# Section 10.6 Answers - Mathematics

1. ({f y}=c_{1}e^{2t}left[egin{array}{c}{3cos t+sin t}{5cos t}end{array} ight]+c_{2}e^{2t}left[egin{array}{c}{3sin t-cos t}{5sin t}end{array} ight])

2. ({f y}=c_{1}e^{-t}left[egin{array}{c}{5cos 2t+sin 2t}{13cos 2t}end{array} ight]+c_{2}e^{-t}left[egin{array}{c}{5sin 2t-cos 2t}{13sin 2t}end{array} ight])

3. ({f y}=c_{1}e^{3t}left[egin{array}{c}{cos 2t+sin 2t}{2cos 2t}end{array} ight]+c_{2}e^{3t}left[egin{array}{c}{sin 2t-cos 2t}{2sin 2t}end{array} ight])

4. ({f y}=c_{1}e^{2t}left[egin{array}{c}{cos 3t-sin 3t}{cos 3t}end{array} ight]+c_{2}e^{2t}left[egin{array}{c}{sin 3t+cos 3t}{sin 3t}end{array} ight])

5. ({f y}=c_{1}left[egin{array}{c}{-1}{-1}{2}end{array} ight]e^{-2t}+c_{2}e^{4t}left[egin{array}{c}{cos 2t-sin 2t}{cos 2t+sin 2t}{2cos 2t}end{array} ight]+c_{3}e^{4t}left[egin{array}{c}{sin 2t+cos 2t}{sin 2t-cos 2t}{2sin 2t}end{array} ight])

6. ({f y}=c_{1}left[egin{array}{c}{-1}{-1}{1}end{array} ight]e^{-t}+c_{2}e^{-2t}left[egin{array}{c}{cos 2t-sin 2t}{-cos 2t-sin 2t}{2cos 2t}end{array} ight]+c_{3}e^{-2t}left[egin{array}{c}{sin 2t+cos 2t}{-sin 2t+cos 2t}{2sin 2t}end{array} ight])

7. ({f y}=c_{1}left[egin{array}{c}{1}{1}{1}end{array} ight]e^{2t}+c_{2}e^{t}left[egin{array}{c}{-sin t}{sin t}{cos t}end{array} ight]+c_{3}e^{t}left[egin{array}{c}{cos t}{-cos t}{sin t}end{array} ight])

8. ({f y}=c_{1}left[egin{array}{c}{-1}{1}{1}end{array} ight]e^{t}+c_{2}e^{-t}left[egin{array}{c}{-sin 2t-cos 2t}{2cos 2t}{2cos 2t}end{array} ight]+c_{3}e^{-t}left[egin{array}{c}{cos 2t-sin 2t}{2sin 2t}{2sin 2t}end{array} ight])

9. ({f y}=c_{1}e^{3t}left[egin{array}{c}{cos 6t-3sin 6t}{5cos 6t}end{array} ight]+c_{2}e^{3t}left[egin{array}{c}{sin 6t+3cos 6t}{5sin 6t}end{array} ight])

10. ({f y}=c_{1}e^{2t}left[egin{array}{c}{cos t-3sin t}{2cos t}end{array} ight]+c_{2}e^{2t}left[egin{array}{c}{sin t+3cos t}{2sin t}end{array} ight])

11. ({f y}=c_{1}e^{2t}left[egin{array}{c}{3sin 3t-cos 3t}{5cos 3t}end{array} ight]+c_{2}e^{2t}left[egin{array}{c}{-3cos 3t-sin 3t}{5sin 3t}end{array} ight])

12. ({f y}=c_{1}e^{2t}left[egin{array}{c}{sin 4t-8cos 4t}{5cos 4t}end{array} ight]+c_{2}e^{2t}left[egin{array}{c}{-cos 4t-8sin 4t}{5sin 4t}end{array} ight])

13. ({f y}=c_{1}left[egin{array}{c}{-1}{1}{1}end{array} ight]e^{-2t}+c_{2}e^{t}left[egin{array}{c}{sin t}{-cos t}{cos t}end{array} ight]+c_{3}e^{t}left[egin{array}{c}{-cos t}{-sin t}{sin t}end{array} ight])

14. ({f y}=c_{1}left[egin{array}{c}{2}{2}{1}end{array} ight]e^{-2t}+c_{2}e^{2t}left[egin{array}{c}{-cos 3t-sin 3t}{-sin 3t}{cos 3t}end{array} ight]+c_{3}e^{2t}left[egin{array}{c}{-sin 3t+cos 3t}{cos 3t}{sin 3t}end{array} ight])

15. ({f y}=c_{1}left[egin{array}{c}{1}{2}{1}end{array} ight]e^{3t}+c_{2}e^{6t}left[egin{array}{c}{-sin 3t}{sin 3t}{cos 3t}end{array} ight]+c_{3}e^{6t}left[egin{array}{c}{cos 3t}{-cos 3t}{sin 3t}end{array} ight])

16. ({f y}=c_{1}left[egin{array}{c}{1}{1}{1}end{array} ight]e^{t}+c_{2}e^{t}left[egin{array}{c}{2cos t-2sin t}{cos t-sin t}{2cos t}end{array} ight]+c_{3}e^{t}left[egin{array}{c}{2sin t+2cos t}{cos t+sin t}{2sin t}end{array} ight])

17. ({f y}=e^{t}left[egin{array}{c}{5cos 3t+sin 3t}{2cos 3t+3sin 3t}end{array} ight])

18. ({f y}=e^{4t}left[egin{array}{c}{5cos 6t+5sin 6t}{cos 6t-3sin 6t}end{array} ight])

19. ({f y}=e^{t}left[egin{array}{c}{17cos 3t-sin 3t}{7cos 3t+3sin 3t}end{array} ight])

20. ({f y}=e^{t/2}left[egin{array}{c}{cos (t/2)+sin (t/2)}{-cos (t/2)+2sin (t/2)}end{array} ight])

21. ({f y}=left[egin{array}{c}{1}{-1}{2}end{array} ight]e^{t}+e^{4t}left[egin{array}{c}{3cos t+sin t}{cos t-3sin t}{4cos t-2sin t}end{array} ight])

22. ({f y}=left[egin{array}{c}{4}{4}{2}end{array} ight]e^{8t}+e^{2t}left[egin{array}{c}{4cos 2t+8sin 2t}{-6sin 2t+2cos 2t}{3cos 2t+sin 3t}end{array} ight])

23. ({f y}=left[egin{array}{c}{0}{3}{3}end{array} ight]e^{-4t}+e^{4t}left[egin{array}{c}{15cos 6t+10sin 6t}{14cos 6t-8sin 6t}{7cos 6t-4sin 6t}end{array} ight])

24. ({f y}=left[egin{array}{c}{6}{-3}{3}end{array} ight]e^{8t}+left[egin{array}{c}{10cos 4t-4sin 4t}{17cos 4t-sin 4t}{3cos 4t-7sin 4t}end{array} ight])

29. ({f U}=frac{1}{sqrt{2}}left[egin{array}{c}{-1}{1}end{array} ight],quad {f V}=frac{1}{sqrt{2}}left[egin{array}{c}{1}{1}end{array} ight])

30. ({f U}approx left[egin{array}{c}{.5257}{.8507}end{array} ight],quad {f V}approx left[egin{array}{c}{-.8507}{.5257}end{array} ight])

31. ({f U}approx left[egin{array}{c}{.8507}{.5257}end{array} ight],quad {f V}approx left[egin{array}{c}{-.5257}{.8507}end{array} ight])

32. ({f U}approx left[egin{array}{c}{-.9732}{.2298}end{array} ight],quad {f V}approx left[egin{array}{c}{.2298}{.9732}end{array} ight])

33. ({f U}approx left[egin{array}{c}{.5257}{.8507}end{array} ight],quad {f V}approx left[egin{array}{c}{-.8507}{.5257}end{array} ight])

34. ({f U}approx left[egin{array}{c}{-.5257}{.8507}end{array} ight],quad {f V}approx left[egin{array}{c}{.8507}{.5257}end{array} ight])

35. ({f U}approx left[egin{array}{c}{-.8817}{.4719}end{array} ight],quad {f V}approx left[egin{array}{c}{.4719}{.8817}end{array} ight])

36. ({f U}approx left[egin{array}{c}{.8817}{.4719}end{array} ight],quad {f V}approx left[egin{array}{c}{-.4719}{.8817}end{array} ight])

37. ({f U}= left[egin{array}{c}{0}{1}end{array} ight],quad {f V}= left[egin{array}{c}{-1}{0}end{array} ight])

38. ({f U}= left[egin{array}{c}{0}{1}end{array} ight],quad {f V}= left[egin{array}{c}{1}{0}end{array} ight])

39. ({f U}=frac{1}{sqrt{2}}left[egin{array}{c}{1}{1}end{array} ight],quad {f V}=frac{1}{sqrt{2}}left[egin{array}{c}{-1}{1}end{array} ight])

40. ({f U}approx left[egin{array}{c}{.5257}{.8507}end{array} ight],quad {f V}approx left[egin{array}{c}{-.8507}{.5257}end{array} ight])

### Homeroom: The Pandemic’s Potential Silver Lining for Kids

In general, there is no more evidence of “understanding” in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way. How do we know, for example, that a student isn’t simply repeating an explanation provided by the teacher or the textbook, thus exhibiting mere “rote learning” rather than “true understanding” of a problem-solving procedure?

Math learning is a progression from concrete to abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities—entities like dollars, percentages, groupings of pencils. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. That is, information and procedures that have been become automatic frees up working memory. With working memory less burdened, the student can focus on solving the problem at hand. Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of “mathematical understanding,” in other words, can impede the “doing” of actual mathematics.

Explaining the solution to a problem comes when students can draw on a strong foundation of content relevant to the topic currently being learned. As students find their feet and establish a larger repertoire of mastered knowledge and methods, the more articulate they can become in explanations. Children in elementary and middle school who are asked to engage in critical thinking about abstract ideas will, more often than not, respond emotionally and intuitively, not logically and with “understanding.” It is as if the purveyors of these practices are saying: “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.” That may be behaviorally interesting, but it is not mathematical development and it leaves them behind in the development of their fundamental skills.

The idea that students who do not demonstrate their strategies in words and pictures or by multiple methods don’t understand the underlying concepts is particularly problematic for certain vulnerable types of students. Consider students whose verbal skills lag far behind their mathematical skills—non-native English speakers or students with specific language delays or language disorders, for example. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain—whether orally or in written words—how they arrived at their answers.

Most exemplary are children on the autism spectrum. As the autism researcher Tony Attwood has observed, mathematics has special appeal to individuals with autism: It is, often, the school subject that best matches their cognitive strengths. Indeed, writing about Asperger’s Syndrome (a high-functioning subtype of autism), Attwood in his 2007 book The Complete Guide to Asperger’s Syndrome notes that “the personalities of some of the great mathematicians include many of the characteristics of Asperger’s syndrome.”

And yet, Attwood added, many children on the autism spectrum, even those who are mathematically gifted, struggle when asked to explain their answers. “The child can provide the correct answer to a mathematical problem,” he observes, “but not easily translate into speech the mental processes used to solve the problem.” Back in 1944, Hans Asperger, the Austrian pediatrician who first studied the condition that now bears his name, famously cited one of his patients as saying that, “I can’t do this orally, only headily.”

Writing from Australia decades later, a few years before the Common Core took hold in America, Attwood added that it can “mystify teachers and lead to problems with tests when the person with Asperger’s syndrome is unable to explain his or her methods on the test or exam paper.” Here in Common Core America, this inability has morphed into an unprecedented liability.

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one? And, to the extent that it isn’t a necessary criterion, should verbal explanation be the way to gauge comprehension?

Measuring understanding, or learning in general, isn’t easy. What testing does is measure “markers” or byproducts of learning and understanding. Explaining answers is but one possible marker.

Another, quite simply, are the answers themselves. If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child’s verbal abilities. But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way. At best, verbal explanations beyond “showing the work” may be superfluous at worst, they shortchange certain students and encumber the mathematics for everyone.

As Alfred North Whitehead famously put it about a century before the Common Core standards took hold:

It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.

Go Math Answer Key for Grade 3 aids teachers to differentiate instruction, building, and reinforcing foundational mathematics skills that alter from the classroom to real life. With the help of Go Math Primary School Grade 3 Answer Key, you can think deeply regarding what you are learning, and you will really learn math easily just like that. So, download chapterwise Go Math Answer Key Pdf for Grade 3 from the accessible links available over here and make your learning more efficient.

### Grade 3 HMH Go Math – Extra Practice Questions and Answers

• Chapter 1: Addition and Subtraction within 1,000 Assessment Test
• Chapter 2: Represent and Interpret Data Assessment Test
• Chapter 3: Understand Multiplication Assessment Test
• Chapter 4: Multiplication Facts and Strategies Assessment Test
• Chapter 5: Use Multiplication Facts Assessment Test
• Chapter 6: Understand Division Assessment Test
• Chapter 7: Division Facts and Strategies Assessment Test
• Chapter 8: Understand Fractions Assessment Test
• Chapter 9: Compare Fractions Assessment Test
• Chapter 10: Time, Length, Liquid Volume, and Mass Assessment Test
• Chapter 11: Perimeter and Area Assessment Test
• Chapter 12:Two-Dimensional Shapes Assessment Test

Following are some bunch of benefits that students and teachers can avail by referring to the HMH Go Math Solutions Key for Grade 3:

• Teachers can learn new techniques and educate students very well during their mathematics classes.
• GO Math Solution Key provides a seamless way to guarantee that students can access the content at proper levels of depth and rigor.
• An in-depth explanation of solutions helps students to succeed in high-stakes assessments.
• Go Math Grade 3 Answer Key pdf of All Chapters are well illustrated with solved examples and practice questions and support to strengthen basic mathematical concepts.

You can even more get some other benefits too with Go Math Solution Key for Grade 3. Hence, mathematics educators advise students to check out the HMH Go math primary school grade 3 answer key and score better marks. Access the respective chapter link provided here and practice more and more for clarifying all your queries at one go.

### FAQs on Go Math Solution Key for Grade 3

1. How do I solve mathematics questions easily?

Solving math problems easily requires more practice and a strong foundation on primary mathematics concepts. Learning math topics from the Go Math Primary School Solution Key is the smartest way. Even you can easily answer all types of questions during preparation or in the exams.

2. How can I find HMH Go Math Answer Key for Grade 3?

You can easily attain HMH Go math grade 3 Anwer key from our page along with step by step solutions for each question of all chapters.

3. Where can I get the solutions for all chapters Go Math Grade 3 Questions?

Go Math Solution Key is the best guide to get all solved questions and answers as well as extra practice questions. Avail the grade 3 Go Math Answer Key and get a good grip on all primary mathematics concepts.

4. Which is the best website to get free Go Math Answer Key for Grade 3?

## Know Each Student Better

ALEKS has patented a machine learning technology called ALEKS Insights (U. S. Patent No. 10,713,965) to promptly alert educators to at risk students. ALEKS Insights provides email notification to instructors calling attention to students (a) who are not succeeding, (b) who cease succeeding, (c) who are excessively procrastinating, or (d) who are learning unrealistically fast. These formative insights enable instructors to take timely action to help the students that need it the most.

Courses available for grades 3-12 can be implemented as a core or supplemental curriculum.

## Quick Link for All Order of Operations Worksheets

Click the image to be taken to that Order of Operations Worksheet.

## GMAT Quantitative Section Breakdown

The table below lists GMAT quant concepts in order of most-to-least frequent. (The most frequent concepts are obviously the most important!) To measure the frequency of GMAT math topics, I analyzed 766 official questions from the official GMATPrep tests 3 and 4, and the Official Guide for the GMAT Review so you don’t have to!

Note, of course, that the figures below are estimates based on a large number of questions, and may not reflect the exact proportions on an individual test.

GMAT Quant conceptPercentage frequencyWhat's it about?
Word Problems58.2%Interpreting the math in stories and descriptions
Integer properties and arithmetic31.1%Interpreting the math in charts and tables
Algebra16.3%Includes both “pure algebra,” and algebra as applied to other GRE quant concepts
Percents, ratios, and fractions13.7%
Two dimensional geometry10.6%Shapes, lines, and angles on the coordinate plane
Statistics6.3%Mean, median, standard deviation, etc…
Powers and roots6.3%
Probability and combinatronics5%Permutations, total number of possibilities, odds of an event happening, etc.
Inequalities4.7%
Sequences3.2%
Coordinate geometry2.9%
Data interpretation0.9%Math problems based on tables, charts, and graphs. You will also find these in the GMAT Integrated Reasoning section.
Three dimensional geometry0.8%
Functions0.4%

Note: Some questions tested multiple concepts and were thus counted more than one time in more than one category. As a result, the percentages in the chart above add up to more than 100%.

• Now that you know the most frequent GMAT math questions, check out the hardest (and easiest) ones in our latest video!

Grade 4 HMH Go Math Solutions Key gives students unlimited practice and feedback. You can understand the Primary School concepts easily by referring to our point to point explained Go Math Grade 4 Answer Key. Look no further and use the handy solutions of Class 4 for the concepts in Go Math Textbooks. Utilize the Chapterwise Solutions prevailing and prepare the concepts as per your requirement.

You will find the 4th Standard Go Math Answer Key extremely helpful to assess your preparation level. Identify areas you are lagging and allot time to those particular concepts. Score high grades and build Math skills by going through the concepts given in crystal clear format. Prepare the corresponding chapter you wish to practice by clicking on the concerned links.

### Grade 4 Homework Practice FL.

Common Core – Grade 4 – Practice Book

(Pages 1- 20) (Pages 21 – 47) (Pages 49- 65) (Pages 67 – 93) (Pages 95 – 109) (Pages 111 – 129) (Pages 131 – 153) (Pages 155- 167) (Pages 169- 185) (Pages 187- 204) (Pages 205- 217) (Pages 219- 244) (Pages 245- 258)

Both students and teachers can have a plethora of benefits in their study with 4th Grade Go Math Answer Key PDF. Practice using the Step by step solutions provided for Go Math 4th Standard Questions. They are as follows

• Go Math 4th Grade Answer Key will be of extreme help for candidates of Primary School to lay a deeper understanding of fundamentals.
• Solved Examples and Various Practice Questions covered as a part of Go Math Answer Key gives you insight into different kinds of questions.
• 4th Standard Go Math Solution Key provided encourages students anywhere and anytime mathematical thinking.
• Solutions prepared for 4th Grade Math are as per the Leading Curriculum and helps you score better grades in the exams.

### FAQs on Go Math 4th Standard Answer Key

1. How do I use the 4th Standard Go Math Answer Key to help me teach math?

You can use the Grade 4 Go Math Answer Key to teach maths by practicing from them. Solve numerous questions before the exam and score well in the exams.

2. Where can I get Grade 4 Go Math Answer Key PDF?

3. Where can I find the 4th Grade Go Math Answer Key for all Chapters?

You can find the 4th Grade Go Math Answer Key for all Chapters from our page. Click on the corresponding chapter and prepare as per your need.

You just need to tap on the quick links available to access the Go Math 4th Class Answer Key. Thereafter, you will be directed to a new page having the download option. Download and save them to use during your preparation.

## Section 10.6 Answers - Mathematics

The exponent of 10 tells you how many places to move the decimal point to the right for positive exponents or left for negative exponents. These rules come in especially handy for writing very large or very small numbers.

Since you will be working with very large and very small numbers, use scientific notation to cut down on all of the zeroes you need to write. Proper scientific notation specifies a value as a number between 1 and 10 (called the mantissa below) multiplied by some power of ten, as in mantissaexponent . The power of ten tells you which way to move the decimal point and by how many places. As a quick review:

10 = 1 × 10 1 , 253 = 2.53 × 100 = 2.53 × 10 2 and 15,000,000,000 = 1.5 × 10 10 which you will sometimes see written as 15 × 10 9 even though this is not proper scientific notation. For small numbers we have: = 1 × 10 -1 , × 10 -2 or about 0.395 × 10 -2 = 3.95 × 10 -3 .

When you divide two values given in scientific notation, divide the mantissa numbers and subtract the exponents in the power of ten. Then adjust the mantissa and exponent so that the mantissa is between 1 and 10 with the appropriate exponent in the power of ten. For example: × 10 10-23 = 0.5 × 10 -13 = 5 × 10 -14 .

Notice what happened to the decimal point and exponent in the examples. You subtract one from the exponent for every space you move the decimal to the right. You add one to the exponent for every space you move the decimal to the left.

Most scientific calculators work with scientific notation. Your calculator will have either an EE'' key or an EXP'' key. That is for entering scientific notation. To enter 253 (2.53 × 10 2 ), you would punch 2 . 5 3 EE or EXP 2. To enter 3.95 × 10 -3 , you would punch 3 . 9 5 EE or EXP 3 [ key]. Note that if the calculator displays 3.53 -14'' (a space between the 3.53 and -14), it means 3.53 × 10 -14 NOT 3.53 -14 ! The value of 3.53 -14 = 0.00000002144 = 2.144吆 -8 which is vastly different than the number 3.53吆 -14 . Also if you have the number 4 × 10 3 and you enter 4 × 1 0 EE or EXP 3, the calculator will interpret that as 4 × 10 × 10 3 = 4 × 10 4 or ten times greater than the number you really want!

One other word of warning: the EE or EXP key is used only for scientific notation and NOT for raising some number to a power. To raise a number to some exponent use the y x '' or x y '' key depending on the calculator. For example, to raise 3 to the 4th power as in 3 4 enter 3 y x or x y 4. If you instead entered it using the EE or EXP key as in 3 EE or EXP 4, the calculator would interpret that as 3吆 4 which is much different than 3 4 = 81.

### Unlimited Questions

Once you have created an assignment, you can regenerate all of its questions with a single click. The new questions will conform to the same parameters as the original questions, but they will be completely new. This feature is at the heart of our software and is what makes it so powerful: you choose the properties of the questions, not the questions themselves. When a question is replaced, you get a new one that is similar to the original question. How it works. You can regenerate entire assignments, particular question groups, or individual questions.

### Easy Spacing

Respace the entire assignment to the desired length with one click. Easily give your students enough room to show their work by increasing the spacing. Or you can save paper by decreasing the spacing.

Spacing can also be controlled manually.

### Presentation Mode

Very useful as a teaching aid when used in combination with an LCD projector or other display system. One to four questions at a time are shown on the screen.

Use this feature while you teach. Prepare your examples with the software, and then use a projector to display the questions on the board. This saves time during planning and during the lesson, and it makes it very easy to present long questions or questions with graphs and diagrams. With one question displayed, you can:

• Change the zoom level -- so students in the back can read it
• Draw lines beside the question to help you organize your work if you solve the question
• Show / hide the question number and the directions.

### Multiple-Version Printing

Print multiple versions of an assignment. You control how each new version is created: scramble the choices, scramble the questions, or make completely new questions. You can also save each new version after it is created.

### Scale Assignment

Proportionally increase or decrease the number of questions in the assignment. This is very useful when planning a lesson. You can create a few questions to use as examples, and then scale up the number of question to create a homework assignment. The questions on the homework will be completely new, yet follow precisely from the lesson--and you don't need to design the questions again.

### Export Questions

Export questions as bitmap images and paste them into your favorite word processing software. Questions created with our products can be added to existing assignments you have created with other programs. Or you can freshen old assignments by replacing old questions with new ones.

All questions are available for export.

### Good Multiple-Choice Questions

Every question you create can be toggled between free-response and multiple-choice format. Multiple-choice questions come with smart, potentially misleading choices. Some are based on common mistakes students make while others are just random but near the correct answer.

You control the number of choices each question has, from two to five.

### Merge Assignments

Merge two or more assignments into one. Easily create quizzes, tests, and reviews by merging the assignments from the unit and then scaling the total to an appropriate length. The questions will be new while following exactly from what you taught.

### Diagrams Drawn to Scale

Diagrams are all accurately drawn, except if the answer would be given away. If an angle is labeled as 30°, then it really is 30°. If a triangle's sides are labeled 3, 4, and 5, then its lengths truly are in a 3:4:5 ratio. Seeing accurate diagrams helps students gain an intuitive understanding of angles and measurements.

When you print an assignment, you choose how the answers are reported:

• On an answer sheet with just the odds
• In context (next to or within the question)

### Graphing and Graph Paper Utility

Supplement your lessons with high-quality graphs and graph paper of any size. Each graph can have zero to two functions graphed on it. Graphs can be of any logical and physical size. You can also tile graphs across the page to maximize your paper use.

### Custom Directions and Custom Questions

Enter your own directions to create new types of problems. Shown on the left was a standard order of operations question that has been modified to be more analytical. You can alter the directions on any question type.

From time to time, you will need to enter your own question. That's what custom questions are for. They can be either free response or multiple-choice and can contain math formatted text (equations, expressions, etc).

### Modify Automatically-Generated Questions

Most automatically-generated questions can be modified manually. If there is a choice you don't like, you can change it. If you wish a question was slightly different, you can change it.

### Paper Size and Margins

Print assignments on any sized paper that your printer supports. If you decide to print an assignment on legal-sized paper, no problem. The questions will automatically be repositioned for you--no cutting and pasting the assignment back together just to use a different paper size. You also have control over the margins, page numbering, and paper orientation.