but ten times as broad, lying on his side, and leaning his head against his hand, with his eyes fixed earnestly upon me. I returned his look with equal observation, and found my senses struck with admiration, observing every part of him full of strength and comeliness, and comparing him with the Lilliputians, whose smallness had rendered their beauties almost imperceptible to me.

Based on the text, how does Gulliver respond to the giant figure?

A) He is initially frightened by the figure’s size, even though he later finds the figure’s appearance admirable.

B) He makes eye contact with the figure, even though the figure’s size is intimidating.

C) He expresses curiosity about the figure’s intentions, even though the figure seems to be observing him peacefully.

D) He compares the figure to the Lilliputians, even though the figure’s size is the only apparent difference between them.

26

Warm-Up

Foundation

Approach

26a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Approach (30 minutes) In this section, students will practice using a variety of Exit strategies for reading closely questions.

Do both of those claims hold up?

SLIDE 26.1

He does compare the figure to the Lilliputians, who are smaller. But it’s not clear that’s the only difference. He says some of the Lilliputians’ qualities weren’t visible to him.

When you’ve found an answer to a reading closely question, take a moment to doublecheck your choice before moving on. Make sure it supports the whole claim. But what if you’re stuck between two choices? What should you do?

It can be tricky when parts of your last two answers seem to work. But if part of an answer is wrong, the whole thing is wrong. Which answer choice, B or D, has part that we can confidently say is unsupported?

Choice D. It’s not clear that the figure’s size is the only difference with the Lilliputians.

Break the remaining answer choices into pieces and compare them to find the essential difference.

Test each piece against the passage for accuracy.

Maybe neither option feels like it fully matches the claim. But one of them will offer more support than the other. The more support in the evidence, the more likely you’ve found the right answer.

We can eliminate answer choice D. Earlier when we looked at that answer choice, we didn’t feel like we had slam dunk evidence to prove it was right. So where does the elimination of answer choice D leave us?

With choice B as the last answer left.

SLIDE 26.2

For instance, take a peek at question 7. What is it asking?

How does Gulliver respond to the giant figure?

Sometimes questions will be like that. You may not love the correct answer, but as long as the other answers are worse, what can you still be confident of?

It’s the right answer.

We’ve got answer choices B and D left as options. If we start by taking answer choice B and breaking it up into pieces, what ideas do we have?

Gulliver makes eye contact with the figure. The figure’s size is intimidating.

In this case, that’s answer choice B.

Do you see evidence for the first part? Does it ever tell us that Gulliver makes eye contact with the figure?

Yes, it says the giant fixed his eyes on Gulliver and that Gulliver returned his look.

Is there evidence the figure’s size is intimidating?

Maybe. He’s really big, in any case.

We’ll keep choice B in mind as we test answer choice D. What do we get when we break choice D into pieces?

Gulliver compares the figure to the Lilliputians.

The figure’s size is the only apparent difference between them.

26b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Strategy PREDICT THE FUTURE When you’re struggling to find evidence, the Predict the Future strategy is an excellent way to guide you toward a correct answer.

1. Cover the answers.

2. Read the question and come up with your own answer.

3. Select the answer that best matches your prediction.

Review question 12 with your teacher. Use the Predict the Future strategy to answer the question as a class.

12

The following text is from John Keats’s 1819 poem “Ode to a Nightingale.”

Thou wast not born for death, immortal Bird!

No hungry generations tramp thee down; The voice I hear this passing night was heard In ancient days by emperor and clown: Perhaps the self-same song that found a path Through the sad heart of Ruth, when, sick for home, She stood in tears amid the alien corn; The same that oft-times hath Charm’d magic casements, opening on the foam Of perilous seas, in faery lands forlorn.

Based on the text, what quality does the nightingale symbolize?

A) The fleeting nature of life and beauty B) The transformation brought by the passage of time C) The timeless and unchanging nature of beauty D) The complexity of human emotions and relationships

30

Warm-Up

Foundation

Approach

30a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Strategy (30 minutes) In this section, students will practice using the Predict the Future strategy to answer a reading closely question.

The poet is making some kind of connection between this bird and immortality, which essentially means living life without having to worry about dying. Cover your answer choices and look back over the poem with that in mind.

What quality could this nightingale symbolize?

SLIDE 30.1

It’s easy to get bogged down in the details of the passage when you’re working on a reading closely question. What are some reasons that might happen?

Ask 1–2 students to share.

Something timeless or without end.

One possibility is you can’t find a good key word in the question. When you can’t find any evidence to either support or eliminate your answers, you can use the Predict the Future strategy. Granted, this isn’t as good as finding real evidence in the passage, but it’s much better than just a random guess.

Which answer choice fits with that?

Choice C. How does that one match with our general prediction?

It’s the only answer choice that mentions anything timeless or enduring.

Practice using the Predict the Future strategy to answer questions 13 and 14. It’s okay if your predicted answer is short and sweet. You’re not trying out to be an SAT test writer. The more general it is, the easier it will be to match it up with a choice.

SLIDE 30.2

Let’s try it with question 12. What’s the first step to Predict the Future?

Cover the answers. Using your hand to cover the answers on the computer screen lets you focus more.

This not only lets you focus more on the question, but it also keeps you from wasting time overanalyzing your options in search of the perfect one. What’s our next step?

Read the question and come up with your own answer.

As you do this, try to keep it loose. Making the answer too specific causes at least a couple of problems. For one, it will just take you longer.

What else could go wrong if you’re getting too specific?

It could be so specific that it sounds like one of the answer choices.

With all that in mind, what descriptive word gives us some information about this nightingale or bird?

Immortal.

30b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

READING CLOSELY

Strategy Work on your own using the Predict the Future strategy to answer questions 13 and 14.

13

14

Nikola Tesla, a Serbian American inventor, played a pivotal role in developing alternating current (AC) electrical systems, foundational to modern AC power transmission. His inventive prowess wasn’t limited to electricity; he also ventured into early wireless communication concepts and imagined machines with a degree of automation. While not directly the father of robotics or computer science, his forward-thinking ideas hinted at these future fields. Tesla’s groundbreaking work undeniably shaped the trajectory of technological advancement, leaving an indelible mark on the world.

The following text is adapted from Kate Chopin’s 1899 novel The Awakening.

Mrs. Pontellier was beginning to realize her position in the universe as a human being, and to recognize her relations as an individual to the world within and about her.

This may seem like a ponderous weight of wisdom to descend upon the soul of a young woman of twenty-eight—perhaps more wisdom than the Holy Ghost is usually pleased to vouchsafe.

Based on the text, what is true about Mrs. Pontellier’s self-awareness?

Which choice best describes Tesla’s impact on various fields, as presented in the text?

A) She is hesitant to explore her thoughts and emotions.

A) His influence extended only to electrical systems.

B) She is overwhelmed by the sudden surge of introspection.

B) His work directly led to the development of modern telecommunications systems.

C) She is gaining a deeper understanding of her place in the world.

C) He explored early concepts of electricity, wireless communication, and automated machines.

D) She is resentful of the wisdom being granted to her by the Holy Ghost.

D) He collaborated extensively with other inventors to develop new technologies.

Warm-Up

Foundation

Approach

Strategy

31a

Wrap-Up

Review

31

Curriculum for the Digital SAT®

READING CLOSELY

Strategy (30 minutes) Give the class 6 minutes to complete questions 13 and 14. Review the answers when time has elapsed.

SLIDE 31.1

SLIDE 31.2

13. The correct answer is C. Using the Predict the Future strategy, begin by reading the question and coming up with your own answer: Tesla’s impact includes developments in electricity, wireless communication, and automated machines.

14. The correct answer is C. Using the Predict the Future strategy, begin by reading the question and coming up with your own answer: Mrs. Pontellier is gaining a deeper understanding of her place in the world.

Choice A is incorrect because it says she is hesitant to explore her thoughts and emotions, but our answer doesn’t mention hesitation and neither does the text.

Choice A is incorrect because it says his influence extended only to electrical systems.

But we know from our answer that Tesla’s impact affected a variety of other kinds of technology, including wireless communication and automated machines.

Choice B is incorrect because neither our answer nor the text implies that she’s overwhelmed; the passage states that she’s beginning to understand her position.

Choice B is incorrect because it says his work directly led to the development of modern telecommunications systems, but that doesn’t particularly align well with our answer, which describes a variety of innovations. Additionally, while the passage says Tesla ventured into early wireless communication concepts, it does not go as far as choice B does.

Choice D is incorrect because neither our answer nor the text mentions any resentment.

Rather, the passage discusses her gaining wisdom and understanding.

Choice C is the correct answer. The text describes Mrs. Pontellier’s growing realization of her individuality and her relationship with the world. The passage clearly indicates her developing insight into her own existence and role, underscoring the idea that she is, indeed, gaining a deeper understanding of her place in the world.

Choice D is incorrect because it says he collaborated extensively with other inventors to develop new technologies, but our answer doesn’t reflect that kind of collaboration, and the text doesn’t mention any other inventors.

Choice C is the correct answer. It says he explored early concepts of electricity, wireless communication, and automated machines.

These three categories of innovation match with our answer.

31b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Strategy SCANNING FOR EVIDENCE Answer reading closely questions more accurately and quickly by identifying the most relevant evidence in the passage.

1. Find the key word or short phrase in the question that seems to be the most important. Use the electronic highlighter to mark it.

2. In the passage, look for synonyms and concepts related to the words you highlighted.

3. Choose the answer that aligns with the meaning of the sentence that includes the key word.

Review question 15 with your teacher. Use the Scanning for Evidence strategy to answer the question as a class.

15

People can estimate the passage of time without the use of clocks or watches, relying on their internal sense of time. Many psychologists, including John Wearden, have explored this phenomenon. In various experiments, participants were tasked with judging durations based on their internal clocks, often without external cues. The findings generally indicate that participants can estimate time intervals with reasonable accuracy, although deviations can occur. This suggests that humans possess an innate ability to gauge time even without external reference points.

Based on the passage, how did the researchers determine the participants’ ability to estimate time in the studies?

A) They asked each participant to judge the passage of time without using any tools.

B) They examined how participants reacted to changes in the environment.

C) They studied the accuracy of participants pressing a button after a specific duration.

D) They tracked the participants’ physical movements during the experiment.

32

Warm-Up

Foundation

Approach

32a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Strategy (30 minutes) In this section, students will practice using the Scanning for Evidence strategy to evaluate answer choices while answering a reading closely question.

Answer choice A says they asked each participant to judge the passage of time without using any tools. What idea have we mentioned that fits with that?

Judging durations based on their internal clocks.

SLIDE 32.1

The passages for reading closely questions aren’t wildly long. Many of them are under a hundred words. What do we call our strategy that will help us sift through those hundred-or-so words to find the exact ones we need?

The fourth sentence has an exact match for our phrase, but how is the overall information in that sentence just a little different from what the question is asking?

The fourth sentence talks about the results of the experiment. The question is asking about how the research was conducted.

Scanning for Evidence.

What’s our first step?

Find the word or short phrase in the question that seems to be the most important. Use the electronic highlighter to mark it.

Let’s start with key words or phrases in the question. What do you see?

Estimate time.

And the third sentence also describes the research. Even though judging durations isn’t a word-for-word match, do we have the evidence we need to feel confident in our answer?

Yes.

The correct answer is almost certainly going to have something to do with estimating time.

You can underline the phrase in your book. On the actual test, you’ll use a digital highlighter.

Moving to step 2, what does it say?

In the passage, look for synonyms and concepts related to the words you highlighted.

We can see the importance of step 2, look for synonyms and concepts, not just exact word matches. Estimating time means the same thing as judging durations. So, we have the evidence we need.

Let’s turn to our passage now. How close of a match can you find for estimate time?

The third sentence mentions judging durations.

And the fourth sentence directly says estimate time.

Use these tips to help you answer questions 16 and 17 on your own.

What does it say about how they were judging durations?

It says their judging of durations is based on their internal clocks.

And what does the fourth sentence say about estimating time?

It says a lot of people can do it with reasonable accuracy.

Step 3 says to choose the answer that aligns with the meaning of the sentence that includes the key word. Does that apply to answer choice A? Yes, it fits with answer choice A.

32b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

READING CLOSELY

Strategy Work on your own using the Scanning for Evidence strategy to answer questions 16 and 17.

16

17

Hopi-Tewa potter Nathan Begaye reimagines vessels of clay by intertwining both traditional Hopi and Navajo techniques. These methods incorporate Hopi coiling and Navajo corrugated patterns, creating textures that evoke the rich artistic traditions found in many Native communities. Moreover, Begaye integrates natural clay pigments, capturing the vibrant and earthy hues cherished in many Native art pieces. In this fusion, he not only challenges the distinctions between Hopi and Navajo cultures but also redefines pottery craftsmanship, blurring the lines between distinct tribal styles and techniques.

The following text is adapted from Upton Sinclair’s 1906 novel The Jungle.

Which choice best describes Begaye’s approach to pottery, as presented in the text?

A) They become successful through their dedication and hard work.

Jurgis had come there, and thought he was going to make himself useful, and rise and become a skilled man; but he would soon find out his error—for nobody rose in Packingtown by doing good work. You could lay that down for a rule—if you met a man who was rising in Packingtown, you met a knave.

Based on the text, what is true about people who succeed in Packingtown?

B) They struggle to find meaningful work but are ultimately rewarded.

A) He combines techniques from both Hopi and Navajo pottery traditions to create his unique pieces.

C) They focus on their individual goals and ignore the needs of others.

B) He focuses solely on Hopi coiling methods while ignoring other pottery techniques.

D) They are likely using dishonest means to achieve their success.

C) He disregards the use of natural clay pigments and minerals in favor of synthetic materials.

D) He is equally influenced by Native American pottery and contemporary Western ceramics.

Warm-Up

Foundation

Approach

Strategy

33a

Wrap-Up

Review

33

Curriculum for the Digital SAT®

READING CLOSELY

Strategy (30 minutes) Give the class 6 minutes to complete questions 16 and 17. Review the answers when time has elapsed.

SLIDE 33.1

SLIDE 33.2

16. The correct answer is A. Using the Scanning for Evidence strategy, begin by underlining the most important word or short phrase in the question. In this case, it’s Begaye’s approach to pottery.

17. The correct answer is D. Using the Scanning for Evidence strategy, begin by underlining the most important word or short phrase in the question. In this case, it’s people who succeed in Packingtown.

Next, look for synonyms and concepts related to the words you underlined in the passage.

Examples include:

Next, scan the passage for synonyms and concepts related to the phrase you underlined in the question. For example, nobody rose in Packingtown by doing good work and if you met a man who was rising in Packingtown, you met a knave.

• reimagines vessels of clay • incorporate Hopi coiling and Navajo corrugated patterns

Choice A focuses on people becoming successful through their dedication and hard work. This contradicts the statement about how only knaves, or dishonest people, were able to rise in Packingtown, so it can be eliminated.

• integrates natural clay pigments • blurring the lines between distinct tribal styles Choice B says that Begaye focuses solely on Hopi coiling methods, but we found evidence that Begaye also incorporates Navajo patterns, so this answer choice can be eliminated.

Choice B focuses on people ultimately being rewarded for their hard work, but this also contradicts the idea that only knaves were rising or seeing rewards. This choice can be eliminated.

Choice C focuses on the idea that he disregards the use of natural clay pigments.

This contradicts the passage, which states that he integrates natural clay pigments. It can also be eliminated.

Choice C focuses on those who are attentive to their individual goals and ignore the needs of others. The passage doesn’t mention how successful people in Packingtown interact with others, so it can be eliminated.

Choice D focuses on his influence from Western ceramics, but as we found, the passage only focuses on his fusion of Hopi and Navajo techniques.

Choice D is correct because it focuses on the fact that people are likely using dishonest means to achieve their success. This answer aligns with the meaning of the sentence that includes the key word (knave) and is supported by the passage’s statement about nobody rising in Packingtown by doing good work.

Choice A is correct because the text emphasizes that Begaye combines both Hopi and Navajo pottery techniques in his work, including the use of Hopi coiling and Navajo corrugated patterns, as well as the integration of natural clay pigments. This choice best captures Begaye’s approach to pottery, which involves blending elements from these two distinct Native American traditions to create unique, innovative pieces.

33b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Wrap-Up

Complete the wrap-up. When you have finished, double-check your work.

4

5 The following text is from Jane Austen’s 1813 novel Pride and Prejudice. Elizabeth lives with her parents, Mr. and Mrs. Bennet, and her four sisters at Longbourn estate.

Oregon hosts Mount Mazama, a volcanic crater that now contains Crater Lake, the deepest lake in the United States. Formed approximately 7,700 years ago by the eruption and subsequent collapse of Mount Mazama, Crater Lake has a depth of 1,949 feet. Geologists are intrigued by the lake’s clarity and purity, as it is solely fed by rain and snowfall. They believe that studying Crater Lake could provide valuable insights into the natural filtration processes of water.

Elizabeth, feeling all the more than common awkwardness and anxiety of his situation, now forced herself to speak; and immediately, though not very fluently, gave him to understand that her sentiments had undergone so material a change since the period to which he alluded, as to make her receive with gratitude and pleasure his present assurances.

The happiness which this reply produced was such as he had probably never felt before; and he expressed himself on the occasion as sensibly and as warmly as a man violently in love can be supposed to do.

According to the text, why are geologists intrigued by Crater Lake?

A) It is unusually deep for a lake in the United States.

B) It was formed by a unique volcanic event that took place 7,700 years ago.

Based on the excerpt, what is true about Elizabeth?

C) The water from rain and snowfall that fills the lake is exceptionally clear and pure.

A) Elizabeth is usually hesitant to express her emotions to others.

D) It provides a habitat for rare species of fish and plants.

B) Elizabeth has always been open about her feelings toward the man in question.

C) Elizabeth believes the man’s happiness is superficial and temporary.

D) Elizabeth had experienced a significant change in her emotions.

36

Warm-Up

Foundation

Approach

36a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

READING CLOSELY

Wrap-Up (16 minutes) Give students 9 minutes to complete the wrap-up. Review the answers afterward.

SLIDE 36.1

SLIDE 36.2

4. The correct answer is D. Choice A is incorrect because the text does not provide enough information to support the idea that Elizabeth is generally hesitant to express her emotions to others. It only describes her awkwardness and anxiety in this specific situation.

5. The correct answer is C. Choice A is incorrect because although the text mentions that Crater Lake is the deepest lake in the United States, it doesn’t specifically state that geologists are intrigued by its depth.

Choice B is incorrect because while the text does explain that Crater Lake was formed by a unique volcanic event, it doesn’t state that this event is the reason geologists are interested in the lake.

Choice B is incorrect because the excerpt actually indicates the opposite. Elizabeth informs the man that her sentiments had undergone a significant change since the period he referred to. This suggests that her feelings toward him were not always as positive as they are at the moment of their conversation.

Choice D is incorrect because the text does not mention anything about rare species of fish and plants being present in Crater Lake or being a reason for geologists’ interest.

Choice C is incorrect because the text does not provide any information to suggest that Elizabeth believes the man’s happiness is superficial and temporary. Instead, it describes the man’s happiness in response to Elizabeth’s assurances.

Choice C is correct because the text specifically states that geologists are intrigued by the lake’s clarity and purity, as it is solely fed by rain and snowfall. This indicates that the exceptional clarity and purity of the water from rain and snowfall is the reason for their interest in Crater Lake.

Choice D is correct because the text clearly states that Elizabeth’s sentiments had undergone so material a change. This indicates that her emotions or feelings toward the man in question had significantly changed, leading her to receive his present assurances with gratitude and pleasure. The passage doesn’t indicate the nature of this change in sentiment, but it is clear that a substantial change has occurred.

36b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: READING

MASTERY FOR THE DIGITAL SAT: READING

Review ENTRANCE Reading closely questions can be identified by this feature:

• a question that asks for specific information from the passage ... or one of the following phrases:

• according to the text • based on the text EVIDENCE AND ELIMINATION To answer a reading closely question, follow these steps: 1. Identify the key words and phrases in the question.

2. Scan the passage for related key words.

3. Eliminate answer choices not supported by evidence from the passage.

EXIT If you’ve found the answer ...

If you have no idea what the answer is ...

Check to make sure that the answer matches the details of the passage. Mark your answer and continue to the next question.

Go with your gut. Select the choice that is the most consistent with the context and sounds best.

If you used a strategy or skipped a question ...

If you’re stuck between two choices ...

Break the remaining answer choices into pieces and compare them to find the essential difference.

Test each piece against the passage for accuracy.

Mark the question so you can come back to it after you answer the remaining questions. On a second pass, double-check that you guessed an answer or try to answer the question.

STRATEGY Predict the Future When you’re struggling to find evidence, the Predict the Future strategy is an excellent way to guide you toward a correct answer.

38

Warm-Up

Foundation

Scanning for Evidence Answer reading closely questions more accurately and quickly by identifying the most relevant evidence in the passage.

Approach

38a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

WRITING

Table of Contents How Mastery for the Digital SAT Works......................................................................................................................................i Chapter 1: Writing Introduction.................................................................................................................................................... 1 Chapter 2: Verbs ............................................................................................................................................................................... 13 Chapter 3: Nouns and Pronouns ...............................................................................................................................................33 Chapter 4: Sentence Formation................................................................................................................................................. 55 Chapter 5: Commas ........................................................................................................................................................................ 75 Chapter 6: Semicolons and Colons .......................................................................................................................................... 97 Chapter 7: Sentence Boundaries ..............................................................................................................................................117 Chapter 8: Misplaced Modifiers ...............................................................................................................................................137 Chapter 9: Transitions ....................................................................................................................................................................161 Chapter 10: Writer's Goal .............................................................................................................................................................181 Contributors ......................................................................................................................................................................................213

Curriculum for the Digital SAT®

Parenthetical Elements

CLAIMS PARENTHETICALILLUSTRATING ELEMENTS VERBS

CONTENT DIMENSIONS

CHAPTER 2

Verb tense, mood, and voice The student will recognize and correct

Verbs

inappropriate shifts in verb tense, voice, and mood within and between sentences.

This chapter explains how to recognize verbs questions on the SAT and introduces a brief, step-

LEARNING TARGETS

by-step process to eliminate incorrect answers. This

1. Identify verbs and their correct tenses.

chapter also demonstrates how to apply the Tune

2. Implement the basic approach for verb tense questions.

Out and Sounds Wrong Is Wrong strategies to answer questions about verb tense and usage.

3. Identify time references when dealing with conflicting tenses.

SCORE DIFFERENTIATION

2 CHAPTER

Skip the Foundation segment if ...

Verbs

LEARNING TARGETS 1. Identify verbs and their correct tenses.

2. Implement the basic approach for verb tense questions.

3. Identify time references when dealing with conflicting tenses.

13

13b13

SAT

Reading and Writing above 500 No subscore

PSAT

Reading and Writing above 480 No subscore

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Warm-Up

Complete the warm-up. When you have finished, double-check your work.

11

33

In the 1930s, a man named Wallace Carothers was struggling to find a breakthrough in synthetic materials at DuPont when his colleague Julian Hill advised him to develop a new kind of fiber from their experimental polymer, leading to the creation of a strong, elastic material. In addition, Hill ______ marketing the material for use in clothing and other industries under a memorable name: nylon.

In the Large Hadron Collider, protons are accelerated close to the speed of light until they ______ a specific energy level, at which point they are guided to collide with each other, enabling scientists to study the fundamental particles and forces of nature.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) will attain

Which choice completes the text so that it conforms to the conventions of Standard English?

B) attain

A) proposed

C) had attained

B) proposes

D) are attaining

C) had proposed D) was proposing

2 The card game Uno, developed in 1971 by Merle Robbins in Ohio, is a variation of the classic card game Crazy Eights. In both games, players take turns to match cards with similar attributes, such as color or number, and using specific cards like “Draw Two” or “Skip” cards _______ a player's chances of emptying their hand and winning the game.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) improves B) are improving C) have improved D) improve

14

Warm-Up

Foundation

Approach

14a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

VERBS

Warm-Up (12 minutes) Give students 6 minutes to complete the warm-up. Review the answers briefly before beginning the lesson.

SLIDE 14.3

SLIDE 14.1

1. The correct answer is A. Choice B is incorrect because proposes is in the present tense, while the rest of the passage is in the past tense.

3. The correct answer is B. Choice A is incorrect because will attain is in the future tense. In this context, a present tense verb is more suitable because it describes a general, ongoing action that occurs during the operation of the Large Hadron Collider.

Choice C is incorrect because had proposed is in the past perfect tense, which would indicate that the action happened before another past action, but there is no other action in the passage that would require this tense.

Choice C is incorrect because had attained is in the past perfect tense. This tense is typically used to describe an action completed before another action in the past. Since the sentence describes a recurring, present action in the operation of the collider, the past perfect tense is not appropriate.

Choice D is incorrect because was proposing is in the past continuous tense, which would indicate that the action was ongoing, but there is no indication in the passage that the proposing was ongoing or continuous; it is more logical to assume that Hill proposed the idea once.

Choice D is incorrect because are attaining is in the present continuous tense. The action of attaining, or reaching, an energy level is something that happens at a single point in time, rather than an ongoing or continuing action. Once the protons attain the level, they can't keep attaining it over and over again.

Thus, the simple present makes more sense than the present continuous tense here.

Choice A is correct because proposed is in the simple past tense, which is consistent with the tense used in the rest of the passage.

SLIDE 14.2

2. The correct answer is A. Choice B is incorrect because are improving is in the present continuous tense, which suggests an ongoing action. In this context, we need a verb form that shows the general effect of using specific cards, not an ongoing action.

Choice B is correct because attain is in the simple present tense, which is appropriate for the context since the rest of the sentence is also in the present tense.

Choice C is incorrect because have improved is in the present perfect tense, which is used to describe an action that began in the past and affects the present. In this context, we need a verb form that shows the general effect of using specific cards, not a past action that affects the present.

Choice D is incorrect because although improve is in the present tense, which is correct, it does not match the singular subject of the sentence, using.

Choice A is correct because improves is in the simple present tense, which is appropriate for this context since the action of using specific cards has a general effect on a player’s chances of emptying their hand and winning the game. The verb improves correctly pairs with using to complete the sentence in Standard English.

14b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

VERBS

Foundation Work with your teacher to fill in chart number 1.

Subject

Past

Present

Future

I

kangarooed

kangaroo

will kangaroo

he/she/it

kangarooed

kangaroos

will kangaroo

they

kangarooed

kangaroo

will kangaroo

Subject

Past

Present

Future

I

learned

1. Verb: learn

he/she/it

learns

they

will learn

Fill in chart number 2 with your favorite verb following the same model.

2. Write your favorite verb: ___________________________________________ Subject

Past

Present

Future

I he/she/it they

Warm-Up

Foundation

Approach

Strategy

15a

Wrap-Up

Review

15

Curriculum for the Digital SAT®

VERBS

Foundation (26 minutes) In this section, students will complete the tables with verbs in different tenses.

It’s the same word that’s at the top of the column.

Past tense verbs usually end with –ed, and that’s definitely true of learned. What about that middle column? Did you copy what’s in the middle row and put learns for both of those?

SLIDE 15.1

Picture two animals side by side: a kangaroo and a snail. Which one would you associate with action?

The kangaroo.

No, those should be learn, not learns.

Kangaroos jump, they kick, and they run. Snails, not so much. In that spirit, we've turned the word kangaroo into a verb. Why is that a little strange?

Only with he, she, or it—the singular subjects— would we use learns. No s needed at the end for pairing with I and they. What about future tense?

What did you put for that?

Will learn both times.

A kangaroo is an animal. It’s a noun, not a verb.

The word kangaroo isn’t actually a verb, but that's fine for this exercise because it makes the verb tense patterns clear. What do you notice about all the “kangaroo” verbs in the past tense? What do they have in common?

That helping word will usually tells us we’re dealing with the future. Moving on to table 2, choose your favorite verb and complete it. Use the patterns we’ve learned but know there are exceptions.

They all end with –ed.

Give students 3 minutes to complete the task.

That –ed ending is super common on verbs that describe things that took place in the past. What about things that are going to happen in the future? What do all of our future kangaroo verbs have in common?

Take a look at the verbs you just used to fill out table 2. Does the pattern generally follow what came before in the tables above?

Yes.

They all have the word will at the beginning.

On the other hand, not every verb forms its tenses in the exact same way. Does anyone have an example of that?

The word will is about as strong a clue as you can get that the verb tense is in the future. It’s describing something that’s going to happen.

Looking to move ahead, which tense haven’t we talked about yet?

Present.

Ask 1–2 students to share.

Since we all have different responses for table 2, trade with a classmate to check answers. Raise your hand if you need my help.

Here, we’re dealing with things that are happening right now. What slight difference do you notice between the verb forms in that middle column?

Two of the examples are just the word kangaroo.

But if the subject is he, she, or it, then the verb has the letter s on the end.

Give students 2 minutes to complete this task.

The bottom line is that verb tenses can seem complicated. But if you think in three big categories—past, present, and future—it will make your life a lot easier. Look for clues whether you're dealing with things that have already happened, are happening now, or are yet to happen in the future.

With these patterns in mind, fill in table 1.

Give students 1 minute to complete the task.

TABLE 1: ANSWER KEY

Let’s look first at the past tense verbs. What did you put in the two blank spaces in that column?

Learned.

Subject

Past

Present

Future

I

learned

learn

will learn

he/she/it

learned

learns

will learn

they

learned

learn

will learn

Table 2: Answers will vary depending on the verb chosen by each student.

15b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Foundation Circle the correct “clue” verb in each numbered sentence in the paragraph. Complete the paragraph by selecting the correct conjugation of the missing verbs.

(1) When I first began taking singing lessons at a studio near my house, I _______________ (was/is/will be) excited. I couldn’t wait to practice my scales, record short tracks, and basically become the next big pop star.

The excitement of singing in a semi-professional environment, however, soon wore off. (2) The daily practice, which involved singing the same pieces over and over again, _______________ (became/becomes/will become) a chore, and all that late-night practicing annoyed a lot of my family members, and my cat. (3) After six months, I felt no closer to my dream of becoming a major recording artist than when I first _______________ (started/start/will start). Luckily, though, I was about to get my big break.

16

Warm-Up

Foundation

Approach

16a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

VERBS

Foundation (26 minutes) In this section, students will identify “clue” verbs to determine the correct tenses in a paragraph.

What is the verb that gives us a clue in sentence 3? What is its tense?

SLIDE 16.1

When we talk about verb tense, we’re talking about the action described by the verb that is taking place. In broad strokes, we’re talking about past tense, present tense, and—what’s the other main category?

Felt. Past tense.

Based on that, what is our missing verb here?

Started.

Look for “clue” verbs when you need to find the correct tense on SAT verb tense questions.

Practice the skills we’ve covered in this section on the following practice set.

Future tense.

One key to finding the correct tense in a question is to get in the habit of spotting other verbs in the sentence and the paragraph. Why might this help?

The verbs we need for our answers are probably in the same tense.

If you can figure out the tense of the verbs you’re given, you won’t feel so, well, tense, when you’re trying to answer this kind of question. Sorry, I couldn’t help myself. Look in your book. Would someone please read sentence 1 aloud?

When I first began taking singing lessons at a studio near my house, I ______________ excited.

What’s the first verb you see in that sentence?

It will give us a clue about the tense we should pick.

Began.

Is began a past, present, or future tense verb?

Past tense.

Look at the options for the missing verb that will go in the first blank. We have was/is/will be. The correct verb tense of the missing verb should match the tense of began, our “clue” verb. Which of our options is also a past tense verb?

Was. Write the verb was in the blank after number 1.

Using these same steps, complete sentences 2 and 3 on your own.

Give students 1 minute to complete the exercise. Afterward, review the answers.

What is a “clue” verb in sentence 2? What is its tense?

Involved or annoyed. Past tense.

So, what is the missing past tense verb?

Became.

16b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Practice Set 1

Complete the practice set. When you have finished, double-check your work.

1

3 The Civil Rights Movement in the United States was driven by various organizations and activists who fought for racial equality. Among the prominent figures __________ Rosa Parks, a key participant in the Montgomery Bus Boycott.

When writing her Pulitzer Prize-winning novel To Kill a Mockingbird (1960), author Harper Lee drew on her own childhood experiences growing up in the Deep South. The classic book was Lee's first published novel, but her writing __________ recognized even before its publication. In 1956, she received a generous Christmas gift from friends to support her writing for a year, allowing her to complete the manuscript.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) was B) were

Which choice completes the text so that it conforms to the conventions of Standard English?

C) are

A) were

D) have been

B) have been C) had been D) are

22

In his 1917 work Fountain, French-American artist Marcel Duchamp challenged the concept of art by presenting a signed urinal as a sculpture.

Today, Duchamp __________ considered a pioneer of conceptual art.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) will be B) had been C) was D) is

18

Warm-Up

Foundation

Approach

18a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

VERBS

Practice Set 1 (12 minutes) Give students 6 minutes to complete the practice set. Review the answers before moving to the next section. verb in the present tense is needed.

SLIDE 18.1

Choice C is incorrect because was is in the past tense. Although Duchamp’s action of presenting the urinal as a sculpture occurred in the past, the sentence is describing his current reputation. Therefore, a present tense verb is required.

1. The correct answer is A. Choice B is incorrect because were is the plural past tense form of the verb to be. In this sentence, the word order is inverted. It can be rearranged to Rosa Parks was among the prominent figures. Then, it's clear that the subject is Rosa Parks, which requires the singular past tense verb was.

Thus, the verb should be in singular form, making were an inappropriate choice.

Choice D is correct because is is in the present tense. The sentence is discussing Duchamp’s current standing in the world of art. He is considered a pioneer of conceptual art now, as people continue to evaluate and appreciate his contributions to the art world. The use of the present tense is appropriate for indicating a present state or condition.

Choice C is incorrect because are is in the present tense. The sentence is discussing historical events and figures, specifically the Civil Rights Movement and Rosa Parks, which requires a past tense verb to indicate that these events and individuals were prominent in the past.

SLIDE 18.3

Choice D is incorrect because have been is in the present perfect tense. This tense is not appropriate here because the sentence is referring to a specific historical context, requiring a simple past tense verb to align with the time frame of the Civil Rights Movement.

3. The correct answer is C. Choice A is incorrect because were is a plural past tense verb. The subject her writing is singular, so a singular verb form is needed for correct subject-verb agreement.

Choice B is incorrect because have been is a plural present perfect tense verb. The singular subject her writing requires a singular verb.

Additionally, the sentence is discussing a past event, specifically the recognition of Lee's writing before the publication of her novel, so a past tense verb form would be more appropriate.

Choice A is correct because was is the singular past tense form of the verb to be. It agrees with the singular subject Rosa Parks and is appropriate for discussing historical contexts, such as the Civil Rights Movement. In this sentence, was effectively links the singular noun Rosa Parks to the phrase among the prominent figures, indicating her inclusion in that group in the past.

Choice D is incorrect because are is a present tense verb and it’s plural. The sentence is discussing past events and requires a singular verb to agree with the singular subject her writing.

SLIDE 18.2

2. The correct answer is D. Choice A is incorrect because will be is in the future tense. The sentence is making a statement about Duchamp’s current reputation, not something that will happen in the future, so the future tense is not appropriate here.

Choice C is correct because had been is in the past perfect continuous tense, which is used to describe an action that began in the past, continued for a period of time, and was completed in the past. It is singular, matching the singular subject her writing. In this context, it suggests that the recognition of Lee’s writing began before the publication of To Kill a Mockingbird and continued until the book was published.

Choice B is incorrect because had been is in the past perfect tense. It’s used to describe an action completed before another action in the past. In this context, we are discussing Duchamp’s present status in the art world, so a

18b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Approach ENTRANCE Verb questions can be identified by this feature:

• variations of similar verbs in the answer choices Review question 4 with your teacher. Do not answer the question.

ENTRANCE

ELIMINATION

EXIT

4

A survey conducted by political scientist Robert Putnam in his book Bowling Alone: The Collapse and Revival of American Community found that the level of civic engagement in the United States, measured by participation in community organizations, experienced a decline, with parent-teacher associations witnessing a drop in membership of about 61% between 1965 and 1995, indicating that social ties among Americans __________ significantly weaker.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) becoming B) to become C) to have become D) had become

20

Warm-Up

Foundation

Approach

20a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

VERBS

Approach (24 minutes) In this section, students will practice identifying a verbs question by considering the answer choices.

SLIDE 20.1

Finish the well-known phrase: “Lights, camera, ...”

Action.

Whether or not you’ve ever been on the set of a movie, you can spot action words in a question’s answer choices. Sometimes they have to do with physical action like running or jumping or flying.

What do we call words that describe things like that?

Verbs.

And of course, not all verbs describe actions that are visible to us. What are some actions you could take that are invisible?

Ask 1–2 students to share. (Examples include verbs describing mental or emotional processes like think, believe, know, love, or want.)

There are also state-of-being verbs like the word is that more or less tell you that someone or something, well, exists. And a state-of-being verb can be connected to a descriptive word, like in the sentence She is tall. Why would we bother thinking through all these kinds of verbs?

We’ll need to be able to spot them to recognize verb questions.

If your answer choices are filled up with one verb or verb phrase after another, it’s gotta be a verb question. There are a couple of different kinds of verb questions the SAT may throw at you, but we can spare you those details for now. The bottom line: how do you know you’re dealing with a verb question?

You’ll see verbs for every answer choice.

Keep an eye out for the action words—or verbs of another kind—in the answer choices, and this step won’t trip you up!

20b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

VERBS

Approach Review questions 5–8 on your own. Circle the two questions that test knowledge of verbs.

5

7

An economic analysis by economist Thomas Piketty in his book Capital in the Twenty-First Century revealed that the wealth inequality in the United States in 2010, measured by the share of total income going to the top 1% of earners, __________ 1.6 times greater than the level of inequality observed in 1980.

The internet has made communication faster and more accessible than ever before. ____________, it has also led to concerns about privacy and security in the digital age.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) However B) For instance

A) being

C) Consequently

B) was

D) In the same vein

C) to have been D) will be

8

The Women’s Suffrage Movement in the United States saw the emergence of many influential women leaders, advocating for women’s right to vote. Among these leaders __________ Susan B. Anthony, cofounder of the National Woman Suffrage Association.

6

Photography was originally a laborious process, requiring hours of exposure and careful handling of chemicals. ____________, early photographers had to be both skilled artists and chemists.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) was B) were

A) Therefore

C) are

B) Conversely

D) have been

C) Similarly D) On the contrary

Warm-Up

Foundation

Approach

Strategy

21a

Wrap-Up

Review

21

Curriculum for the Digital SAT®

VERBS

Approach (24 minutes) Give the class 1 minute to circle the identifying language in questions 5–8. Review the answers afterward.

SLIDE 21.1

SLIDE 21.3

5. Question 5 is a verbs question. Each answer choice offers a different tense for the verb to be.

7. Question 7 is not a verbs question. The question instead tests transitional words and phrases.

SLIDE 21.2

SLIDE 21.4

6. Question 6 is not a verbs question. The question instead tests transitional words and phrases.

8. Question 8 is a verbs question. Each answer choice offers a different tense for the verb to be.

21b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Approach ELIMINATION To answer a verbs question, follow these steps: 1. Determine what kinds of verbs appear in the answer choices.

2. If you’re dealing with different verb tenses, eliminate answers that don’t reflect the tense of other verbs in the sentence.

3. If you’re looking at singular and plural verbs, eliminate answers that don’t agree with the subject.

Answer question 4 with your teacher.

ENTRANCE

ELIMINATION

EXIT

4

A survey conducted by political scientist Robert Putnam in his book Bowling Alone: The Collapse and Revival of American Community found that the level of civic engagement in the United States, measured by participation in community organizations, experienced a decline, with parent-teacher associations witnessing a drop in membership of about 61% between 1965 and 1995, indicating that social ties among Americans __________ significantly weaker.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) becoming B) to become C) to have become D) had become

22

Warm-Up

Foundation

Approach

22a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

VERBS

Approach (24 minutes) In this section, students will practice applying the steps in Elimination to a verbs question.

We want to know, at least in a basic sense, what tense those verbs are. One way to think about that is to ask: do the actions that most of them describe take place in the past, the present, or the future?

SLIDE 22.1

Verbs questions can be a little puzzling because there are two different kinds you can run into.

But our Elimination step will help us cut right through the fog. What does our first step say?

Determine what kinds of verbs appear in the answer choices.

They’re mostly in the past.

Other than witnessing and indicating, they’re all past tense verbs. Knowing that, what do we need to do next?

Eliminate answer choices that aren’t in the past tense.

What kind of verbs might we find? Steps 2 and 3 will help us to see what we’re looking for. We’re talking about answer choices that fall into two different buckets. What are you going to see with the first kind of answer choices?

Different verb tenses.

Choice A is becoming. What should we do with that choice?

And the second kind?

It isn’t in the past, so we can eliminate it.

They have singular and plural verbs.

Choice A is out. What about choice B? What does it say, and what should we do with it?

It says to become. That doesn’t sound like it’s describing something in the past, so we can eliminate it too.

Our answer choices should point us in one of two directions. Look in your book at question 4.

Specifically, look over the answer choices. Which of the two kinds of verbs questions do we have here?

We’ve eliminated choices A and B. That leaves us with choice C, to have become, and choice D, had become. Which of those two answer choices most clearly describes something happening in the past?

Choice D, had become.

Different verb tenses.

How do you know they’re different tenses?

They all have to do with the same verb, to become. But some have to do with different times, like the present and the past.

Choice C might be tempting because it describes something that takes place partly in the past. But the kind of action it describes continues up to the present, or at least happened very recently.

Why does choice D seem to line up better with the other verbs in the sentence?

Look at our second Elimination step. It begins with: If you’re dealing with different verb tenses.

We’ve decided that’s what we’re dealing with, so what does step 2 say we should we do next?

Eliminate answers that don’t reflect the tense of other verbs in the sentence.

It’s totally in the past.

We’re dealing with a long sentence here. Wind your way through it looking for the verbs that are already there. What do you see?

Conducted. Found. Measured. Experienced.

Witnessing. Indicating.

Let’s make sure to check it. If you plug choice D, had become, into the sentence, does it make sense when you read it?

It sounds good.

Choice D is correct. When you run into a verbs question on your test, ask: are the answer choices offering up different verb tenses? If they are, checking the tense of the other verbs in the sentence can give you a big clue to the answer.

22b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

VERBS

Approach Answer questions 5 and 8 on your own. Focus on eliminating incorrect answers.

5

7

An economic analysis by economist Thomas Piketty in his book Capital in the Twenty-First Century revealed that the wealth inequality in the United States in 2010, measured by the share of total income going to the top 1% of earners, __________ 1.6 times greater than the level of inequality observed in 1980.

The internet has made communication faster and more accessible than ever before. ____________, it has also led to concerns about privacy and security in the digital age.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) However B) For instance

A) being

C) Consequently

B) was

D) In the same vein

C) to have been D) will be

8

The Women’s Suffrage Movement in the United States saw the emergence of many influential women leaders, advocating for women’s right to vote. Among these leaders __________ Susan B. Anthony, cofounder of the National Woman Suffrage Association.

6

Photography was originally a laborious process, requiring hours of exposure and careful handling of chemicals. ____________, early photographers had to be both skilled artists and chemists.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) was B) were

A) Therefore

C) are

B) Conversely

D) have been

C) Similarly D) On the contrary

Warm-Up

Foundation

Approach

Strategy

23a

Wrap-Up

Review

23

Curriculum for the Digital SAT®

VERBS

Approach (24 minutes) Give the class 4 minutes to answer questions 5 and 8. Review the answers afterward.

SLIDE 23.1

SLIDE 23.2

5. The correct answer is B. First, we determine whether the answer choices present different verb tenses or whether they alternate singular and plural verbs. Since the answer choices are being, was, to have been, and will be, they offer different verb tenses.

8. The correct answer is A. First, we determine whether the answer choices present different verb tenses or whether they alternate singular and plural verbs. Although not all answer choices are in the same tense, the first two answers, was and were, are both in the past tense. Because of that, we’re looking at singular and plural verbs.

Our next step is to eliminate answers that don’t reflect the tense of other verbs in the sentence.

Since the verbs in the sentence are revealed and measured, we need to eliminate answers that are not in the past tense.

Our next step is to eliminate answers that don’t agree with the subject.

Choice B is incorrect and can be eliminated because, while Susan B. Anthony is one person and therefore singular, were is a plural verb.

Choice A is incorrect and can be eliminated because being is not in the past tense. It’s a present participle, and it does not provide a clear or complete thought in this context.

Choice C is incorrect and can be eliminated because are is a plural verb.

Choice C is incorrect and can be eliminated because to have been is not in the past tense.

It’s in the present perfect tense and does not provide a clear or complete thought in this context.

Choice D is incorrect and can be eliminated because have been is a plural verb.

Choice A is correct because was is a singular verb that fits grammatically with Susan B. Anthony.

Choice D is incorrect and can be eliminated because will be is in the future tense, not the past tense. In addition, the sentence is discussing a comparison of wealth inequality levels between two specific years in the past (1980 and 2010). Using the future tense is not appropriate for this context.

Choice B is correct because was completes the sentence by providing a simple past tense verb that aligns with the time frame being discussed. It creates a clear and concise statement that the level of wealth inequality in 2010 was 1.6 times greater than that in 1980.

The sentence is discussing a completed action in the past, so the simple past tense is the appropriate choice.

23b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Approach EXIT If you’ve found the answer ...

Plug it into the sentence and confirm that it sounds right. Mark your answer and continue to the next question.

If you’re stuck between two choices ...

Go with your gut. Select the choice that is most consistent with the context and sounds best.

If you have no idea what the answer is ...

Watch out for doubles in your answer choices. Look for the choice that stands out from the rest.

• If you see more than one answer with the same helping verb, you can likely eliminate both.

• If you see two past tense options and one present tense, choose the present tense one.

• If you see two singular options and one plural option, pick the plural one.

Review questions 9 and 10. Work with your teacher to choose the correct answer from the remaining choices.

ENTRANCE

ELIMINATION

9

EXIT

10

Similar to other Arctic insects, the Arctic woolly bear caterpillar (Gynaephora groenlandica) cannot produce its own heat in the extreme cold; therefore, during the freezing winter months, it __________ by producing natural antifreeze called a cryoprotectant in its body, which prevents its cells from freezing and rupturing.

American inventor Richard James developed a well-known playful novelty item. In the 1940s, he invented the Slinky, which __________ of a traditional metallic spring.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) were a flexible, walking version B) are flexible, walking versions

A) had adapted

C) were flexible, walking versions

B) adapted

D) is a flexible, walking version

C) would adapt D) adapts

24

Warm-Up

Foundation

Approach

24a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

VERBS

Approach (24 minutes) In this section, students will practice answering verbs questions using various Exit strategies, depending on how many answers they are initially able to eliminate before becoming stuck.

Choice D is correct. Down to those two possibilities, just plugging the remaining answers in and going with our gut led to the right answer. Take a look at question 10. What’s the difference between the two answer choices that are left?

SLIDE 24.1

When you’ve found an answer to a verbs question, take a moment to double-check your choice before you move on. Plug the answer into the sentence and confirm that it sounds right.

But what if you can’t narrow your options down to one answer choice? What should you do if you get stuck between two options?

Go with your gut. Select the choice that is most consistent with the context and sounds best.

One says are flexible. The other says is a flexible.

The sentence is talking about the toy called the Slinky. Does it sound right to say that a Slinky is a flexible, walking version or that a Slinky are flexible, walking versions? Go with your gut.

You might not have known there’s a direct connection between your ear and your gut. Okay, you might not want to put that on a biology test.

But with verbs questions, it’s okay to let your ear and your gut be your guide.

The first one sounds right: A Slinky is a flexible, walking version.

And which choice does that make correct?

Choice D. Once again, choice D is correct. When it comes to this kind of question, the convenient thing is that we use words every day. Even just talking, without knowing it, we’re practicing using verbs correctly. So, any time you get stuck after a couple of answer eliminations, try the ear-andgut check. Usually, the answer that sounds right really is!

SLIDE 24.2

For instance, take a look at question 9. What’s the subject of this long, winding sentence?

The Arctic woolly bear caterpillar.

I wonder if it turns into an Arctic woolly bear butterfly. Well, that’s beside the point. We can see that choices A and B have been eliminated— looks like that was because they’re both past tense. We’re down to options C and D. Try plugging each one in and reading the sentence to yourself. Which one passes the gut test?

Choice D. Choice C would have us say that during the freezing winter months this caterpillar would adapt. That almost makes it sound like it stopped adapting at some point along the way. But choice D just says it adapts. That describes a process that it continues to do, which only makes sense with natural behavior. Are you feeling confident about choice D? Yes.

24b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

VERBS

Practice Set 2

Complete the practice set. When you have finished, double-check your work.

11

13 Electrons in a cathode ray tube, a component used in older television sets, are accelerated by electric fields until they __________ a desired velocity, at which point they are directed to strike a phosphorescent screen, creating an image.

In his renowned series of water lily paintings from the late 1800s and early 1900s, French Impressionist painter Claude Monet sought to depict the ethereal quality of his own garden in Giverny, employing dappled brushstrokes and vivid colors to render the water’s surface and the floating vegetation. Interestingly, the iconic water lilies themselves __________ the central focus in many of these works.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) will achieve B) achieve

Which choice completes the text so that it conforms to the conventions of Standard English?

C) had achieved

A) are

D) are achieving

B) have been C) were D) is

12 Established in 1945 to promote international cooperation and maintain peace and security, the United Nations initially consisted of 51 member countries. By 1980, the organization __________ its original membership.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) has tripled B) had tripled C) triples D) will triple

Warm-Up

Foundation

Approach

Strategy

25a

Wrap-Up

Review

25

Curriculum for the Digital SAT®

VERBS

Practice Set 2 (12 minutes) Give students 6 minutes to complete the practice set. Review the answers before moving to the next section.

Choice D is incorrect because will triple is in the future tense, which is used to describe actions that will take place after the present moment. However, the text is referring to a past event, so the future tense is not appropriate.

SLIDE 25.1

11. The correct answer is B. Choice A is incorrect because will achieve is in the future tense. In this sentence, a present tense verb is needed to indicate an action that occurs whenever electrons are accelerated by electric fields in the context of a cathode ray tube’s operation.

Choice B is correct because had tripled is in the past perfect tense, which is used to describe actions that were completed before another past action or event. In this case, the United Nations tripled its membership by 1980 after its establishment in 1945, so the past perfect tense is the most appropriate choice.

Choice C is incorrect because had achieved is in the past perfect tense. This tense is used to describe an action that was completed before another past action. However, the sentence describes a process that happens in general terms whenever a cathode ray tube is in operation, so a simple present tense verb is appropriate.

SLIDE 25.3

13. The correct answer is A. Choice B is incorrect because have been is in the present perfect tense, suggesting a condition or state that originated in the past and is still relevant or continuing into the present. While it’s technically correct to refer to the paintings in the present perfect since they still exist and are observed today, the use of present simple can be more straightforward and customary in discussing the content or thematic elements of artwork.

Choice D is incorrect because are achieving is in the present continuous tense. It suggests an ongoing process, but in this context, the simple present tense is more appropriate to describe a general fact related to the functioning of a cathode ray tube.

Choice B is correct because achieve is in the simple present tense. This choice is appropriate because it describes an action that occurs once in the operation of a cathode ray tube. Electrons are accelerated until they achieve a desired velocity as a regular part of the device’s operation, making the simple present tense the correct option.

Choice C is incorrect because were is in the past tense. While it might seem appropriate given that the paintings were made in the past, discussing the thematic and content elements of artworks, especially those that continue to be viewed and studied, is often done in the present tense.

SLIDE 25.2

12. The correct answer is B. Choice A is incorrect because has tripled is in the present perfect tense, which is used to describe actions that have been completed at an unspecified time in the past. However, the text is referring to a specific time period by 1980, so the present perfect tense is not appropriate.

Choice D is incorrect because is doesn’t agree in number with the plural subject water lilies.

We need a plural verb to match the plural subject for correct subject-verb agreement.

Choice A is correct because are is in the present tense and agrees with the plural subject water lilies. When discussing the content, themes, or elements within artworks, it is conventional to use the present tense, as these aspects of the art continue to be observed and interpreted in the present every time the works are viewed.

Choice C is incorrect because triples is in the simple present tense, which is used to describe habitual or general actions. However, the text is referring to a specific event that happened in the past, so the simple present tense is not appropriate.

25b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Strategy TUNE OUT Use the Tune Out strategy to ignore unnecessary words, which will make it easier to apply grammar rules.

You can use this strategy on verbs questions when you have a complicated sentence with several words between the subject and the verb. Follow these steps: 1. Cross out all the words in the sentence that come between the subject and the verb.

2. Plug each answer into the underlined portion.

3. Eliminate answers that don’t agree with the subject.

Review question 14 with your teacher. Use the Tune Out strategy to answer the question as a class.

14

Irish playwright Samuel Beckett’s Waiting for Godot, a tragicomedy originally written in French, __________ scholars’ understanding of the bilingual context.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) has expanded B) are expanding C) have expanded D) expand

26

Warm-Up

Foundation

Approach

26a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

VERBS

Strategy (28 minutes) In this section, students will practice using the Tune Out strategy to answer a verbs question.

How does that sound?

SLIDE 26.1

Pretty good.

These verbs questions from the SAT can feel like they’re scrambling your brain. The answer choices usually have only small differences and sound a lot alike. Fortunately, we’ve got a strategy that “cuts to the chase,” so to speak.

It will help you focus on just the parts of the question you really need. What’s this strategy called?

Plug in each of the other three answer choices.

See how those sound.

Give students 1 minute to complete the task.

Did any of those other answer choices sound good?

No, they sounded weird.

Tune Out.

Step 3 for our strategy says Eliminate answers that don’t agree with the subject. Did choices B, C, and D sound wrong enough that you’d be confident to eliminate them?

The bottom line is that we want to tune out all the words that come between the subject and the verb. That’s the first step of our strategy. Why do you think we’d want to do that?

It will make it clearer which verb fits with the subject.

Yes. And you’re feeling good about choice A as the correct answer?

Totally.

Let’s practice using question 14. What’s the subject of the sentence?

Waiting for Godot.

Choice A is the correct answer. It’s the only answer that is a singular verb—meaning it fits with one single play—and it’s also the right verb tense. Yet we arrived at the answer without focusing too much on those details. By tuning out the words we didn’t need, it became clear which verb fit with our subject.

That’s kind of an unusual subject. What is Waiting for Godot?

It’s the name of a play.

Since our verb is going to go in the blank, what do we need to cross out or tune out?

The phrase between the commas, a tragicomedy originally written in French.

Keep in mind that when you’re actually taking the SAT, you’ll be on a computer. On a screen, you won’t be able to cross words out the way you can in your book. What do you think you might have available that you could write on?

Scratch paper?

You’ll have scratch paper, so one option would be to jot down the words that remain after you “tune out” the phrase we don’t need. If you did that, what might you write down?

Waiting for Godot (blank) scholars’ understanding.

What does step 2 of our strategy say to do?

Plug each answer into the underlined portion.

What do you get when you plug choice A in?

Waiting for Godot has expanded scholars’ understanding.

26b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

VERBS

Strategy Work on your own using the Tune Out strategy to answer questions 15 and 16.

15

16

Spanish composer Isaac Albéniz’s Suite Española (1886), a collection of piano pieces influenced by Spanish folk tunes, __________ researchers’ understanding of the crosscultural and innovative musical styles with which Albéniz worked.

Vera Rubin, an American astronomer, __________ data about galaxy rotation curves when she made a groundbreaking discovery: evidence of the existence of dark matter, an invisible substance that makes up a significant portion of the universe’s mass.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) are deepening

A) analyzed

B) has deepened

B) has been analyzing

C) have deepened

C) will analyze

D) deepen

D) was analyzing

Warm-Up

Foundation

Approach

Strategy

27a

Wrap-Up

Review

27

Curriculum for the Digital SAT®

VERBS

Strategy (28 minutes) Give students 4 minutes to complete questions 15 and 16. Review the answers afterward.

SLIDE 27.1

SLIDE 27.2

15. The correct answer is B. Using the Tune Out strategy, cross out all the words in the sentence that come between the subject and the verb.

That leaves us with Suite Española (blank) researchers’ understanding of the cross-cultural and innovative musical styles.

16. The correct answer is D. Using the Tune Out strategy, cross out all the words in the sentence that come between the subject and the verb.

That leaves us with Vera Rubin (blank) data about galaxy rotation curves when she made a groundbreaking discovery.

Next, plug each answer into the underlined portion.

Next, plug each answer into the underlined portion.

Choice A is incorrect because it does not make logical or grammatical sense to say Suite Española [are deepening] researchers’ understanding of the cross-cultural and innovative musical styles.

Choice A is incorrect because although both actions (analyzing data and making the discovery) did occur in the past, using simple past for both verbs (analyzed and made) doesn't quite work. The intended meaning of this sentence is that Rubin made her discovery (a one-time occurrence) while she was in the process of analyzing the data (an ongoing action), and using simple past for both verbs does not convey this idea.

Choice C is incorrect because it does not make logical or grammatical sense to say Suite Española [have deepened] researchers’ understanding of the cross-cultural and innovative musical styles.

Choice B is incorrect because has been analyzing implies an action that is continuing up to the present time, but that doesn't work with the simple past verb made a discovery, which describes an action that was completed in the past.

Choice D is incorrect because it does not make logical or grammatical sense to say Suite Española [deepen] researchers’ understanding of the cross-cultural and innovative musical styles.

Choice B is correct because it makes sense to say Suite Española [has deepened] researchers’ understanding of the cross-cultural and innovative musical styles. Furthermore, the singular noun a collection of piano pieces pairs correctly with the singular helping verb has.

Choice C is incorrect because will analyze describes an action that will take place in the future, but that doesn't work with the simple past verb made a discovery, which describes an action that was completed in the past.

Choice D is correct because it clearly and correctly conveys the idea that Rubin was in the process of analyzing data (an ongoing action that took place in the past) when she made a groundbreaking discovery (a one-time action that occurred in the past).

27b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

VERBS

Strategy Work on your own using the Sounds Wrong Is Wrong strategy to answer questions 18 and 19.

18

19

Throughout the 1930s and 1940s, American artist Georgia O’Keeffe famously depicted the rugged landscapes of New Mexico in her art, using bold shapes and rich colors to represent the region’s distinct geological features and expansive skies. The evocative animal skulls that often __________ a recurring motif in these paintings are particularly striking.

American architect Alfred M. Butts invented a popular word game. In 1938, he developed the game of Scrabble, which __________ of the simpler word-based board games of that era.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) were an enhanced, scoring version

Which choice completes the text so that it conforms to the conventions of Standard English?

B) are enhanced, scoring versions C) were enhanced, scoring versions D) is an enhanced, scoring version

A) are B) have been C) were D) is

Warm-Up

Foundation

Approach

Strategy

29a

Wrap-Up

Review

29

Curriculum for the Digital SAT®

VERBS

Strategy (28 minutes) Give students 4 minutes to complete questions 18 and 19. Review the answers afterward.

SLIDE 29.1

SLIDE 29.2

18. The correct answer is A. Using the Sounds Wrong Is Wrong strategy, plug each answer choice into the blank and see if it sounds wrong or incomplete.

19. The correct answer is D. Using the Sounds Wrong Is Wrong strategy, plug each answer choice into the blank and see if it sounds wrong or incomplete.

Choice B is incorrect because it reads The evocative animal skulls that often [have been] a recurring motif in these paintings are particularly striking. This creates a verb tense mismatch between have been and are.

Choice A is incorrect because it reads In 1938, he developed the game of Scrabble, which [were an enhanced, scoring version] of the simpler word-based board games of that era. It creates a mismatch between the singular Scrabble and the plural verb were.

Choice C is incorrect because it reads The evocative animal skulls that often [were] a recurring motif in these paintings are particularly striking. This creates a verb tense mismatch between were and are.

Choice B is incorrect because it reads In 1938, he developed the game of Scrabble, which [are enhanced, scoring versions] of the simpler word-based board games of that era. It creates a mismatch between the singular Scrabble and the plural verb are.

Choice D is incorrect because it reads The evocative animal skulls that often [is] a recurring motif in these paintings are particularly striking. This creates a subject-verb agreement mismatch.

Choice C is incorrect because it reads In 1938, he developed the game of Scrabble, which [were enhanced, scoring versions] of the simpler word-based board games of that era. It creates a mismatch between the singular Scrabble and the plural verb were.

Choice A is correct because it creates a complete, correct sentence: The evocative animal skulls that often [are] a recurring motif in these paintings are particularly striking.

Choice D is correct because it creates a complete, correct sentence: In 1938, he developed the game of Scrabble, which [is an enhanced, scoring version] of the simpler word-based board games of that era. The singular noun Scrabble fits correctly with the singular noun is.

29b

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

MASTERY FOR THE DIGITAL SAT: WRITING

Practice Set 3

Complete the practice set. When you have finished, double-check your work.

22

20

British archaeologist Howard Carter’s discovery of Tutankhamun’s tomb (1922), one of the most intact ancient Egyptian burial sites ever found, __________ significant insights into the art, burial customs, and royal history of the New Kingdom period in ancient Egypt.

A study led by Dr. Jane Lubchenco, a marine ecologist, found that the rate of kelp forest recovery in marine protected areas __________ three times faster than in areas open to fishing, highlighting the effectiveness of these conservation zones in promoting ecosystem resilience.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) provide

A) is

B) provides

B) being

C) have provided

C) to be

D) providing

D) will be

21 American composer Amy Beach’s Gaelic Symphony (1896), the first symphony composed by an American woman, __________ scholars’ understanding of the significant contributions of women to classical music in the late 19th century.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) has broadened B) are broadening C) have broadened D) broaden

30

Warm-Up

Foundation

Approach

30a

Strategy

Wrap-Up

Review

Curriculum for the Digital SAT®

MASTERY FOR THE DIGITAL SAT: WRITING

VERBS

Wrap-Up

Complete the wrap-up. When you have finished, double-check your work.

4

6 Architecture and the sensation of space are central to Frank Lloyd Wright’s designs. In Fallingwater, for example, the integration of the house with its natural surroundings creates a seamless connection to the outdoors, while in the Guggenheim Museum, the character of the spiral layout __________ a form of spatial experience whereby the visitor’s path evokes a sense of continuous movement.

When developing the revolutionary e-commerce platform Amazon (1994), entrepreneur Jeff Bezos was inspired by his interest in the internet’s potential. The successful businessman has founded multiple companies, but his entrepreneurial spirit __________ evident early on. At the age of 12, he started his first business venture, a summer camp for young students.

Which choice completes the text so that it conforms to the conventions of Standard English?

Which choice completes the text so that it conforms to the conventions of Standard English?

A) were

A) presented

B) was

B) has presented

C) have been

C) presents

D) are

D) will be presenting

5 Inventor Thomas Edison and entrepreneur George Westinghouse were key players in the “War of the Currents,” a period of intense competition that __________ the development and adoption of alternating current (AC) and direct current (DC) electrical power systems.

Which choice completes the text so that it conforms to the conventions of Standard English?

A) have influenced B) were influencing C) influenced D) influence

Warm-Up

Foundation

Approach

Strategy

31a

Wrap-Up

Review

31

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