The Math Price is Right

Perhaps some of the readers of this page will not appreciate the unique reference being made in the title of this activity to a famous TV game show, called “The Price Is Right“. If you are one of those, here is a brief description of that program, so that the math activity presented below will make sense.

On the program the contestants won nice prizes if they could guess the monetary value of the object in question: TV sets, stereos, or other valuable items. There were often three persons competing for the same prize simultaneously. Each would state his or her best estimate
of the price. The winner was the person whose estimate came the closest without going over! Simple idea, but effective. It depended highly on an individual’s number sense (a hot topic these days in the mathematics literature) and general good sense about the value of material objects
in today’s economy.

Now for the “Math” Price…

We can turn this basic idea of closest without going over into a math class activity that uses higher level thinking, calculators, and the concept of squaring a number (something very necessary when one enters algebra and advanced math). It goes like this:

The class is told that they’re going to play a game much like the TV show. They will do two things:

1. Choose a number.

2. Multiply it by itself. (This is the squaring idea.
And where the calculator comes in.)

If one’s result is the closest to some pre-set TARGET number, announced before the selection process of Step #1, then the goal has been accomplished.

Initially, only whole numbers would be used, as I am assuming that we are playing this game with say, 4th grade students. So a game may have gone something like this:

TARGET NUMBER: 500

1. Bob chooses 21 and Ralph chooses 22.

2. Bob’s square number is 441 whereas Ralph’s is 484.

Hence Ralph is the winner.

NOTE: if Ralph had chosen 23, his square of 529 would have been closer than Bob’s value, but it was over 500, hence could not win.

After play has gone on for some time, and the students are becoming more adept at playing it, it is recommended to start extending the game into other dimensions. One thing that can be done while still working with whole numbers is to use the concept of the “cube” of a number. This merely means that one uses the selected number three times as a factor in the multiplication step.

For example: 1728 is the cube of 12 because

12 × 12 × 12 = 1728

Obviously, larger target numbers need to be selected now. But that’s okay; the computation is not hard due to the use of the calculator. The hard part is the thinking! (Hmmm… but that’s good, too.)

A second thing that can be tried is the use of decimals. Even at the 4th grade level this should cause no great difficulty. We are, after all, talking about money here. And most primary school students are familiar with prices such as $12.95 and the sort. Returning to the squaring version of the game, we can proceed in this way:

Let’s use Bob and Ralph again. In trying to come close to 500 again, Bob might try 22.3, whereas Ralph chooses 22.4. Now when Bob squares his number he gets “497.29“. (Very close.) But poor Ralph! His square of “501.76” went over the target this time. So, he loses. What is nice about this feature of the game is that the squares of numbers in the “tenths” are numbers in the “hundredths“, which merely resemble money amounts. It is also important for students to see a fundamental pattern here, namely,

ab.x2 = cde.yy

[The reader is to understand that my focus is on the “x” and “y” parts;

a number with one decimal place has a square with two decimal places. It’s shocking how many students don’t observe this.]

Finally, the game can be turned into a single-person activity in this way:

How close can you come to a given target number, using the squaring procedure, if you are allowed as many guesses as you wish?

This takes the idea away from its competitive setting and puts

it in a problem solving one. This brings us back to recording our investigations in our old friend, the “T-chart“. Let’s see how it might look for a target of 200.

                         n   |    n2
                   ----------|------------
                       14    |  196  too low
                       15    |  225  too high
                     --------|---------
                       14.2  | 201.64  too high
                       14.1  | 198.81  too low

It is clear that 14.1 produces the winning value this time. If students can handle it, one could proceed to values of n that have 2 decimals places. The principal change here will be that the squares will have 4 decimal places, that’s all.

ONE FINAL COMMENT

Not to be overlooked in this work is that we are actually preparing the student for the concept of “square root” (and “cube root”) which will be confronted in the future, concepts that need careful development prior to their formal use in higher mathematics. If some groundwork is laid in the early years, then things will go more smoothly later on.

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