Lesson Note on Cumulative Frequency: Tables, Histogram and Frequency Polygon for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: Third Term
Week: 5
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Statistics
Previous Lesson: Statistics: Mean, Median and Mode of Grouped Data.
Topic: CUMULATIVE FREQUENCY I
Subject Matter: Construction of cumulative frequency table including class intervals tally frequencies class boundaries, drawing of histogram and frequency polygon, deduction of frequency polygon from histogram, drawing of frequency polygon using mid-value and frequency.

Specific Objectives

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

Cognitive Domain:

• Define cumulative frequency, class boundary, histogram, and frequency polygon.
• Construct a cumulative frequency table from given data.
• State the steps for drawing a histogram and a frequency polygon.

Affective Domain:

• Appreciate the importance of cumulative frequency in data analysis.
• Participate actively in class discussions and practical drawing exercises.

Psychomotor Domain:

• Construct class boundaries from class intervals.
• Draw a histogram accurately from a frequency distribution table.
• Deduce and draw a frequency polygon from a histogram.
• Draw a frequency polygon using mid-values and frequencies directly.

Social Domain:

• Work collaboratively with peers to solve problems related to cumulative frequency.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2
• https://www.mathsisfun.com/data/cumulative-frequency.html
• https://byjus.com/maths/cumulative-frequency-curve/

Instructional Materials

The teacher will teach this lesson with the aid of:

• Cumulative frequency charts
• Graph board
• Graph books
• Pencils
• Rulers
• Erasers
• Calculators

Rationale for the Lesson

This lesson helps pupils to organize and interpret large sets of data more effectively. Understanding cumulative frequency, histograms, and frequency polygons provides essential skills for analyzing statistical information encountered in daily life and further studies.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of basic statistics, including frequency distribution, class intervals, and how to draw bar charts.

Lesson Content/Board Summary

CUMULATIVE FREQUENCY I

Cumulative Frequency Table

Cumulative frequency is the running total of frequencies. It is found by adding each frequency to the sum of its predecessors. A cumulative frequency table helps to organize data to easily see the number of observations that fall below a certain value.

Steps to Construct a Cumulative Frequency Table:

• List the class intervals.
• Record the frequency (f) for each class interval.
• Calculate the class boundaries.
• Calculate the cumulative frequency (cf) by adding the frequency of each class to the cumulative frequency of the previous class.

Example 1: Construct a cumulative frequency table for the following data:

Solution:

Class Boundaries

Class boundaries are the true limits of a class interval. They are obtained by subtracting 0.5 from the lower class limit and adding 0.5 to the upper class limit (for whole number data). For data with one decimal place, subtract/add 0.05, and so on.

Example: For the class interval 10-19, the lower class boundary is 10 – 0.5 = 9.5 and the upper class boundary is 19 + 0.5 = 19.5.

Histogram

A histogram is a graphical representation of a frequency distribution in the form of rectangles with class boundaries on the x-axis and frequencies on the y-axis. The bars are joined together without gaps.

Steps to Draw a Histogram:

• Draw the x-axis (horizontal) and label it with class boundaries.
• Draw the y-axis (vertical) and label it with frequencies.
• Choose appropriate scales for both axes.
• Draw rectangular bars for each class interval. The width of each bar corresponds to the class interval, and the height corresponds to its frequency.
• Ensure there are no gaps between the bars.

Frequency Polygon

A frequency polygon is a line graph that shows the distribution of frequencies. It can be drawn by connecting the midpoints of the tops of the bars of a histogram or by plotting mid-points against frequencies directly.

Steps to Draw a Frequency Polygon from a Histogram:

• Draw a histogram.
• Mark the midpoint of the top of each bar.
• Connect these midpoints with straight lines.
• Close the polygon by connecting the first and last midpoints to the midpoints of the imaginary preceding and succeeding classes on the x-axis (where frequency is zero).

Steps to Draw a Frequency Polygon Using Mid-values Directly:

• Calculate the mid-point of each class interval (Mid-point = (Lower limit + Upper limit) / 2).
• Plot the mid-points on the x-axis and the corresponding frequencies on the y-axis.
• Join the plotted points with straight lines.
• Close the polygon by extending the line to the mid-points of the imaginary preceding and succeeding classes on the x-axis.

Example 2: Draw a histogram and a frequency polygon for the data from Example 1.

Solution:

Step 1: Draw the axes.
Draw a horizontal axis (x-axis) for class boundaries and a vertical axis (y-axis) for frequency.

Step 2: Label the axes and choose scales.
Label the x-axis from 9.5 to 59.5, with appropriate intervals. Label the y-axis from 0 up to 12 (the highest frequency), with appropriate intervals.

Step 3: Draw the histogram.
Draw bars for each class interval:

• Bar for 9.5-19.5 (height 5)
• Bar for 19.5-29.5 (height 8)
• Bar for 29.5-39.5 (height 12)
• Bar for 39.5-49.5 (height 10)
• Bar for 49.5-59.5 (height 7)

Step 4: Draw the frequency polygon (from histogram).
Mark the midpoints of the tops of the histogram bars (e.g., at (14.5, 5), (24.5, 8), etc.). Connect these points with straight lines. To close the polygon, extend the line from (14.5, 5) to the midpoint of the imaginary preceding class (4.5, 0) and from (54.5, 7) to the midpoint of the imaginary succeeding class (64.5, 0).

Step 5: Draw the frequency polygon (using mid-values directly).
Plot the points (mid-point, frequency): (14.5, 5), (24.5, 8), (34.5, 12), (44.5, 10), (54.5, 7). Connect these points with straight lines. Close the polygon as in Step 4.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils, reviews the previous lesson on frequency distribution, and introduces the concept of cumulative frequency by asking how they would find the total number of pupils who scored below a certain mark.
Pupils’ Activity: Pupils respond to the teacher’s questions and recall previous knowledge on frequency distribution.
Learning Point: Pupils connect new concepts to prior knowledge.

Step 2: Cumulative Frequency Table

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains what cumulative frequency is and leads pupils through constructing a cumulative frequency table, including class intervals, frequencies, class boundaries, and cumulative frequencies, using an example on the board.
Pupils’ Activity: Pupils observe, ask questions, and copy the example, contributing to the calculation of cumulative frequencies.
Learning Point: Pupils learn to construct a cumulative frequency table.

Step 3: Class Boundaries

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains the concept of class boundaries and demonstrates how to calculate them from given class intervals.
Pupils’ Activity: Pupils listen and practice calculating class boundaries for different intervals.
Learning Point: Pupils understand and can calculate class boundaries.

Step 4: Drawing a Histogram

Time: 10 minutes
Teaching Skill: Demonstration/Guidance
Teacher’s Activity: The teacher demonstrates on the graph board how to draw a histogram, emphasizing the use of class boundaries on the x-axis and frequencies on the y-axis, ensuring no gaps between bars.
Pupils’ Activity: Pupils pay attention, ask questions, and begin to draw their own histograms in their graph books following the teacher’s guidance.
Learning Point: Pupils learn the correct procedure for drawing a histogram.

Step 5: Deducing Frequency Polygon from Histogram

Time: 5 minutes
Teaching Skill: Demonstration
Teacher’s Activity: The teacher shows how to deduce a frequency polygon by marking the midpoints of the tops of the histogram bars and connecting them with straight lines, closing the polygon.
Pupils’ Activity: Pupils follow along, marking midpoints on their drawn histograms and connecting them to form a frequency polygon.
Learning Point: Pupils learn to draw a frequency polygon from a histogram.

Step 6: Drawing Frequency Polygon using Mid-values

Time: 5 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains how to calculate mid-points for each class interval and demonstrates drawing a frequency polygon by directly plotting mid-points against frequencies.
Pupils’ Activity: Pupils calculate mid-points and draw the frequency polygon using this alternative method.
Learning Point: Pupils learn an alternative method for drawing a frequency polygon.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. What is cumulative frequency?
2. How do you calculate class boundaries for the interval 20-29?
3. List two steps involved in drawing a histogram.
4. How can a frequency polygon be deduced from a histogram?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the main points of the lesson, reiterating the importance of accurately constructing cumulative frequency tables, histograms, and frequency polygons. The teacher then gives homework.
Pupils’ Activity: Pupils listen, ask final questions, and copy the assigned homework.
Learning Point: Pupils consolidate their understanding and apply learned concepts through homework.

Lesson Keywords

• Cumulative frequency – The running total of frequencies in a frequency distribution.
• Class interval – A range of values used to group data in a frequency distribution.
• Class boundary – The true limits of a class interval, used to ensure continuity in histograms.
• Histogram – A bar chart-like graph showing frequency distribution with continuous bars.
• Frequency polygon – A line graph representing frequency distribution, often derived from a histogram or mid-points.
• Mid-value – The midpoint of a class interval.

Differentiation

For struggling learners, the teacher will provide pre-calculated tables and guide them step-by-step through the drawing process. Advanced learners will be challenged with more complex data sets or asked to interpret information from existing graphs.

Note for teachers using this lesson plan

Ensure that pupils have graph books and necessary drawing instruments. Emphasize accurate scaling and labeling of axes. Provide ample practical examples and allow pupils to draw independently to reinforce understanding.