Square Roots Learning Targets: Interpret and simplify the square of a number. Determine the square root of a perfect square. Dominique Wilkins Middle School is holding its annual school carnival. Each year, classes and clubs build game booths in the school gym. This year, the student council has asked Jonelle’s math class for help in deciding what size the booths should be and how they should be arranged on the gym floor. The class will begin this work by reviewing some ideas about this area. Example A: Find the area of this rectangle. The rectangle has been divided into squares. Assume that the length of each side of the small square is 1cm. Step 1: Find the length and width of the rectangle. This rectangle is 5cm long and 3 cm wide. Step 2: To find the area of the rectangle multiply the length times the width. A=l*w A= 5*3 15cm 2 Solution: The area of the rectangle is 15cm 2 . Note that the units are cm 2 . Try These A: Before deciding on how to arrange the booths, the student council needs to know the area of the gym floor. Several students went to the gym to measure the floor. They found that the length of the floor is 84 feet and the width of the floor is 50 feet. a. Find the area of the gym floor. Explain how you found the area and include units in your answer. b. Explain the area of the gym floor mean?
Transcript
Square Roots Learning Targets:
Interpret and simplify the square of a number. Determine the square root of a perfect square.
Dominique Wilkins Middle School is holding its annual school carnival. Each year, classes and clubs build game booths in the school gym. This year, the student council has asked Jonelle’s math class for help in deciding what size the booths should be and how they should be arranged on the gym floor. The class will begin this work by reviewing some ideas about this area.
Example A: Find the area of this rectangle. The rectangle has been divided into squares. Assume that the length of each side of the small square is 1cm. Step 1: Find the length and width of the rectangle. This rectangle is 5cm long and 3 cm wide. Step 2: To find the area of the rectangle multiply the length times the width. A=l*w A= 5*3 15cm2 Solution: The area of the rectangle is 15cm2. Note that the units are cm2.
Try These A:
Before deciding on how to arrange the booths, the student council needs to know the area of the gym floor. Several students went to the gym to measure the floor. They found that the length of the floor is 84 feet and the width of the floor is 50 feet.
a. Find the area of the gym floor. Explain how you found the area and include units in your answer.
b. Explain the area of the gym floor mean?
Now the class is going to focus on the area of the squares, because this is the shape of he base of many of the game booths.
Example B: Find the area of the square below. Do you need to know the length and width to be able to determine its area? 4 cm Step 1: The length of one side is give and since this is a square all sides are the same. Step 2: To find the area multiply the length by itself. 4*4= 16cm2 Solution: The area of the square is 16cm2.
Try These B: This drawling shows the floor space of one of the carnival
s booths. It is a square with the length of one side labeled with the letter s. The s can be given any number value since the booths are (side going to be different sizes. Length) Complete the table by finding the areas of some of the different sized booths. The lengths of a side are in feet represented by s, as the drawing above shows. Include your units for the area in the last column.
Length of side (in feet)
Picture Expanded Form (calculation)
Area of the Square UNITS!!!
S= 3
S=6
S=8
S=11
S=4.2
S= 10
A (Area)
For each calculation in Try These B, you found the product of a number times itself. The product of a number times itself can be written as a power with a base and an exponent.
Example C: Write 5*5 as a power with a base and an exponent. Step 1: Identify the base. The base is 5 Step 2: Identify the exponent. The exponent is the number of times the base is being multiplied by itself. 2 Step 3: Write steps 1 and 2 in Exponential Form. 52 Solution: 5*5 written in exponential form is 52
Guided Practice: Write the following in exponential form.
1) 3*3*3*3*3
2) 4.1*4.1*4.1
3) ½ * ½ * ½
4) x*x*x*x*x*x*x
5) y*y*y*y*y
Individual Practice: Write the following in exponential form. Identify the base and exponent.
1*1*1*1 Exponential Form: Base: Exponent:
7.1*7.1*7.1
Exponential Form: Base: Exponent:
¾* ¾* ¾
Exponential Form: Base: Exponent:
m*m*m*m*m*m Exponential Form: Base: Exponent:
t*t*t*t Exponential Form: Base: Exponent:
Math Terms: A POWER is a
number multiplied by itself. A number
or expression written with an exponent is in
exponential form.
Be
Example D:
s The area of the floor of a square booth is 36 ft2. What is the length of the side s of this booth? Step 1: The area is the side length squared. To find s, find the square root of 36ft2.
The symbol for the square root is .
Step 2: To solve , think about which number times itself equals 36. Also, you can put it in your calculator. Solution: The side length is 6 ft (The units on the side length is just units not square units!)
Guided Practice: Find each square root.
Area (square units)
Visual Representation
Side Length (units)
Area (square units)
Visual Representation
Side Length (units)
49 ft2
12 ft
9 ft
121 ft2
Math Tip: Square root of a number is when a number multiplied by itself it gives you that number.
Perfect Square is when you take the square root of a number and end up with a number without a decimal.
Squaring a number is when you take that number and multiply it by itself.
36 ft2
Individual Practice: Find the area or side length of each booth. Area
Interpret and simplify the cub of a number. Determine the cube root of a perfect cube.
The student council with all the work the class has done on the carnival so far. The class has found the areas of the floors and the side lengths of the booths. One concept remains for the class to review before completing the needed work. The booths do not just take up floor space; they also have height. The diagram represents a cubic foot. Its dimensions are 1ft * 1ft * 1ft
Example A: Find the volume of the cube if it has an edge length
of 2 units each. Find the volume in exponential form and in cubic units. el 2 units “edge
length” Step 1: The volume of a cube is found by multiplying Length times width times height.
V=l*w*h V=2*2*2 Step 2: To find the volume in exponential form, write the base, which is the edge length and volume is cubed so the exponent is a 3 for three dimensions.
V=2*2*2 V=23
Step 3: Use your calculator to figure out the math! To enter 23 in your calculator push 2 ^ 3 = it will tell you the answer is 8 because it did: 2*2*2 . Solution: The volume of a cube with an edge length of 2 is 8 u3.
Guided Practice: Complete this table to show the volume of each booth in expanded, exponential and standard form. Edge Length Visual
Representation Expanded
Form Exponential
Form Standard
From
2
V 2
2*2*2 23 8ft3
4
5
6
8
V Volume
V
The exponent used in each exponential expression of volume in the guided practice is the same. This exponent can be used for the volume of a cube. Using
this exponent is known as CUBING A NUMBER .
If you know the volume of a cube you can determine the edge length of the cube. The operation used to find the edge length is called the cube root. The
symbol is
. Individual Practice: Aaron says to find volume you must use l*w*h. Amy says you can use l3. Who is correct and why? _______________ ________________________________________________________________________________ Volume (cubic units)
Simplify expressions with powers and roots. Follow the order of operations to simplify
expressions. When terms with exponents and roots appear in expressions, use the correct order of operations to simplify.
Always go from left to write. You may skip an operation if you do not see it in the
expression. You can do division before multiplication if it comes
first (left to right). You can do subtraction before addition if it comes
first (left to right).
Example B:
Use order of operations to evaluate the expression: 250-(3*5)2+7 Step 1: Use order of operations and start (look for) parentheses and simplify.
250-152+7 Step 2: Use order of operations to look for exponents/roots then simplify.
250-225+7 Step 3: Use order of operations and look for multiplication and/or division from left to right.
250-225+7 --- None so skip these operations Step 4: Use order of operations and look for addition and/or subtraction from left to right. Whatever comes first is the operation you do!
250-225+7 subtraction came first so subtract 250-225= 25 25+7 now finish with addition. 25+7 = 32
Solution: 250-(3*5)2+7 = 32
Order of Operations: 1) Parentheses 2) Exponents/Roots (left
to right) 3) Multiplication
Or Division (left to right)
4) Addition or Subtraction
(left to right) Please Excuse My Dear Aunt Sammy. PEMDAS
Guided Practice: Use order of operations to evaluate each expression. Steps must be shown for each step.
5*5 – 6*2 24/6 * 2 3 + 5(7-5) 18 - 5*3
4(3+2)2 - 7 2[9(6 -4)] + 4
6 – 2 + 19+36 32
[10-(4-1)]*9 2[50-8(2+3)] 6(5+3) (-2)4 52*24
Jane and Bill were given the expression 3+24 2*3 . They both solve it differently. Who is correct, explain. ____________________________ Jane: 3+24 2*3 Bill: 3+24 2*3 3+24 6 3+12*3 _____________________________ 3+4 3+ 36
7 39 _____________________________
Fractions, Decimals, and Percents Learning Target:
Convert between fractions, decimals and percents. Model fractions graphically.
Fraction :
Decimal: 0.3
Percent: 30%
Guided Practice: Convert to the following forms. Fraction Decimal Percent Visual
Representation
.59
.4
78%
3.23
4
.08%
75%
Individual Practice: Determine which is greater:
30% or 1/3 .9 or 88% 5/6 or 80%
.7 or 7% 80% or 4/5 1/3 or 33%
In the United States in 1990, about one person in twenty was 75
years old or older. Write this fraction as a percent. ________________________________
Rational/ Irrational/Terminating Learning Target:
Define and recognize rational, irrational, and terminating numbers.
Math Terms Terminating Decimals: a decimal that ends. (stops) Repeating Decimals: a decimal that never ends and repeats in a pattern. Rational Numbers: a number that can be represented as a fraction or a terminating/repeating decimal. Irrational Numbers: numbers that can’t be written as a fraction. Their decimal is not terminating and does not repeat.
State if the following are terminating or repeating decimals.
State if the following are rational or irrational numbers.
Is the following a repeating decimal? Explain why.
a) .545454212121…. _________________________________________________________
b) .23232323….. _________________________________________________________ c) .12344444444…. _________________________________________________________
d) Which number is greater than
? A)
b) 28% c)
Estimating Non Perfect Squares *** Get other hand out ***
Individual Practice: Estimate the following non-perfect square roots to the nearest hundredth. NO CALCULATORS!
Embedded Assessment 1 Representing Rational and Irrational Numbers
Weather or Not? Natural disaster can help anywhere in the world. Examples of natural disasters include tornados, earthquakes, hurricanes, and tsunamis. Two of the most well-known natural disasters are Hurricane Katrina (2006), which hit New Orleans, Louisiana, and Japan’s tsunami (2011).
1. After Hurricane Katrina,
of the city of New Orleans was flooded.
Show the number of the city that was flooded in the following ways: Decimal: Visual Representation: Percent:
2. Only 58% of the people in the cost areas of Japan took the warning system seriously
that a tsunami was coming and evacuated the area. Represent this number in the following ways: Decimal: Fraction:
The area of land that is affected by a natural disaster can vary greatly. Thinking of this destruction as a perfect square can give you a good visualization of how much area was affected.
3. The total square miles affected by the natural disasters are given below. Find the side length of the area affected if it was a square. Hurricane Katrina: 90,000 square miles Japan Tsunami: 216 square miles
4. Given that a storm has a destruction area in the shape of a square, give the total area affected if the side length of the square was:
312.2 Miles
30
kilometers
When natural disasters occur, organizations such as the Red Cross help by sending crates of supplies to those who are affected. Those crates contain first aid, food, drinks, and other supplies.
5. Explain how when given the edge length of the cubical crate you can find the volume. Provide an example.
6. Given the following volumes of crates determine the edge length.
8ft3 27 ft3
** Completing this ahead of time before your assessment date will earn you extra credit!**
Properties of Exponents Learning Targets:
Understand and apply properties of integer exponents. Simplify multiplication expression with integer
exponents. Simplify division expression with integer exponents.
As I was going to St. Ives, I met a man with seven wives.
Every wife has seven sacks, And every sack had seven cats.
Every cat had seven kittens. Kittens, cats, sacks, wives,
How many were going to St. Ives? The Guinness Book of World Records claims this is the oldest mathematical riddle in history, this can be used to explain exponent work. Use this table to determine the number of kittens in the riddle—in expanded form, exponential form and standard form.
Expanded
Form Exponential
Form Standard
Form Base
Wives
Sacks
Cats
Kittens
Do you notice any patterns?
Math Terms: Expanded Form: is expressing the number in terms of multiplication. (writing it out the long way!) Exponential Form: expressing the number with a base and an exponent. Standard Form: is the answer
Multiplying Exponents
Example A:
Simplify 7*7*7*7*7*7*7 in various ways. Then compare your results. Step 1: Use the associate property of multiplication to rewrite the expanded form:
(7*7*7)(7*7*7*7) or (7*7)(7*7*7*7*7) Step 2: Rewrite each expression in exponential form:
(7*7*7)(7*7*7*7)= 73* 74 (7*7)(7*7*7*7*7)= 72 * 75 Step 3: Simplify each power. Notice that the exponents are being added and the base stays the same.
73* 74= 7 3+4=77 72 * 75= 72+5=77
Solution: In expanded form 7*7*7*7*7*7*7 is the same as 73* 74 =77 in exponential form, which is the same as 823,543 in standard form.
Guided Practice: Simplify the expressions.
a) 39 * 33 b) 92 * 912 c) 63
* 6-1 d) 5-4 * 57
e) a7 * a4 f) t2 * t5 g) h2* 62 * h6 h) 83 * y3
Dividing Exponents Example B:
Simplify:
Step 1: Write
in expanded form.
7*7*7*7 7*7*7 Step 2: Simplify by crossing out the same base in the numerator and denominator.
7*7*7* 7 = 7 (You can cross 3 sevens out on the top because you 7*7*7 have 3 sevens on the bottom) Step 3: You can also simply by taking the TOP exponent minus the BOTTOM exponent:
= 74-3=71 or 7
** Step 2 and 3 are the exact same thing just in a different way**
Solution:
= 71 or 7
Guided Practice: Simplify the expressions.
a)
B )
C)
d)
e)
f)
Individual Practice: Complete the following table by simplifying the following expression in expanded, exponential, and standard form.
Expression Expanded Exponential Standard
* Sam and George are examining the expression
. Sam says the answer is George
knows the answer is Who is correct and why? Explain to the other people why they are incorrect. ______________________________________________________________________________________ ____________________________________________________________________________________________________ ____________________________________________________________________________________________________
Negative Exponents Learning Targets:
Understand and apply properties of integer exponents. Simplify expressions with negative exponents.
Example A: *********All powers must have positive exponents. *********
Write as an equivalent expression without a negative exponent.
Step 1: Write as a fraction.
(all whole numbers are always over 1)
Step 2: Move the exponent to the opposite side of the fraction bar. This will CHANGE the exponent from negative to positive.
=
Solution: =
What is a reciprocal? _____________________________________________________________________________ Guided Practice: Rewrite these expressions without a negative exponent.
a) b) c)
d)
Example B:
Consider the expression
. Simplify the expression by dividing and by writing the
numerator and denominator in expanded form. Compare the results. Step 1: Divide by subtracting the TOP exponent minus the BOTTOM exponent.
= 43-8 = 4-5
Step 2: Write the numerator and denominator in expanded form and then simplify.
=
=
Step 3: Compare the two results:
4-5 and
Solution: The solutions in Step 3 are the exact same answer!! The fraction is the correct way to write the answer as you saw in EXAMPLE A
Guided Practice: Simplify these expressions. Your final answer should NOT have negative exponents.
a)
b)
c)
Individual Practice: Simplify the following expressions
a) b)
c)
d)
e) f)
g)
h) i)
Power of Zero Learning Target:
Understand and apply properties of integer exponents. Simplify expressions with zero as the exponent.
Although the riddle As I was going to St. Ives has only one narrator, the number 1 can also be written as a base of 7. To see how this works you can examine several ways to express the number 1.
Example A:
Simplify
using exponents and explain the results.
Step 1: Rewrite the fraction by expressing the numerator and denominator in exponential form.
=
Step 2: Simplify the expression by dividing. (Subtract the top minus the bottom)
=
=
Solution:
is equal to 1 and also equal to
ANYTHING to the ZERO power is 1
Guided Practice: Simplify these expressions.
a) b)
c)
Power to a Power Learning Target:
Simplify expressions with exponents raised to a power. Recall from the riddle that the man had 7 wives, each wife had 7 sacks, each sack had 7 cats, and each cat had 7 kittens. Suppose that each kitten had 7 stripes. Now assume each stripe on each kitten contains seven spots. The situation is becoming more complicated, and the need for using exponents has grown…. Exponentially.
The number of spots can be written as a power raised to another power.
When an exponential expression is raised to a power, multiply the exponents to simplify.
(This always has parentheses ( ) in it!)
Example B:
Show that =
Step 1: = The exponent 2 means you have two groups of Step 2: Solution: = Guided Practice: Simplify these expressions. Write your answer in exponential form.
a) b) c) d)
Individual Practice: Simplify these expressions. Write your answer in exponential form.
a) b) c)
d) e)
f)
g) h) i)
Exponent Rules: Rule Example
Mu
ltip
lyin
g E
xpo
nen
ts
D
ivid
ing
Exp
on
ents
Neg
ativ
e E
xpo
nen
ts
Po
wer
to
a
Po
wer
Po
wer
of
Zer
o
Scientific Notation Learning Target:
Express numbers in scientific notation. Convert numbers in scientific notation to standard form. Use scientific notation to write estimates of quantities.
The story of Gulliver’s Travels describes the adventures of Lemuel Gulliver, a ship’s doctor, who becomes stranded in many strange places. In Lilliput, Gulliver finds that he is a giant compared to the people and the world around him. During another voyage Gulliver is stranded in another land, Brobdingnag, where he is as small to the people as the Lilliputians were to him. The story never says how tall Gulliver is, but it does tell how the heights of the Lilliputian people and the people from Brobdingnag compare to Gulliver’s height. The many descriptions of size in this tale provide ways to explore the magnitude, or size, of numbers. Powers of 10 will be used to express these very large and very small numbers. For this activity assume that Gulliver is 5 feet tall.
Example A: A person from Brobdingnag is 10 times as tall as Gulliver. Determine the height of the person. Step 1: Gulliver is 5 feet tall, and the person from Brobdingnag is 10 times as tall, so multiply to find the height of the person. 5* 10= 50 feet tall Step 2: Write an expression using Gulliver’s height and power of 10 to represent the height of 50 feet. (Power of 10 means the base is 10 and has an exponent) 5* 101=50 Solution: A person from Brobdingnag is 50 feet tall or in terms of Gulliver’s height the person is 5* 101 feet tall.
A tree in Brobdingnag is 100 times the height of Gulliver. How tall is the tree? Write the height of the tree as an expression in terms of Gulliver’s height.
#
between 1-9.999
Notice the expression you have written shows the product of the factor and a power of 10 with an exponent that is a positive integer. This is written
in the form of scientific notation, where the factor can equal 1 but must be less than 10 ( 1 <#<10 )
Example B: Convert the following number into scientific notation: 234000 Step 1: Note where the decimal is in a number. If you do not see one it is always behind the last digit on the left like a period in a sentence.
234000. Step 2: Move the decimal to the right or the left so that you are making a “new” number that is between 1 and 9.999.
234000. = 2.34000 Step 3: Now you fill in the scientific notation form: 2.34 (you can leave off the extra zeros on the end) 2.34 * 10 The exponent comes from how many times we moved the decimal. 2.34 * 105
Solution: 234000 = 2.34 * 105 It is a positive 5 because we want the number to be large when we are finished, even though we moved to the left.
*
Example C: Convert the following number into scientific notation: .0045300 Step 1: Note where the decimal is in a number. If you do not see one it is always behind the last digit on the left like a period in a sentence. .0045300 Step 2: Move the decimal to the right or the left so that you are making a “new” number that is between 1 and 9.999.
.0045300 = 004.5300 Step 3: Now you fill in the scientific notation form: 4.53 (you can leave off the extra zeros in the front and on the end) 4.53 * 10 The exponent comes from how many times we moved the decimal.
4.53 * 10-3
Solution: .0045300 = 4.53 * 10-3 It is a negative 3 because we want the number to be smaller when we are, even though we moved to the right.
Guided Practice: Write the following numbers in scientific notation. Show your work.
a) 567,000
b) 42 c) 120,340,000,000
d) .0000000067
d) .876 e) .9999
Individual Practice: Write the following in scientific notation. Show all work.
a) .00000000000026 b) 7 c) 10.4
d) 893 e) .072 f) .0000603
g) 304,605,000,000 h) 4,000,000,000,000 i) 60,000
j) 99 k) .001 l) 56.702
Example D: Rewrite the diameter of the sun, 1.39 * 109 in standard form. Step 1: Identify the exponent. Is it positive or negative. Positive the number will get larger. Negative the number will get smaller. 1.39 * 109 The exponent is a POSITIVE 9
Step 2: Move the decimal the number of times of the exponent in the direction based off the exponent then fill in with zeros. Positive exponent move to the right (bigger) Negative exponent move to the left (small) * think number line* ** Move it 9 times to the right**
1.390000000= 1,390,000,000
Solution: 1.39 * 109 = 1,390,000,000
Example E:
The width of a red blood cell is 3 * 10-4 . What is the width of a red blood cell in standard form? Step 1: Identify the exponent. Is it positive or negative. Positive the number will get larger. Negative the number will get smaller.
3 * 10-4 The exponent is negative 4 Step 2: Move the decimal the number of times of the exponent in the direction based off the exponent then fill in with zeros. Positive exponent move to the right (bigger) Negative exponent move to the left (small) * think number line*
* Move 4 times to the right 0003. = .0003
Solution: 3 * 10-4 = . 0003
Guided Practice: Convert the following from scientific notation to standard form.
a) 5.2 * 10-4 b) 2.5 * 106 c) 4.23 * 10-6
d) 1.2 * 103 e) 9.9 * 107 f) 6.034 * 10-2
Individual Practice: Complete the following table. 1)
Standard Form Scientific Notation Rank least to greatest
2,300,000,000
3.4 * 103
0.00009
1.7* 10-3
6.99 * 10-7
.00086
70
8.92 * 108
6.07* 100
2) Wire 1 has a diameter of 9 * 10-2 inches. Wire 2 has a diameter of 2.4 * 10-3 in. Wire 3 has a diameter of .0023 inches. Order the diameters in order from least to greatest.
3) Is 10.2 * 104 in scientific notation? Explain why or why not. Thinking abstractly: Place the following expressions in the appropriate columns.
Guided Practice: Simplify each expression. Write the answer in scientific notation. Show all steps, and work.
a) (2*105)(3*104) b) (1.6*108)(3*104) c) (4*1012)(6*105)
Try this one: (9*105)(3*104) What happened? __________________________________________________________________________________
How to adjust scientific notation
43.2* 103
15*103 22*105
33* 10-3
52*10-6 82*10-1
Rule when exponent is positive Rule when exponent is negative
Try these:
( 4.2* 108)(3*10-5)
(4*101015)(6*10-7)
Dividing Scientific Notation Learning Target:
Divide numbers expressed in scientific notation.
Example A:
Jupiter is the largest planet with a mass of about 1.9008*1027 kg. and Mercury is
the smallest planet with a mass of about 3.3* 1023 kg. Using the numbers in scientific notation determine how many times larger Jupiter is than Mercury.
Step 1: Regroup the problems.
=
Step 2: Divide
Use the dividing exponents rule and subtract the
top exponent minus the bottom exponent. Step 3: Rewrite correctly in scientific notation. 576 * 104 5.76 * 103
Solution:
What do you notice about division with scientific notation compared to what you did with multiplication with scientific notation?
Guided Practice: Simplify each expression. Write each answer in scientific notation.
a)
b)
Individual Practice: Simplify each expression. Write each answer in scientific notation.
a)
b)
c)
d) (7.5
e)
f)
(
Embedded Assessment 2 Exponents and Scientific Notation Contagious Mathematics While checking her e-mail, Lisa stumbles across a cryptic message from something named 5up3r H4xx0r. In the message, 5up3r H4xx0r claims to have developed a computer virus and is set to release it on the Internet. Once the virus has infected two computers, the potential exists for it to spread exponentially, because each infected computer has a chance to pass it along to the next computer it connects with. The only way for the virus to be stopped, says the hacker is if Lisa correctly answers each of the following questions.
1. The pattern of the spread of the virus will be 1, 2,4,8, …. Identify the next three numbers in this pattern.
2. Express the first seven numbers in the pattern as a power of 2.
3. Describe how the 18th term in the pattern could be determined.
4. Describe how to simplify each of the following expressions. Simplify each
expression and leave your answer in exponential form.
5. Replace the variables with numbers in the expressions so that the expression
would result in the ANSWER 26.
6. Write each number in scientific notation.
20,000,000 2,400
7. Simplify each expression. Leave your answer in scientific notation.
