Deal or No Deal to Teach Probability

5th, 6th, 7th, Code/Programming, First Day Activities, Lessons, Probability, Statistics and Probability , , 0 Comments

Oftentimes, the class periods right before breaks are so crazy that it is hard to get much done. I have gotten in the habit of using these periods to introduce mathematical topics that might not otherwise be covered. This year, before thanksgiving, I set up a game of deal or no deal, so I could introduce my students to probability and expected value.

Deal or No Deal is a press-your-luck style game show where contestants try to win money. A contestant is shown a bunch of cases that contain between $1 and $1 million dollars. At the beginning of the game the contestant chooses a case but does not see its contents. Then they open other cases, each time revealing more cases which cannot be the one they chose. After opening each case they are offered a deal and they can take that deal or continue opening cases. Here's Howie's explanation:

I always found this show fascinating and thought it would be a fun game to play with students. I wanted the students to be playing for something tangible, so instead of imaginary money, I offered them skittles. I prepared 10 cases and blindly labeled them with letters. They represent 1, 5, 10, 25, 50, 100, 200, 400, 800, and 1000 skittles.[2]Logistically, counting more than 100 skittles is a waste of time. So instead, I found the mass of an average skittle and then used a scale to measure out the winning numbers. As you'll see below, I didn't have to do this often. In case you are curious, each skittle weighs about 0.0388 ounces. The image below shows sets of "cases" for each of my four classes.

IMG_20151119_112931052_HDR

This game was fun. Really fun! The kids had a blast and it was really exciting. Here's how they did:

6th Grade - Group 1
Highest offer: 351 skittles
Result: 10 skittles
Percent of highest offer: 2.84 %

7th Grade - Group 2
Highest Offer: 183 skittles
Result: 25 skittles
Percent of highest offer: 13%

6th Grade - Group 3
Highest Offer: 155 skittles
Result: 5 skittles
Percent of highest offer: 31%

6th Grade - Group 4

Highest Offer: 560 skittles
Result: 400 skittles
Percent of highest offer: 71.4%

Not a single group took a deal. Instead they kept opening cases until only their case was left. At each decision point, the class had to vote, so students used varying rationals to convince their peers. Some students used the gambler's fallacy while others used probability, expected value, and game theory. A few students even argued that risking a large amount of skittles would be worth it because they could gloat if they won a lot. These discussions presented lots of great opportunities to discuss probability topics.

I knew I needed a computer program to offer the deals or my students would jump on my case -- saying that's not a fair offer. So I wrote a program in python (you can try it out here). In the end, my program was probably a little too cruel when it came to offering deals. I had the program calculate the expected value based on the cases left and offer a percentage of the EV. At first I only offered 50% of the EV, but this climbed to 90% of the EV by the final deal. The next time I try this -- and there will definitely be a next time -- I am going to have the percentage of the EV climb more quickly.

Let me know if you have ever tried Deal or No Deal with your kids. I am also happy to help with the coding if you want to try this in your class.

Others' Ideas

Sarah wrote about using Deal or No Deal with her students. I really like that she had her students thinking about not just expected value, but the probability of earning more than a specific threshold: 100,000. My students were thinking about thresholds a lot. Most groups made some version of the argument, "Let's keep opening until we all get at least 10 skittles."

Thanks to Dan Meyer, Mr. Lucchese wrote about the mathematics of Deal or No Deal. He argues that the "banker" is trying to minimize the amount paid out by extending each contestant's time playing the game since more contestants means more money paid out.

Distance Model - Subtracting

2nd, 3rd, 4th, 5th, 6th, 7th, Algorithms/Models/Strategies, Computation, Content, Decimals, Estimated Grade Level, Fractions, Whole Number Add. & Sub. 1 Comment

Many students come to understand subtracting as taking away. For example, \(10 - 3\) becomes known as "ten take away three." But, this is not the only context subtraction can represent. It can also be used to find the distance or difference of two numbers.

Screen Shot 2015-10-31 at 2.04.22 PMTake-away models often lead to lots of regrouping. To solve \(3000-199\), most students (and frankly adults) do a lot of regrouping.

This is a complicated procedure that is hard for many student to make sense of. Teachers often use base-10 blocks to model the regrouping (for example 1 thousand becomes 10 hundreds) but my experience has been that many students still struggle to apply and justify this algorithm even in older grades.

While it is important to know how to regroup, modeling subtraction as a distance can provide students with an alternative method for solving subtraction problems. So, what does modeling subtraction as a distance look like?

Distance Model for SubtractingWhen you think of the result of a subtraction problem as the distance (really it's a vector) between the two given numbers (minuend - subtrahend) it is easy to see that you can adjust both the minuend and subtrahend by the same value and the difference will not change.[9]For visual proof of this, see the above gif. As the left and right end of the line segment increase or decrease by the same amount, the length of the line does not change. Put more simply, you can increase both the numbers in a subtraction problem by the same amount and the difference remains the same.[10]Technically you are increasing the first number and subtracting more with the second number.

Algebraically, it looks like this:

Statements Reasons
\(a-b=a-b\) Given
\(a-b+0=a-b\) Additive Identity[11]Any number plus 0 equals the given number.
\(a-b+(1-1)=a-b\) \(0 = 1-1\)
\(a+1-b-1=a-b\) Commutative Property[12]By treating \(-1\) as \(+(-1)\) you can move subtraction operands around as well.
\((a+1)-(b+1)=a-b\) Factoring[13]\(-b-1=-(b+1) \) because I factored out a \(-1\). I simply grouped \((a+1)\) to make the result easier to see.

 

Being able to envision subtraction this way yields efficient ways of subtracting all sorts of numbers. Here is @bstockus demonstrating it on twitter for whole numbers:

This process works well for whole numbers, but it really shines with fractions. Consider for a moment the challenge of this subtraction problem:

\(3\frac{1}{8}-1\frac{3}{4}\)

Solving this requires making common denominators, then regrouping from minuend by turning one whole into eighths or by making an improper fraction.

Either \(2\frac{9}{8}-1\frac{6}{8}\) or \(\frac{25}{8}-1\frac{6}{8}\)

If instead I increase both the minuend and subtrahend by the same amount, I can turn this into a much easier problem; one that does not require borrowing.

\((3\frac{1}{8}+\frac{2}{8})-(1\frac{6}{8}+\frac{2}{8})\)

\(3\frac{3}{8}-2=1\frac{3}{8}\)

The same move can be accomplished with decimals too. In summary, this strategy is really handy any time it is easier to do a bit of adding (or subtracting) instead of subtraction with lots of regrouping.

Strengths

  • Supports student understanding of a lessor used interpretation of subtraction
  • Provides an alternative to regrouping in subtraction algorithms
  • Can be used with whole numbers, fractions, and decimals
  • Requires introducing students to distance model of subtraction

Weaknesses

  • Requires that students are familiar with distance model of subtraction
  • Not as efficient in all circumstances

Sources[14]Leave a note in the comments if you know of another source where this model appears.

  • Ben shares that EMP and EngageNY use this model (and he shares some other great thoughts in the comments).

Brownie Pan Model - Multiplying Fractions

5th, 6th, Algorithms/Models/Strategies, Computation, Content, Fractions , , , , , 0 Comments

The Brownie Pan Model is an area model for multiplying two fractions. It is often used to introduce students to the idea of finding a part of a part in the context of sectioning a pan of brownies.

Strengths

  • Helps to illustrate why the traditional algorithm works \(\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}\)
  • Highlights part-of-part meaning of multiplication of fractions

Weaknesses

  • Sectioning can be challenging when fractions have large denominators or for students whose work is messy
  • Can lead to confusion when students try to model fractions greater than 1 with a single brownie pan or square[17]For example, to show \(3\frac{1}{2}\times\frac{3}{4}\) one would need \(3\frac{1}{2}\) squares/pans each with \(\frac{3}{4}\) shaded. Students who model this using 1 pan often lose track of what the whole is for each fraction.

Sources[18]Leave a note in the comments if you know of another source where this model appears.

  • Connected Mathematic Project 3 - Grade 6 - Let's Be Rational: In problem 2.1, the text uses this model to introduce the concept of finding part of a part

How It Works

This model helps to visually represent finding a part of a part. It is often introduced with story problems like this:

Stephen eats \(\frac{3}{4}\) of a brownie pan that has only \(\frac{2}{5}\) left. What fraction of the whole brownie pan does Stephen eat?

Brownie Pan-01Once students recognize that dividing the whole square into an array where the denominators are the dimensions, the process can be shortened into fewer steps.

In the shorter process below, students shade \(\frac{2}{5}\) and \(\frac{3}{4}\) of the whole by using horizontal and vertical lines. The overlapping region describes the product of the fractions.

In practice, this is done only in the final square (see: bottom square), but I showed an intermediate step for visual clarity.

Brownie Pan-02

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