POLYNOMIAL CONCEPTS: Test
WORKSPACE:
(e + 3j)(e - 3j)
2 2
e - 3ej + 3ej - 9j
2 2
e - 9j
***
2 4
n - 49t
2 2
(n + 7t)(n - 7t)
***
2 2
(2p + 4t)(3p - 3t)
2 2 2 4
6p - 6pt + 12pt - 12t
2 2 4
6p + 6pt - 12t
***
NULL
***
2 2
(2x + 3y)(x - 5y)
4 2 2 2
2x - 10xy + 3xy - 15y
4 2 2
2x - 7xy - 15y
***
2
(x - 3)(x + 2x - 6)
3 2 2
x + 2x - 6x - 3x - 6x + 18
3 2
x - x - 12x + 18
***
7 3
-2m + 72m
3 2 2
-2m(m - 6)(m - 6)
(answer b)
***
ANS:
b , a , d , b , c , a , d , NOT KNOWN , c , c
PERMUTATIONS, COMBINATIONS, & FACTORALS: Pre-Test/Test
This lesson will evaluate your knowledge of permutations and combinations and your skill in manipulating factorals.
NOTES:
WORKSPACE:
ANS:
NOT:25 , NOT:4 , NOT:5 , NOT:15 , NOT:1 , NOT:729 , NOT:125 , NOT:10 , NOT:13 , NOT:64
MEASUREMENT PERMUTATIONS
NOTES:
In this activity you make a tree diagram and find permutations of objects in a set.
When you answer a question wrong, you will get a second try. If you answer correctly on the second try, you will get a reduced score for that question.
Each question in this activity is scored!
***
FRIENDS PHOTOGRAPH:
Mike wants to take a picture of his friends Erin, Patti, and Craig. He wants to know how many different ways there are to arrange them in a tow for the picture. ANS:3
A diagram can help Mike find the number of different arrangements for Erin, Patti, and Craig.
There are three different ways for him to choose someone for the left position.
Once the left position is filled, how many ways can Mike choose someone for the middle position? ANS:2
After the left position is filled, there are two different ways to choose someone for the middle position.
For example, with Erin on the left, either Patti or Craig can go in the middle.
How many different ways can Mike arrange his friends in the left and middle positions? ANS:6
After the left and middle positions are filled, how many ways can Mike fill the right position? ANS:1
For example, if Erin and Patti are in the left and middle, only Craig can go on the right.
There are 6 different ways to arrange the 3 friends.
Each of these arrangements is called a permutation of the 3 friends.
The order of the members of the permutation is important. Erin---Patti---Craig is a different permutation than Patti---Erin---Craig.
WAY TO FIND THE NUMBER OF PERMUTATIONS:
A way to compute the number of permutations is to multiply the number of possible choices for each person.
3 * 2 * 1 = 6
***
COUNTY CLERK CANDIDATES:
A county clerk wants to know how many ways she can list 4 candidates for an office on a ballot.
How many ways can she choose a candidate for the first space on the list? ANS:4
How many different ways can she choose candidates for the first and second spaces? ANS:12
The number of permutations of the 4 candidates can be found by multiplying the number of possible choices for each space.
4 * 3 * 2 * 1 = 24
***
CLASS VICE-PRESIDENT/PRESIDENT:
The teacher wants to know how many different ways he can choose a president and vice-president from the class.
How many ways can he choose a student to be a president of the class? ANS:10
How many ways can the teacher choose students to be the president and the vice-president of the class? ANS:20
Why? 10 * 9 = 90. There are 90 ways.
***
WORKSPACE:
ANS:
3 , 2 , 6 , 1 , 4 , 3 , 12 , 2 , 24 , 1 , 10 , 9 , 90 , 336 , 120
MEASUREMENT: Practice Permutations
NOTES:
In this activity you solve problems about arrangements of objects.
***
WORKSPACE:
ANS:
120 , 24 , 210 , 3024 , 6840
MEASUREMENT COMBINATIONS
In this activity you make a tree diagram and find permutations of objects in a set.
NOTES:
MOLLY:
Molly's assignment was to read any 2 books of the 3 books on the reading list. She wants to figure out how many different ways she can do the assignment.
Call the books A, B, and C.
There are 3 ways to do the assignment:
AB, AC, and BC
Each of these choices is called a combination.
The order of the objects is not important in combinations. BA is the same combination as AB.
MOVIE:
Monty has time to watch any two of four movies. He wants to know how many different ways he can do this.
Call the movies A, B, C, and D.
These are some ways he can see two movies:
AB, AC, and AD
How many ways can Monty see two movies from a total of four movies? ANS:6
MELINDA & ICE CREAM:
Melinda can choose any three of five flavors of ice cream for a sundae, She wants to know how many different combinations she can make a sundae.
Call the flavors A, B, C, D, and E.
Here are some ways to make a sundae:
ABC, ABD, ABE, ACD, ACE, and ADE.
How many different ways can Melinda make a sundae? ANS:10
WORKSPACE:
ANS:
AC , BC , BC , 6 , BCD , 10
MEASUREMENT FORMULAS
In this lesson you learn about factorials and formulas for fidning permutations and combinations. You may use a calculator in this activity.
NOTES:
Formulas for permutations and combinations use a concept called factorials.
"!" is a sepcial symbol used to show the product of consecutive numbers.
1! = 1
1! is read as "1 factorial."
2! = 2 * 1
2! is read as "2 factorial."
3! = 3 * 2 * 1
3! is read as "3 factorial."
n! = n(n - 1)(n - 2)(n - 3)...* 3 * 2 * 1
n! is read as "n factorial."
ROW PICTURE:
Moe, Joe, and Flo want to sit in a row for a picture. The photographer wants to know how many ways they can do this.
He can find the number of permutations by making a list... but that is time consuming and it's much better to multiply to find the permutations:
3 * 2 * 1 = 6 permutations
This is the general formula for the number of permutations of n objects taken at n at a time:
P = n!
n n
CLASS ELECTIONS:
Mr. Bonnell must choose a president, vice-president, and secretary from his class of 12 students. He wants to know how many ways he can do this.
This is the formula for finding the number of permutations:
12!
P = --------
12 3 (12 - 3)
/ \
/ \
total number of
number of objects
objects selected
12!
= ---
9!
12 * 11 * 10 * 9!
= -----------------
9!
= 12 * 11 * 10
This is the general formula for the number of permutations of n objects taken r at a time:
n!
P = --------
n r (n - r)!
/ \
/ \
total number in
number the subset
This is the general formula for the number of combinations of n objects taken r at a time:
n!
C = ----------
n r r!(n - r)!
/ \
/ \
total number in
number the subset
WORKSPACE:
ANS:
6 , 720 , 40320 , 30 , 144 , 600 , 336 , 36 , 24 , 120 , 35 , combin. , 792 , perm. , 336 , combin. , 120
MEASUREMENT: COMBINATION FORMULAS
In this activity you solve problems about making choices.
NOTES:
WORKSPACE:
ANS:
10 , 10 , 56 , 126 , 210
MEASUREMENT: USING COMBINATION FORMULAS
A problem that asks about the number of arrangements of objects is solved using permutations.
A problem that asks about the number of selections of objects is solved using combinations.
NOTES:
WORKSPACE:
ANS:
5 , 15120 , 56 , 10 , 56 , 210 , 60 , 5040 , 20 , 12
MEASUREMENT: PERMUTATION FORMULAS
In this activity you will solve problems using permutations and combinations.
NOTES:
WORKSPACE:
ANS:
360 , 20 , 362880 , 120 , 10
PERMUTATIONS, COMBINATIONS, & FACTORALS: Test
This lesson will evaluate your knowledge of permutations and combinations and your skill in manipulating factorials.
NOTES:
WORKSPACE:
ANS:
720 , 720 , 126 , 967680 , 5034 , NOT:120 , 35 , 126 , 462 , 504
SIMILARITY 1: Pre-Test/Test
This lesson will evaluate your knowledge about similarity.
NOTES:
WORKSPACE:
ANS:
1/2 , b , b , a , a , c , c , c , NOT:c , 24
CONGRUENCY 1: Pre-Test/Test
This lesson will evaluate your knowledge of congruency.
NOTES:
WORKSPACE:
ANS:
a , c , c , d , c , b , 46 , d , d , 120
CONGRUENCE: Introduction
In this unit, you will study methods for identifying and reproducing congruent triangles.
These methods can also be used to prove whether or not two given triangles are congruent.
NOTES:
Products that are mass-produced must be IDENTICAL in size and shape.
In geometry, figures which have the same shape and size are congruent.
WORKSPACE:
ANS:
CONGRUENCE FUNDAMENTALS: Definition of Congruence
NOTES:
The Copy Cat Company uses transformations to decide when polygons are congruent.
They know that segments with the same lenght are congruent.
They also know that angles with the same measure are congruent.
Let's look at what happens to segment length and angle measure when a polygon is translated, reflected, or rotated.
The measure of a segment is not changed by a translation, reflection, or rotation.
We say that translations, reflections, and rotations preserve length.
Now, let's look at angles as they are transformed.
The measure of an angle is not changed by a translation, reflection, or rotation.
We say that translations, reflections, and rotations preserve angle measure.
The Copy Cat company uses translations, reflections, and rotations to see if one polygon can be made to coincide with another polygon.
Now you can practice drawing congruent figures...
Now let's practice finding congruent polygons using transformations...
WORKSPACE:
ANS:
YES , NO , YES , YES
CONGRUENCY 1: Pre-Test/Test
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