5 x 5 odd Matrix Magic Square

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  Class 8 NCERT bridge course Answers Activity W 5.6  Watch It Rise: Exploring Exponent Graphs

Activity W 5.6 

Students may be engaged in identifying and continuing patterns related to the exponents of a number. They may plot the graphs and check for themselves how much the graph rises with larger exponents.

Objective:
Students will investigate how exponential functions behave by plotting the powers of different base numbers. They will visually observe the pattern of exponential growth and compare how different bases grow over time.

 Material Required 

 Graph paper or online graphing tools. 

 Rulers or measuring tools.

Procedure 

 Plot the values of each exponent of numbers, such as, 2, 3, 4 on a graph paper. 

Plot exponents (0, 1, 2, 3, etc.) on the horizontal axis and the corresponding values 2n, 3n, 4n etc., on the vertical axis. 

 Students may be asked to observe the shape of the graph. 

For some bases, it will be a smooth curve that grows slowly at first and then rapidly.

 They may observe the pattern in which the graphs grow and explain it. 

They may tell in which it was the steepest and why. 

 Exponent versa corresponding values.

1. Choose Base Numbers
   Select base numbers such as 2, 3, and 4.

2. Create a Table of Values
   For each base number, compute the value of:  a where n=0,1,2,3,4,5 

   Example for base 2:

   • $2^0 = 1$
     

   • $2^1 = 2$
     

   • $2^2 = 4$
     

   • $2^3 = 8$
     

   • $2^4 = 16$
     

   • $2^5=\mathrm{32<span\; class="base"><br></span>}$

3. Plot the Graphs

   • Place exponents (n = 0, 1, 2, ...) on the x-axis

   • Place the values of $a^n$ on the y-axis

   • Plot the points for each base and connect them smoothly

4. Observe the Graphs

   • Note how the curves grow

   • Identify which graph is the steepest and discuss why

   • Compare the curves of different bases

5. Extension (Optional):
   Plot values of fractional bases (e.g., 0.5, 1.5) and negative exponents (e.g., $2^{-1}, 2^{-2}$)

Reflections – Sample Student Answers

Following questions may be discussed: 

1. How do the numbers increase with each step?

   • The values increase multiplicatively, not additively.

   • With each step, the number is multiplied by the base again, which leads to rapid growth.

2. Plot graphs for exponents of numbers which are fractions or negative numbers.

   • For a base like 0.5:

     • $0.5^0 = 1$, $0.5^1 = 0.5$, ${0.5}^2=\mathrm{0.25,\; etc.}$

     • The graph decreases with increasing exponent.

   • For negative exponents:

     • $2^{-1} = 0.5$, $2^{-2} = 0.25$, etc.

     • The graph approaches zero but never touches it.

3. Which base grows the fastest and slowest?

   • Base 4 grows the fastest: its curve is steepest.

   • Base 2 grows slower compared to 3 and 4.

   • Fractional bases decrease over time.

   • The greater the base, the faster the growth.

4. How are exponents used in real life?

   • Computing power: Moore’s law suggests computing power doubles every 2 years → exponential growth.

   • Population growth: Under ideal conditions, populations grow exponentially.

   • Interest rates: Compound interest involves exponential growth of money.

   • Physics and Chemistry: Radioactive decay and reactions follow exponential models

   • (for example, computing power, population growth models, interest rates, etc.) 

     
     

 Class 8 NCERT bridge course Answers Activity W 5.5 exploring numbers. 

 The activity will help in developing the habit of exploring numbers. 

Download the worsheet: Click Here

Part 1: Two-Digit Number Reversal and Subtraction

Procedure 

1. Ask the students to write a two-digit number. 

2. They should reverse the digits and form a new number. 

3. Subtract the smaller of these numbers from the larger one. 

4. Using the result, repeat the process. 

5. Students may observe when the process stops. 

Discuss about it. 

Example: 

1. Choose a two-digit number (e.g., 52).

2. Reverse the digits → 25.

3. Subtract the smaller from the larger:
   → 52 - 25 = 27

4. Reverse again: 72 - 27 = 45

5. Continue the process:
   → 54 - 45 = 09
   → 90 - 09 = 81

6. Observations:
   

   • Eventually, the number 81 appears.

   • If we check 81 ÷ 9 = 9, and 8 + 1 = 9 → There's a link with the table of 9.

   • Any two-digit number (with different digits) tends to reach a multiple of 9.

   • Numbers like 81, 63, 27 may appear—all are multiples of 9.

   
   

Answer to Q1: Is there any link with the table of 9?

 Yes, the numbers generated through this process often end up being multiples of 9. This happens because reversing and subtracting two-digit numbers preserves a difference divisible by 9.

Answer to Q2: What if the two digits are the same?

• Example: 33
  → Reversed = 33
  → 33 - 33 = 0

• The process stops immediately.
  So, when digits are the same, the result is always 0, and the process ends.

 

Part 2: Three-Digit Number Digit Arrangement and Subtraction

Procedure Recap (Example: 256):

1. Descending: 652

2. Ascending: 256
   → Subtract: 652 - 256 = 396

Then:
→ 963 - 369 = 594
→ 954 - 459 = 495
→ 954 - 459 = 495 again

Observation:

 The process reaches a stable number, 495, and continues to repeat it.

This is known as a Kaprekar constant (for 3-digit numbers).

Discuss 

Answer to Q1: What if any digits are the same?

• Example: 232
  → Descending: 322
  → Ascending: 223
  → 322 - 223 = 099
  → 990 - 099 = 891
  → 981 - 189 = 792
  → 972 - 279 = 693
  → 963 - 369 = 594
  → 954 - 459 = 495

 The process still ends at 495, although it might take longer.

Answer to Q2: What if all three digits are the same?

• Example: 555
  → Descending: 555
  → Ascending: 555
  → 555 - 555 = 0

 Just like in the two-digit case, the process stops immediately.

Extension: Three-digit Number Reverse and Add

Procedure

1. Think of a 3-digit number in which the first and the last digits differ by at least 

 2. Reverse its digits and subtract the smaller number from the larger of the two. 

3. Add the resulting number and its reverse. 

Example:

1. Choose a 3-digit number with first and last digits differing by at least 2
   Example: 321

2. Reverse it: 123

3. Subtract smaller from larger:
   → 321 - 123 = 198

4. Reverse 198 = 891

5. Add both:
   → 198 + 891 = 1089

Observation:

 The final result is 1089 — a famous number trick!

Try others:

• 532 → reverse: 235
  → 532 - 235 = 297
  → 297 + 792 = 1089

This works for most such numbers.

What do you find?

Summary of Key Findings

Activity	Observation
2-digit process	Leads to multiples of 9 (e.g., 81)
2-digit same digits	Results in 0 immediately
3-digit process	Always reaches 495 (Kaprekar constant)
3-digit same digits	Results in 0
3-digit reverse/add	Final result is always 1089

Class 8 NCERT bridge course Answers Activity W 5.4 Making different stories for the same number sentence

 Activity W 5.4: Making different stories for the same number sentence 

Traditionally, word problems appear in textbooks or in classroom teaching at the end of a chapter. 

Often, little time and attention is spent on making sense of these word problems.

 Students often get confused with the words and the message the sentences convey. 

Let the students create their own stories, or word problems; 

narrate a mathematical sentence like

 5 + 6 = 11

 can help to build an understanding of the mathematical ideas and lead to greater problem solving skills. 

It can help the students overcome the difficulties of making sense of the context of the word problems because they will construct their own context and focus on making the story fitints mathematics. 

In that way, it also helps them with identifying, which mathematical representation to use.

This activity is directed towards forming such contexts using number statements. 

Procedure 

 Give the students a number statement say, 5 + 7 = 12. 

This statement can be represented by several mathematical relationships, such as: 

Number Sentence: 5 + 7 = 12

Step 1: Write Different Mathematical Relationships

• Adding 5 and 7 together makes 12.

• 5 more than 7 gives 12.

• The total number of things is 5 + 7 = 12.

• 7 less from something leaves 5 (i.e., 12 - 7 = 5).

• 12 minus 5 equals 7.

• 12 is the sum of 5 and 7.

•  Give the students more such numerical statements and ask them to write different mathematical relationships for them. 
•  Now, ask the students to formulate a story or word problem for each of these relationships.
•  Encourage them to use their imagination! 

• Example: For the statement: Adding 5 and 7 together make 12;

•  it could be— In a cricket match, Kalyani and Shreya are batting together. Kalyani made 5 runs, whereas Shreya made 7. What is the total number of runs they made?

Step 2: Create Different Word Problems or Stories

1. Adding 5 and 7 together makes 12:
   In a cricket match, Kalyani and Shreya are batting together. Kalyani made 5 runs, whereas Shreya made 7. What is the total number of runs they made?

2. 5 more than 7 gives 12:
   A fruit seller had 7 apples in one basket. He put 5 more apples into the basket. How many apples are there now?

3. The total number of things is 5 + 7 = 12:
   There are 5 red balloons and 7 blue balloons at a party. How many balloons are there in total?

4. 7 less from something leaves 5 (i.e., 12 - 7 = 5):
   Rina had some toffees. She gave 7 toffees to her brother and was left with 5. How many toffees did she have originally?

5. 12 minus 5 equals 7:
   There were 12 books on a shelf. If 5 books were taken away, how many books are left on the shelf?

6. 12 is the sum of 5 and 7:
   A gardener planted 5 rose plants in one row and 7 rose plants in another row. How many rose plants did he plant in total?

Example 2
Let's take the number sentence 8 + 4 = 12 and go through the same steps.

Number Sentence: 8 + 4 = 12

Step 1: Different Mathematical Relationships

1. Adding 8 and 4 together makes 12.

2. 8 more than 4 gives 12.

3. The total number of things is 8 + 4 = 12.

4. 4 less from something leaves 8 (i.e., 12 - 4 = 8).

5. 12 minus 8 equals 4.

6. 12 is the sum of 8 and 4.

Step 2: Word Problems or Stories

1. Adding 8 and 4 together makes 12:
   Ravi has 8 pencils and his friend gives him 4 more. How many pencils does Ravi have in total?

2. 8 more than 4 gives 12:
   A basket had 4 oranges. Someone added 8 more oranges. How many oranges are there now?

3. The total number of things is 8 + 4 = 12:
   There are 8 boys and 4 girls in a group. How many children are there altogether?

4. 4 less from something leaves 8 (i.e., 12 - 4 = 8):
   Rita had some flowers. She gave 4 flowers to her friend and was left with 8. How many flowers did she have at the beginning?

5. 12 minus 8 equals 4:
   There were 12 candies in a box. After giving 8 candies to her brother, how many does Anu have now?

6. 12 is the sum of 8 and 4:
   There are 8 green frogs and 4 brown frogs on a lily pad. How many frogs are there in total?