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How to Make Change for a Dollar? Ah, Let Me Count the Ways!

 

Hi All,


This is a very interesting exploration for grades 4 - 6. It features algebraic thinking, patterns, logical thinking, money, addition, and subtraction. I had a great time doing this one myself. (But then again, I'm a bit of a nerd for stuff like this.) This is a great problem for partners or small groups and I would highly recommend making that an option. Also having coins available as manipulatives is a good idea as well.

We begin with the question, “How many ways are there to make change for $1.00?” At first it seems simple and it is fun to begin by having students estimate how many ways they think are possible. There are actually 293. Most students estimate far less.

Talk about an opportunity for logical thinking! Where should we start? Are there patterns? Let's slow down. Here's a reasonable sequence you may want to follow with your students.

a. List All the Coins: Before they begin exploring, ask students to name all the coins and list these on board or screen; penny, nickel, dime, quarter, half dollar, dollar coin.

b.  Examples of Making Change: Ask for an example of one way to make change for a dollar. Someone might say, "Four quarters." Write this down. As for another that uses pennies. Write this down.

c. Explain the Challenge: Now tell students, "There is definitely more than one way to make change for a dollar, but how many do you think there are?" Record estimates. Explain, "Today and probably tomorrow too, I'd like you to find all the ways to make change for a dollar and to record them."

d. How to Record? : Ask if there is an easier way to record an answer other than writing it all out with words. Some students like to draw circles with a number value inside. Others like to just give the coin a letter symbol ie; N = nickel. Share these ideas together.

e. How to Approach the Problem? : There's going to be a lot to keep organized. Lined notebook paper works well. Some students like graphic paper. Neat work will help students not to miss any ways.

The challenge in this for students is in trying to develop a logical, systematic approach to the problem instead of just randomly listing combinations they know well.  There is no one right way to approach this, but being systematic is key. 

f. Share Strategies: After about ten minutes of work, have students share strategies for creating their combinations as well as how they are checking and staying organized to be sure they don’t miss a possible combination.

Before you teach this one, you might want to try it out yourself or with another teacher or spouse (see how much fun you can have at home?!) You'll find some interesting patterns and strategies if you do.

 All the best,

Bob


 

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