Wednesday, March 26, 2014

4!

Me: I need two volunteers. You have no idea what you're doing. Thanks Brianna and Jesus. Go stand in front of the whiteboard on the side of the room. You are the two contestants in today's Spelling Bee.
This is how I opened today's lesson. Wait. A Spelling Bee in math class? I address the audience:
Me: I need your help. I am going to ask you a question. The answer is a number. I am not interested in any categories like gender, height, age, birthday, first name, last name, etc. For my Spelling Bee, I need you to take my contestants and order them for me. What's the maximum amount of ways I could order these two contestants?
Students have time to think and some quickly raise their hand to say, "Two."
Me: Show me. Tell us what they are.
Student: Right now Brianna is first. Jesus is second. We could switch them and Jesus goes first.
Me: [looking at Brianna and Jesus] Do what she said.
Brianna and Jesus switch order.
Me: Have I maxed out all the possible combinations for ordering Brianna and Jesus?
Class: Yes!
For a little comic relief, I toss Jesus an easy word to spell.
Me: Jesus, spell "cat".
Jesus: C-A-T
Me: Wait. What?
I learned today that most kids don't know how a spelling bee works, so I call on a few kids to explain the three steps:
  1. Say the word.
  2. Spell the word.
  3. Repeat the word.
Me: Jesus, let's try this again. Spell "cat".
Jesus: Cat. C-A-T. Cat.
Me: Bri, spell "discombobulate".
Brianna: Ughhhhhhh. What?!
Me: Okay, can I get a third contestant for our spelling bee? Jesus, since you're the winner, please pick someone. 
Standing in front of the audience, I now have Jesus, Brianna, and Garry.
Me: Okay, let's say their current order is one possible combination. Let's keep Jesus first. Can you get any other combinations with Jesus being first?
Student: Yea, switch Bri and Garry. 
I look at Bri and Garry.
Me: Do it! Okay we now have two possible combinations. Have we maxed out the possible combinations with Jesus being first or can we get more?
Class: We're maxed out.
Me: Okay, someone give me a new combination.
Student: Put Brianna first this time. Then Jesus. Then Garry.
Me: Okay, we now have three combinations. Can we get more where Brianna is first?
I repeat this process until the class has agreed we maxed out our combinations with six total. Great. I toss this information in a table like this to keep track of it.
Me: So what if I add a fourth contestant to the spelling bee? 
Sarah: No!
Me: Really Sarah? What? Are we going to have more or less combinations?
Sarah: More.
Me: Gimme some guesses everyone. Toss something out there for fun. How many combinations could we get with four people in the spelling contest?
Students tell me 8, 10, 9, 12, 16, 13 and I write all of them up on the board. I ask for some quick reasoning behind the guesses.
Me: Ok, thanks. You all can't be right. Instead of moving people around, let's do this instead. 
I gave each group a sandwich bag with four different colored snap cubes: red, green, blue, yellow. Students were to work in their groups to figure out all the possible combinations of four colors. They were to write it down in their notes for the day. I circulated the room, noticing student work.

For groups that think they're done, but wrong (like only 12 combinations):
I zone in on one combination and keep their two colors fixed, "Have you maxed out all the combinations with these two at the front?" Usually this is the only nudge they need to get closer to the correct number of combinations.

For groups that are on track:
I make it obvious I note their work, or ask for a quick explanation, or I quickly move to another group.

Groups that finish and have the correct answer:
I have them explain their work, organization, process, and reasoning. I ask if they feel confident and usually they do. I'm not going to string them along. I respond, "That makes sense to me." followed by:
Me: So what if I gave you a fifth color?
Student: [typical response] Ughhh. 
Me: Oh, what's wrong?
Student: That's a lot of work.
Me: I know, right? I'm right there with ya. I wouldn't want to write out all those possible combinations either. So, your job is to try and figure out a shortcut. In other words, if I just gave you four colors right now, how could we quickly get 24 combinations without writing them all out. If I'm now giving you five colors, what would be a quick way to figure out all the possible combinations?
Once I see that most groups have reached the magic number (24), I show them this and have them count.
Me: One clap on three for the closest guess. 
1-2-3 CLAP!

Many kids see that 4 groups of six combinations yields 24 combinations. I toss 24 into our table and ask the whole class about finding the possible combinations for five colors. Typically, the students want to avoid this nonsense and express some noise of rebellion.
Me: What's wrong? You guys don't want to write out all the combinations? Well, let's try and find a shortcut. Do we see anything from our table that might help us?
To my pleasant surprise, at least one kid in each of the three participating classes found the following relationship:
Abraham, Brianna, and Daisy: You take the previous "Combos" result and multiply it by the diagonal "Colors" amount to get the new amount of "Combos."
Me: Let's see if that works.
It does. Great!
Me: Okay hot shots! This is a great shortcut. What if our principal walked in and gave us 13 colors. How would I quickly figure out the total number of combinations since I don't have the number of combinations from 12 colors?
Here's where I introduced the use of factorials. Yes, I could have spent time getting the kids to look for this pattern, but I simply didn't have or make the time. I felt it was a good place to show them that putting the factorial symbol after a number means to multiply it by all of the natural numbers less than the given number.

4! = 4 x 3 x 2 x 1 = 24
Me: So if our principal walked in and said, "Find all the combinations of 13 colors." we'd go thirteen...
Class: ...times twelve, times eleven, times ten, times nine...
In reflection, this lesson created more successes for my students than I anticipated. Some include:

  • Discovering patterns and relationships within a table,
  • Creating a need for the factorial of a number,
  • Adding another vocabulary term to our tool belt, and
  • Finding combinations more efficiently.

This lesson started with a low-entry of two students and two combinations.
We built in the next part by finding six combinations for 3 students.
We built in a guess for the combinations of four students so they can invest in the question and look for patterns.
We manipulated four colors, organized our combinations, made conjectures, and arrived at a reasonable answer that maxed out the combinations.
We pushed those students who finished early to discover a shortcut on their own.
We created a need for avoiding excessive work with larger numbers and a need for some type of formula (factorials) that will get us the same result.

I came into this lesson with a rusty understanding of factorials, probability, and combinations. Anyone who is against Common Core State Standards, think again! It's making math teachers know their content better, so they can better serve their students. It's opening the door for students to reason their way in math class. I'm not blogging to get into the importance of CCSS right now. However, I'm convinced this was way better than me standing in front of the students telling them to put an exclamation point after 4 (like this 4!) and to just multiply 4 by 3 by 2 by 1 to get all the possible combinations of four somethings. Instead, the students discovered the relationship (pattern) within the table and felt confident in discovering the total combinations of five colors without drawing them all out.

Factorial,
848!

11 comments:

  1. Andrew, you and your students might enjoy the short video my friend Justin Solonynka made with his class. They were looking for all the possible ways to arrange Justin's daughter's train puzzle. It's pretty fantastic. http://tinyurl.com/trainmath

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  2. I love the way you built a context for factorial. Simply introducing it by itself without understanding where it comes from is much less powerful. Great job.

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    1. I agree with Robert, and you really created a want to know the math. Thanks for sharing.

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  3. 72659532352331669465387317964541490327158603407862497819051505167825212554092719825854319518063147027513610184785376207903389097321213097332694262924197459034702554574380465837160150990371301177373649597858306887704291472881011725163932784636620821649210484706132426658314255823136841392895137831883319666065284532341893484879148185259318038571097249178263188727301290486974419425716246412938022509147382599642683189959864993819223344172114834546575875156081714079173827947196082060615343004503615966063413925926301221456673919719127522236470716834832340449878752888211875239276474344027834195149760630064452549229837172849969447280465584626913608835465592004523949856883499197900133721283181599046422696956058427924744558162627684162701811188071678070411066072710587640976813426254877943334464341936581387645078195485722746514055555539273360079229026308832278833977481295747506349955791323351785671037426041360127299901330120015187889890377161297280037348999275284463582829909489287314801425772015957342345105572075099637006077183155337807264695539104871880890989949497097631895018352151203056561945027828634296885592206765976815358134204874664973096676231581429335640709739179834598238236938828532649711609963136405375200042665756350748532740089500069708175027520823536024600878141904329406389592595194159418391405554004186072496281101093069602296381270877911049732416398127330127568840478116642497274529274667014893743677353031567876890476191264149324344394821579054104905749129538546919362695992627695105862331341735043600368288312912728971554097264729920456994303923494565877428348011989724490225195748535466620449658740748556715401450120006869210681849861817149375735439438715842682010657157370379070049494587056522597331005526812705244760849401074331281864704304029392357276085948343295746838337566519294143034426906312344586977858432466673977151238081557159113028782126382687927051632571942632737996800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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  4. Thanks for sharing your approach. Did you share with students that factorials are a kind of permutation and that in math combinations are groupings where the order doesn't matter? Just wondering how deep you took this in subsequent lessons.

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  5. I did this today, so I wanted to stop by and say "TTTHHHAAANNK YYOUU!!!". FACTORIAL!!!!
    I passed out strips of paper with color dots instead of the cubes, which is quick and cheap alternative. I also found a website that has previous Spelling Champion words, and then dictionary.com helped with the pronunciation, and hilarity ensued.
    We have been knee deep in quadratics lately, so it was a welcomed break for the students. I've been as CCSS as I can be with quadratics, gallery walks and the such, but it still hits a point of dragging. An added bonus was when my high achieving students immediately went to write and equation, and realized that the patterns they are used to searching for didn't exist.
    I finished the lesson talking about how important FACTORIALS!!!!! are in quantum physics. I didn't spend a lot of time with that aspect of it, but I wanted to cement the fact that this new tool they learned is important.

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  6. I think this a good approach and the very helpful article regarding factorial of 4. I think this article help the student to learn more about the factorial. I appreciated your work. I would like to say Thanks for sharing this article.

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