Liberia - Grade 9 - Mathematics - Measures of Variability

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Subject: Mathematics

Semester: 2

Period: 6

Week: 31

WEEK 31

Class: Grade 9
Age: 14 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Measure of Variability
Focus: Range, Variance, Standard Deviation and Applications

SPECIFIC OBJECTIVES:

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

Define variability and explain its importance in statistics.
Compute the range of a given data set.
Compute the variance of a data set using the formula.
Compute the standard deviation of a data set.
Apply measures of variability in real-life situations.

INSTRUCTIONAL TECHNIQUES:

Question and answer
Guided demonstration
Step-by-step problem solving
Group activities
Discussions

INSTRUCTIONAL MATERIALS:

Worksheets with data sets
Whiteboard and marker
Flashcards with data values
Graph paper (optional for application)
Calculator (optional)

PERIOD 1 & 2: Introduction to Variability and Range

PRESENTATION:

Step	Teacher’s Activity	Pupil’s Activity
Step 1 – Anticipation (Warm-up)	Teacher asks: “Why do we need to know how spread out data is?” Shows two groups of test scores with the same mean but different spread.	Pupils compare the data sets and respond.
Step 2 – Building Knowledge (Explanation)	Defines variability as the measure of how spread out or scattered the values in a data set are. Explains the importance (e.g., comparing exam results, analyzing business sales).	Pupils listen and take notes.
Step 3 – Demonstration	Introduces Range: largest value – smallest value. Works through examples: 1. Data: 5, 8, 12, 15 → Range = 15 – 5 = 10 2. Data: 20, 22, 24, 30 → Range = 30 – 20 = 10	Pupils solve along with teacher.
Step 4 – Consolidation (Practice)	Gives pupils short exercises to compute range of small data sets.	Pupils calculate individually or in pairs.

NOTE ON BOARD:

Variability shows how data values differ from each other.
Range = Highest value – Lowest value
Example: For data 2, 4, 6, 8 → Range = 8 – 2 = 6

EVALUATION (5 exercises):

Find the range of: 2, 4, 6, 10.
Find the range of: 12, 15, 18, 25.
Scores: 30, 40, 50, 60 → range = ?
Data: 5, 7, 9, 12, 15 → range = ?
Why is range an incomplete measure of variability?

CLASSWORK (5 questions):

Find the range of: 8, 12, 15, 19.
Find the range of: 40, 42, 47, 50.
Data: 5, 10, 20, 25 → range = ?
Heights (in cm): 150, 160, 170, 180 → range = ?
Explain one limitation of range.

ASSIGNMENT (5 tasks):

Collect the ages of 5 classmates and find the range.
Find the range of: 2, 3, 5, 10, 12.
Find the range of: 45, 55, 60, 70, 80.
Create your own data set of 6 numbers and calculate the range.
Write one real-life example where range is useful.

PERIOD 3: Variance

PRESENTATION:

Step	Teacher’s Activity	Pupil’s Activity
Step 1 – Anticipation	Teacher recalls range, then asks: “Is range enough to describe variability?” Explains why we need variance.	Pupils respond.
Step 2 – Building Knowledge (Explanation)	Defines variance as the average of the squared differences from the mean. Writes formula: σ²=∑(x−xˉ)²ⁿ	Pupils copy formula.

Step 3 – Demonstration	Example: Data = 2, 4, 6 Mean = (2+4+6)/3 = 4 Variance = [(2–4)² + (4–4)² + (6–4)²] / 3 = (4+0+4)/3 = 8/3 = 2.67	Pupils follow calculation.


Step 4 – Consolidation	Provides another example: 5, 7, 9. Guides students to compute variance.	Pupils practice in groups.

NOTE ON BOARD:

Variance measures the average squared deviation from the mean.
Formula: σ²=∑(x−xˉ)²ⁿ

EVALUATION (5 exercises):

Find the mean and variance of: 1, 2, 3.
Data: 10, 12, 14 → variance = ?
Data: 4, 8, 12 → variance = ?
Why is variance always positive?
Why do we square deviations in variance?

PERIOD 4: Standard Deviation

PRESENTATION:

Step	Teacher’s Activity	Pupil’s Activity
Step 1 – Anticipation	Teacher explains that variance is useful but not in original units. Introduces standard deviation (SD).	Pupils listen.
Step 2 – Building Knowledge (Explanation)	Defines SD as the square root of variance. Formula: σ=√∑(x−xˉ)²ⁿ	Pupils copy formula.
Step 3 – Demonstration	Example: Data = 2, 4, 6 Variance = 2.67	Pupils follow step by step.
	SD = √2.67 ≈ 1.63

Step 4 – Consolidation	Gives another example: 3, 5, 7. Guides pupils to compute SD.	Pupils solve with guidance.

NOTE ON BOARD:

SD = √Variance
Example: Data = 2, 4, 6 → Variance = 2.67, SD ≈ 1.63

EVALUATION (5 exercises):

Find the SD of 1, 2, 3.
Data: 10, 20, 30 → SD = ?
Why is SD more useful than variance?
Compute SD of: 5, 5, 5.
Compute SD of: 2, 3, 5, 7.

PERIOD 5: Real-Life Applications of Variability

PRESENTATION:

Step	Teacher’s Activity	Pupil’s Activity
Step 1 – Anticipation	Teacher asks: “Where can we use variability in real life?”	Pupils brainstorm examples (sports, business, exam results).
Step 2 – Building Knowledge (Explanation)	Explains: Business (sales fluctuation) Sports (performance comparison) Weather (temperature changes) Exams (student performance spread)	Pupils listen.
Step 3 – Demonstration	Gives real-life data: Daily sales = 20, 25, 30, 35, 40. Compute mean, range, variance, SD	Pupils solve in groups.
Step 4 – Consolidation	Discusses results and interpretations.	Pupils share answers.

EVALUATION (5 exercises):

Why is variability important in exam scores?
A company’s daily sales are 50, 60, 70. Find the range and SD.
Daily temperatures: 28°C, 30°C, 31°C, 29°C. Find the variance.
How does SD help in comparing data sets?
Give two examples of variability in sports.

CLASSWORK (5 questions):

Calculate variance and SD for: 2, 4, 6.
Find the range, variance, SD of: 5, 10, 15.
Find the SD of: 10, 12, 14, 16.
Write one real-life use of SD.
Why is SD preferred over variance?

ASSIGNMENT (5 tasks):

Collect the ages of 5 family members and compute range, variance, and SD.
Find the SD of: 2, 3, 4, 5, 6.
Collect exam scores of classmates and compute variability.
Research how weather forecasters use SD.
Write a short note on why SD is the best measure of variability.