Week 8­ Series.notebook
September 27, 2013
CAN WE ADD AN ARITHMETIC SEQUENCE?
Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question
Week 8, Lesson 1
1. Warm­up
2. ICA­ Scavanger Hunt
3. Notes­ Arithmetic
Series
51
Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up
Warm­up: Find the explicit formula for the following arithmetic sequence, then find the 12th term
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ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
Week 8­ Series.notebook
September 27, 2013
ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
You have 15 minutes to complete the scavenger hunt. You will now be participating in a scavenger hunt. You will start off at a specific question. The answer to that question will be at the top of the paper for another problem. That is the next problem you will do. Continue until you finish at the problem you originally started with. EXAMPLE
1
0
2 + 2
2
9
8 ­ 3
3
4
3 + 6
4
5
3 ­ 3
Does anyone have any more questions or points of confusion they would like to go over?
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Week 8­ Series.notebook
September 27, 2013
Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes
Arithmetic Series
Arithmetic Series­ An arithmetic series is the sum of the terms from an arithmetic sequence
i.e.­ 7 + 14 +21 +28
i.e.­ 42 + 39 + 36 + ... + 18
In order to properly use the formula below, you will usually need to find the last term that you are looking for. ­You will need to find the nth term
The Sum of an Arithmetic Series
Summary:
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Week 8­ Series.notebook
September 27, 2013
Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes
Arithmetic Series
Example 1­ Find the sum of the following arithmetic series
14 + 17 + ... + 26
Example 2­ Find the sum of the following arithmetic series
62 + 51 + 40 + ... + 7
Example 3­ Find the sum of the first 12 terms of an arithmetic series where a1 = 16 and d = ­3
Summary:
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Week 8­ Series.notebook
September 27, 2013
QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ Quiz Like Question
Find the sum of the first 100 natural numbers
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Week 8­ Series.notebook
September 27, 2013
HOW IS THE FORMULA DIFFERENT FOR GEOMETRIC SERIES?
Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question
Week 8, Lesson 2
1. Warm­up
2. Notes­ Geometric
Series
3. ICA­ QUIZ like Question
53
Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up
Warm­up: Find the sum of the first 8 terms of an arithmetic series when a1 = 32 and a2 = 26
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Week 8­ Series.notebook
September 27, 2013
Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes
Geometric Series
Geometric Series­ A geometric series is the sum of the terms of a geometric sequence
i.e.­
4 + 8 + 16 + 32
i.e.­
128 + 64 + 32 + ... + 1
Just like an arithmetic series, many times you will need to find the nth term.
For GEOMETRIC series, you will also need to find r
The Sum of a Geometric Series
Summary:
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Week 8­ Series.notebook
September 27, 2013
Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes
Geometric Series
Example 1­ Find the sum of the first 7 terms of a geometric series when a1 = 243 and r = 1/3
Example 2­ Find the sum of the first 10 terms of the following geometric series 400 , 300 , 225 , ... Example 3­ Look at the series below. For what value of n will the sum of the series first exceed 384.4?
15 + 22.5 + 33.75 + ...
Summary:
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Week 8­ Series.notebook
September 27, 2013
QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ QUIZ Test Like Question
Consider the geometric sequence 512 , 256 , a , 64 , b , ...
Find a and b
Find the explicit formula for the sequence
Find a10
Find S10 For what value of n will the series first exceed 1023.75?
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Week 8­ Series.notebook
September 27, 2013
HOW CAN WE USE SEQUENCES AND SERIES IN WORD PROBLEMS?
Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question
Week 8, Lesson 3
1. Warm­up
2. Notes­ Applying Series
3. ICA­ Allowance
Scam
55
Warm­up: Find the sum of the first 6 terms of the geometric sequence where a1 = 56 and r = 2/3
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Week 8­ Series.notebook
September 27, 2013
Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes
Series as a Word Problem
Clara wants to buy some land. She can choose between two different payment options.Both options require her to pay for the land in 20 monthly installments.
Option 1: The first installment is $2500. Each installment is $200 more than the one before.
(a) Write the explicit formula for Option 1
(b) Write down the values of the second and third installments
(c) Calculate the value of the final installment;
(d) show that the total amount that Clara would pay for the land is $88,000.
Summary:
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Week 8­ Series.notebook
September 27, 2013
Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes
Series as a Word Problem
Clara wants to buy some land. She can choose between two different payment options.Both options require her to pay for the land in 20 monthly installments.
Option 2: The first installment is $2000. Each installment is 8% more than the one before.
If Clara chooses option 2,
(a) find the value of the second installment;
(b) show that the value of the fifth installment is $2721.
(d) Clara knows that the total amount she would pay for the land is not the same for both options. She wants to spend the least amount of money. Find how much she will save by choosing the cheaper option.
Summary:
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ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
Week 8­ Series.notebook
September 27, 2013
ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
Julian decides that he wants to make some money off of his parents. He makes the deal with his parents that he will do any chores that they want, as long as they pay him $1 on the first day of the month, and each day give him one more dollar than the day before.
How much would his parents give him on the last (31st) day of the month?
How much would his parents have paid him in total over the entire month?
His parents laughed at him because they know that is too expensive, so he comes up with a different plan. He said instead, they need to give him $0.01 on the first day, and all they need to do is double it. His parents agree. How much do they owe him on the 7th day? How much in total have they paid him on the 7th day?
How much do they owe him on the last (31st) day of the month? How much in total have they paid him?
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ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
Week 8­ Series.notebook
September 27, 2013
ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
Consider the geometric series where Find S4
Mentally estimate the following:
Calculate the following
using the formula
S6
S6
S10
S10
S50
S50
S1000
S1000
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Week 8­ Series.notebook
September 27, 2013
Lesson Plan: Week 1, Lesson 4
Content Objectives:
Determine the theoretical probability of events, estimate probabilities using experiments, and compare the results.
Use concepts and formulas of area to calculate geometric probabilities.
Determine the number of possible outcomes of an event.
Apply appropriate means of computing the number of possible arrangements of items using permutations where order matters, and combinations where order does not matter.
Apply the addition and multiplication principles of counting and represent these principles algebraically using factorial notation.
­ ­ ­ ­ ­ ­ ­ ­ ­ ­ Solve applied problems using the attributes of similar triangles.
Identify similar polygons.
Show that two triangles are similar using AA, SSS, and SAS Theorems.
Use the triangle proportionality and mid­segment theorem.
Solve problems using ratio and proportions.
­ ­ ­ ­ ­ ­ ­ ­ ­ ­ Know that after any of transformation (rotation, reflection, and translation), the shape still has the same size, area, angles and side lengths.
Know that if one shape can become another using rotation, reflection, and translation, then the two shapes are called congruent.
Understand that during transformation called a dilation, (enlargement or reduction), the shape becomes bigger
or smaller.
Dilation does not result in congruent shapes, the shapes will be similar. Same shape, yet different sizes are
similar shapes. Discover the line of reflection, the center of rotation, and the center of dilation.
­ ­ ­ ­ ­ ­ ­ ­ ­ ­ Understand that the term “solving the triangle” means that if we start with a right triangle and know any two sides, we can find or solve for the unknown side. Investigate the fundamental concepts behind trigonometry: three basic trig functions and how to determine
which trig function to use.
Know that recognizing special right triangles (30, 60 and 90) in geometry can help you to problem solve.
Use SOHCAHTOA to memorize the three main trigonometric functions.
­ ­ ­ ­ ­ ­ ­ ­ ­ ­ Identify and name parts of a circle
Find the circumference and area of circles
Use properties of tangents
Use properties of arcs and central angles; identify and name
Solve problems by applying the relationship between radii, diameters, chords, and tangents.
Determine measures of central and inscribed angles and their intercepted arcs.
­ ­ ­ ­ ­ ­ ­ ­ ­ ­ Identify how many vertices, sides, edges a polygon has.
Compare and contrast polygons two­dimensional figures.
Identify and classify polygons using manipulatives and create two­dimensional figures.
Identify and describe polygons (concave, convex, regular, pentagon, hexagon, n­gonal, circles and sector areas).
Identify and describe properties of a circle, kite, trapezoid, parallelogram, rectangle, square, and rhombus.
Use congruent relationships of two­dimensional figures to determine unknown values, such as angles, side lengths, perimeter or circumference and areas.
Apply the interior and exterior angle sum of convex polygons to solve problems.
­ ­ ­ ­ ­ ­ ­ ­ ­ ­ Identify how many vertices, sides, edges, and/or faces of a three­dimensional figure.
Compare and contrast three­dimensional figures.
Use manipulatives to create three­dimensional figures
Apply surface area and volume formulas for prisms, pyramids, cylinders, cones and spheres.
Draw three­dimensional figures with appropriate labels and make a three­dimensional model from a net.
Draw, describe, and analyze solid geometry figures.
Language Objectives:
compare
discuss
explain
label
paraphrase
restate
classify
describe
experiment
justify
name
rephrase
compose
edit
identify
list
predict
rewrite
contrast
elaborate
interview
listen
present
state
Colored Pens
Index Cards
Scissors Tape
Compass
Navigator
Staplers
Tape Measure
define demonstrate
evaluate
investigate
match
present your point of view
summarize
Materials Needed:
Calculators
Graph Paper
Protractor Straws
Worksheet
Colored Pencils Hi­Lighters
Ruler
String
Yard Stick
Flash Light
Pattern Blocks
Staple Remover
Tangrams
Activities/Directions:
Closure:
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Week 8­ Series.notebook
September 27, 2013
CAN YOU DEMONSTRATE YOUR KNOWLEDGE OF SEQUENCES AND SERIES?
Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question
Week 8, Lesson 4
1. Warm­up
2. Sequences and Series
Quiz
3. Midterm Review
57
Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up
Warm­up: Find a10 and S10 of the arithmetic sequence 17 , 22 , 27 , ...
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