Once upon a time there was a father who wanted to save money for his children. 

He decided to open up a bank account for each one. 

He needs help deciding which of the three options he should use. 

Option 1 ONE SOLUTION

They both get initial amounts of $100 plus open accounts.

But they will get different amounts (older brother gets $1 more per month) 

$5 a month deposit

$100 to start

$6 a month deposit

$100 to start

Baby Ben

Toddler Timmy

Dad believes that giving them both $100 to start makes it fair. He doesn't believe the extra dollar will make a difference.        

Can you write the equations? 

What would be the difference in 10 years? (120 months?                                            

Since the amounts of the deposits are different. The two accounts will only be the same at one point. (at the beginning. Doesn't matter how long. Every month Timmy will get more money and get farther away from being fair. 

ONLY FAIR AT ONE POINT IN TIME!!! NOT FAIR IN THE END!!!

Option 2 NO SOLUTION

He feels bad about the monthly deposits not being fair. He decides to give them the same monthly deposit ($10). He adds $20 extra to Timmy to make up for not starting an account earlier5

$5 a month deposit

$20 to start

$5 a month deposit 

$40 to start

Baby Ben

Toddler Timmy

Dad believes that giving them both the same deposit makes it fair. He doesn't believe the extra money Timmy gets to start will make a difference.        

Can you write the equations? 

What would be the difference in 10 years? (120 months?                                            

Since the amounts to start are  different. They two accounts will never be the same at any point.  Since they both start at different points and grow the same they will never catch each other.  

NEVER FAIR!! NEVER EQUAL!!!!

Option 3  MANY SOLUTIONS

Dad just wants things to be fair. So he decides to try everything equal. He gives them both $20 to start the account and deposits $7 in each account. 

$7 a month deposit

$20 to start

$7 a month deposit 

$20 to start

Baby Ben

Toddler Timmy

Dad believes that giving them both the same initial amount and same deposit will make it fair. He can see giving anything different can change the fairness so he would to test something new.

Can you write the equations? 

What would be the difference in 10 years? (120 months?                                            

Since both the amounts to start  are the same. The starting point will be the same. The growth every month will be the same. Both accounts are identical in every way!

ALWAYS EQUAL, ALWAYS FAIR!!!

Dad finally gets it! The slope is the monthly deposit! Every month the amount rises. The initial amount is the y-intercept. That's where it starts! Dad has this saving thing under control!

I got this!!!

Oh honey!