Example 3.6.1.
Given the exact length of a book is be 28.4 cm what is the approximate length of the book?
Section 3.6 Approximations and Errors
Approximation is when we give a value that is close to the exact value, but not exact. We use approximation when:
\(\textbf{Why Do We Approximate?}\)
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Some values are difficult to measure exactly.
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Tools like rulers or weighing scales are not always perfectly precise.
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It saves time and is good for rough planning or daily use.
\(\textbf{What is error in measurement?}\)
Whenever we approximate or measure something, there’s a difference between the actual (true) value and the measured value. This difference is called an error.
There are two main types of error:
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Absolute Error: This is the difference between the actual value and the measured value. For example, if the actual length of a pencil is 10 cm and you measure it as 9.5 cm, the absolute error is 0.5 cm.- Therefore absolute error = ∣Measured Value−True Value∣
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Relative Error: This tells us how big the error is compared to the true value.- Relative Error = Absolute Error ÷ True Value
Example 3.6.2.
A bag of sugar is labeled 1 kg, but the actual weight is 1.05 kg.
Subsection 3.6.1 Approximation of quantities in measurements
\(\textbf{a) Approximating Length}\)
Activity 3.6.1.
Tip for Learners: Before starting, measure your palm, foot, or stride length in cm.For example:
-
palm length: 8 cm
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foot length: 25 cm
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stride length: 75 cm
In these approximations we will use the palms of our hands, feet or strides
Measure the lemgth of your palm using a well labeled ruler.
What did you get?
What about the length of your feet? Measure also and estimate the length of one stride.
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Materials Needed:
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A ruler or measuring tape
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A variety of classroom objects
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A notebook or paper for recording measurements
-
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Use your palm, foot, or stride to estimate the length of each object e.g the black board or hite board or the teacher’s table.
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Count how many units of palm-widths or foot-lengths or stride lengths it takes to span the object.
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Multiply by the average length of your palm/foot to get your estimated length in cm.
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Record your estimated lengths in a table.
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Now, measure the actual length of each object using a ruler or measuring tape.
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Calculate the absolute and percentage error.
| Object part used |
Count | Estimated Length (cm) |
Total EStimated Length (cm) |
Measured Length |
Absolute error |
Percentage error |
|---|---|---|---|---|---|---|
| Desk | Palm | 5 | 5 × 8 = 40 | 42 | 2 cm | 4.76% |
| Hallway | Stride | 6 | 6 × 60 = 360 | 370 | 10 cm | 2.70% |
| Book | Palm | 2 | 2 × 8 = 16 | 15 | 1 cm | 6.67% |
Example 3.6.4.
One palm length of Keith is 8cm long. He measures the length of his desk as 5 palm lengths. What is the approximate length of his desk in cm?
Solution.
To find the approximate length of Keith’s desk, we multiply the number of palm lengths by the length of one palm:
\begin{align*}
\text{Approximate length} = \amp \text{length of one palm} \times \text{number of palm lengths}
\end{align*}
\begin{align*}
= \amp = 8 \text{ cm} \times 5 \\
= \amp = 40 \text{ cm}
\end{align*}
b) Approximating mass
Activity 3.6.2.
Estimate the mass of various classroom objects, then measure their actual mass using a digital or mechanical scale. Compare your estimates with the actual values and calculate the absolute and percentage error.
Choose at least three of the following objects: chalk, textbook, water bottle, eraser, pencil case, or lunchbotatement
Record your estimates and measurements in the table belotatement
| No. | Object | Estimated Mass (g) |
Measured mass (g) |
Absolute error (g) |
Percentage error (\%) |
|---|---|---|---|---|---|
| i) | |||||
| ii) | |||||
| iii) |
After completing the table, calculate the absolute and percentage errors for each object.
Use these formulas to calculate errors:
-
\(\text{Estimated value- Measured value}\) for Absolute Error
-
\((\text{Absolute Error ÷ Measured} × 100\) for Percentage Error
Reflect on which items you estimated most accurately. What strategies helped? Why is approximating mass useful in real-life situations like shopping, cooking, or science?
\(\textbf{a) Approximating area}\)
Activity 3.6.3.
Estimate the area of flat surfaces using square units, then measure and calculate the actual area using appropriate tools (e.g., ruler or measuring tape). Compare the estimate with the actual value and find the percentage error.
Choose three flat surfaces such as a desk top, book cover, exercise book page, floor tile, or windowpane. Estimate the area by visualising or using hand-sized units (like palm-sized squares). Then measure the length and width to calculate the actual area.
| Item number |
Object | Estimated area (cm²) |
Measured length (cm) |
Measured width (cm) |
Actual area (cm²) |
Absolute error (cm²) |
Percentage error (\%) |
|---|---|---|---|---|---|---|---|
| a) | |||||||
| b) | |||||||
| c) |
Use this formula to calculate the actual area: \(Area = Length × Width\)
After completing the table, calculate the absolute and percentage errors for each object.
Use these formulas to calculate errors:
-
\(Estimated - Actual\) for Absolute Error
-
\((Absolute Error \div Actual) × 100\) for Percentage Error
Reflect on which items you estimated most accurately. What strategies helped? Why is approximating area useful in real-life situations like home design, gardening, or art?
Example 3.6.7.
\(\textbf{a) Approximating volume}\)
Activity 3.6.4.
Estimate the volume of irregular objects by using water displacement. This method allows you to find the actual volume by measuring how much water is pushed out of the way when the object is fully submerged.
Fill a measuring jug with water and record the initial water level. Carefully submerge an object into the water. Record the new water level. The difference between the two levels gives the actual volume of the object.
Before submerging each object, estimate its volume in millilitres (mL). After measuring, calculate the absolute error and percentage error of your estimate.
| No. | Object | Estimated Volume (mL) | Initial Water Level (mL) | Final Water Level (mL) | Actual Volume (mL) | Absolute Error (mL) | Percentage Error (%) |
|---|---|---|---|---|---|---|---|
| 1. | |||||||
| 2. | |||||||
| 3. |
Use this formula to calculate the volume of the object: \(Volume = Final Level − Initial Level\)
To determine errors:
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\(\displaystyle Absolute Error = Estimated − Actual\)
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\(\displaystyle Percentage Error = (Absolute Error ÷ Actual) × 100\)
Example 3.6.9.
Activity 3.6.5.
Working in small groups, estimate the total amount of water (in millilitres or litres) required to fill a variety of classroom containers. Then measure the actual capacity of each item using a measuring jug or graduated container. Finally, calculate the absolute and percentage error for each estimate and determine which group made the most accurate overall predictions.
Suggested items: a chalk box, teacher’s water bottle, flower vase, handwashing bucket, and a plastic cup.
Record your data in the table below and fill in the estimated values based on your group’s discussions. After measuring, calculate the absolute and percentage errors for each item.
| Item | Group Estimate (mL) | Measured Capacity (mL) | Absolute Error (mL) | Percentage Error (%) |
| Example: Vase | 1,200 | 1,000 | 200 | 20% |
| i. | ||||
| ii. | ||||
| iii. |
Example 3.6.11.
\(\textbf{a) Approximating time}\)
Activity 3.6.6.
Estimate how long it takes to perform common classroom tasks, such as walking to the board, passing out books, or sharpening a pencil. Then use a stopwatch or phone timer to measure the actual time taken. Compare your estimates with actual timings and calculate the absolute and percentage error.
Work in pairs. One student performs the task while the other times and records the data. Then switch roles.
| Task | Estimated Time (seconds) | Measured Time (seconds) | Absolute Error (s) | Percentage Error (%) |
| Walk to board and write a sentence | ||||
| Sharpen a pencil | ||||
| Distribute exercise books to a row | ||||
| Open and arrange textbooks |
Use appropriate formulas to compute the accuracy of your estimates
Discuss: Which task was the hardest to estimate? Why might people misjudge time? How can accurate time estimation be useful in exams, presentations, or careers?
Example 3.6.13.
\(\textbf{a) Approximating temperature}\)
Activity 3.6.7.
Estimate the temperatures of various classroom items or surfaces in degrees Celsius. Then measure the actual temperature using a thermometer. Record your estimates and measured values, then calculate the absolute and percentage errors to evaluate your accuracy.
Suggested items: a cup of tap water, the palm of a student’s hand, a shaded classroom desk, and a closed bottle of water left in the sun.
| Item | Estimated Temperature (°C) | Measured Temperature (°C) | Absolute Error (°C) | Percentage Error (%) |
| Tap water | ||||
| Palm of hand | ||||
| Shaded desk | ||||
| Sun-warmed bottle |
Use the following formulas to calculate estimation errors:
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\(\displaystyle Absolute Error = |Estimated − Measured|\)
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\(\displaystyle Percentage Error = (Absolute Error ÷ Measured) × 100\)
Discuss with your group: Which item was most difficult to estimate? What influenced your guesses? How does this activity help you in understanding temperature in real-life settings such as weather, body health, or food safety?
\(\textbf{Tips for Estimating and Measuring Quantities}\)
Estimating and measuring quantities accurately is a valuable skill in everyday life. Here are some tips to help you improve your estimation and measurement skills:
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Always estimate before measuring to develop a sense of approximation and improve reasoning skills.
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Use appropriate instruments such as thermometers, measuring jugs, or rulers to validate your estimates.
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Calculate absolute and percentage errors to evaluate how close your estimates were to the actual values.
Example 3.6.15. Approximating Temperature Using Estimation.
A student places their hand on a metal surface that was exposed to sunlight. They estimate the temperature of the surface to be 38°C based on how warm it feels. Later, they use a digital thermometer and record the actual temperature as 41°C. Approximate the absolute and percentage error in the student’s estimation.
Solution.
Estimated temperature = 38°C
Measured temperature = 41°C
\begin{align*}
Absolute error = \amp 38 − 41 \\
= \amp 3° C
\end{align*}
\begin{align*}
\text{Percentage error} = \amp \frac{3}{41} \times 100 \% \\
= \amp 7.32 \%
\end{align*}
Therefore, the estimated temperature was off by 3°C, which is approximately a 7.32% error.
\(\textbf{Conclusion}\)
In this section, we learned how to approximate quantities in measurements and understand the concept of error. We practiced estimating lengths, masses, areas, volumes, temperature and time durations using everyday objects and methods. We also calculated absolute and percentage errors to evaluate the accuracy of our estimates.
Subsection 3.6.2 Determine errors using estimations and actual meaasurements of quantites
Activity 3.6.8.
Choose an object (e.g., a pencil, an apple, a cup).
Estimate its length, weight, or volume without measuring it using easuring tools.
Now, measure the actual length, weight or volume using a ruler, scale, or measuring cup.
Record both the estimated and measured values.
Calculate the absolute error by subtracting the estimated value from the actual value.
Activity Table Template
| Item | Quantity Type | Estimated Value | Measured Value | Absolute Error |
|---|---|---|---|---|
| Pencil | length (cm) | 10 cm | ||
| Apple | weight (g) | 12 | ||
| Cup of water | volume (ml) | 250 ml | ||
Discuss the results with your classmates. How close were your estimates to the actual measurements? What factors might have influenced your estimations?
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Rank the items from least to greatest relative error.
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Predict next time using learned error patterns.
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Discuss how professionals (engineers, chefs, scientists) rely on accurate measurements in their work.
\(\textbf{Extended Activity}\)
Now, repeat the activity with a different object or quantity.
This time, try to improve your estimation skills based on what you learned from the first activity.
Record your new estimates and actual measurements in the same table format.
a) Was your estimate higher or lower than the actual value?
b) Which type of quantity (length, weight, or volume) was easier to estimate? Why?
c) Why is it important to understand error in real-life situations (e.g., cooking, building, shopping)?
d) What can we do to improve our estimation skills?
\(\textbf{Key Takeaway}\)
1. Approximation helps when exact values aren’t necessary.
` 2.All measurements have some error.
3. Estimating quantities is a useful skill in everyday life.
4. Knowing the actual measurement allows you to see how close your estimate was.
5. The absolute error is the difference between the estimated and actual values.
6. Knowing the error helps you see how accurate or inaccurate your result is.
Checkpoint 3.6.17. True/False.
True.
- False! Relative error is a percentage, and it depends on the size of the actual value
False.
- False! Relative error is a percentage, and it depends on the size of the actual value
If the statement correct choose true and if its wrong choose false. \({\color{blue}\text{Relative error is always smaller than absolute error. }}\)
Checkpoint 3.6.18.
True.
- True!
False.
- True!
If the statement correct choose true and if its wrong choose false. \({\color{blue}\text{A good measurement has a very small error.}}\)
Activity 3.6.9.
-
If the statement correct choose true and if its wrong choose false. \({\color{blue}\text{All measurements are always 100% accurate.}}\)
True.
- False! All measurements are not always 100% accurate, there is always a margin of error.
False.
- False! All measurements are not always 100% accurate, there is always a margin of error.
Activity 3.6.10.
Example 3.6.19.
Without using a ruler, Jane estimates the length of her English textbook to be 30 cm. She then uses a ruler and finds the actual length is 28.5 cm. Calculate the Absolute Error.
Subsection 3.6.3 Percentage errror
Activity 3.6.11.
\(1. \textbf{Materials Needed:}\)
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Ruler or tape measure (for length)
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Different weighing scales (for weight/mass) i.e:
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Kitchen scale (for small weights)
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measuring cup or jug (for volume)
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Stopwatch or timer (for time)
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Calculator (for calculations)
Students should rotate through stations where they estimate and then measure quantities and finally calculate the percentage error for each task.
| Station | Task | Tools provided |
|---|---|---|
| 1. | Estimate and measure length | Ruler/tape measure |
| 2. | Estimate and weigh an object | Kitchen scale |
| 3, | Estimate and measure volume | Measuring cup/jug |
| 4. | Guess time to complete a task (e.g., jumping 10 times) |
Stopwatch or timer |
2. Instructions for each station:
a. Station 1 (Length): Estimate the length of a pencil, then measure it with a ruler. Record both values.
b. Station 2 (Weight): Estimate the weight of a bag of sugar, then weigh it using a kitchen scale. Record both values.
c. Station 3 (Volume): Estimate the volume of water in a cup, then measure it using a measuring cup or jug. Record both values.
d. Station 4 (Time): Estimate how long it takes to complete a task (e.g., jumping 10 times or breathing in deeply and estimating time in seconds you can hold your breathe), then use a stopwatch or timer to measure the actual time. Record both values.
3. At each station:
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Estimate the quantity first.
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Then they measure it accurately.
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Finally, calculate: \(\text{Absolute error = Estimated measurements - Actual measurements} \)\(\text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{Estimated Value}}{\text{Measured Value}} \right) \times 100\%\)
4. Groups record their data in a tracking sheet.
5. After rotating through all stations, students return to their seats and reflect.
\(\textbf{ Tracking Sheet Example:}\)
| Station | Quantity type |
Estimated Value |
Actual Value |
Absolute error |
Percentage error |
| 1. | Length in cm |
60 | 58 | 2 | 3.45% |
| 2. | Weight in g |
350 | 370 | 20 | 5.41% |
| 3. | Volume in ml |
250 | 240 | 10 | 4.17% |
| 4. | Time in seconds |
5 | 6 | 1 | 16.67% |
Discussion questions:
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Which station had the highest error?
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How did your estimates compare to the actual measurements?
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What was the largest percentage error you encountered?
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Why might estimating volume or weight be harder than length?
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How close were your estimates to the actual measurements?
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What factors might have affected your estimates?
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How can you improve your estimation skills in the future?
Example 3.6.22.
A student estimates the length of a pencil to be 30 cm, but the actual measurement is 28.5 cm.
Solution.
To find the absolute error and percentage error, we use the formulas:
\begin{align*}
\text{Absolute Error} = \amp \text{Estimated Value - Actual Value} \\
= \amp 30 \, \text{cm} - 28.5 \, \text{cm}\\
= \amp 1.5 \text{cm}
\end{align*}
\begin{align*}
\text{Percentage Error} = \amp \frac{\text{Absolute error}}{\text{Actual value}} \times 100%\\
= \amp \frac{1.5 \, \text{cm}}{28.5 \, \text{cm}} \times 100%\\
\approx 5.26\%
\end{align*}
Therefore, the absolute error is 1.5 cm and the percentage error is approximately 5.26%.
Example 3.6.23.
Amina estimates the weight of a bag of flour to be 2 kg, but the actual weight is 1.95 kg. Calculate the flour’s pecentage error.
Solution.
To find the absolute error and percentage error, we use the formulas:
\begin{align*}
\text{Absolute Error} = \amp \text{Estimated Value - Actual Value} \\
= \amp 2 \, \text{kg} - 1.95 \, \text{kg}\\
= \amp 0.05 \text{kg}
\end{align*}
\begin{align*}
\text{Percentage Error} = \amp \frac{\text{Absolute error}}{\text{Actual value}} \times 100%\\
= \amp \frac{0.05 \, \text{kg}}{1.95 \, \text{kg}} \times 100%\\
\approx 2.56\%
\end{align*}
Therefore, the absolute error is 0.05 kg and the percentage error is approximately 2.56%.
