[size=150]Elena and Han are discussing how to write the repeating decimal [math]x=0.13\overline{7}[/math] as a fraction. [br]Han says that [math]0.13\overline{7}[/math] equals [math]\frac{13764}{99900}[/math]. “I calculated [math]1000x=137.77\overline{7}[/math] because the decimal begins repeating after 3 digits. Then I subtracted to get [math]999x=137.64[/math]. Then I multiplied by [math]100[/math] to get [br]rid of the decimal: [math]99900x=13764[/math]. And finally I divided to get[math]x=\frac{13764}{99900}[/math]." [br]Elena says that[br][math]0.13\overline{7}[/math] equals [math]\frac{124}{900}[/math]. “I calculated [math]10x=1.37\overline{7}[/math] because one digit repeats. Then I subtracted to get [math]9x=1.24[/math]. Then I did what Han did to get [math]900x=124[/math] and [math]x=\frac{124}{900}[/math]."[/size][br][br]Do you agree with either of them? Explain your reasoning.
How are the numbers [math]0.444[/math]and [math]0.\overline{4}[/math] the same? How are they different?
[size=150][math]2.2^2=4.84[/math] and [math]2.3^2=5.29[/math]. This gives some information about [math]\sqrt{5}[/math].[/size][br][br]Without directly calculating the square root, plot [math]\sqrt{5}[/math] on all three number lines using successive approximation.