Spreading Secrets (Investigation)

Have you ever told your friends a secret, only to find out that secret spread very quickly?! Imagine you told two of your friends a secret. A day later, each of your friends told two new people a secret. One day after that, each of those people told two new people the secret, and so on! By the third day, how many people know your secret?

Objectives

• To visualise indices (or powers)
• To confirm why the Multiplication Law is true
• To confirm why the Division Law is true
• To visualise the zero index

Materials

• 2 pieces of paper
• Pencil/Pen

Procedure

1. Take one sheet of paper and fold it in half width wise.
2. Without unfolding the paper fold it in half width wise again.
   

    

3. When you unfold the paper you should have 4 equal spaces on your page. Do not label the top box. Label the boxes below it 1-3 as shown in the picture.
4. Draw a stick figure to represent yourself in the box that is not numbered.
   
5. Suppose you have a very personal secret and you decide to share this secret with your two friends. Draw two lines from yourself to two stick figures in the box labeled with a 1.
   
6. It turns out that the friends are not good at keeping secrets. The next day your two friends each tell two more people the secret. In the box labeled with a 2 draw two lines from each person to the two new people they told as shown in the picture.
7. The people your friends told aren’t good at keeping secrets either. On the third day each one of the people who your friends told tell two more people. In the box labeled with a 3 draw two lines from each of the people your friends told to the people they told the secret to.

Questions

1. How many people were told the secret on day 3 alone?
2. By day 3 how many total people, including yourself, knew your secret?
3. This scenario shows how indices (or powers) work. The number of people who you told the secret to on day 1 is your base number. What number do you think represents the power in this scenario? Why?
4. Write out the amount of people who are told the secret each day index notation.
5. How many people will be told the secret on day 4 if this continues?
6. How many people will be told the secret on day 5 if this continues?
7. Before day 1, how many people including yourself knew your secret? What power do you think this represents the base being raised to? Why?
8. Multiply the amount of people who were told the secret on day 1 by the amount of people who were told the secret on day 2. Write this multiplication problem out in index notation. What do you notice about the result of the multiplication? Write out the result in index notation as well. Compare it to the amount of people told the secret on day 3. What do you notice?
9. Divide the amount of people who were told the secret on day 1 by the amount of people who were told the secret on day 3. Write this division problem index notation. What do you notice about the result of the division? Write out the result in terms exponential notation. Compare it to the amount of people told the secret on day 2. What do you notice?
10. What would happen if you had told 5 people your secret on day 1? How many people would know your secret by day 3?

Optional

1. Pick a number of people you want to tell your secret to on day one. Draw this scenario out on a large piece of paper as you did in the original case.
2. How many people know by day 4?
3. Compare and contrast this situation with the original.