Three coin problem

Pedagogical suggestions

Three coin problems allow learners an opportunity to enter and explore a problem at multiple levels to make discoveries and enjoy their learning.

The goal is to explore and find several possible combinations. To extend from there they can review coin values, add and compare total amounts, and make conjectures about what amounts are possible and not possible.

Materials

Coins - quarters, dimes, nickels, and pennies (?) dollar (?); sack or bag.

Vocabulary

• Combination is a selection of a given number of elements from a larger number without regard to their arrangement.
• Theoretical probability is equal to number of the selected outcomes out of the total number of possible outcomes. Or fraction.
• Experimental probability is the number of times the selected outcome occures out of the total number of trials. Or fraction.
• If the order you pick them in matters (which changes the problem from a combination to a permutation).

Background information

Decide on an initial problem.
The coins that might be in your pocket: quarters, dimes, nickels, and pennies and how many of each.

With the end of minting pennies, it won't be as realistic as in the past, but asking what combinations of three coins isn't the same.

Might consider using using a dollar coin in place of the penny.

Or might consider using paper money: one, five, ten, and twenty. Probably not going to want to use real paper money as the manipulative,

Procedure

Exploration

1. Presnt the problem. Start with with four coins - penny, nickel, dime, and quarter, but don't say how many or what kind to start.
   
2. Shake your pocket, purse, or sack and say: I searched the house for different coins and they are in here. I wonder what I will get if I take three coins out? (all three at one time)
3. Learners may ask how many coins? or what kinds of coins did you find. Say you don't want to give any more information now. That you will later.
4. Accept all answers and write them for all to see.
5. Explain. Today we are going to explore combinations of American coins: quarters, dimes, nickels, and pennies and make a list of the possible different combinations for three coins at a time.
6. Decide if they will use the picture on the worksheet and reason logically all possibkle combinatins (theoretical) or if they will have a bag with the four coins in to select three from (experimental).
7. Provide time to explore in pairs and record their findings.

Invention

1. Bring learners together and have them display their findings.
2. Record them and discuss if each is possible and label them as possible combinations of three coins.
3. For lower grades a list of possible combinations is a good place to agree upon before moving on to the values of different combinations of three coins. Discovering all possible combinations isn't necessary for younger learners.
4. For more advanced learners you might explore what might be all of the possible combinations of three different coins and strategies to discover them. Combining them systematically - p,n,d; p,n,q; p,d,q; n,d,q .
5. Another way of thinking about it is -
   Since there are only four distinct coins that can be left out, there can only be exactly four distinct groups of three.

Extensions

What are the different total values for the different combinations of three coins?

1. p,n,d;
2. p,n,q;
3. p,d,q;
4. n,d,q

Rather than picking three coins at a time could select one coin at a time (record it), replace it, select a second (record it), replace, it and finally a third.