Infinite Decimal Expansions: IM 8.8.15
[url=https://im.kendallhunt.com/MS/teachers/3/8/15/preparation.html]“Infinite Decimal Expansions”[/url] from IM Grade 8 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Taula de continguts
- Lesson 8.8.15
- IM 8.8.15 Lesson: Infinite Decimal Expansions
- Practice 8.8.15
- IM 8.8.15 Practice: Infinite Decimal Expansions
IM 8.8.15 Lesson: Infinite Decimal Expansions
[size=150]In the applet below, 3 digits after the decimal for the decimal expansion of [math]\frac{3}{7}[/math] have been calculated. Find the next 4 digits.[/size]
Here is a set of cards for your group. Each card will have a calculations side and an explanation side which you can see by clicking the cards.
[size=150]The cards show Noah’s work calculating the fraction representation of [math]0.4\overline{85}[/math]. Arrange these in order to see how he figured out that [math]0.4\overline{85}=\frac{481}{990}[/math] without needing a calculator.[br][br]Use Noah’s method to calculate the fraction representation of:[/size][br][br][math]0.1\overline{86}[/math]
[math]0.7\overline{88}[/math]
Use this technique to find fractional representations for [math]0.\overline{3}[/math] and[math]0.\overline{9}[/math].
Why is [math]\sqrt{2}[/math] between 1 and 2 on the number line?
Why is [math]\sqrt{2}[/math] between 1.4 and 1.5 on the number line?
How can you figure out an approximation for [math]\sqrt{2}[/math] accurate to 3 decimal places?
Label all of the tick marks. Make sure to add arrows from the second to the third number lines.
Elena notices a beaker in science class says it has a diameter of 9 cm and measures its circumference to be 28.3 cm. What value do you get for using these values and the equation for circumference, [math]C=2\pi r[/math]?
Diego learned that one of the space shuttle fuel tanks had a diameter of 840 cm and a circumference of 2,639 cm. What value do you get for using these values and the equation for circumference, [math]C=2\pi r[/math]?
Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of pi and plot that number on all three number lines.
How can you explain the differences between these calculations of [math]\pi[/math]?
IM 8.8.15 Practice: Infinite Decimal Expansions
[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.