Lesson 7.3: Four-Option Test-Taking Strategies

Introduction

In this lesson, students will explore effective test-taking strategies specifically designed for the ACT Mathematics section. As a reminder, the ACT Mathematics section consists of a range of questions aimed at measuring mathematical skills and understanding. Among these skills, problem-solving and modeling are crucial as they reflect students' abilities to analyze and interpret mathematical information in real-world contexts. Today, we will focus on strategies involving four-option questions, which are common in standardized tests like the ACT.

Learning Objectives:

• Understand and apply the technique of backsolving from answer choices and substituting convenient numbers to find solutions more efficiently.
• Grasp the concepts of elimination and estimation tailored to scenarios with four answer options.
• Develop strategic methods for solving problems using backsolving and substitution.
• Learn to eliminate implausible options quickly while using estimation to confirm answers.
• Familiarize yourself with key ideas and terminology related to backsolving, elimination, and estimation strategies.

Backsolving from Answer Choices

Backsolving is a powerful technique used in standardized tests that is particularly effective when you are faced with multiple-choice questions. The strategy involves starting with the answer choices provided and working backward to see which option satisfies the conditions of the problem posed. This method is often faster than calculating an answer from scratch.

When to Use Backsolving

You should consider backsolving when:

1. The question involves complex calculations that might lead to errors if computed directly.
2. You have a clear understanding of the potential answer values provided in the choices.
3. The question seems to focus heavily on numerical values rather than conceptual reasoning.

Example of Backsolving

Consider the following question:

What is the solution to the equation $3x + 4 = 22$?

The answer choices are:

A) 4

B) 5

C) 6

D) 7

Let's begin by evaluating each option:

• If $x = 4$:

$$3(4) + 4 = 12 + 4 = 16 \quad \text{(Not correct)}$$

• If $x = 5$:

$$3(5) + 4 = 15 + 4 = 19 \quad \text{(Not correct)}$$

• If $x = 6$:

$$3(6) + 4 = 18 + 4 = 22 \quad \text{(Correct)}$$

• If $x = 7$:

$$3(7) + 4 = 21 + 4 = 25 \quad \text{(Not correct)}$$

Thus, the correct answer is C) 6.

Key Concepts:

• Backsolving: Start with the answer choices and substitute them back into the original equation. Pick the choice that results in a true statement.
• This method often saves time and reduces the likelihood of making calculation errors.

Substituting Convenient Numbers

Another effective strategy is to substitute numbers that seem 'convenient' into the problem to help streamline the process of finding the correct answer. This works well especially in word problems or when dealing with unknown quantities.

When to Use Substitution

You may want to use substitution when:

1. The problem involves variables without providing concrete values.
2. Real numbers can simplify complex expressions.
3. You can devise straightforward numerical scenarios to evaluate relationships or outcomes.

Example of Substituting Convenient Numbers

Consider the following problem:

If $x + 2y = 10$ and $y = 3$, what is the value of $x$?

Here, we can substitute $y = 3$:

$$x + 2(3) = 10$$

$$x + 6 = 10$$

$$x = 10 - 6 = 4$$

Hence, the solution is $x = 4$.

Key Concepts:

• Substitution: Replacing variables in the problem with numbers to simplify calculations and enhance understanding of relationships.
• Taking the time to substitute a few realistic values can often lead you towards the correct insight quickly.

Elimination Strategies

Elimination is a technique used to rule out certain answer choices based on logical reasoning or mathematical principles. This can enhance overall performance by helping students focus on the most plausible options.

How to Employ Elimination

1. Identify Clearly Wrong Answers: Look for options that do not make sense mathematically.
2. Use Contradictory Information: If an option contradicts the conditions of the problem, eliminate it.
3. Compare and Contrast: If two options are similar and one definitely leads to a logical inconsistency, eliminate it.

Example of Elimination

Consider this question:

If the total of four numbers is 36, which of the following could be the largest number?

A) 25

B) 20

C) 18

D) 15

• If the largest number is A) 25, then the total of the other three must be $36 - 25 = 11$, which can be possible.
• For option B) 20, the total for the other three would be $36 - 20 = 16$, which is feasible.
• For option C) 18, we would have $36 - 18 = 18$ for the other three. This is impossible to have all three add up to 18 while being each less than or equal to 18.
• For D) 15, the sum of the remaining numbers would be $36 - 15 = 21$, which makes sense.

This means that only C) 18 must be eliminated.

Key Concepts:

• Elimination: The process of discarding incorrect answer choices based on logical deductions, mathematical principles, or the context of the question.
• Efficient elimination can quickly narrow down choices to the correct answer.

Estimation Techniques

Estimation is a crucial skill in the ACT Mathematics section that allows students to quickly arrive at answers by approximating values rather than calculating exact answers. This skill is especially useful in questions that deal heavily with real-world scenarios or when exact numbers are less critical.

When to Estimate

• In situations where precise calculations are complex and time-consuming.
• When the answer choices are close together, making rough estimates useful to figure out which option provides the most realistic approximation.

Example of Using Estimation

Consider a problem that asks, What is the approximate value of $12.7 + 19.5$? The answer choices could be:

A) 32

B) 30

C) 28

D) 26

To estimate, we can round:

$$12.7 \approx 13 \quad \text{and} \quad 19.5 \approx 20$$

So,

$$13 + 20 \approx 33$$

The closest option is A) 32.

Key Concepts:

• Estimation: Approximating values to simplify calculations, helping in decision making when faced with closely spaced answer choices.
• It is essential in cases where calculation complexity would lead to excessive time consumption.

Conclusion

In this lesson, students learned various strategies to tackle four-option questions effectively in the ACT Mathematics section. You have explored backsolving techniques that allow you to work backwards from the answer choices, the use of convenient number substitution to simplify problem-solving, elimination strategies to quickly discard implausible answers, and the importance of estimation for time management. Mastering these techniques can improve accuracy and speed on the exam. With sustained practice and application of these strategies, improved performance in the ACT Mathematics section is within reach.

Study Notes

• Backsolving: Work from the answer choices back to the problem.
• Substitute convenient numbers to simplify solving equations.
• Eliminate clearly incorrect options using logic.
• Use estimation to quickly narrow down answer choices.
• Practice these strategies to enhance exam performance and confidence.