If you start from the left end of a piano keyboard, you will first find four keys. Continuing to the right, you will find a series of octaves, each containing twelve keys.
Starting from the left on a piano, how many keys must you count in order to reach the end of the third octave?
After you have counted fifty-two keys, how many full octaves have you counted?
Most modern pianos have seven full octaves to the right of the first four keys. Counting from the left to the end of the seventh octave, how many keys are there?
Complete the rows for the number of keys per octave and the number of separate keys at the left end of the keyboard. Then, enter a variable for the number of octaves and use this variable to write an expression for the number of keys.
Use the Worksheet to complete the problem. The column labels describe the quantity from the problem scenario that the column is about. The row labels describe the type of value that goes in the row: quantity descriptions, units of measure, algebraic expressions, or numeric values to answer a question.
Quantity Name
Octaves
Keys
Unit
octaves
keys
Starting from the left on a piano, how many keys must you count in order to reach the end of the third octave?
Question 1
After you have counted fifty-two keys, how many full octaves have you counted?
Question 2
Most modern pianos have seven full octaves to the right of the first four keys. Counting from the left to the end of the seventh octave, how many keys are there?
Question 3
Keys per octave
keys per octave
Keys not belonging to a full octave
keys
Expression
1) To reach the end of the third octave, you must count 36 keys (12 keys per octave x 3 octaves).
2) After counting 52 keys, you have counted 4 full octaves (12 keys per octave x 4 octaves = 48 keys).
3) There are 88 keys in a standard piano, so there are 52 keys from the left end to the end of the seventh octave (88 total keys - 4 keys at the beginning = 84 keys in full octaves, and then 4 more keys in the seventh octave).
Keys per octave: 12, 12, 12, 12, 12, 12, 12, 4
Keys not belonging to a full octave: 4, 0, 0, 0, 0, 0, 0, 0
Let n represent the number of octaves. The expression for the number of keys is 4 + 12n.