Rational and Irrational Numbers
Warm Up Write each fraction as a decimal. 2 5 4. 2 3 8 7 10 5. 3 3 4 8 9 6. 5 6
Work with positive rational and irrational numbers. Make connections among the real numbers by converting fractions and decimals and approximating irrational numbers. Understand that every number has a decimal expansion. Convert a repeating decimal to a rational number. Evaluate square roots of perfect squares and cube roots of perfect cubes. Estimate an irrational number. Extend the positive rational and irrational numbers to include negative numbers and compare and order real numbers.
A square garden has an area of 20 square feet A square garden has an area of 20 square feet. Explain why the side length cannot be rational. Approximate the length od each side of the garden to the nearest tenth and to the nearest hundredth.
Write each fraction as a decimal. 2 5 5 9
Write each decimal as a fraction in simplest form. 0.355 0. 43
To express a rational number as a decimal, divide the numerator by the denominator. To take a square root or a cube root of a number, find the number that when squared or cubed equals the original number. To approximate an irrational number, estimate a number between to consecutive perfect squares.
How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal? The decimal will terminate if the denominator is an even number, a multiple of 5, or a multiple of 10.
How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator? Start by identifying the place value of the decimal's last digit, and then use the corresponding power of 10 as the denominator of the fraction.
How can you tell if a decimal can be written as a rational number? If the decimal is a terminating or repeating decimal, then it can be written as a rational number.
Some decimals may have a pattern but still not be a repeating decimal that is rational. For example, in 3.12112111211112…, you can predict the next digit, and describe the pattern. (There is one more 1 each time before the 2.) However, this is not a terminating decimal, nor is it a repeating decimal, and it is therefore NOT a rational number.
Solve each equation for x. 𝑥 2 =324 343= 𝑥 3 𝑥 2 = 25 144 𝑥 3 = 125 512
Compare the values for 13 2 and 1.3 2 .
How do you know whether 2 will be closer to 1 or closer to 2?
The word irrational, when used as an ordinary word in English, means without logic or reason. In mathematics, when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers.
An artist wants to frame a square painting with an area of 400 square inches. She wants to know the length of the wood trim that is needed to go around the painting. If x is the length of one side of the painting, what equation can you set up to find the length of a side? How many solutions does the equation have? Do all of the solutions that you found make sense in the context of the problem? Explain. What is the length of the wood trim needed to go around the painting?
Solve each equation for x. Write your answer a radical expression Solve each equation for x. Write your answer a radical expression. Then estimate to one decimal place, if necessary. 𝑥 2 =14 𝑥 3 =1331 𝑥 2 =144 𝑥 2 = 29
To find 15 , Beau found 3 2 = 9 and 4 2 = 16 To find 15 , Beau found 3 2 = 9 and 4 2 = 16. He said that since 15 is between 9 and 16, 15 must be between 3 and 4. He thinks a good estimate for 15 is 3+4 2 =3.5. Is Beau’s estimate high, low, or correct? Explain.
Exit Ticket 1. Write as a decimal: 2 5 8 and 1 7 12 2. Write as a fraction: 0.34 and 1. 24 Solve 𝑥 2 = 9 16 for x. Solve 𝑥 3 =216 for x. Estimate the value of 13 to one decimal place without using a calculator.