Thought for the day: Wondering how many ways there are to do something can lead to many opportunities for deep mathematical learning.

Concepts: recognizing and counting coins, place value, one- and two-digit addition, recognizing and extending patterns

Beginning

I notice a dime and a penny.
I notice that the penny shows heads and the dime shows tails.
I notice the amount "47 cents" written below the coins.
I notice that I can see only part of the date on the penny.
I notice a torch and some branches on the dime. (Students may be curious to learn more about these!)
I wonder why the total value of the coins is different than the number of cents written.

Exploring

I notice that pennies and dimes are place value amounts (1 and 10).
I wonder why the picture doesn't show nickels or quarters.
I wonder how I can make 47¢ using only dimes and pennies.
I wonder how many ways there are to make 47¢ using only dimes and pennies.
I wonder if there is an easy way to predict how many ways there are to make different amounts using only dimes and pennies.
I wonder how many ways there would be if I had nickels (and/or quarters), too.
I wonder how many ways there are to make amounts greater than \$1.00 if I use dollar bills, dimes, and pennies.

Creating

As they notice and wonder, students may create their own problems, strategies, and ideas for organizing their data. Many of their ideas will flow from things that they have wondered about. For example, they may

Create tables or other formats for organizing their data about the different combinations of coins.
Create new problems by changing the number of cents or coins that they use to try to make the total.
Create problems extending beyond coins into paper money—or even values of imaginary coins!

Reflecting and Extending

I notice that making tables of my data helps me to discover patterns.
I notice that writing the numbers in order also helps me to discover patterns.
I notice
that there are 5 ways to make 40¢, 41¢, 42¢, 43¢, 44¢, 45¢, 46¢, 47¢, 48¢, and 49¢.
I notice that the number of ways to make amounts between 1¢ and 99¢ is always 1 greater than the tens digit.
I wonder what causes this pattern.
I notice that I can think of 1-digit amounts as having a tens digit of 0.
I notice that I get the same answers for pennies and dimes that I would get for the ways to make 1- and 2-digit numbers from 1s and 10s.
I notice that using pennies, dimes, and dollar bills is like using the place value groups of 1, 10, and 100.
I wonder how many ways there are to make amounts between 1¢ and \$9.99 using only pennies, dimes, and dollar bills.

Notes

Linda Jensen Sheffield has a similar activity (called "How Many Ways?") about coin combinations in her excellent book, Extending the Challenge in Mathematics.

There are five ways to make 47¢ using only dimes and pennies.

0 dimes and 47 pennies
1 dime and 37 pennies
2 dimes and 27 pennies
3 dimes and 17 pennies
4 dimes and 7 pennies

By experimenting with different amounts and collecting and organizing their data, students may discover that they can make any number of cents in the 40s in five different ways. In fact, the number of ways you can make any amount less than \$1.00 (with only dimes and pennies) is always equal to one more than the tens digit in the amount. Students may challenge themselves to understand and explain why this is the case.

The answers to this question relate to the process of renaming numbers using place value. For example, you may name 47 as

0 tens and 47 ones
1 ten and 37 ones
2 tens and 27 ones
3 tens and 17 ones
4 tens and 7 ones

Searching for a pattern in the number of ways to make amounts greater than \$1.00 using pennies, dimes, and dollar bills brings in three place values (ones, tens, and hundreds) and may be an excellent challenge for the strongest students!