Mathematics

Answer Equivalence in Calculus Courseware

HawkesLearning 2 Comments

We know that oftentimes in calculus, there’s more than one way to solve a problem. While some online systems don’t allow for multiple correct answers, Hawkes Learning’s courseware was built by subject matter experts who painstakingly went through examples to ensure students are given credit for equivalent answers.

Marvin, one of our lead calculus content editors, explained why it’s so important to include equivalent answers in the courseware: “There are often different methods of solving, and we don’t want to penalize students for getting a correct answer. When that happens, students get frustrated and doubt themselves. We want to boost their confidence.”

Our calculus subject matter experts Marvin and Claudia shared a few examples that show our courseware giving credit for correct alternative answers.

Sample Problem from Trigonometric Integrals

The first two correct answers are generated using Method 1 of solving, while the next three are generated using Method 2 of solving.

Problem

Evaluate the indefinite integral ∫ 7tan(4x)sec6(4x)dx. Use C for the constant of integration. Write the exact answer. Do not round.

Correct Answer 1

Method 1: We can use u-substitution with u = sec(4x) after rewriting the integral as
7 ∫ sec5(4x) · sec(4x)tan(4x)dx. Note that the answer has the fraction 7/24 as the coefficient of the secant function.

clc3-1.png

Correct Answer 2

Method 1: We can use u-substitution with u = sec(4x) after rewriting the integral as
7 ∫ sec5(4x) · sec(4x) tan(4x)dx. Note that the answer has the secant function as part of the numerator of the answer.

clc3-2.png

Correct Answer 3

Method 2: We can use u-substitution with u = tan(4x) after rewriting the integral as
7 ∫ tan(4x) [1 + tan2(4x)]2sec2(4x)dx. Note that the answer has several terms with tangent and fractional coefficients.

clc3-3.png

Correct Answer 4

Method 2: We can use u-substitution with u = tan(4x) after rewriting the integral as
7 ∫ tan(4x) [1 + tan2(4x)]2sec2(4x)dx. Note that the answer has the fraction 7/8 factored out.

clc3-4.png

Correct Answer 5

Method 2: We can use u-substitution with u = tan(4x) after rewriting the integral as
7 ∫ tan(4x) [1 + tan2(4x)]2sec2(4x)dx. Note that the answer has the fraction 7tan2(4x)/8 factored out.

clc3-5.png

Correct Answers 6 & 7

If students rewrite the integrand in terms of sine and cosine and work it out correctly, credit is also given.

Below are two examples of a student answering the problem using cos(4x).

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Sample Problem from The Chain Rule

This question shows the application of the Chain Rule, and the correct answer can be written in different ways as shown below.

Problem

Find the derivative of the function F(x) = – 3(13 + 2√x)-5.

Correct Answer 1

The student applies the Chain Rule and writes the last factor as 1/√x.

clc1-1.png

Correct Answer 2

The student applies the Chain Rule and writes the last factor as x -1/2.

clc1-2.png

Correct Answer 3

The student applies the Chain Rule and rewrites the square root of x in terms of fractional exponents.

clc1-3.png

Correct Answer 4

The student applies the Chain Rule and rewrites the whole answer as one fraction using the positive exponent 6 for the expression in parentheses.

clc1-4.png

Correct Answer 5

The student applies the Chain Rule and rewrites the answer as one fraction using the exponent of negative 6 for the expression in parentheses.

clc1-5.png

Sample Problem from Integration by Parts

Problem

Evaluate the integral ∫(t + 1)e4tdt. Use C for the constant of integration. Write the exact answer. Do not round. (Hint: Use an alternative method if integration by parts is not required.)

Correct Answer 1

The student applies integration by parts and writes the answer obtained by evaluating
uv – ∫ v du.

clc2-1.png

Correct Answer 2

The student applies integration by parts and writes the answer as one fraction with the common denominator and e4t factored out.clc2-2.png

Correct Answer 3

The student applies integration by parts and writes the answer with e4t factored out but no common denominator for the fractions.clc2-3.png

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Interested in seeing more of the calculus courseware? Contact us today at info@hawkeslearning.com or 1-800-426-9538 to get free access to the student courseware!

Inside Our Calculus Courseware: Trigonometric Substitutions

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The word “Trigonometric” by itself scares students. Combining it with “Substitutions and Evaluation” is downright terrifying. After all, the student must select the appropriate substitution, transform the integrand from an algebraic to a trigonometric expression, make the appropriate change in limits of integration (or rewrite their antiderivative in terms of the original variable), and finally evaluate the antiderivative. There are pitfalls everywhere along the way. One thing students often fail to do is carry out the last step and evaluate the integral because they’re so relieved to have found the antiderivative.

Sample Problem #1

Below is an example of this problem type and ways we show students how to avoid those common pitfalls:

CLC-CLCSV-7.4-1.png

Step-by-Step

In the Practice mode, students have access to learning aids to help them understand how to tackle each problem. For example, they can choose Step-by-Step in the Tutor area.

This tool provides a step-by-step breakdown of the problem, walking the student through the problem in manageable pieces. While it provides plenty of guidance, the Step-by-Step portion does ask the student to input the results of each step so they are learning as they go.

In Step 1, since the integrand does not exactly match any of the expressions corresponding to a trigonometric substitution, specifically the expression under the radical, the student is asked to identify the equivalent form of that expression after it has been rewritten by completing the square.

CLC-CLCSV-7.4-2.png

In Step 2, the student will identify the limits of integration after the first change in variable.

CLC-CLCSV-7.4-3.png

In Step 3, the student will identify the trigonometric substitution.

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In Step 4, the student calculates the differential dt in terms of the new variable θ after the substitution in Step 3.

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In Step 5, the student will identify the limits of integration in terms of θ resulting from the trigonometric substitution.

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In Step 6, the student is prompted to simplify the integrand if the absolute value can be removed. The condition for which this is possible is verified.

CLC-CLCSV-7.4-7.png

In Step 7, the student will find and evaluate the antiderivative. There is no need to rewrite the antiderivative in terms of the original variable since the limits of integration have been rewritten at each step in terms of the new variables when new variables were introduced. Because of this, taking the antiderivative and evaluating it is straightforward.

CLC-CLCSV-7.4-8.png

 

Sample Problem #2

Students often are so relieved at finally having found the antiderivative, they fail to take the final step and evaluate that antiderivative for a definite integral. The following Explain Error example notes when this occurs and prompts the student to take that final step.

The correct but unevaluated antiderivative is entered.

CLC-CLCSV-7.4-9.png

Students can select the Explain Error option to receive precise feedback from the system’s artificial intelligence. This tool anticipates and diagnoses specific errors, stopping students in their tracks and showing them not only that their answer is incorrect, but why it is incorrect.

Here, we note the correct but unevaluated antiderivative has been entered as the answer.

CLC-CLCSV-7.4-10.png

The student then returns to Practice mode, evaluates the result at the limits of integration, and completes the question.

CLC-CLCSV-7.4-11.png

 

Calc Book and Computer

 

Interested in seeing more of the calculus question bank? Contact us today at info@hawkeslearning.com or 1-800-426-9538 to get free access to the student courseware!

Contextualized Learning in Hawkes’ Discovering Statistics and Data Corequisite Course

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Students in your corequisite course have most likely seen these lessons before—some even two or three times. Yet, it’s just not sticking, and students are feeling frustrated.

What can you do?

Contextualize the prerequisite content for your corequisite students.

Updates to the Discovering Statistics and Data + Integrated Review courseware include new Making Connections and Looking Ahead sections in review lesson modules. These sections provide examples and videos connecting the foundational concepts to the credit-bearing material.

The Making Connections section informs students at the beginning of the lesson why they need to learn the upcoming review content.

Check out the example from the “Addition with Real Numbers” lesson:

dis3-4-r-1-intro

Students then walk through the instructional content of the lesson to get familiar with the concepts. At the end, they encounter the new Looking Ahead section, which shows students how to apply what they’ve learned and how it will help them understand the next lesson:

dis3-4-r-1-outro

 

Explore another example from our “Absolute Value Inequalities” lesson. Before students delve into the material, they get a brief introduction:

dis3-10-r-2-intro

Once students are acquainted with the lesson, they can look ahead to what’s next:

dis3-10-r-2-outro1dis3-10-r-2-outro2dis3-10-r-2-outro3

With this contextualized approach to learning, students will gain a greater sense of why they’re being taught this information, making it more important to them.

dis3ripad 3d

 

Interested in seeing more of this course? Contact us today at info@hawkeslearning.com or 1-800-426-9538 to get free access to the student courseware!

Contextualized Learning in Hawkes’ Precalculus Corequisite Course

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Students in your corequisite course have most likely seen these lessons before—some even two or three times. Yet, it’s just not sticking, and students are feeling frustrated.

What can you do?

Contextualize the prerequisite content for your corequisite students.

Updates to the Precalculus + Integrated Review courseware include new Making Connections and Looking Ahead sections in review lesson modules. These sections provide examples and videos connecting the foundational concepts to the credit-bearing material.

The Making Connections section informs students at the beginning of the lesson why they need to learn the upcoming review content.

Check out the example from the “Addition and Subtraction with Fractions” lesson:

prcr-1-r-4-intro

Students then walk through the instructional content of the lesson to get familiar with the concepts. At the end, they encounter the new Looking Ahead section, which shows students how to apply what they’ve learned and how it will help them understand the next lesson:

prcr-1-r-4-outro

 

 

Explore another example from our “Greatest Common Factor or Two or More Terms” lesson. Before students delve into the material, they get a brief introduction:

prcr-4-r-1-intro

Once students are acquainted with the lesson, they can look ahead to what’s next:

prcr-4-r-1-outro

With this contextualized approach to learning, students will gain a greater sense of why they’re being taught this information, making it more important to them.

prcripad 3d

 

Interested in seeing more of this course? Contact us today at info@hawkeslearning.com or 1-800-426-9538 to get free access to the student courseware!