From Experimental to Theoretical Probability

Probability is an interesting topic.  Really, it’s different from most mathematics our students learn.  It’s the only topic in K-8 mathematics that doesn’t follow patterns.  For example, we might know that flipping a coin 10 times should result in 5 heads and 5 tails, but in reality, it is quite likely that we will get some other result than this.

Because of this, I believe that we need to spend our time playing with tasks and making predictions, a lot, before we ever broach the concept of theoretical probability.  We need to have students play the same games several times, where they can change their predictions based on previous experiments, before we provide opportunities for our students to understand the theoretical.


Big Ideas of Probability

When planning a problem in any topic it is always a good idea to consult Marian Small’s or Van de Walle’s “Big Ideas”.  Here are 2 big ideas we looked at as we constructed the problem below:

  • An experimental probability is based on past events, and a theoretical probability is based on analyzing what could happen. An experimental probability approaches a theoretical one when enough random samples are used.  – John Van de Walle
  • The relative frequency of outcomes (of experiments) can be used as an estimate of the probability of an event. The larger the number of trials, the better the estimate will be.  – Marian Small

The Task Context

We explained to students that we had created a new game and wanted to test it out.  While each group would play the same game, the board for each game was slightly different.  We explained that it was their job to play the game and tell us which game board we should keep.


Rules for the Game
  • The game is for 3 players
  • 2 coins are flipped each turn
  • Player 1 wins if 2 Heads are flipped
  • Player 2 wins if 2 Tails are flipped
  • Player 3 wins if one of each are flipped
  • The result is placed on the game board (H for 2 Heads, T for 2 Tails, B for both – one of each)
  • The winner of game is determined by the player with the most wins in that game.
  • Once a game is finished the entire game is coloured 1 colour (Red for Heads, Blue for Tails, Green for Both).  If a tie exists, 2 colours are used equally.

The Game Boards

Each student is given a game board with 100 sections, however, each individual game consists of a different number of trials.

Game board 1 – 1 game of 100 trials

Game board 2 – 2 games of 50 trials each

Game board 3 – 5 games of 20 trials each

Game board 4 – 10 games of 10 trials each

Game board 5 – 20 games of 5 trials each

Download copies of these game boards here


Results

After playing several games using each game board here is what we found:

IMG_6496
1 game of 100
IMG_6502
2 games of 50
IMG_6495
5 games of 20
IMG_6500
10 games of 10
IMG_6499
20 games of 5

What do you Notice?  What do you Wonder?

Students noticed:

  • The player who was chosen as “Both” won far more than the other two
  • Game boards with more trials (game of 100 or 50 or 20) were coloured all/mostly green.
  • Game boards with less trials (game of 10 or 5) had more red and blue sections than other boards, but still mostly green

Students wondered:

  • Why did “Both” keep winning?
  • Why is a game board with more trials more likely that green will win?

In the end, several students came to the conclusion that getting a Heads and a Tails must be more probable.  Here is what they came up with:

heads tails.JPG


Some things to think about

If a Standard/Expectation tells us that our students need to understand that experimental probability approaches theoretical probability with more trials, we can’t just tell students this.  We need to set up situations where students are actually experimenting (including: making predictions, performing an experiment, adjusting predictions/making conjectures, re-testing the experiment…..).

As with anything in mathematics, our students need ample time and the right experiences to make sense of things before we rush to a summary of the learning.


I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

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An Unsolved Problem your Students Should Attempt

There are several great unsolved math problems that are perfect for elementary students to explore.  One of my favourites is the palindrome sums problem.
In case you aren’t familiar, a palindrome is a word, phrase, sentence or number that reads the same forward and backward.


The problem itself comes out of this conjecture:

If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome.

Let’s take a look at the numbers 12, 46 and 95:

Palindrome adding1


Palindrome adding2


Palindrome adding3


As you can see with the above examples, some numbers can become palindromes with 1 simple addition, like the number 12.  12 + 21 = 33.  Can you think of others that should take 1 step?  How did you know?

Other numbers when added to their inverse will not immediately become a palindrome, but by continuing the process, will eventually, like the numbers 46 and 95.  Which numbers do you think will take more than 1 step?


After going through a few examples with students about the process of creating palindromes, ask your students to attempt to see if the conjecture is true (If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome).  Have them find out if each number from 0-99 will eventually become a palindrome. Ask students if they need to find the answer for each number?  Encourage them to make their own conjectures so they don’t need to do all of the calculations for each number.


Mathematically proficient students look closely to discern a pattern or structure. They notice if calculations are repeated, and look both for general methods and for shortcuts. (SMP 7)

Some students will notice:

  • Some numbers will already be a palindrome
  • If they have figured out 12 + 21… they will know 21 + 12.  So they don’t have to do all of the calculations.  Nearly half of the work will have already been completed.
  • A pattern emerging in their answers … 44, 55, 66, 77, 88, 99, 121… and see this pattern regularly (well, almost)
  • A pattern about when numbers will have 99 as a palindrome, or 121… 18+81 = 27+72 = 36+63…

Thinking about our decisions:
  • What is the goal of this lesson?  (Practice with addition?  Looking for patterns?  Perseverance?  Making conjectures? …)
  • Do we share some of the conjectures students are making during the time when students are working, or later in the closing of the lesson?  
  • How will my students record their work?  Keep track of their answers?  Do I provide a 0-99 chart or ask them to keep track somehow?
  • Will students work independently / in pairs / in small groups?  Why?
  • Do I allow calculators?  Why or why not?  (think back to your goal)
  • How will I share the conjectures or patterns noticed with the class?
  • Are my students gaining practice DOING (calculations) or THINKING (noticing patterns and making conjectures)?  Which do you value?

The smallest of decisions can make the biggest of impacts for our students!  


So, at the beginning of this post I shared with you that this problem is currently unsolved.  While it is true that the vast majority of numbers have been proven to easily become palindromes, there are some numbers that require many steps (89 and 98 require 24 steps), and others that have never been proven to either work or not work (198 is the smallest number never proven either way).


Some final thoughts about this problem…

After using this problem with many different students I have noticed that many start to see that mathematics can be a much more intellectually interesting subject than they had previously experienced.  This problem asks students to notice things, make conjectures, try to prove their conjectures and be able to communicate their conjectures with others…  The problem provides students with the opportunity to both think and do.  It offers students from various ability levels access to the problem (low floor), and many different avenues to challenge those ready for it (high ceiling).  It tells students that math is still a living, growing subject… that all of the problems have not yet been figured out!  And probably the most important for me, it sends students messages about what it means to really do mathematics!

 

palindrome2
Taken from Marilyn Burns’ 50 Problem Solving Lessons resource