Lesson Notes By Weeks and Term v3 - Primary 6
Indices (power)
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Indices (power)
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Subject: General Mathematics
Class: Primary 6
Term: 2nd Term
Week: 4
Theme: Basic Operations
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Write numbers in in dex form; Solve problems in volving powers (in dices ) Solve problem on quantitative reasoning in volving in dices.
or equation related to indices. Worked
Examples:
1. Find the missing number: $2, 4, 8, \_, 32$.
Observe the pattern: $2 = 2^1$ $4 = 2 \times 2 = 2^2$ $8 = 2 \times 2 \times 2 = 2^3$ The pattern is powers of
2. The next term should be $2^4$. $2^4 = 2 \times 2 \times 2 \times 2 = 16$.
Check: $2^5 = 32$. The missing number is 16.
2. If $3^x = 27$, find the value of x.
Think: How many times do we multiply 3 by itself to get 27? $3^1 = 3$ $3^2 = 3 \times 3 = 9$ $3^3 = 3 \times 3 \times 3 = 27$ Therefore, $x = 3$.
3. Which number completes the series: $1, 4, 9, 16, \_$.
Observe the pattern: These are square numbers. $1 = 1 \times 1 = 1^2$ $4 = 2 \times 2 = 2^2$ $9 = 3 \times 3 = 3^2$ $16 = 4 \times 4 = 4^2$ The next number should be $5^2$. $5^2 = 5 \times 5 = 25$. The missing number is
2
5. Definition of Indices (Powers)
Indices (singular: index) or powers provide a concise way to represent a number multiplied by itself a specific number of times.
Base: The number that is being multiplied repeatedly.
Index (or Exponent/Power): The small number written above and to the right of the base, indicating how many times the base is multiplied by itself.
Index Form (or Power Form): The entire expression consisting of the base and its index (e.g., $2^3$).
Example: In the expression $5^3$: 5 is the base. 3 is the index (or exponent/power). $5^3$ is the index form. It means $5 \times 5 \times 5$.
Reading Index Forms: $2^2$ is read as "2 squared" or "2 to the power of 2." $3^3$ is read as "3 cubed" or "3 to the power of 3." $4^5$ is read as "4 to the power of 5." Objective 1: Writing Numbers in Index Form To write a number in index form, the teacher should guide students to identify a base number that, when multiplied by itself repeatedly, results in the given number. Prime factorization can be a useful tool, though for Primary 6, direct identification of repeated factors is often sufficient for common numbers. Worked
Examples:
1. Write 8 in index form using base
2. Think: What power of 2 gives 8? $2 \times 2 = 4$ $2 \times 2 \times 2 = 8$ Therefore, $8 = 2^3$.
2. Express 81 in index form using base 3. $3 \times 3 = 9$ $3 \times 3 \times 3 = 27$ $3 \times 3 \times 3 \times 3 = 81$ Therefore, $81 = 3^4$.
3. Write 100 in index form using base 10. $10 \times 10 = 100$ Therefore, $100 = 10^2$.
Objective 2: Solving Problems Involving Powers (Indices)
This involves two main types of problems: a) Evaluating the value of numbers in index form: This means calculating the actual numerical value. b) Simple multiplication of numbers in index form with the same base: This introduces the basic idea of the product rule for indices without formally stating the rule. Worked
Examples:
1. Evaluate $4^3$. $4^3$ means $4 \times 4 \times 4$. $4 \times 4 = 16$ $16 \times 4 = 64$ Therefore, $4^3 = 64$.
2. Evaluate $2^5$. $2^5$ means $2 \times 2 \times 2 \times 2 \times 2$. $2 \times 2 = 4$ $4 \times 2 = 8$ $8 \times 2 = 16$ $16 \times 2 = 32$ Therefore, $2^5 = 32$.
3. Multiply $3^2 \times 3^3$. (This is a simplified introduction to the product rule for indices) $3^2$ means $3 \times 3$. $3^3$ means $3 \times 3 \times 3$. So, $3^2 \times 3^3 = (3 \times 3) \times (3 \times 3 \times 3)$. This is $3 \times 3 \times 3 \times 3 \times 3$, which is 3 multiplied by itself 5 times.
Therefore, $3^2 \times 3^3 = 3^5$. (Note for teacher: Emphasise that for multiplication, the base must be the same, and the indices are added, but do not formally introduce the "product rule formula" as this is Primary 6).
4. Multiply $5^1 \times 5^2$. $5^1$ means $5$. $5^2$ means $5 \times 5$. So, $5^1 \times 5^2 = 5 \times (5 \times 5) = 5 \times 5 \times 5$.
Therefore, $5^1 \times 5^2 = 5^3$.
Objective 3: Solving Problems on Quantitative Reasoning Involving Indices This involves identifying patterns or finding missing values in a sequence or equation related to indices. Worked
Examples:
1. Find the missing number: $2, 4, 8, \_, 32$.
Observe the pattern: $2 = 2^1$ $4 = 2 \times 2 = 2^2$ $8 = 2 \times 2 \times 2 = 2^3$ The pattern is powers of
2. The next term should be $2^4$. $2^4 = 2 \times 2 \times 2 \times 2 = 16$.
Check: $2^5 = 32$. The missing number is 16.
2. If $3^x = 27$, find the value of x.
Think: Teaching Materials: Whiteboard/Chalkboard, markers/chalk, charts displaying examples of index forms, flashcards with numbers to convert to index form and vice versa.
Introduction (10 minutes): Teacher Activity: Begin by reviewing repeated multiplication. Ask students to calculate $2 \times 2 \times 2 \times 2 \times 2$. Then ask for $10 \times 10 \times 10 \times 10$.
Student Activity: Students solve the repeated multiplication problems.
Teacher Activity: Introduce the challenge of writing very long repeated multiplications. "Is there a shorter way to write $2 \times 2 \times 2 \times 2 \times 2$ or $10 \times 10 \times 10 \times 10$?" Introduce the term "indices" or "powers" as a shorthand. Explain the base and the index with simple examples like $2^3$. Explanation and Demonstration (20 minutes): Teacher Activity: Clearly define base, index, and index form using examples. Write $3^4$ on the board, pointing out the base and index.
Demonstrate Objective 1: Writing numbers in index form. Use the worked examples from the Key Concepts section (e.g., 81 as $3^4$, 100 as $10^2$). Emphasize finding the repeated factor.
Demonstrate Objective 2: Solving problems involving powers. Show how to evaluate expressions like $4^3$ and $2^5$ step-by-step. Introduce simple multiplication of indices with the same base, explaining $3^2 \times 3^3 = 3^5$ by expanding it out (i.e., $(3 \times 3) \times (3 \times 3 \times 3)$).
Student Activity: Students listen, ask questions, and take notes. They may be asked to orally identify the base and index in new examples provided by the teacher. Guided Practice / Class Activity (20 minutes): Teacher Activity: Divide students into small groups (3-4 students). Provide each group with a few numbers (e.g., 27, 64, 125, 16, 49) and ask them to: a) Write each number in index form using a specified base (e.g., 27 with base 3, 64 with base 2). b) Evaluate expressions like $5^3$, $3^4$. c) Solve simple multiplication problems like $2^2 \times 2^3$. Circulate among groups, providing support and clarification.
Student Activity: Students work collaboratively in their groups, discussing and solving the problems. They write their solutions on exercise books or on small whiteboards/slates if available.
Teacher Activity: Bring the class together. Ask representatives from each group to present their solutions on the board. Facilitate discussion and peer correction. Correct any misconceptions. Quantitative Reasoning Activity (10 minutes): Teacher Activity: Present one or two quantitative reasoning problems involving indices (e.g., "Find the missing number: $10, 100, 1000, \_$" or "If $4^x = 16$, find x"). Guide students to identify the underlying pattern of powers.
Student Activity: Students attempt to solve the problems individually or in pairs.
Teacher Activity: Discuss the solutions as a class, explaining the reasoning step-by-step.
Conclusion (5 minutes): Teacher Activity: Summarise the key concepts: what indices are, how to write numbers in index form, how to evaluate them, and how to perform simple multiplication of indices with the same base. Remind students of the importance of indices in simplifying mathematics.
Student Activity: Students participate in the summary, asking any final questions. The teacher should work through these examples with the students, explaining each step clearly.
Question: Write the number 125 in index form with base
5. Solution: We need to find out how many times 5 is multiplied by itself to get 125. $5 \times 5 = 25$ $25 \times 5 = 125$ So, 5 is multiplied by itself 3 times.
Therefore, $125 = 5^3$.
Commentary: This directly addresses Objective 1 and Evaluation Guide 2, reinforcing the process of converting a number to its index form.
Question: Evaluate $6^3$.
Solution: $6^3$ means $6 \times 6 \times 6$. First, $6 \times 6 = 36$. Next, $36 \times 6$: ``` 36 x 6 216 ``` Therefore, $6^3 = 216$.
Commentary: This addresses Objective 2, focusing on calculating the value of an index expression.
Question: Multiply $2^3 \times 2^4$.
Solution: $2^3$ means $2 \times 2 \times 2$. $2^4$ means $2 \times 2 \times 2 \times 2$. So, $2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2)$. Counting the total number of 2s being multiplied, there are $3 + 4 = 7$ factors of
2. Therefore, $2^3 \times 2^4 = 2^7$.
Commentary: This addresses Objective 2 and Evaluation Guide 3, introducing the concept of multiplying indices with the same base by expanding the terms.
Question: If $10^z = 10,000$, find the value of z.
Solution: We need to find out how many times 10 is multiplied by itself to get 10,000. $10^1 = 10$ $10^2 = 10 \times 10 = 100$ $10^3 = 10 \times 10 \times 10 = 1,000$ $10^4 = 10 \times 10 \times 10 \times 10 = 10,000$ Therefore, $z = 4$.
Commentary: This addresses Objective 3 and Evaluation Guide 4, a quantitative reasoning problem requiring students to deduce the power.
Write 8 in index form using base
2. Think: What power of 2 gives 8?
$2 \times 2 = 4$
$2 \times 2 \times 2 = 8$
Therefore, $8 = 2^3$.
Express 81 in index form using base 3.
$3 \times 3 = 9$
$3 \times 3 \times 3 = 27$
$3 \times 3 \times 3 \times 3 = 81$
Therefore, $81 = 3^4$.
Write 100 in index form using base 10.
$10 \times 10 = 100$
Therefore, $100 = 10^2$.
Objective 2: Solving Problems Involving Powers (Indices)
This involves two main types of problems:
a) Evaluating the value of numbers in index form: This means calculating the actual numerical value.
b) Simple multiplication of numbers in index form with the same base: This introduces the basic idea of the product rule for indices without formally stating the rule.
Worked
Examples:
Evaluate $4^3$.
$4^3$ means $4 \times 4 \times 4$.
$4 \times 4 = 16$
$16 \times 4 = 64$
Therefore, $4^3 = 64$.
Evaluate $2^5$.
$2^5$ means $2 \times 2 \times 2 \times 2 \times 2$.
$2 \times 2 = 4$
$4 \times 2 = 8$
$8 \times 2 = 16$
$16 \times 2 = 32$
Therefore, $2^5 = 32$.
Multiply $3^2 \times 3^3$. (This is a simplified introduction to the product rule for indices)
$3^2$ means $3 \times 3$.
$3^3$ means $3 \times 3 \times 3$.
So, $3^2 \times 3^3 = (3 \times 3) \times (3 \times 3 \times 3)$.
This is $3 \times 3 \times 3 \times 3 \times 3$, which is 3 multiplied by itself 5 times.
Therefore, $3^2 \times 3^3 = 3^5$.
(Note for teacher: Emphasise that for multiplication, the base must be the same, and the indices are added, but do not formally introduce the "product rule formula" as this is Primary 6).
Multiply $5^1 \times 5^2$.
$5^1$ means $5$.
$5^2$ means $5 \times 5$.
So, $5^1 \times 5^2 = 5 \times (5 \times 5) = 5 \times 5 \times 5$.
Therefore, $5^1 \times 5^2 = 5^3$.
Population Growth and Economic Data: Indices are used to express large numbers. For example, if a village's population doubles every 10 years, the population after 'x' decades can be represented using powers of
2. Similarly, large national budgets or debts are often expressed using powers of 10 (e.g., 'billions' is $10^9$). This helps students appreciate how indices simplify understanding vast numbers in demography or economics within Nigeria.
Area and Volume Calculations: When calculating the area of a square farm plot with sides of length 'L', the area is $L \times L = L^2$. For a cube-shaped water tank with sides of length 'S', the volume is $S \times S \times S = S^3$. This provides a concrete, visual application of "squared" and "cubed" in local contexts like land measurement or water storage.
Digital Storage and Technology: While perhaps slightly advanced, the concept of computer memory (e.g., kilobytes, megabytes, gigabytes) being powers of 2 ($2^{10}$ for a kilobyte, $2^{20}$ for a megabyte) can be introduced simply. This connects indices to everyday technology items like phones and computers, which are increasingly common in Nigerian homes.