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
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.
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.
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?
Intro: I would play this movie to catch the attention:
Review of previously learned material consists of the following problems: ***You will have to decide how many people are lining up and how many are men/women together as a class**
1) In the video how many different ways can the people line up with
a) no restrictions
b) The men behind the women.
c) the women standing together
d) the women aren't standing together
2) Later, 3 cab drivers pull up simultaneously. Determine the number of distinct permutations you could have if each person travels alone.
3) The next day, 2 cabs pull up and there are 30 different ways they can drive the number of people present with one person in each cab. Determine the number of people waiting in line.
4) How many different possible values of r could there be if we were calculating 8Pr?
Go through the questions, after giving some time for the students to work on them cooperatively.
Body:
A thief was trying to break into a keypad and he sprayed it with Luminol. A chemical which brightens when it comes into contact with the oils left by a finger. He noticed that there were a fingerprint on the number 2 and 7, and two fingerprints on the number 4. How many different possibilities are there for the code to the safe?
Let students work and
struggle through, and lead them towards the identity of:
Organizing a objects
where there are n repetitions of one object, there are a!/n! distinct
permutations.
Questions:
1) Hertz Rental Cars has 3 identical SUVs, 5 identical cars, and 6 identical trucks. Determine the number of arrangements Hertz can have using all their vehicles.
2) How many PIN numbers can you create if they must be 5 digits long where 3 digits must be the same?
3) Create a scenario where you would calculate 8!/(3!2!) to arrive at the solution.
I used to believe there was a dichotomy between science and religion. It seemed in the past, that when science could not explain why something was true, we turned to religion and simply attributed to God. There have also been arguments against the existence of a God, from The problem of Evilto Michael Martins proof, A Disproof of Gods Existence. Even after growing up in a Catholic home, I believed that one person was either religious or logical and scientific, but not both; a Venn-diagram with no overlap. 8 years ago, this all changed when I started teaching at Notre Dame; a Catholic high school. When I make references to Church and Religion, I will be referring to the Catholic Church. Once I started working at Notre Dame, I met many Catholic Science teachers and was shown that you can be a Catholic logical thinker. My eyes were opened to the reality that this Venn-Diagram does have an overlap. I believe, like others in the past, I did not understand the implications that Religion has on Science and also how Science impacts Religion. I will show, through my own stories how Science and Religion can, and do, coexist.
When I first started teaching Science, I was worried I would go against the teachings of the Church when I started to address the age of our Planet. Before, I took the literal sense of the Word in the book of Genesis which stated that the Earth was formed in six days, some 6000 years ago. Science has shown that the Universe is around 15 Billion years old. An obvious contradiction! After some research, on how I will address this in my Science Class, I read Frank Sheed (1982) say
"one shouldnt be forced to choose between evolution and creation." and he continues on to say that "Creation answers the question why does everything exist, why there isnt nothing? While evolution, is a theory, as to how come the Universe did develop once it existed." (Pg. 58) As well, when Genesis was written, humanity did not fully understand the workings of the universe and these six days just corresponded with the Babylonian creation myth Enuma Elish, and does not really mean God created the Earth in six day at all. Religion now fully accepts the fact the Earth is not 6000 years old and that uses science to determine when God created the Universe One discussion which arose in my class was around the idea of evolution for Apes versus the creation of humans by God. Again, before looking deeper I believed these were two contradictory ideas, but instead they are not. The Church has said it is not against the idea of evolution, and in fact evolution exists in our current era (Some people will not grow molars), and that the evolution of the body is an almost certain fact. However, religion teaches us that the soul, in which is inside our body, was not the result of any certain evolution but instead infused by God Himself; again not contradictory, but two theories from the same thread of truth. Due to the limit of words I will not go further into detail, but science and religion also agree on when life starts, how pre maritial sex will lead to increased divorce rates, and many other theories. Science and religion are both logical, deal with science and fact, and truthfully it is
because of Science that my faith is strong.
Students will learn how to apply the Fundamental Counting Princple with restrictions and Permutations.
Intro: Show the following video:
Since there is an abundant amount of different ways these books are arranged, simply ask
"How would we determine the total number of ways the book shelf could be arranged?"
Next ask
"What would we need to know?"
After some discussion, show the picture of the top left shelf as seen below:
Then I would ask the following questions:
1) How many different ways can these books be arranged, if the spine of the book must be facing out, and:
a) The shelf will only have 4 books and a pink book must be on each end?
b) The shelf will only have 8 books the pink books must be next to each other?
c) All books are on the shelf with no restrictions?
The last question is there to assist with introducing factorial notation.
Explain that: "!" is the symbol for factorial notation and can be used on any non-negative integer n. The formal defintion is:
n!=n(n-1)(n-2)(n-3)...(3)(2)(1)
As well as how to get use it on the calculator.
I would then go through operations on factorials showing that (n!)(x!) does not equal (nx)! nor does it hold for any other operation. As well that 0! = 1
Next ask for the previous picture:
2) How many different ways can you arrange the books on the shelf if you only want:
a) 4 books on the shelf?
b) 7 books on the shelf?
c) 13 books on the shelf?
Using the process, you can introduce nPr, and how the previous answers can be solved with n!/(n-r)!, with guiding the students towards this answer.
The formal definition being:
When permutating n objects picking r at a time we would write nPr = n!/(n-r)!
Ask the students how many diffeerent ways to arrange the letter FILE, then FILL, then FLLL, then LLLL.
Let students work and struggle through, and lead them towards the identity of:
Organizing a objects where there are n repetitions of one object, there are a!/n! distinct permutations.
From here I have yet to find real life scenarios and would give various questions and words to rearrange such as MISSISSIPPI. (Would love ideas here if you have any...)
Here is my lesson plan for my first lesson in Math 30-1 on Permutations and Combinations. Which covers the outcome: Apply the fundamental counting principle to solve problems.
(My students sit in groups of 4 and 5)
First show the following funny video,
Next, show this video:
After which, ask the question "How many different possible pin numbers could there be?"
Give the students about 2-3 minutes to discuss and then re-ask the last question "Is there any more information you need?"
This is where you can go in any direction you please. In my lesson, the gentleman in the movie has either a 4 or 5 digit PIN number (we don't know).
Now allow students to work for approximately 10 minutes.
After this is done, ask the class:
How did you arrive at your number? Here I would actually have students come up and solve on the board What assumptions did you make? Is there a way we could arrive at the answer more efficiently?
Take this time to discuss that you should multiply the different possibilities of having a 5 digit pin, and the possibilities of having a 4 digit pin, and then you should ADD these answers together.
Next show the students the following picture of a hand knit mitten, and explain the following:
Jennifer makes this mitten out of four different parts, the fabric of the entire mitten, the middle "tree or leaf" part, the bead, and the strings which tie them together, and has multiple different colours for each part.
Then ask: How many different mittens can Jennifer make?
I would let students talk as long as they needed until they realized they are missing a lot of vital information. Ask for any questions or information they might need (just like the previous question) and provide them with the following: (You can change as you see fit)
5 different colours of yarn for the mitten
3 different tree/leaf colours
2 different beads
5 different colours for the string.
Give time to solve and then ask
How did you arrive at your number? Here I would actually have students come up and solve on the board
What assumptions did you make?
Is there a way we could arrive at the answer more efficiently?
Next, you can go the link https://order.bostonpizza.com/EN#content=/Menu/ViewMenu/&CategoryItemsContainer=/Menu/CategoryGroup/dfa5509b-935b-4776-b157-bfefef2ab654
Which shows that Boston Pizza currently has 4 different types of wings with 21 different flavours of each type of wing.
The problem: Red Deer Rebels (or whichever local hockey team you want) is having dinner and orders 8 different double orders of chicken wings, how many different combinations could there be?
Again, using the same process students will need to know if you can have more than 1 flavour, and you can have up to 2 flavours PER double order, or they could be the same flavour as well.
After, ask the three crucial questions again, with some leading if needed.
If you have more time I would ask the following question:
Should Alberta, currently, be concerned with the number of phone numbers in the province and truly needed to add the 3rd area code (587)?
Following the same procedure of asking if they require more information and then the three crucial questions of debrief.
