Download understand real numbers - White Plains Public Schools

Topic: Test Review
Objectives: SWBAT review topics for their next test:
Recognize and state the value of the square root of a perfect square (up to 225)
Determine the square root of non-perfect squares using a calculator.
Classify irrational numbers as non-repeating/non-terminating decimals.
Recognize the difference between rational and irrational numbers.
Identify the two consecutive whole numbers between which the square root of a non-perfect square
whole number less than 225 lies (with and without the use of a number line)
Distinguish between the various subsets of real numbers (counting/natural numbers, whole numbers,
integers, rational numbers, and irrational numbers).
Place rational and irrational numbers (approximations) on a number line and justify the placement of the
numbers
Simplifying Radicals & Operations with radicals
Standards:
(Stated in objective)
Do Now:
See slide one
Lesson: Test Review
Students will be review for their 1st Test of the quarter. The class will review the topics
presented in the power point. They will they work on the teacher made “HOW TO” to
determine which areas they need the most help. Tomorrow students will be working at stations
on the area(s) they feel they need the most review.
Classwork attached
HW
Take Home quiz due tomorrow. Finish all “how to” questions. Circle the topics you would like to
have additional review.
“HOW TO” UNDERSTAND REAL NUMBERS: Whole Numbers
Integers
0, 1, 2, 3, …
... -3, -2, -1, 0, 1, 2, 3, …
Rational Numbers
A rational number is any number written as a simple
fraction, or any decimal that repeats or terminates.
Examples:
0.25, 0.464646…, 25 ,
4
3
Irrational Numbers
An irrational number is a decimal that does not
repeat or terminate.
Examples:
2 , 10 , , 0.121121112…
TEST REVIEW QUESTIONS
1. Which is a rational number?
Explain why you think your choice is correct?
a)
b)
c)
d)
19
49
29
π
_______ is rational b/c ___________________
_____________________________________.
2. Which number is irrational?
a.
b.
c.
d.
.75
.5
.323232…
.123456…
Explain why you think your choice is correct?
_______ is irrational b/c __________________
_____________________________________.
3. Which of the following is an
irrational number?
Explain why you think your choice is correct?
a)
16
b)
17
c)
25
d)
36
_______ is irrational b/c __________________
_____________________________________.
4. Which number is rational?
Explain why you think your choice is correct?
a.
b.
c.
d.
_______ is rational b/c ___________________
75
87
100
_____________________________________.
70
5. Which number is rational?
Explain why you think your choice is correct?
a. 
_______ is rational b/c ___________________
b. 0.25
c.
7
d.
10
_____________________________________.
6. Which number is an irrational number?
2
a.
b. 
c.
Explain why you think your choice is correct?
4
3
_______ is irrational b/c __________________
_____________________________________.
-1.8
d.  9
7. The set below contains only which type of numbers?
{–2, 6, 3½, 12, 1.25, 40,
a) mixed numbers
b) rational numbers
c) whole numbers
64 , 200}
Explain why you think your choice is correct?
_________ numbers are _________________
_____________________________________.
d) counting numbers
SQUARES AND SQUARE ROOTS
Perfect Squares – the product of a number times itself. Numbers that form perfect
square have equal side lengths. Perfect Squares represent the number of squares needed
to fill the inside of a square.
The Square Root of a Number A number which is multiplied by itself to give the
square number. It also represents the length of the side of a square.
1. List the first fifteen perfect squares.
___, ___, ___, ___, ___, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____
SOLVE – NO CALCULATOR !!!!!
2) How many whole numbers are between
20 and 50 ?
_________
Show your work or explain why:
3) Find the value of each:
25
81
________
 12    5  5 
2
4) Find the values of n:
a)
n = 8
b) n2 = 196
n = ________
n = _____ or ______
_______
5) Which number below is an integer?
a)
36
25
c)
25
36
b)
d)
c) n2 – 16 = 9
Show your work or explain why:
6) Between which two consecutive whole
numbers is 54 ?
7) Which product is irrational?
3 3 3
2  2 or
36
4
4
36
_______ and _______
Show your work or explain why:
_________
Explain why:
8) Write the value for letters a and b shown on the number line:
6
9
b
a
a _________
9) Graph
15
b _________
(Write the value for b as a square root.)
on a number line and estimate to the nearest whole number.
You may use a calculator on the questions below!
10) If the perimeter of a square is 60 units long, what is the area of the
square?
11) If the perimeter of a square is 4 11 units long, what is the area of the
square?
12) Áine’s room is square-shaped. The room has an area of 112 ft2.
How long is each wall? (Round your answers to the nearest tenth of a foot.)
Simplifying Radicals
To simplify means to find another expression with the same value. It does not mean to
find a decimal approximation. Think of simplifying as “reducing” the radical.
METHOD 1: Finding the largest perfect square factor.
Find the largest perfect square that will divide evenly into the number under the radical
sign.
1
4
9
16
25
36
49
64
81
Step 1 Write the number under the radical as the product of the largest perfect square
factor and another factor from dividing.
72  36  2
Step 2
Give each number in the product its own radical sign.
Step 3
Simplify the “perfect” square under the radical.
72  36  2  36 2
72  6 2
METHOD 2: Using Prime Factorization
72
8
2 3
4
2
9
3
2
72  2 2 2 3 3
72  2  2  3
72  2  3 2
72  6 2
You Try! Simplify each radical.
24
72
45
75
800
54
100
Operations with Radicals
Adding/Subtracting Radicals
You can simplify radicals by combining like terms. If the terms are not like,
simplify the radicals so that the radical parts are the same!
Example 1:
3 12
 2 3
3 4 3  2 3
4 is the largest
perfect square
factor of 12.
32 3
6 3
You try it!
2 98

162
 2 3
 2 3
4 3
Multiplying Radicals
Multiply coefficient by coefficient and radicand by radicand. Simplify where
possible.
Example 2:
15  5
You try it!
105  3
3 5 5
Switch, so
that the
coefficient
is in front of
the radical!
3 5
5 3
Dividing Radicals
Divide coefficient by coefficient and radicand by radicand. Simplify where
possible.
Example 3:
Divide 9 into 27 and
3 into 75. Leave your
answer as a fraction
until simplified.
27 75
9 3
3 25
1
35
1
15
You try it
16 162
2 18
You Try! Simplify each radical.
9 40
30 6
5 8 7 3
4 28
5 35
-4 8  2 26
45
5
24
2
72
3
96
2