Lesson 7.2: Estimation, Accuracy and Checking

Introduction

In this lesson, we will explore the essential skills of estimation, accuracy, and checking, which are integral to successful problem-solving in mathematics. The ability to estimate answers helps set realistic expectations before diving into computations. Understanding rounding and significant figures is crucial for presenting answers with the appropriate level of precision. Additionally, checking work through substitution and reverse calculations ensures that we can trust our results. By the end of this lesson, students will be able to apply these techniques effectively in various mathematical contexts.

Learning Objectives

• Estimate an answer before computing to set expectations.
• Use sensible rounding and significant figures in an answer.
• Check work through substitution, reverse calculation, and judging reasonableness.
• Produce a quick estimate to anticipate the size of an answer.
• Round and present an answer to an appropriate accuracy.

Estimating an Answer Before Computing

Estimation is the process of finding an approximate value that is close enough to the actual answer for practical purposes. It allows us to set expectations and gauge whether our final answer makes sense.

Techniques for Estimation

1. Rounding Numbers: This involves simplifying numbers to make them easier to work with. For example, instead of using $248$, we might round it to $250$ which is easier to multiply.

1. Using Compatible Numbers: This technique involves selecting numbers that are easy to compute mentally. For example, when calculating $47 \times 6$, we can estimate it by rounding $47$ to $50$, making the multiplication easier: $50 \times 6 = 300$.

Example 1: Estimating a Sum

Let’s estimate the sum of $128 + 256$.

Step 1: Round the numbers to the nearest hundred.

• $128 \approx 100$
• $256 \approx 300$

Step 2: Add the rounded numbers.

• $100 + 300 = 400$

Thus, we estimate that $128 + 256 \approx 400$.

Rounding and Significant Figures

Rounding refers to the process of adjusting the digits in a number to make it simpler, while significant figures are the digits that carry meaningful information about its precision. Understanding how to round correctly and identify significant figures is essential for accuracy in mathematics.

Rounding Rules

1. Identify the last significant digit: If rounding to the nearest ten, look at the units place.
2. Consider the next digit: If it is $5$ or greater, round up; if less than $5$, round down.

Example 2: Rounding a Decimal

Let’s say we want to round $56.789$ to three significant figures.

Step 1: Identify the third significant figure, which is $7$.

Step 2: Look at the next digit, $8$. Since $8$ is greater than $5$, we round the $7$ up to $8$.

Thus, $56.789$ rounded to three significant figures becomes $56.8$.

Checking Work: Substitution and Reverse Calculation

Checking our work is just as important as performing the initial calculations. There are several methods to verify that our answers are correct:

1. Substitution: Substituting values back into the equation can confirm whether our computations are accurate.
2. Reverse Calculation: This involves performing the operation backward to see if we arrive at the original numbers.
3. Judging Reasonableness: By considering the context and the numbers involved, we can assess if our answer is reasonable.

Example 3: Checking a Solution for an Equation

Let's solve the equation $x + 5 = 12$.

Step 1: Solve for $x$:

$$ x = 12 - 5 $$

{$ = 7$}

Step 2: Check by substituting $7$ back into the original equation:

$$ 7 + 5 = 12 $$

{(True)}

This confirms that our solution is correct.

Producing a Quick Estimate

Creating a quick estimate involves using the techniques of rounding and compatible numbers to gauge how large or small our answer might be before calculating.

Example 4: Estimating a Product

Let’s estimate the product of $34 \times 19$.

Step 1: Round the numbers:

• $34 \approx 30$
• $19 \approx 20$

Step 2: Multiply the rounded numbers:

$$ 30 \times 20 = 600 $$

We estimate that $34 \times 19 \approx 600$.

Conclusion

In this lesson, we have covered key concepts of estimation, rounding, significant figures, and methods for checking our work. These skills not only help us perform calculations more effectively but also enhance our ability to understand the mathematics we are working with. As we progress, remember to always estimate first, round appropriately, and check your answers for accuracy. Practice these techniques regularly to improve your mathematical problem-solving skills.

Study Notes

• Estimation helps set expectations before calculations.
• Round numbers for easier arithmetic.
• Significant figures indicate precision and accuracy in answers.
• Check work by substitution and reverse computation.
• Always judge the reasonableness of answers derived through calculations.