Download Rational and Irrational Numbers

Rational and Irrational Numbers
Warm Up
Write each fraction as a decimal.
1.
2
5
2.
7
10
3.
8
9
4.
3
2
8
5.
3
3
4
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. 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
25
𝑥 =
144
125
𝑥 =
512
2
3
Compare the values for
2
2
13 and 1.3 .
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. Then
estimate to one decimal place, if
necessary.
𝑥 2 = 14
𝑥 3 = 1331
𝑥 2 = 144
𝑥 2 = 29
2
3
To find 15, Beau found = 9
2
and 4 = 16. He said that since 15
is between 9 and 16, 15 must be
between 3 and 4. He thinks a good
3+4
estimate for 15 is
= 3.5. Is
2
Beau’s estimate high, low, or
correct? Explain.
Exit Ticket
1. Write as a decimal:
5
2
8
and
7
1
12
2. Write as a fraction: 0.34 and 1. 24
2
3. Solve 𝑥 =
9
for
16
x.
4. Solve 𝑥 3 = 216 for x.
5. Estimate the value of 13 to one decimal
place without using a calculator.