A 1 Megohm variable resistor that can change the frequency and an SPDT (single pole double throw) switch that controls the frequency range from 22.4663 Hz to 15,000 kHz and from 1.5296 Hz to 1.0213 kHz.

The 555 timer frequency equals to:

f = 1.44 / ((R1 + 2*R2)*C)

(Where R2 = R2a + R2b)

Therefore:

f = 1.44 / ((R1 + 2*(R2a+R2b))*C)

R1 = 1000 ohms

R2a = 1000 ohms

R2b = 1 Megohm (variable)

For C = 32 nF:

At maximum frequency setting R2b = 0:

f = 1.44 / ((1000 ohms + 2*(1,000 ohms))*(32*10^-9 F))

 = 1.44 / ((3000 ohms)*(32*10^-9 F))

 = 15,000 Hz

 = 15 kHz

At minimum frequency setting R2b = 1,000,000 ohms:

f = 1.44 / ((1000 ohms + 2*(1,000,000 ohms + 1,000 ohms))*(32*10^-9 F))

 = 1.44 / ((2003000 ohms)*(32*10^-9 F))

= 22.4663 Hz

For C = 470 nF:

At maximum frequency setting R2b = 0:

f = 1.44 / ((1000 ohms + 2*(1,000 ohms))*(470*10^-9 F))

 = 1.44 / ((3000 ohms)*(470*10^-9 F))

 = 1,0213 Hz

 = 1.0213 kHz

At minimum frequency setting R2b = 1,000,000 ohms:

f = 1.44 / ((1000 ohms + 2*(1,000,000 ohms + 1,000 ohms))*(470*10^-9 F))

 = 1.44 / ((2003000 ohms)*(470*10^-9 F))

 = 1.5296 Hz