Category Archives: Middle School

Counting Principles and “The Price is Right”

I have a confession to make…..it’s really quite embarassing…

I’m a Price is Right nerd.

{sigh} wow, feels so good to get that off my chest.

priceSince I was a little kid, I loved watching the Price is Right.  I know all the games, many of the prices, and of course can name the “back in the day” models without batting an eye (Dian, Holly, Janice).  I even made the pilgramage to Television City a few years back to see Bob Barker in his last years of hosting.

Now, as a stat teacher, I have used a number of Price is Right games in the classroom as probability lessons.  I’ve given a number of talks using Plinko as the centerpiece of a lesson.  Almost all of Price’s games have some probability element.  Here are a few games you can discuss on your classes, starting with basic ideas, and moving up to more complex counting principles.

DOUBLE PRICES

DoublePricesThis is the most simple probability game on the show.  The contestant is shown a prize, and two possible prices for the prize.  If the contestant guesses blindly, they then have a 50% of choosing the correct price, and winning the prize.  In all of these games, the pricing aspect is a “clue” to the player, which hopefully increases their chance of winning.  But in all of these examples, we will look at the games as random chance experiments.

ONE WRONG PRICE

This game is only slightly more  difficult than Double Prices.  In it, the contestant is shown three prizes, each with a price tag, one of which is an incorrect price.  If the contestant identifies the incorrect price, they win all 3 prizes.  Given random guessing, a contestant has a 1/3 chance of winning.

SAFE CRACKERS

Here’s where we start looking at some more interesting counting methods.  In this game, a contestant can win a large prize and a smaller prize by correctly giving the price to the smaller prize.

SafeThe smaller prize has 3 distinct digits in its price, which the contestant is given.  They must place the digits in the correct order to find the price.  With 3 digits to place in order, we have 3! = 6 possible prices.  BUT, in this game, the price always ends in zero (they don’t tell you this, but it’s always true), which means this is essentially a 50-50 game.  For example, if the 3 given digits are 0, 9 and 5 – then there are only two possible prices, $950 or $590.

BONKERS

A contestant has a chance to win a prize with 4-digits in its price.  A “dummy” price is given, like $5447, and the contestant must determine if each digit in the actual price of the prize is higher or lower than the digit in the dummy price.

BonkersTo make their guesses, the contestant places markers either above or below each digit in the dummy price.  If they are correct with all 4 digits, they win the prize.  If they are wrong, they can go back and make changes.  A total of 30 seconds is given to make as many guesses as they can, running back and forth between the game board and the guessing buzzer.

Each digit has 2 outcomes, higher or lower.  Since there are 4 digits, there are 2 x 2 x 2 x 2 = 16 different outcomes.  With only 30 seconds to make guesses, this game often comes down to how well the contestant uses their time to maximize the number of guesses out of the 16.

TEN CHANCES

In this game, a contestant can win 3 different prizes: one with 2 -digits in its price, one with 3-digits, and a car with a 5-digit price.

The contestant is first shown 3 digits, 2 of which make up the price of the first prize.  The goal is to use as few “chances” as possible to get the correct price, which allows the player to move on the the next prize.  With 3 digits to choose from, there are in theory 3 x 2 = 6 possibilities.  But like Safecrackers above, the price will always end in zero; so there are only 2 real choices.

For example: if the given digits are 0, 3 and 5, then the only real possibilities are $30 or $50.  Often, the frustration in this game is associated with contestants who don’t know that all the prices always end in zero…which causes me to yell at my television.

ChancesMoving onto the next prize, 4 digits are given, 3 of which make up the next price.  In theory, there would be 4 x 3 x 2 = 24 choices here.  But again, given that the price will always end in zero, there are only 3 x 2 = 6 viable choices.

The goal in this game is to economize your Chances, so that you have a good number left to play for the car.  5 digits are given, all of which must be used in the car price.  In theory, this gives 5 x 4 x 3 x 2 = 120 choices.  But there are 2 ideas at play here: the price will end in zero AND the price will always begin wth 1 or 2.  Depending on the assortment of digits given, this reduces the number of possible prices a player needs to assess.

THREE STRIKES

DisksThis is one of the most difficult games on the show to win, and is often played for a luxury car.  The game is played with 8 wooden disks, which are placed in a bag and shaken.  5 of the disks have number on them; digits in the price of the car.  The other 3 disks have red “strikes” on them.  A disk is drawn, and if a number is drawn, the player must tell which position in the car’s price the number represents.  If they are correct, the disk is removed from circulation.  And if the player is able to complete the price of the car before drawing all of the strikes, they win the car.

This game is a bit tricky to analyze, because often numbers are drawn repeatedly, as the contestant tries to narrow down where digits go in the price.  For an in-class analysis, let’s assume that the player knows the price, and is trying to just draw the digits.

If you are assuming perfect play, then we could simply list all of the possible ways to arrange the numbers and strikes.  Let N indicate a number and X indicate a strike.  So you could have:

NNNXXNXN (Loss, since 3 X’s occur before all N’s are drawn)
XNXNNNNX (Win)
NNXNNXXN (Loss)…..

The number of ways to play the game is then 8C3, which is 56.  Now, this may seem like a small number, but I am treating the numbers in the bag as similar objects, since we are assuming the contestant places them correctly.

BoardNow, I could go through and count the number of these 56 that produce wins, but I think it might be simpler than that.  The game really comes down to the last item on the list.  If the last item is an X, then you will have won the game.  If the last item is an N, then you have lost.  The chance that the last item is an X 3/8, so the probability of winning the game, assuming perfect play, is 3/8.

So, this game only has  37.5% chance of victory IF the contestant plays perfectly.  Add in that often the contestant often must struggle to position the digits, and you see why the game is so difficult to win.

LINE ‘EM UP

LineUpThis game is played for a car, with 3 smaller prizes as well.  The 3 smaller prizes have prices with 3 -digits, 2 digits, and 3-digits.  The prices of these smaller prizes are used to fill in the middle 3-digits in the price of the car, as shown on the game board here.  This give 3 x 2 x 3 = 18 possible outcomes.  The nice part about this game is that the contestant is given a second chance, and is told how many digits they have correct after the first attempt.  If the player needs their second attempt, it would be interesting to analyze how many choices of the 18 remain, given that they have 0, 1 or 2 of the digits correct.

There are plenty of other games on the show which also have basic counting prinicple ideas worth exploring.  Some quick hits:

  • Balance Game – how many different ways can 2 bags be chosen from the given 3.
  • Dice Game – how likely is it to roll correct digits?  When should I choose higher or lower?
  • Golden Road – how difficult is it to advance to and win the big price at the end of the Golden Road?
  • Let Em Roll – how many different ways can the 5 special dice be rolled?
  • Make Your Move – how many different possibilities exist for moving tre sliders?
  • Race Game – how many ways can the price tags be placed?
  • Take Two – how many ways are there to choose 2 prizes from the given 4?

And some in-class ideas:

  • Let students choose a game to analyze.  Create a poster and share with the class.
  • Start each day of your probability unit with “a Game a Day”.  Start with the easy games, and move it to the more complex ones.
  • Have a contest where students design there own pricing games.

Thanks to my friends at Golden-Road.net for the fun pictures.

Adventures in Common Denominators

At my high school, I host “math lab” on alternate days.  The lab is an open room where students can obtain math help during their lunch or free period.  I like this assignment because I get to see a lot of different kids during the week, gain some insight into the approaches of my colleagues, and get my hands dirty in many math courses.

Last week one of my “regulars”, who often leaves class early for sports, came to the lab for Algebra 2 help.  He missed out on a adding/subtracting rational expressions lesson, but had the notesheets handy.  We wrote the first problem on the board:

Problem 1

Nothing too fancy. But in math lab, I don’t know the students as well as my own, so I need to do a quick check for some background before diving into new material. A spot check for understanding of fraction operations was in order:

Add Fracts

The student approached the board confidently and “added” the fractions:

adding

…sigh…. sometimes there isn’t enough coffee in the world. But all is not lost, and after a stare-down, the student recognized he had acted too quickly, and completed the problem correctly. This led to another problem. This time, I asked the student to just tell what me what the common denominator would be:

Problem 4

Without hesitation, the student knew the correct denominator to be 24. But why is it 24? The student could not defend his answer, but was absolutely sure 24 was the LCD. On the one hand, I am happy that the student has achived enough fluency with his number sense to confidently find the denominator. But, on the other hand, has lack of a process is going to hurt us now when we try to apply LCD’s to algebraic expressions.

TRY THIS WITH YOUR STUDENTS

When I start my lesson on adding rational expressions, I hand out index cards to every student.  I give students 2 minutes to repond to the following prompt:

How do you find a least common denominator? Provide directions for finding an LCD to somebody who does not know how to find one.

I collect all of the cards, shuffle them, and choose a few randomly to share under the document camera.  We will discuss the validity of the explanations, and use parts of the explanations to come up with a class-wide definition of an LCD.  Here is what you can expect to get back on the cards:

  • Some students will recognize that factors play a role, but won’t recognize that the powers of the factors are imporant.
  • Many students will attempt to use an example as their definition.  This allows for a discussion of a mathematical definition.  Is one example helpful in establishing a rule?  How about 2 examples?  How are examples helpful, if the reader does not know how to find an LCD?
  • Some students will provide a hybrid of the last two bullets.
  • If a student does provide a suitable definition, it’s time for you to play dumb.  Let the class assess the language and verify that the definition is, or is not, suitable.  In one class, I “planted” a working definition in with the student cards to see if they could identify a working definition.

The Least Common Denominator of two or more fractions is the product of the factors of all denominators, raised to the highest power with which they appear in any denominator.

HOW DO EARLIER MATH EXPERIENCES PROVIDE A SUITABLE BACKGROUND FOR RATIONAL EXPRESSIONS IN ALGEBRA?

I am curious how math teachers approach Least Common Denominators in earlier grades, and how these approaches translate to algebra success.  Here’s how a few online math sites approach LCD’s.

First up, mathisfun (search for “Least Common Denominator):

Mathisfun

This is the approach I suspect many teachers take to help students find LCD’s.  It works for manageable denominators, but becomes cumbersome when we have 3 or more denominators to consider, and certainly is not helpful in our algebra world.  Also, if you get too confused, you can use the “Least Common Multiple Tool” this website provides.  I suppose it’s not an inappropriate method, but a more algebra-friendly process should eventually develop.

Next up, Everyday Mathematics at Home website:

EM

The good news – we have a formal definition!

The bad news – you have to know what an LCM is to use it.

This is a more formal version of the Mathisfun example.  We could adapt it for use in algebra, but again, a definition of LCM is required here.

Khan Academy starts by using lists of multiples and provides and example with a trio of numbers for which we want the LCM:

Khan

The factor trees and the color verification that all 6, 15, and 10 are all factors of 30 is nice, but this example conveniently leaves out any scenario where a number is a factor multiple times, and this is the only example given.

Finally, let’s check out how PurpleMath tackles LCM’s, with a non-intuitive example:

PM

Now we are getting someplace.  Not only does this method stress the importance of factors, it shows the importance of include all powers of those factors.  And I could transfer this method easily to algebra class!

Have any insihgts into teaching LCD’s, either for a fractions unit, or in algebra?  Would enjoy hearing ideas, feedback and reflections!

Globe-Trotting With Random.org

The Common Core places increased imporance on statistics in middle school, beyond the tasks of creating simple data displays often encountered in middle-school texts.  The new standards require that students be able to describe distributions, compare samples to populations, and design simulations:

  • CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered
  • CCSS.Math.Content.7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

It’s all great table-setting for AP Statistics down the road, and working with authentic, interesting data.  In this activity, students use an online resource to perform a simulation in order to find the proportion of the earth’s surface covered with land (as opposed to water).  This is not a new activity, as a number of teachers suggest using an inflatable globe and some classroom tossing to reach estimates.  But I think the method below uses some web tools in a novel way, and encourages some authentic geography discussions.

GATHERING SOME INITIAL IDEAS

Before diving into any simulation, I like to gather ideas from my students, to see if they have any initial estimates or backround about the problem.  Using sites like Padlet or TodaysMeet are great for encouraging and archiving discussions and participation, or you can go old-school and just record initial estimates on the board.  Asking an initial questions like: “Do you know the percentage of Earth which is covered by water” to start the discussion.  The start a discussion of sample size: “Would it be better to sample 20 points on earth, or 40 points, or 100 points?  What factors would be part of your decision?”

COLLECTING DATA 

In this simulation, students will sample a random point on the earth’s surface, record whether the point is covered by land or water, and repeat for a given sample size.  For this, we will use the site Random.org, which generates random events, mostly things like numbers and dice, and their Random Coordinate Generator, which chooses a random location on earth and displays it using Google Earth.

Google Earth

The site works quickly, and the water/land data can often be determined without issue.  It is also easy to zoom in and out to take care of those “close calls”.  Quite a more accurate measure than the thumb-check data from the inflatable globes.  Very few issues arose in my trials with this method, but the biggest snag is Antarctica, which is land, but often appears light blue on Google Earth.  Also, a few rare occasions produced a data point above the North Pole on the map, which I discarded.  For your class, have each student generate a sample of size 10, and look at the proportion of land hits.  Below, I did 10 trials for 4 different sample sizes:

Samples

The next steps depend on the sophisitcation and grade-level of your class.  But in general, we want to know which sample size provides the best estimates.  How do you know?  Have students write explanations which defend a particular sample size.

Multiple sources (Circle graph from ChartsBin,NOAA Information) verify that about 29% of the Earth’s surface is land.  Do our trials verify this?  How often were our trials within 10% of this 29% mark?

Within 10%

As more data is collected, free site like StatKey can be used to generate appropriate graphs and statistics.

StatKey

FOR AP STATISITCS

I see this as an improvement of an existing AP Stats exploration, and opening activity for Confidence Intervals for proportions, and extension into the behaviors of CI’s.  For those playing along at home, here are the calculations for 2 standard deviations (Margin of Error) for my given samples:

2sds

And the corresponding intervals, showing how often my sample proportions were captured within each interval:

Intervals