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Error: #f88
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#backstageCloak {background:[[ColorPalette::Foreground]]; opacity:0.6; filter:'alpha(opacity=60)';}
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ol ol ol {list-style-type:lower-roman;}
ol ol ol ol {list-style-type:decimal;}
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ol ol ol ol ol ol {list-style-type:lower-roman;}
ol ol ol ol ol ol ol {list-style-type:decimal;}

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.externalLink {text-decoration:underline;}

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/* the 'a' is required for IE, otherwise it renders the whole tiddler in bold */
a.tiddlyLinkNonExisting.shadow {font-weight:bold;}

#mainMenu .tiddlyLinkExisting,
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.whenBackstage {display:none;}
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StyleSheet for use when a translation requires any css style changes.
This StyleSheet can be used directly by languages such as Chinese, Japanese and Korean which need larger font sizes.
body {font-size:0.8em;}
#sidebarOptions {font-size:1.05em;}
#sidebarOptions a {font-style:normal;}
#sidebarOptions .sliderPanel {font-size:0.95em;}
.subtitle {font-size:0.8em;}
.viewer table.listView {font-size:0.95em;}
@media print {
#mainMenu, #sidebar, #messageArea, .toolbar, #backstageButton, #backstageArea {display: none !important;}
#displayArea {margin: 1em 1em 0em;}
noscript {display:none;} /* Fixes a feature in Firefox where print preview displays the noscript content */
<div class='header' macro='gradient vert [[ColorPalette::PrimaryLight]] [[ColorPalette::PrimaryMid]]'>
<div class='headerShadow'>
<span class='siteTitle' refresh='content' tiddler='SiteTitle'></span>&nbsp;
<span class='siteSubtitle' refresh='content' tiddler='SiteSubtitle'></span>
<div class='headerForeground'>
<span class='siteTitle' refresh='content' tiddler='SiteTitle'></span>&nbsp;
<span class='siteSubtitle' refresh='content' tiddler='SiteSubtitle'></span>
<div id='mainMenu' refresh='content' tiddler='MainMenu'></div>
<div id='sidebar'>
<div id='sidebarOptions' refresh='content' tiddler='SideBarOptions'></div>
<div id='sidebarTabs' refresh='content' force='true' tiddler='SideBarTabs'></div>
<div id='displayArea'>
<div id='messageArea'></div>
<div id='tiddlerDisplay'></div>
<div class='toolbar' macro='toolbar [[ToolbarCommands::ViewToolbar]]'></div>
<div class='title' macro='view title'></div>
<div class='subtitle'><span macro='view modifier link'></span>, <span macro='view modified date'></span> (<span macro='message views.wikified.createdPrompt'></span> <span macro='view created date'></span>)</div>
<div class='tagging' macro='tagging'></div>
<div class='tagged' macro='tags'></div>
<div class='viewer' macro='view text wikified'></div>
<div class='tagClear'></div>
<div class='toolbar' macro='toolbar [[ToolbarCommands::EditToolbar]]'></div>
<div class='title' macro='view title'></div>
<div class='editor' macro='edit title'></div>
<div macro='annotations'></div>
<div class='editor' macro='edit text'></div>
<div class='editor' macro='edit tags'></div><div class='editorFooter'><span macro='message views.editor.tagPrompt'></span><span macro='tagChooser excludeLists'></span></div>
To get started with this blank [[TiddlyWiki]], you'll need to modify the following tiddlers:
* [[SiteTitle]] & [[SiteSubtitle]]: The title and subtitle of the site, as shown above (after saving, they will also appear in the browser title bar)
* [[MainMenu]]: The menu (usually on the left)
* [[DefaultTiddlers]]: Contains the names of the tiddlers that you want to appear when the TiddlyWiki is opened
You'll also need to enter your username for signing your edits: <<option txtUserName>>
These [[InterfaceOptions]] for customising [[TiddlyWiki]] are saved in your browser

Your username for signing your edits. Write it as a [[WikiWord]] (eg [[JoeBloggs]])

<<option txtUserName>>
<<option chkSaveBackups>> [[SaveBackups]]
<<option chkAutoSave>> [[AutoSave]]
<<option chkRegExpSearch>> [[RegExpSearch]]
<<option chkCaseSensitiveSearch>> [[CaseSensitiveSearch]]
<<option chkAnimate>> [[EnableAnimations]]

Also see [[AdvancedOptions]]
This site was created by [[Eric Fitton|http://www.efitton.net]]. I am a math teacher at [[Bellevue Community Schools|http://www.bellevue-schools.com]] and a Masters Student at [[Michigan State University|http://www.msu.edu]] in the [[MAET Program|http://edutech.msu.edu/masters.html]]. There are two primary reasons for this website's existence. First, this website exists for an assignment in my CEP 810 course. Secondly, to help my students, Bellevue Students and any other students with probability, specifically counting problems.

[>img[[Front of the Dairy Store|DairyStoreFrontWeb.JPG]] I can be contacted during the school year at efitton at bellevue-schools dot com and I also check efitton at efitton dot net on a semi-regular basis. This dairy problem first went live on June 27th 2011 while [[efitton.net|http://efitton.net]] launched March 9th, 2008 and is where I host my [[GeoGebra Applets|http://www.geogebra.org]]. You can see from the timeline on the right the last time the site was modified.

For fellow educators, as you can see I am trying to engage my students in problem solving with these tasks. While there is clearly guidance in both identifying problem solving strategies and choosing one particular problem solving strategy, there is also a chance for students to problem solve on their own. In addition to locating content, evaluating the videos, evaluating their strategies, and hopefully synthesizing; I see a role for students working together once they have made progress on an individual task. In your classroom this could be whole class discussion, small groups, or pairs. You could also try blogging or another 21st Century technology. Anyhow, I wish you and your students well.

In the future I plan on adding some amount of additional content. I may very well add an additional task of counting ice cream scoops without replacement. I see this as a way to use inquiry to generate combinations and permutations. Other alternate tasks might be double scoops with five different flavors so students cannot directly get the answer from the videos. I might add a Dave video in the future with an additional problem solving technique showcased. In the meantime, make this website work for your students. If you don't like a task, change it for your class. If you want to make a video, send me a copy and I will probably add it to this site. I strongly suggest a [[CopyLeft|http://www.gnu.org/copyleft/]] or [[Creative Commons|http://creativecommons.org]] license so future collaboration is possible.

I took all the photos on the website and generated all of the content except for the clipart of the single, double and triple scoop cones used in the Alice Video. The clipart of the cones were all found in the [[public domain|http://www.clker.com/clipart-13319.html]].
<html><a rel="license" href="http://creativecommons.org/licenses/by-sa/3.0/"><img alt="Creative Commons License" style="border-width:0" src="http://i.creativecommons.org/l/by-sa/3.0/88x31.png" /></a><br /><span xmlns:dct="http://purl.org/dc/terms/" property="dct:title">Counting Problem Website</span> by <a xmlns:cc="http://creativecommons.org/ns#" href="http://dairy.efitton.net" property="cc:attributionName" rel="cc:attributionURL">Eric Fitton</a> is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/3.0/">Creative Commons Attribution-ShareAlike 3.0 Unported License</a>.<br />Based on a work at <a xmlns:dct="http://purl.org/dc/terms/" href="http://dairy.efitton.net" rel="dct:source">http://dairy.efitton.net</a>.<br />Permissions beyond the scope of this license may be available at <a xmlns:cc="http://creativecommons.org/ns#" href="http://dairy.efitton.net" rel="cc:morePermissions">http://dairy.efitton.net</a>.</html>
Alice [[draws a diagram|http://www.teachervision.fen.com/math/problem-solving/48931.html]].

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http://www.cs.odu.edu/~toida/nerzic/content/problem_solving/problem_solving.html#example2 gives an example of what it means to use prior knowledge.

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Below are three different videos showing how to solve [[Task 1]]. All three "students" solved the problem with very different strategies. Even if you solved Task 1 with no difficulty, it might be a good idea to watch all three videos to help you think about how to solve the final tasks.

Videos will loop in a new "tiddler"

Alice decided to draw a diagram.
[[Alice Makes a Diagram|Alice Diagram]]
Bob made a table.
[[Bob Makes a Table|Bob Table]]
Carol used prior knowledge.
[[Carol Prior Knowledge|Carol Prior Knowledge]]

When you are finished, and you can always come back, proceed to [[Task 2]].
[>img[Mr. Fitton with ice cream|Mrfittonatdairyweb.JPG]][[Michigan State University|http://www.msu.edu]] is a [[land-grant university|http://en.wikipedia.org/wiki/Land-grant_university]]. One of the best parts of the farming tradition of MSU is that MSU has it's own [[dairy store|http://dairystore.msu.edu/default.asp]] where they sell both ice cream and cheese. Your rotund teacher is honestly more interested in the ice cream than the cheese. He notices that they have many more flavors of ice cream than they can actually have available at the store but is able to find on their website both how many flavors they serve at the [[Anthony Hall Diary Store|http://maps.google.com/maps/place?q=anthony+hall+michigan+state+university&hl=en&cid=7969287400933964607]] and how many flavors they serve at the [[Union Store|http://www.msuunion.com/index.html]].


Your task by the end of this project is to determine how many different one scoop, two scoop and three scoop ice cream cones you could possibly buy at the Dairy Store location and at the Union location. True, the Dairy store doesn't sell a triple scoop, but Mr. Fitton might possibly buy a double scoop and then a single scoop and add the single scoop to the top of double.

To start this project you must first find how many flavors of ice cream that are sold both at Anthony Hall and at the Union.

Once you have found how many different flavors are available at both the Dairy Store and at the Union Store and you have ''reread the introduction'' and made sure you understand the overarching task, then you may open [[Problem Solving]]. 
[[Problem Solving]]
[[Task 1]]
[[Different Approaches]]
[[Task 2]]
[[Task 3]]
[[About This Website]]
Clearly finding the number of single scoops is easy. How many flavors of ice cream are there at each location?

At this point in your academic career you have used many [[problem solving strategies|http://library.thinkquest.org/25459/learning/problem/?tql-iframe#strategy]]. Problem solving is complicated enough that many people and institutions have addressed it. Some are [[math specific|http://www.thesingaporemaths.com/stratf.html]] and [[some are not|http://en.wikipedia.org/wiki/Problem_solving]].

|[[Strategies|http://fcit.usf.edu/fcat8m/resource/fcat/strat.htm]]|[[Problem Solving Approach|http://www.cs.odu.edu/~toida/nerzic/content/problem_solving/problem_solving.html]]|
|Make a table|1. Understanding the problem|
|Make an organized list|2. Making a plan of solution|
|Look for a pattern|3. Carrying out the plan|
|Guess and check|4. Looking back i.e. verifying|
|Draw a picture or graph or diagram|
|Work backwards|
|Solve a simpler problem|
|Sort out information that is not needed|

Dealing with thirty some flavors is a difficult way to start. I am imposing that all of you use the problem solving strategy of [[solving a simpler problem|http://www.glencoe.com/sites/common_assets/mathematics/math_review/PS_Simpler_Problem.pdf]]. Dealing with one or two flavors is too simple. I am suggesting, ''and by suggesting I mean I will be assigning a grade'', that you first try and solve this problems with four flavors. To be consistent in our writing, sharing and collaborating I will choose the flavors:
* Vanilla
* Chocolate
* Strawberry
* Mint
Your [[first task|Task 1]] is to determine how many different ways you can order 'two scoops' of ice cream given ''only'' four choices.

[img[picture of cheese at the dairy store|cheeseweb.jpg]]
Picture taken at Anthony Hall Dairy Store.
by [[Mr. Fitton|http://www.efitton.net]] using [[MSU's|http://www.msu.edu]] [[Dairy Store|http://dairystore.msu.edu/default.asp]]
A Counting Problem
Your first task is to spend the next thirty minutes trying to determine how many ways you can order a double scoop of ice cream; but first we hit a snag.

Justin and Janice both work for an ice cream shop. When a customer orders Chocolate and Vanilla from Justin he puts whatever ice cream that is closest on the bottom and the other on the top. Every once in a while someone complains that he put them in the wrong order but Justin just figures some people just like to complain. Janice feels and acts very differently. If someone orders Chocolate and Vanilla she always asks which goes on top and which on the bottom. Janice always likes to start with Chocolate and puts Vanilla on the bottom when she eats ice cream so she ''knows'' that the order matters.

Mr. Fitton has decided that both Justin and Janice have valid points. Please determine how many ways you can order a double scoop with the four flavors both when order does matter, ''and'' when order does not matter. Again, the four flavors are:

If you determine both answers in less than thirty minutes, please think of additional explanations of why you are correct, or try to solve the problem an additional way. Think of you problem solving strategies.If necessary go back to [[Problem Solving]].

You must get approval from Mr. Fitton before going to [[Different Approaches]].

[img[Ice Cream Counter|Icecreamcounterweb.JPG]]
The task is straightforward; getting to the solution may not be.

#How many ways can you order a double scoop of ice cream at Anthony Hall if order matters?
#How many ways can you order a double scoop of ice cream at Anthony Hall if order does not matter?
#How many ways can you order a double scoop of ice cream at Union Hall if order matters?
#How many ways can you order a double scoop of ice cream at Union Hall if order does not matter?

When you are comfortable about your answer, check with Mr. Fitton and then proceed to [[Task 3]].

[img[Chalk/Sales Board at the Anthony Hall Dairy Store|salesboardweb.JPG]]
Picture taken at Anthony Hall by Mr. Fitton.
Hopefully two scoops was pretty easy. Now lets make it interesting.

#How many ways can you order a triple scoop of ice cream at Anthony Hall if order matters?
#How many ways can you order a triple scoop of ice cream at Anthony Hall if order does not matter?
#How many ways can you order a triple scoop of ice cream at Union Hall if order matters?
#How many ways can you order a triple scoop of ice cream at Union Hall if order does not matter?

This is more difficult. I ''//suggest//'' trying to solving this problem with only 4 or 5 flavors first. I do not expect everyone to completely solve this problem on their own. However, I do expect you to make progress on both problems //order mattering ''and'' ordering not mattering//. I will be asking you to post to your blog your thinking, your progress and where you get stuck. You will be placed in groups to finish this project and I expect everyone to make progress to the solution and be a productive member of your group. If you do not contribute to your group, ''being fired'' is a very real possibility.

Work hard. Do math.