Download I2 Unit 1A file

Integrated 2 UNIT 1A Schedule: Real Numbers
Day 1:
Journal 1.1 (Prime, Factor, Prime Factorization)
Topic 1.01: Getting Started. SWBAT: Build on understanding of prime numbers,
integers, rational numbers on a FYTE p.5 #1 – 5. HW1 OYO p. 6 #6-10
Day 2:
Journal 1.2 (Squares, prefect squares, what number squared is___)
Topic 1.02: Defining Square Roots, SWBAT: Define square roots and the symbol
Activity: Graphing y = x2 in Ti-83, FYTE p.9 #1 – 3. HW2 OYO p. 10 #1, 4, 6 – 10
Day 3:
Journal 1.3 (Combining like terms)
Topic 1.03: Arithmetic with square roots, SWBAT: learn basic operations ( + - x / )
involving square roots, combining like terms with square roots, Activity: Arithmetic
Investigation using Ti-83, HW 3 p. 14 1, 4, 5, 9, 11
Day 4:
Journal 1.4 (Squares, prefect squares, perfect square factors)
Topic 1.04: Simplified forms of square roots, “square free” SWBAT: change square
roots into conventional forms, FYTD p.16 #1 – 8 . HW4 OYO p. 17 #1, 5, 7, 8a, 9ab, 10
Day 5:
Journal 1.5 (Ratio, Rational Number)
Topic 1.05: Rational and Irrational Numbers, SWBAT: List the differences between a
Rational and an Irrational Number Activity: Venn Diagram of Z, R, Q, FYTD p.21 #1 – 3.
HW5 OYO p. 27 #9, 10, 12,
Day 6:
Journal 1.6 (cubes, fourth powers, x3 = 31)
Topic 1.06: Roots, radicals, and the nth root, SWBAT: Define and explain the meaning
of radicals and “nth roots”, calculate roots, Activity: Explore graphs of y = x2, y = x 3, y
= x4, FYTD p.29 #1 – 7. HW6 OYO p. 30 #1, 5, 7, 8
Day 7:
Review Day
Day 8:
QUIZ UNIT 1A on _____________
Topic 1.01: Getting Started. SWBAT: Build on understanding of prime numbers, integers, rational
numbers on a FYTE p.5 #1 – 7
FYTD p.16 #1 – 8
1) Hint: what should your scale be? How will you handle 101?
2)   1 
1

Try some values. Habit of Mind: Guess and check with a calculator. Can you graph it?
3)
a
a2
Prime Factorization of a2
How many 2’s ?
Is it possible to find a perfect square with
an odd number of 2’s in its prime
factorization?
4)
b
2b2
Prime Factorization of 2b2
How many 2’s ?
Is it possible to find the double of a
perfect square with an even number of
2’s in its prime factorization?
5) Can a perfect square ever be twice as great as another perfect square?
6) 4 = _________ 9 = _________ 16 = __________________ 25 = _________
36 = ______________________ 49 = _________ 64 = ___________________________
81= _____________________________ 100 = ________________________ 121 = __________________
144 = ____________________________ 400 = ______________________ 900 = __________________
What do you notice about the factors?
7) Determine whether each of the prime factorizations represent perfect squares (without a
calculator!!!) Then explain how you know?
a) _________________________________________________________________________
b) _________________________________________________________________________
c) _________________________________________________________________________
Topic 1.02: Defining Square Roots, SWBAT: Define square roots and the symbol
Activity: Graphing
2
y = x in Ti-83, FYTE p.9 #1 – 3.
In the journal, we asked, “what number squared is 25?” This implies that 5 is the square root of 25 .
Notation: ____________________________________________________________________________
Recognize that the square and the square root are inverses (or mathematical opposites) of eachother.
is a real number, s, such that s ≥ 0 and s2 = r .
Definition: If r ≥ 0 , the square root
r
Example: If 36 ≥ 0 , the square root
36 is a real number, 6, such that 6 ≥ 0 and 62 = 36 .
3) Is the conjecture
x 2  x true for all real numbers?
 x  x ?
2
What about
Reflect: So if x is positive, what do the square and square root do to each other?
3 2  6
2) Show that
Show that
10
 5
2
. Use the definition of a square root. “such that s ≥ 0 and s2 = r ”
. Use the definition of a square root. “such that s ≥ 0 and s2 = r ”
1) Use the graph of y = x2 to approximate
x
y = x2
y
5
. Habit of Mind: GUESS!
5 = ________
(x,y)
-3
-2
-1
0
1
2
3
Hit “Y = “ button
Hit “Zoom” #6
Hit “TRACE” Move Cursor Hit “2nd” “ x2 “
Topic 1.03: Arithmetic with square roots, SWBAT: learn basic operations ( + - x / ) involving square
roots, combining like terms with square roots, Activity: Arithmetic Investigation using Ti-83
Use your calculators to verify which equations are true, and which are false.
7  2  14
7 2 5
8 2  6
6
 3
2
Make generalizations (using variables) for each TRUE operation involving square roots.
_________________________________________
Prove:
_____________________________________
x  y  xy using the definition of a square root, “such that s ≥ 0 and s2 = r ”
Combining like terms with square roots.
Consider the following expression: 3x  5x = ________ “3 boys plus 5 boys makes __________”
3 2  5 2 = ________
Check decimal value of each side _________ = _________ Are they equivalent?
1) 5 7  7 
2) 4 3  2 2 
3) 3 3  4 3 
4) 5 7  2  3 7  4 2 
5) 5 3  5 2 
6) 4 6  5 3  2 
Topic 1.04: Simplified forms of square roots, “square free” SWBAT: change square roots into
conventional forms, FYTD p.16 #1 – 8 .
Read the top half of page 16. Any questions? Write each square root as “square free” .
18 
75 
160 
In groups, you try FYTD #1 – 4 .
32 
92 
20 
Continue reading to the bottom. Why does multiplying by 5 over 5 preserve equality?
Remove the square root in the denominator.
2
3
1
6
4 5
2
2 2
2
FYTD p. 17 #5 – 8
1
3
2
7
3 11
11
2
6
1800 
1 5
 
2 5
Topic 1.05: Rational and Irrational Numbers, SWBAT: List the differences between a Rational and an
Irrational Number Activity: Venn Diagram of Z, R, Q, FYTD p.21 #1 – 3
Ratio: _______________________________________________________________________________
A rational number is ____________________________________________________________________
Is 7 a rational number?
Is -9 a rational number?
Is 0 a rational number?
Is 5.8 a rational number?
Is 2
3
a rational number?
5
FYTD p.21
1) 1.341 =
2) 1.3 + 2.8 =
3) 5
2
=
3
Example: Use the Pythagorean Theorem ___________________________ to find the missing side. The
longest side is called the __________________________________
1
2
3
5
1
So how big is
3
2 ? Use a calculator:
2
5
2 = ______________________________________
What is different here?
Is
2 real?
An Irrational Number is ______________________________________________________________
Prove
2 is irrational by using contradiction and the prime factorization of perfect squares.
Make a venn diagram to house the 4 sets of numbers:
Z = Integers, Q = Rational Numbers, R = Real Numbers , and the rest of the numbers are Irrational
Topic 1.06: Roots, radicals, and the nth root, SWBAT: Define and explain the meaning of radicals and
“nth roots”, calculate roots.
What is the
8 ? Guess ___________ Why did you guess where you guessed?
8 with the inverse operation or squaring and the equation x 2  8 .
You are linking the
Now what is the “3rd root or cube root” of 8 ?
3
8 = __________ Which equation helps now? _______
The cube root or 3rd of 8 is 2 because 23 = 2 x 2 x 2 = 8 .
You try
3
4
25 = __________ Guess ________ Use calculators to get close to solution of x _  25
27 
4
16 
6
1
3
125 
3
 27 
Activity: Explore graphs of y = x2, y = x 3, y = x4, y = x5, y = x6 , y = x7, Sketch a graph of each below.
Hit “Y = “ button
y = x2
Hit “Zoom” #6
y=x3
y = x4
Hit “^” to make exponents
y = x5
What patterns do you see in the shapes of the graphs and the exponents?
Is there a real number that satisfies x5 = -7 ? How do you know?
Is there a real number that satisfies x6 = -7 ? How do you know?
Hit “GRAPH”
y = x6
y = x7