Radical Functions Lesson 1

I had strep throat

Permutations and Combinations Lesson 7

Review:

Have students work on:

1) If you are arranging 3 different Math texbooks, 4 different Science textbooks, and 5 different English textbooks, how many different ways can you organize them if:
a) the books of the same subject must be together?
b) The shelf must start and end with an English book?
c) How many different ways can you pick 2 books from each subject?

2) Explain why 5C2=5C3?


3) Solve: n+1Cn-1=15

4) How much different monetary amounts can you make from 1 penny, 1 dime, 1 quarter, and 1 dollar?

4) Explain everything you know about the expansion of (x+y)^n.

Body:

1) If you know one row of the Pascal's Triangle is

1

8

28

56

70

56

28

8

1

Determine  the next row.

2) Determine the last term of the expansion (x-2y)^8.

3) Determine the third term of (a/b-2)^9

4) One term in the expansion of (x+a)^8 is 448x^6, determine the value of a.

5) Determine the constant term in the expansion of  (2x-1/x^2)^15

Permuatations and Combinations Lesson 6

Review:

1)Provide students with a pathways problem:
2) Create a word where the number of different arrangements is 8!/(3!5!)
3) If you invited 5 people to your party but forgot to ask them to RSVP, how many different arrangements are possible?

This last question is the key to linking past knowledge to new knowledge.

Next show the video:

Then create a plinko board on the board and ask the students to determine the number of ways for the plinko to fall.  Assuming:
1) The Plinko does not come back up
2) It either falls left or right.
(They should create the first couple of rows of Pascal's triangle)

Create enough rows to row 6 (the answer to the last question in the review).  Rewrite the last row in Combination notation instead of numeral notation.

Next, ask them to expand
(x+y)^0, (x+y)^1..., (x+y)^3.  (They should be getting irritated at this point).

Now, ask if they notice a pattern here...linking the co-efficients to the Pascal's Triangle.
Observations:
(x+y)^n has n+1 terms, uses the n+1 row of the triangle, and will have co-efficients of nC0, nC1,...,nCn

Depending on time you can create the binomial theorem with them or just state it:
tk+1=nCk(x)^(n-k)y^k

Now give them some questions:
Determine the ___ term of the expansion (__+___)^___ , where the blanks can be various numbers and variables.

Perms and Combs Lesson 5

Intro:  **I will now start to mix Perms and Combs together**

The options at Harvey's are:
Tomatoes, lettuce, pickels, hot peppers, onions, ketchup and mustard.  Determine how many different burgers are possible.

In University, some professors allow for choices on their exams.  One specific professor gave 5 questions in Part A, and 4 questions in Part B.  She required the class to complete 2 questions from each part.  Determine the number of arrangements possible. 

On the secon exam, the professor allowed for bonus marks and gave 3 questions in each part, and asked the students to do a minimum of 2 questions in each part.  Determine the number of arrangements.

In the class there were 28 people, 15 men and 13 women.  If the professor wanted to choose a president and vice president from both the men and women, determine the number of arrangements possible. 

Lastly, out of the whole class, up to 3 had to be selected to be on the Dean's survey group.  Determine the number of possible arrangements.

Body:  Algebra with Combinations:

1) Solve for n:
a)2(nC2)=n+1C3


b) 720(nC5)=n+1P5

2) Explain why 8C3 = 8C5.

Permutations and Combinations Lesson 4

Review:  Provide students with the following images, and explain the following:

Oreos have evolved largely over the last years, and now have many different types of cookies.  The pictures show only some of the option available to consumers these days. 

The pictures show:

• Chocolate or Vanilla flavoured cookies
• Strawberry, Chocolate, or Vanilla fillings
• A single layer, a double layer, or double layer with an additional cookie in between.

Since I cannot find any Oreo's with a different kind of cookie throughout, we shall assume all the cookies used have to be the same.  Determine the total number of different Oreo's which could be created.

 

Body:

 

Provide students each with the following diagram

Ask them to determine the number of paths from various corners to other corners. **This might take some struggling...but let them struggle!**. Start with points close then further and further away. Leading them towards solving it by which ever method you prefer to teach.

Provide them with different paths and restrictions such as must start at XX go through XX and end up at XX.

 

Now introduce Combinations.  In the recent Olympics, 8 men ran the 100m in the Final medal Race.  First place recieved Gold, Second place recieved Silver, Third placed recieved Bronze.  Determine the number of different ways the men could have placed.

 

Before the Finals there were heats were only the top 3 times in each heat would advance to the next level.  If in the first heat, there were 8 men running, determine the number of different combination of men who could advance to the next round.

 

After the race, the men shake hands to congratulate each other.  Determine the total number of handshakes for the 8 men. 

 

Give students time to work together and lead them towards the idea, and then eventually explain, that when order DOES not make a difference, the number of combinations are

n!/(n-r)!r!

 

Permutations - order makes a difference - n!/(n-r)!

Combinations - order does not make a difference - n!/(n-r)!r!

 

How many different sums of money can you make with $5, $10, a penny, and a dime?