Estimating Length, Capacity, Mass, Age and Time for JSS 1

Estimation

Performance Objectives

By the end of the lesson, students should be able to:

1. estimate the dimensions of objects and distances within the school environment;
2. estimate capacity and mass of some familiar objects;
3. estimate time for some day-to-day activities;
4. solve quantitative aptitude problems involving estimation.

Meaning of Estimation

Estimation is the process of finding an approximate value, size, or quantity that is close enough to the actual value without performing an exact measurement.

In daily life, exact measurements are not always necessary or possible. For instance, a person may need to estimate the amount of water needed to cook a meal, the time it takes to walk to school, or the length of a desk to see if it can fit through a doorway. A good estimate is based on experience, comparison with known standards, and logical reasoning.

Estimating Dimensions and Distances

Dimensions refer to the measurable sizes of an object, such as its length, width, and height. Distance refers to the amount of space between two points or locations.

Before standard measuring tools like rulers and tape measures were invented, people used parts of the human body to estimate lengths. These are known as non-standard units of measurement.

Common non-standard units of measurement include:

1. Handspan: The distance from the tip of the thumb to the tip of the little finger when the hand is fully stretched.
2. Cubit (or Elbow length): The distance from the elbow to the tip of the middle finger.
3. Foot: The length of a person’s foot from the heel to the big toe.
4. Stride (or Pace): The distance covered by taking one large step.

Observe the diagram below to see how these body parts are used for estimation.

Standard units of measurement are globally accepted units. For length and distance, the standard metric units are millimetres ($\text{mm}$), centimetres ($\text{cm}$), metres ($\text{m}$), and kilometres ($\text{km}$).

The relationships between these standard units are:

$$10\text{ mm} = 1\text{ cm}$$
$$100\text{ cm} = 1\text{ m}$$
$$1000\text{ m} = 1\text{ km}$$

Worked Example 1

Estimate the length of a standard classroom blackboard in metres, and then state how a student can verify this estimate using a non-standard unit.

Solution:

A standard classroom blackboard is usually about twice the height of an average JSS 1 student. A reasonable estimate for its length is $3\text{ m}$.

To verify this estimate using a non-standard unit, a student can use cubits. Since one cubit of a JSS 1 student is approximately $35\text{ cm}$ to $40\text{ cm}$, a $3\text{ m}$ blackboard should measure about $8$ to $9$ cubits.

Worked Example 2

Estimate the distance from the JSS 1 classroom door to the school assembly ground in metres.

Solution:

An average human stride or pace is approximately $0.75\text{ m}$ to $1\text{ m}$. If a student walks from the classroom door to the assembly ground and counts $120$ normal paces, the estimated distance is calculated as:

$$\text{Estimated Distance} \approx 120 \times 0.8\text{ m} = 96\text{ m}$$

Therefore, a reasonable estimate for the distance is approximately $100\text{ m}$.

Estimating Capacity and Mass

Estimating Capacity

Capacity is the amount of liquid that a container can hold.

The standard units for measuring capacity are litres ($\text{L}$ or $\text{l}$) and millilitres ($\text{mL}$ or $\text{ml}$). The relationship between them is:

$$1000\text{ ml} = 1\text{ litre}$$

To estimate capacity, it is helpful to compare unknown containers with familiar commercial containers. Common reference containers in Nigeria include:

1. A standard sachet of “pure water” which holds $500\text{ ml}$ (half a litre).
2. A small plastic bottle of soft drink which holds approximately $50\text{ cl}$ or $500\text{ ml}$.
3. A large plastic bottle of dispenser water which holds approximately $19\text{ litres}$.
4. A standard household bucket which holds approximately $10$ to $15\text{ litres}$.
5. A large yellow jerrycan (often called a “Jerry Can” or “Gallon”) which holds $20$ to $25\text{ litres}$.

Look at the diagram below to compare the capacities of these common containers.

Worked Example 3

A school water tank is filled using a $25\text{-litre}$ jerrycan. If it takes $40$ full jerrycans to fill the tank completely, estimate the capacity of the water tank in litres.

Solution:

The capacity can be estimated by multiplying the capacity of one jerrycan by the total number of jerrycans used:

$$\text{Estimated Capacity} = 40 \times 25\text{ litres}$$
$$\text{Estimated Capacity} = 1000\text{ litres}$$

The estimated capacity of the school water tank is $1000\text{ litres}$.

Estimating Mass

Mass is the amount of matter in an object.

The standard units for measuring mass are grams ($\text{g}$) and kilograms ($\text{kg}$). The relationship between them is:

$$1000\text{ g} = 1\text{ kg}$$

To estimate the mass of an object, compare it with known masses. Common reference masses include:

1. A packet of cabin biscuits which has a mass of about $150\text{ g}$.
2. A packet of granulated sugar which has a mass of $1\text{ kg}$.
3. A standard bag of rice which has a mass of $50\text{ kg}$.
4. A standard mathematical set which has a mass of about $200\text{ g}$.

Worked Example 4

Estimate the mass of a JSS 1 student’s school bag containing five textbooks, three exercise books, and a mathematical set.

Solution:

To find a reasonable estimate, the masses of the individual items can be estimated and added together:

1. An empty school bag weighs approximately $0.5\text{ kg}$.
2. Five textbooks weigh approximately $0.4\text{ kg}$ each, which equals $2.0\text{ kg}$.
3. Three exercise books weigh approximately $0.1\text{ kg}$ each, which equals $0.3\text{ kg}$.
4. A mathematical set weighs approximately $0.2\text{ kg}$.

Adding these estimates together:

$$\text{Total Estimated Mass} = 0.5\text{ kg} + 2.0\text{ kg} + 0.3\text{ kg} + 0.2\text{ kg} = 3.0\text{ kg}$$

A reasonable estimate for the mass of the loaded school bag is $3\text{ kg}$.

Estimating Age and Time

Estimating Age

Age is the length of time a person has lived or an object has existed. Estimating age involves looking at physical features, developmental milestones, or historical contexts.

Common age estimations for people in a school environment include:

1. JSS 1 Student: Typically between $10$ and $12$ years old.
2. Secondary School Principal: Typically between $40$ and $60$ years old.
3. A newborn baby: Typically measured in weeks or months, under $1$ year old.

Estimating Time

Time estimation involves determining the duration of an event or identifying the time of day without using a clock.

The time of day or night can be estimated using natural cues:

1. Sunrise: Occurs around $6:00\text{ a.m.}$, indicating early morning.
2. Overhead Sun: When the sun is directly overhead and shadows are shortest, the time is approximately $12:00\text{ noon}$.
3. Sunset: Occurs around $6:30\text{ p.m.}$, indicating early evening.
4. Position of Stars and Moon: Indicates different periods of the night.

Estimating the duration of common activities helps in planning daily schedules. Examples of estimated durations include:

1. Singing the Nigerian National Anthem: Approximately $1$ to $2$ minutes.
2. Walking a distance of $1\text{ km}$ at a normal pace: Approximately $10$ to $15$ minutes.
3. A standard school lesson period: Approximately $40$ minutes.
4. A night of healthy sleep for a JSS 1 student: Approximately $8$ hours.

Worked Example 5

A student leaves home when the sun is just rising and arrives at school when the assembly bell rings. If the assembly bell rings at exactly $7:30\text{ a.m.}$, estimate the duration of the student’s journey.

Solution:

The sun rises at approximately $6:00\text{ a.m.}$ in Nigeria. If the student left home at sunrise, the departure time was approximately $6:00\text{ a.m.}$.

The arrival time is $7:30\text{ a.m.}$.

The estimated duration of the journey is the difference between the departure time and arrival time:

$$\text{Duration} = 7:30\text{ a.m.} – 6:00\text{ a.m.} = 1\text{ hour and } 30\text{ minutes}$$

The estimated duration of the journey is $1.5\text{ hours}$ (or $90\text{ minutes}$).

How to Estimate and Verify Measurements

To build accurate estimation skills, a person must practice estimating first and then measuring with standard instruments to find the error in the estimate.

The required materials for this activity are:

1. A standard $30\text{ cm}$ ruler
2. A measuring tape (at least $5\text{ m}$ long)
3. A kitchen weighing scale
4. A $1\text{-litre}$ graduated cylinder or measuring jug
5. Familiar classroom objects (textbook, desk, water bottle)

The verification procedure involves the following steps:

1. Select the object to be measured (for example, the length of a classroom desk).
2. Write down an estimated value before using any measuring tool.
3. Use the appropriate standard measuring tool (such as a measuring tape) to find the actual measurement.
4. Calculate the estimation error by finding the difference between the estimated value and the actual value:

$$\text{Estimation Error} = |\text{Estimated Value} – \text{Actual Value}|$$

5. Record the results in a table to track improvement over time.

Quantitative Aptitude Problems Involving Estimation

Quantitative aptitude problems test the ability to use mathematical logic, patterns, and relationships to make reasonable estimates.

Problem 1

Study the relationship between the objects and their masses in the list below:

1. $1$ exercise book $\approx 100\text{ g}$
2. $1$ textbook $\approx 400\text{ g}$
3. $1$ mathematical set $\approx 200\text{ g}$

Based on this pattern, estimate the total mass of a package containing $3$ exercise books, $2$ textbooks, and $2$ mathematical sets.

Solution:

Calculate the estimated mass of each group of items:

$$\text{Mass of exercise books} = 3 \times 100\text{ g} = 300\text{ g}$$
$$\text{Mass of textbooks} = 2 \times 400\text{ g} = 800\text{ g}$$
$$\text{Mass of mathematical sets} = 2 \times 200\text{ g} = 400\text{ g}$$

Sum the estimated masses to find the total mass:

$$\text{Total Estimated Mass} = 300\text{ g} + 800\text{ g} + 400\text{ g}$$
$$\text{Total Estimated Mass} = 1500\text{ g} = 1.5\text{ kg}$$

The estimated mass of the package is $1.5\text{ kg}$.

Problem 2

A rectangular school garden is estimated to have a length of $20\text{ m}$ and a width of $10\text{ m}$. If the actual length is $18.5\text{ m}$ and the actual width is $9.2\text{ m}$:

1. Calculate the estimated area of the garden.
2. Calculate the actual area of the garden.
3. Find the difference between the estimated area and the actual area.

Solution:

First, calculate the estimated area using the estimated dimensions:

$$\text{Estimated Area} = \text{Estimated Length} \times \text{Estimated Width}$$
$$\text{Estimated Area} = 20\text{ m} \times 10\text{ m} = 200\text{ m}^2$$

Second, calculate the actual area using the actual dimensions:

$$\text{Actual Area} = 18.5\text{ m} \times 9.2\text{ m} = 170.2\text{ m}^2$$

Third, find the difference between the two areas:

$$\text{Difference} = \text{Estimated Area} – \text{Actual Area}$$
$$\text{Difference} = 200\text{ m}^2 – 170.2\text{ m}^2 = 29.8\text{ m}^2$$

The difference between the estimated area and the actual area is $29.8\text{ m}^2$.

Problem 3

A water tap drips at an estimated rate of $1\text{ ml}$ of water per minute. Estimate how many litres of water will be wasted if the tap drips continuously for one full week ($7\text{ days}$).

Solution:

First, calculate the total number of minutes in one week:

$$\text{Minutes in an hour} = 60$$
$$\text{Minutes in a day} = 60 \times 24 = 1440\text{ minutes}$$
$$\text{Minutes in a week} = 1440 \times 7 = 10080\text{ minutes}$$

Second, calculate the volume of water wasted in millilitres:

$$\text{Water wasted} = 10080\text{ minutes} \times 1\text{ ml/minute} = 10080\text{ ml}$$

Third, convert the volume from millilitres to litres:

$$\text{Volume in litres} = \frac{10080\text{ ml}}{1000}$$
$$\text{Volume in litres} \approx 10.1\text{ litres}$$

The estimated volume of water wasted in one week is approximately $10\text{ litres}$.

Summary

Estimation is a vital mathematical skill that allows people to make quick, sensible approximations of physical quantities. By comparing unknown objects with familiar body parts or standard reference items, reasonable estimates can be made for length, distance, capacity, mass, age, and time. Regular practice of estimating followed by actual measurement helps to reduce estimation errors and improve accuracy.

Questions

Solve the following problems:

1. State three non-standard units of measurement that can be used to estimate the length of a classroom floor.
2. A student estimates the length of a teacher’s table to be $1.5\text{ m}$. The actual measurement is $1.38\text{ m}$. Calculate the estimation error.
3. Which of the following is the most reasonable estimate for the mass of a standard JSS 1 Mathematics textbook?
   1. $5\text{ g}$
   2. $450\text{ g}$
   3. $10\text{ kg}$
   4. $50\text{ kg}$
4. An empty plastic bucket has an estimated capacity of $12\text{ litres}$. If a small cup holds $300\text{ ml}$ of water, estimate how many full cups of water are needed to fill the bucket.
5. A man is twice as old as his son. If the son is a JSS 1 student aged $11$ years, estimate the age of the father.
6. A car travels at an average speed of $80\text{ km/h}$. Estimate the time in hours it will take the car to travel a distance of $240\text{ km}$.
7. A standard sachet of water holds $500\text{ ml}$. If a family drinks $12$ sachets of water daily, estimate the total volume of water consumed by the family in one week, in litres.
8. If a student takes $80$ paces to walk across a football field, and each pace is approximately $0.75\text{ m}$ long, estimate the length of the football field in metres.