Category Archives: Math with Calculators

Math Activities that involve Calculators

Angle Activity

Background Comments

One of the more common lessons in measurement in middle school math classes is that of measuring the number of degrees in an angle. This, of course, implies the use of a fairly technical instrument, for students of this age anyway: the protractor. All those strange numbers going in two different directions is certainly more complicated than an ordinary ruler. But once that aspect is taken care of, there still remains the matter of homework exercises for practice. One thing that has always bothered me is that when a textbook shows several examples of angles to be measured, a big problem emerges. The angles are often rather hard to measure, from a physical standpoint. By this I mean they are often too small to fit the protractor being used; or they are positioned too close to the center of the opened book, hence it is difficult to make the protractor lie flat, etc. The obvious solution is to have the angles drawn on flat pages. This implies one of two things: a separate workbook or lots of single-sheet photocopies for the teacher to prepare. But wait! There’s another way. Why not have the student draw his own angles on his own paper before measuring? It certainly would give him added practice with the concept. But wait again! How would one know if all his work was being done at least reasonably well? With a large class doing such a home-work each with their own angles of all sizes, it would be a nightmare to evaluate, even in a general way. A solution to that dilemma would be: design a way to “force” the students to make angles of a size that the teacher knows in advance what the degree measure will be. This can be done. Watch.

The Lesson

About five years ago, I designed a lesson that involves three unusual allies: graph paper, the protractor, and a scientific calculator. Here is how it’s done.

1. First we will draw our angle on graph paper, using a set of coordinate axes.
2. Next we will measure it with our protractor.
3. Then we will use our calculator to check our measurement accuracy.

Example: Measure the angle formed by the x-axis and a ray that passes through the point (8, 6).

Solution: The angle has been drawn on a pair of coordinate axes, as shown below.

When we measure it, we see that it is approximately 37º.

But how can the student be sure that the answer is correct, or at least reasonably close? It is sometimes difficult to handle that instrument. So I told my students to get out their scientific calculators and do this key sequence:

1. Press 6 [÷] 8 [=] (Result: 0.75)

2. Press [Inv], or [2nd], or [Shift] (depending on one’s model)

3. Press [tan].

The result in the display is 36.869898, which rounds to 37 (to the nearest whole number), or 36.9 (to the nearest tenth). Our work was close enough to be acceptable.

And that, in a nutshell, is all there is to it. Simple, direct, and multi-faceted in approach. We used skills in plotting points in the coordinate plane, measuring with protractors, and utilizing some unusual keys (for middle school students anyway) on a calculator. By the way, no mention is made about the trigonometry concepts that are implied in the calculator checking process. My experience showed that the kids could accept the procedure for what it was: a way to find the number of degrees in the angle. They were instructed that in Step #1 that it was always “divide the y-value of the ordered pair by the x-value“.

Homework Sample

Exercises: Draw the angle first; measure it with your protractor; and finally do a calculator check.

1) (7, 5) 2) (2, 9) 3) (10, 3)

4) (6, 11) 5) (14, 12) 6) (15, 15)

tt(1/29/94)

Calculator Poker

Mathematical games are always a popular adjunct in any teacher’s repertoire of motivational activities. This game promotes learning and fun at the same time by combining two unlikely allies: a deck of ordinary playing cards and a calculator. The material presented here utilizes a simple, four-function calculator with a square root key. With minor adjustments, the more powerful scientific models may be used.

There are many ideas that a student can pick up while playing this game, not the least of which is the added familiarity with the calculator’s utility as a tool. Reading and writing the displayed digits carefully and precisely are skills that need to be developed in certain students. Of particular relevance to the game is the careful identification of which digits produce one’s best hand.

The basic game involves 3 players using 3-digit numbers. Remove the king, queen, jack and ten of any particular suit from the deck of cards. The nine remaining cards (Ace to 9) constitute the playing deck.

The value of the Ace will be 1. Each player uses his/her own calcuator, a recording sheet with headings: Number, Root, Combination, Points, a copy of the scoring chart (Figure 1), and a pencil.



RULES OF PLAY

  1. The dealer shuffles the deck and deals 3 cards face down to each player.
  2. Each player arranges his/her cards left-to-right, without turning them over, in any way desired.
  3. When all players have arranged their cards, the dealer says, “Turn over.” All cards are then turned face up without changing their left-to-right order. The 3-digit number thus produced is entered into the first column of the recording sheet.
  4. All players enter their numbers into their calculators and press the square root key. The displayed value, which represents one’s playing hand, is written in the next column of the recording sheet.
  5. Each player then carefully studies the digits of the square root to determine the highest scoring combination (see Score Chart) which is then written in the third column. For example, the number 142 yields the square root of 11.916375, and the three ones give the highest scoring combination of”three of a kind”.
  6. The points are determined by adding the base score given in the Score Chart to the value of the largest digit involved in the combination. The largest digit in the combination constitutes the Bonus Points. So, in the illustration above, 142 has a square root of 11.916375, for which the base score is 30 points (three of a kind: 1, 1, 1) plus 1 Bonus Point for the largest digit in the combination. After adding, the total (i.e. 31) would bewritten in the Points column of the recording sheet.

    Examples: If one has two pairs (6,6 & 2,2), the score would

    be 20 + 6 = 26 points.

    If one has a full house (4,4,4 & 7,7), the score is 50 + 7

    = 57 points.

    For a run (1,2,3,4,5), the player scores 100 + 5 = 105 points.

  7. For the 2nd and 3rd hands of a game, Rules 1-6 are followed as before. But for the 2nd hand, the square root key is pressed twice; for the 3rd hand, the square root key is pressed three times.
  8. After 3 hands are completed, the points are totaled and the winner is declared.
A Sample Game
Number Root Combination Points
795 28.195744 1 pair: 4, 4 14
641 5.0316972 all different 200
436 2.1376461 2 pairs: 6,6,1,1 26

This player’s score is 240, a good game.



ADDITIONAL SCORING COMMENTS

First, from time to time a perfect square is formed by the cards. Since here a square root (i.e. one press of the key) only has 2 digits and that isn’t enough to form any hand in the first part of the score chart, an award of 100 points is given. And once in a while, all the eight digits that appear in the display are different, or distinct. We feel that such an event is unique enough to merit the award of 200 points! Experience has revealed that occasionally only 7 digits (or even 6) appear, all of which are distinct. So a point value is assigned to each of these cases. However, no bonus points are given in any of the above cases.

When using scientific calculators with 10-digit display capacity, just instruct the players to copy the first 8 digits that appear, that is, “truncate to 8 digits”. While rounding the decimal part to achieve 8 digits is possible, our experience has shown that truncation is simpler to explain to the majority of students.



VARIATIONS OF THE BASIC GAME

There are many possibilities to vary the basic game: use more number cards at once (i.e. 2 sets of Ace through 9), deal more cards for each hand (i.e. 4 cards to form 4-place numbers), include one or

more jokers (they equal zero, or become a “wild card”), allow more hands to constitute a game, etc. Presented below are several more variations that have proved popular and instructive in our classroom. Of course, the reader is encouraged to devise his/her own rules.

  • Decimal Numbers. Once the three cards in the basic game are turned face up and the digits are noted on one’s paper, a decimal point can be inserted between the second and third digits, or prior to the first one. Example: for 382 we would have “38.2” or “0.382”. It is of interest to note that placing the point between the first and second digits does not producea different set of digits for one press of the square root key, as can be demonstrated by the example shown here.The reason behind this is well known to anyone who has studied the properties of radicals in an algebra class. However it is not necessary to go into the details of why in order to play the game. But it could be explained if students express an interest. So turn it into another of those special “teachable moments” we all look for. However, for 2 (or 3) presses of the square root key, different digits do appear.
  • Fractions Game. Form a fraction using the cards dealt to you.For instance, when using 4 cards per deal, let the first two form the numerator and the latter two form the denominator. In most calculators, this only amounts to converting the fraction to its decimal equivalent by dividing before pressing the square root key. For those who have calculators with fraction keys(as the TI Explorer and others), this is made even easier. This game variant, therefore, reinforces the connection between decimals and fractions.
  • Add-to-your-neighbor Game. Your playing number is found by adding your 3-digit “card number” to that of the player to your immediate left (or right) and putting the sum into the first column of the recording sheet. For instance, if the cards gave 329 for A, 416 for B, and 785 for C, then each player’s new number would be 745 for A, 1201 for B, and 1114 for C.
  • Third-Round-Sum Game. You form your playing number for the thirdround by adding the numbers used for the first and second rounds. With 624 for the first round and 179 for the second round, you use 803 for the third round. (This has the advantage of speeding up the game as there’s no need to deal the cards again.)
  • Constant-Multiples Game. Throughout your game you multiply yourcard number by some pre-set number, say your age, the day’s date, a favorite prime, etc. If Player A is 14 years old and his card value is 291, the playing number would be 4074. For students who can appreciate it, this is a good time to introduce the constant operation feature of the calculator and show that pressing the root key does not inactivate the constant multiplier as one goesfrom round to round.


WHAT ELSE?

An important spinoff can involve a statistical analysis of the data that can be collected after many games have been played by the whole class or merely a small group. For example, have the students

compute the arithmetic mean of various final scores: per hand, per game, per group of players, etc. An additional connection to probability can also be explored by constructing frequency histograms of the types of the hands that occurred throughout. The raw data is easily obtained from the students’ recording sheets.

One of our students showed us an important example of that “what-if” curiosity that needs to be promoted more in our classes. She thought: “What if I had arranged my cards differently? What might

have been my score?” So she took the card numbers from several of her games that day and proceeded to form all the various possibilities, then obtained the square roots and points. She was so proud of herself, as we were of her. What makes this more outstanding is that she was a below average performer, yet did this quite independently.

This game, in its own special way, provides a connection between the world of regular mathematics and that of simple gaming pastimes. It’s fun and one learns all the while.


CALCULATOR POKER

Score Chart

           Combination                          Base Score 

	One Pair                                  10 points

	Two Pairs                                 20 points

	Three of a Kind                           30 points

	Three Pairs                               40 points

	Full House (3 of one kind, 2 of another)  50 points

	Two Trios (2 sets of 3 of one kind)       60 points

	Four of a Kind                            70 points

	Five of a Kind                            80 points

	Run (5 consecutive digits)               100 points

	Special Cases:

	*Perfect Square                          100 points

	*All 8 digits different                  200 points

	*Only 7 digits showing, all different    150 points

	*Only 6 digits showing, all different    125 points

	[*No bonus points in these cases.]

				Figure 1

Sample Hands & Scores
Number Root Hand Points
158 12.569805 5, 5 15
145 12.041594 4, 4 & 1, 1 24
142 11.916375 1, 1, 1 31
138 11.74734 1, 1, 4, 4 & 7, 7 47
149 12.206555 5, 5, 5 & 2, 2 55
152 12.328828 2, 2, 2 & 8, 8, 8 68
485 22.022715 2, 2, 2, 2 72
147 12.124355 1, 2, 3, 4, 5 105
169 13 square 100
786 28.035691 all different 200

[Note: The square roots in this table were obtained from a simple, 4-function, 8-digit calculator.]


This article of mine is from MATHEMATICS TEACHING in the MIDDLE SCHOOL. NCTM. Feb. 1998.
pp. 366-8. Reprinted with permission. (See photos following.)


Egyptian Fractions Target Game

Recently I had the opportunity to play around with a novel middle school level calculator that “does fractions”. [For those who are curious, it is the Sharp EL-E300 model.] In fact, not only does it “do” them in the normal vertical format, but also it shows more than one fraction at a time in the display window. “Neat-o torpedo”, I thought. “Now what could I do with this clever instrument that would capture the interest and attention of my students while at the same time teach them some good stuff about this time-honored bug-a-boo of math at this school level?”

After spending some time reflecting on the matter, I decided I wanted to create a game that would involve various concepts: addition of 2 (or 3) common fractions, perhaps their subtraction as well, comparing relative sizes of 2 or more fractions, and decimal equivalents of fractions would be nice, too. Since estimation and number sense are hot topics these days, I wanted to include them also. And there was one more thing: how about a little historical connection thrown in for good measure!

The latter goal was easily met: Egyptian Fractions. (See my page on Egyptian Math for more on this topic.) From there, things just took a natural course, and I developed the little game that you will see described below. By the way, that’s an Egyptian fraction up there on the left side of the title bar; its modern form is given on the right.


Rather than presenting a lot of complicated rule descriptions of how to play, I think I’ll just describe how two students might go about a sample game. Our players are the ubiquitous John and Mary.

Play begins when their teacher announces that the “target fraction” will be “7/19“. The timer is set for 2 minutes. At the word “GO!“, John and Mary must each try to find two Egyptian fractions whose sum is as close to 7/19 as possible, without exceeding it, before time runs out.

If you recall your math history, you will know that John and Mary are looking for two fractions of the form

1
---.
x

Another name for such fractions is unit fractions, perhaps because the numerator is the unit “1”. However you choose to call them, the addition
procedure for such a pair is really quite easy. Observe:

                  1     1      b      a     b + a
                 --- + --- = ---- + ---- = -------
                  a     b      ab     ab     ab

In other words, the sum of two unit fractions is equal to the sum of the two denominators over the the product of the denominators. For more about all this, go to this WTM article: Fraction Addition.

[Recall that in Egyptian fraction work, all denominators must be distinct values, no repeats. So a is NOT equal to b.]


The two minutes are up. Let’s see how John and Mary are doing now. John chose 3 and 30 as his denominators, whereas Mary chose 4 and 11. So their respective sums are these:

          1      1     33     11             1      1     15
         --- + ---- = ---- = ----    and    --- + ---- = ----
          3     30     90     30             4     11     44

The next step is to determine which sum is closer to the target fraction (7/19) without going over. This can be done in two ways: fractionally or decimally. If speed of play is more important to you, then the decimal approach is probably better; otherwise, doing it by fractions could provide a different view of the situation. First, let’s look at the decimal way.

We convert the 3 fractions (target and 2 sums) to their decimal forms by simple division in our calculators, obtaining

target: 7/19 = 0.368421053

John: 11/30 = 0.366666667

Mary: 15/44 = 0.340909091

It’s clear that both players have obtained fractions whose values are less than the target’s value. It is considerably more helpful if we round our decimal expressions to some convenient level. Here I would recommend “to the nearest 1000th“, giving us these numbers. target: 7/19 = 0.368

John: 11/30 = 0.367

Mary: 15/44 = 0.341

Mere inspection, or simple subtraction, allows us to declare a winner here: John. His error was approximately “0.001” (“Very good, John.”) while hers was approximately “0.027” (“Not bad at all, Mary.”) “How about another game, kids? Try 13/47 as your target fraction.”


While they are busy on that, let’s return to the first game and discuss it some more, particularly the other, fractional, way to compare the results. This time we need to subtract the fraction sums from the target fraction. That is not so difficult as you might believe if we use the formula that was presented in the page, Fraction Addition, or rather a minor alteration of what was given there, namely:

                    a	    c	    ad - bc
                   ---  -  ---  =  ---------
                    b	    d	     bd

Using this formula on our target and sums fractions, these results are produced:

 7     11     210 - 209      1
  ---- - ---- = ----------- = -----
  19     30        570        570
  7     15     308 - 285      23
  ---- - ---- = ----------- = -----
  19     44        836        836

Again the results appear rather lopsided, even in fraction form, that 1/570 is smaller than 23/836. Checking it in general can be done in 2 ways: fractional and decimally. And again, decimally wins for ease of computation. The 2 differences have these decimal values:

1/570 = 0.001754385964912 or about 0.002

23/836 = 0.02751196172249 or about 0.028

[These figures agree, allowing for rounding of course, with what we found earlier.]

But to do it by fractions requires a bit more time; here’s how I would proceed. I like the “cross products” test.

                          836          13110
                    1         23
                  -----  <  -----
                   570       836

And since 836 is definitely less than 13,110, the left fraction is smaller than the right fraction. [I leave it up to you to ferret out the why’s and wherefore’s. :>) ]


Well, it’s about time to check in on our game players and see who is the winner this time. How’d you do, kids?

John says he used 4 and 50 as his denominators, while Mary chose 5 and 14. So the sums this time are

1      1      54      27             1      1     19
--- + ---- = ----- = -----    and    --- + ---- = ----
4     50     200     100             5     14     70

Getting right down to the nitty-gritty (i.e. decimal comparison strategy), we have these figures:

target: 13/47 = 0.276595745

John: 27/100 = 0.27

Mary: 19/70 = 0.271428571

While John’s sum produced a nice fraction with a terminating decimal form, and pretty close to the target fraction we might add, it’s easy to see that Mary’s choice is just a little bit better. (Girls get the point here!)


Extensions & other comments

Throughout the whole discussion of this game use of the calculator has been assumed at all moments. If one has access to a model which handles fractions, things are a bit more interesting. But such models are not required; the game can be played with even simple four-function types. The players just have to “know” a bit more mathematics. (And that’s not a bad thing either, is it?)

The time element of the game is enhanced if one’s calculator has a “replay” feature (as does the Sharp EL-E300 refferred to at the start of this page). Changing the numbers to investigate a “better choice” is more effecient with such calculators. And after one has gained a lot of experience there is a “trick” that is useful to recognize to make one’s search more efficient. Namely, after all is said and done, the error value (and not the sum of the fractions) is the most important information to be obtained. So one could combine and streamline the computation by using parentheses, or even not using them. For John’s work in the first game it would look like this:

7/19 – (1/3 + 1/30)
or 7/19 – 1/3 – 1/30

Now, by using the cursor keys and the replay feature, the denominators can be adjusted or changed rapidly, then the difference is more quickly
computed, even in decimal form.

Finally, one way to make this game a bit more challenging would be to require three fractions to be added. Now, things really start to get interesting.

Postscript

While playing around with the target fraction in the first game, I noted this:

1/5 + 1/6 + 1/570 = 7/19, exactly

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.

Distinct Digit Squares

INTRODUCTION

A. When a number is multiplied by itself, the resulting product is called a SQUARE NUMBER, or simply a SQUARE.

12 × 12 = 144 so 144 is a square number.
35 × 35 = 1225 so 1225 is a square number.
133 × 133 = 17,689 so 17689 is a square number.

B. Sometimes a square is made up of digits that are all different, that is, it has “no repeats”. Such a square is called a distinct-digit square (DDS).

Example: 13 × 13 = 169; there are no repeated digits in 169,
so it is a distinct-digit square.

But 21 × 21 = 441; since the 4 is repeated in 441, this is not
a distinct-digit square.

PROBLEM

You are to use your calculator to help you make a list of ten (10) distinct-digit squares. But–one more thing–they must all contain either 5 or 6 digits. That is, they should be “5-place” or “6-place” numbers.


Largest Number Squared

INTRODUCTION

If you multiply 142 by itself, what is the product?  _________
If you multiply 781 by itself, what is the product?  _________
Now look at your two answers.
The first one was a 5-place number, and the second one was a
6-place number, right?
(If not, you made a mistake somewhere.  Do the wrong one(s) again.)

PROBLEM I

You now see than when you multiply a 3-place number by itself, you might get a 5-place or a 6-place product.

Your problem is to use your calculator to find the largest 3-place number that when multiplied by itself gives just a 5-place product.

(Hint: The number is greater than 142.)

PROBLEM II

Compute these two products:

1022 × 1022 = ________
7803 × 7803 = ________

Do you see that the first product is a 7-place number, and the second one is an 8-place number? (If not, check your work as before.)

This time you are to find the largest 4-place number which when multiplied by itself will still only make a 7-place product.

(HINT: It is greater than 1022.)

PROBLEM III

Compute these two products:

17 × 17 = _______
83 × 83 = _______

Do you see that the first product is a 3-place number, and the second one is a 4-place number?

This time you are to find the largest 2-place number which when multiplied by itself will still only make a 3-place product.

(HINT: It is greater than 17.)

PROBLEM IVThe Brainbuster

You have done three problems with your calculator that were almost the same. Each time you had to find the largest number which
when multiplied by itself gave a product with an odd number of places,
right?

Now you will be asked to do the whole thing one more time–this is the BRAINBUSTER!

Find the largest five-place number which when multiplied by itself gives only a nine-place product.

But unfortunately, this time your calculator will not be able to help you; a 9-place number is too big for the calculator’s display area.

However, things are not so bad if you will look at the answers you found for the first three problems. There is an important clue there that will tame this tough problem. Do you see it?
CLUE PATTERN:

The largest 3-place product came from ______;
The largest 5-place product came from ______;
The largest 7-place product came from ______.

Same-Digit Pairs of DDSs

INTRODUCTION

In first section you found several squares that we called DDSs. (Remember: these are squares whose digits are “all different, no repeats”.)

In this section, we will explore something interesting about certain of those DDSs. Look at these squares:

37² = 1369 and 44² = 1936

Both 1369 and 1936 are DDSs, of course. BUT, there is one more thing that is strange: they both contain the same digits, just arranged in a different order.

There are many more cases like this. Before you start the exercise below, make sure you understand this idea by finding the squares for these two numbers: 32 and 49.

EXERCISES

In the groups of numbers below, two of them will give DDSs with the same digits, but arranged in a different order. The other numbers also produce DDSs, but do not have the same digits. Find the correct
pair in each group.

  1. 144, 175, 174
  2. 305, 153, 198
  3. 136, 228, 267, 309
  4. 233, 193, 305, 172
  5. 152, 142, 118, 179, 147

Below is given a large group of numbers that will give “same-digit pairs”, like you found above; some will not. Find the numbers that make this type of pair and put them together.

267 281 186 273 224
213 282 286 226 214

Once in a while we can find three or more DDSs that use the same digits. Look at this example:

36² = 1296     54² = 2916     96² = 9216

Do you see that all three squares contain the same digits, only in a different order. Now this is strange indeed! And it does not happen as oiften as was true for the same-digit pairs. But, as we will see, it can happen several times, if we are patient enough to look.

The following eleven numbers will produce DDSs that can be grouped into three same-digit families. Each family will have at least three members in it, maybe more. Can you separate all of them into their proper families?

181 148 154 128
209 203 269 196
191 302 178 .

So far, all of our DDSs have been only 5-place numbers. But the same thing can happen with 6-place DDSs, too. And, would you believe it? There are even more pairs and family-sized groups than you saw before.

Here are several numbers that will produce DDSs pairs or families. Can you separate them as you did before?

324 353 364 375
403 405 445 463
504 509 589 645
661 708 843 905


NOTE:

This piece was written by me and published in The Oregon Mathematics Teacher, Sept. 1978. At that time calculators with a 10-digit display were not the common models available to students at the elementary or middle school levels. So the “Brainbuster” problem above needs to be adjusted to take that into account, or only permit the use of 8-digit models while
doing this activity.


The M.O.M. Game Part 2

After students have become skilled in determining the LCM of two or more numbers via the calculator, they should be given the opportunity to use this ability for higher level thinking on more advanced situations. Therefore, this article will describe an activity that can be used to achieve this aim. Again teamwork and cooperation should be stressed.

First, instruct the students to set up a “t-chart”, as shown here:


			numbers  |  match
		     ------------|--------
		        3 and 4  |
			         |
		                 |
			         |
			         |

They should then find the “match” (LCM) for 3 and 4 by the method of the first lesson. It is 12, of course; it should be entered into the chart in the second column.

Little by little, one pair at a time the remaining pairs are presented to the students. After a while, the following t-chart will result:

 			numbers  |  match
		     ------------|--------
		        3 and 4  |   12
			4 and 5  |   20
		        5 and 6  |   30
			6 and 7  |   42
			7 and 8  |   56
			8 and 9  |   72
				 |

(If the teacher deems it desireable, a few more pairs may be offered. This is a flexible point.)

By now, the goal of the activity should begin to become apparent, at least to those individuals who have been trained and encouraged to look for patterns in their math work. Since so few have been so trained, especially at lower grade levels, say 4th grade or so, the teacher may
have to direct the students’ attention to the pattern. Here, in this
chart the “big idea” is:

The match for the two numbers is their product!

Or in other words: “the LCM of two consecutive counting numbers is their product.” This means, in practical terms, all the long work was really not necessary after all. But one should not overlook the value of the longer method. It presents the true conceptual meaning behind the term “LCM”, which is the object of our quest, namely to find the lowest (or
least) of the common multiples of the respective numbers, by actually
observing the multiples of those numbers.


Once the students have discussed this pattern, they should be directed to move on to Problem #2. It is shown in its final form below, but of course should produced in a manner similar to the first one.

 			numbers  |  match
		     ------------|--------
		        3 and 5  |   15
			5 and 7  |   35
		        7 and 9  |   63
			9 and 11 |   99
		       11 and 13 |  143
				 |

To be observed or pointed out is that the numbers used this time are consecutive odds. And, perhaps surprisingly (to some students, at least) the match is again the product of the two numbers being considered. The conclusion should be stated in words such as these:

The LCM of two consecutive odd numbers is their product.


At this point, students should be getting the idea that it’s pretty easy to find the LCM of two numbers: just multiply them and presto!, you have it. So it is time to present a situation where such is not the case. And you don’t have to look far to do it. Problem #3 considers
consecutive even numbers.

In order to motivate the search for the truth in this situation, a change to our t-chart is required — a third column is necessary.

 			numbers  |  product |  match
		     ------------|----------|--------
		        4 and 6  |    24    |   12
			6 and 8  |    48    |   24
		        8 and 10 |    80    |   40
		       10 and 12 |   120    |   60
		       12 and 14 |   168    |   84
				 |

One way to proceed is to ask the students to enter the products of the numbers, and then return to the old way to find the LCM. It should produce a bit of a shock that “now the product is NOT the same
as the LCM/match. But then, just what is going on here? Obviously, the
LCM is now half the product. Similar and related though it may be, this
problem is not the same as the first two. And therein lies the point:
before one draws a conclusion about a pattern, one should test and probe
the case thoroughly, double checking things as much as possible along the
way.

[Of course, for those who know a little bit more about number theory, the reason for the different result in #3 was that consecutive even numbers, or any two even numbers, are not relatively prime, whereas the number pairs in the first two problems were.]


To evaluate the learning that the teacher hope is occuring in this work, the following problem could be given:

Use consecutive multiples of three for the number pairs.

Here is the t-chart:

		    numbers  |  product |  match
		 ------------|----------|--------
		    6 and 9  |          |
	            9 and 12 |          |
		   12 and 15 |          |
		   15 and 18 |          |
		   18 and 21 |          |
		             |	        |

What happens now?

tt(9/7/93)


The MOM Game

LCM’s via the Calculator or The “M.O.M.” Game

One of the basic topics of elementary number theory is that of finding the Least Common Multiple of two or more numbers. Testifying to its basic simplicity is the fact that is presented as early as the 4th grade. But too often it is a concept that is only cursorily taught, as evidenced by the confusion that lingers long into one’s educational experience. How many times have you heard a student mix up the ideas of LCM with the Greatest Common Factor (GCF) — in an advanced math class?

Perhaps this confusion is caused by insufficient initial concept development. After an example or two are presented, the learner is expected to find LCMs through a cumbersome process of writing a list of several multiples for each number involved. (Later on an algorithmic procedure, like prime factorization, is introduced.) As long as the numbers are small and/or the LCM is found relatively soon, making a list is not a difficult matter. But let the numbers become a bit larger, and the task becomes something that most students would find tedious, unnecessarily time consuming, and unproductive as a learning activity.

Enter the calculator. Here is a way to make finding LCMs of bigger numbers an adventure. Use two calculators and two students, and it becomes an activity in cooperative learning.

First introduce the whole class how to use the constant add feature of the calculator. Show how this aspect of the calculator produces as many multiples as one needs or desires. (NOTE: some models start producing the next multiple of the number on the first press of the [=] key, others require a second press.)

Next, form pairs of students to work in a cooperative format. Present the two numbers for which the LCM is to be found, say for example, 12 and 14. Each member of the team enters one of the numbers into his/her calculator. Then each presses the [+] key. (Note: for Casio models, the [+] key must be pressed twice, causing a “K” to appear in the display.) From this point on, the [=] key will be pressed.

Team members now “take turns” pressing their own [=] keys according to whomever has the smaller number showing in the display window of his/her own calculator. When the numbers make a match — and they always will match, eventually — the goal has been reached, the LCM has been found. For the example cited above, this occurs with 84.

Turn:	 1    2	   3    4    5    6    7    8    9    10    11
----------------------------------------------------------------
"12":	24        36        48        60        72          84

"14":	     28        42        56        70         84

Of course, sometimes depending on the numbers, one person will press [=] two or more times in a row before the second person has his/her turn. Observe: let’s use the numbers 6 and 16.

Turn:	 1    2    3    4    5    6    7    8    9
-----------------------------------------------------
"6":	12   18        24    30  36        42	48

"16":            32                   48

So the LCM of 6 and 16 is 48.

It should now be clear that the process is basically a simple one, made more enjoyable by the use of a simple technology: the calculator. And students enjoy it more for that reason.

As a class-group activity, both factors of cooperation and gentle competition are involved. Present a pair of numbers for which the whole class is to find the LCM. Each member of each team takes one of the numbers. At the signal “Begin!”, all students start working on the problem. The first team to find the LCM is the winner.

The factor of cooperation is very important in this activity. If one member of a team presses [=] too hastily or incorrectly, his/her team might “overshoot” the LCM. So while speed is important, caution and careful observation of the values in the display window is advisable as well.

A natural follow-up question, after a student has had a certain degree of experience, is: if one continues the basic process after finding the match, when will the next match occur? The answer is, of course, not until twice the first match has been attained. If possible, this should not be revealed too soon to the learners; rather it is hoped that the result would be discovered by them.

One of the nice features of this activity is that there is no need to limit the problem to small numbers. The calculator adds large numbers just as easily and quickly as small ones. It can also be used as a short, 5-minute class-ending activity before the dismissal bell rings. Just have a few exercises prepared in advance. And it doesn’t cause any ‘mess”, as no paper or pencils are needed.

Try it! Your students will like it.

P.S. Why is this activity called the MOM Game? Well, it comes from the phrase of “Matching Our Multiples”. Get it?


Below are charts of prepared values for classroom use.

number pair LCM No. of multiples
14, 18 126 9, 7
20, 35 140 7, 4
16, 56 112 7, 2
26, 65 130 5, 2
28, 63 252 9, 4
33, 39 429 13, 11
45, 51 765 17, 15
56, 72 504 9, 7
55, 88 440 8, 5
45, 57 855 19, 15

This chart has numbers that are relatively prime.

15, 16 240 16, 15
15, 17 255 17, 15

This chart has values that were chosen by students. Note the interesting LCM’s that resulted. 1040 is the famous IRS tax form. And 714 is Jack Webb’s police badge number on his famous TV series “Dragnet”. They were chosen on 8/24/93 and 9/8/94, respectively, by students in the fourth grade of the Escuela Americana, San Salvador, E. S.

65, 80 1040 16, 13
51, 42 714 14, 17

Notice that the MOM Game can be extended to a three-person activity. The basic rules still apply. Here are some prepared values to use.

Number trios LCM No. of multiples
2, 3, 5 30 15, 10, 6
2, 3, 7 42 21, 14, 6
6, 8, 10 120 20, 15, 12
6, 7, 8 168 28, 24, 21

Notice the patterns in this chart.

3, 5, 7 105 35, 21, 15
6, 10, 14 210 same
9, 15, 21 315 same

This chart has a different sort of pattern.

3, 4, 5 60 20, 15, 12
4, 5, 6 60 15, 12, 10

And finally, when three students have a lot of “time on their hands”, or should I say “fingers”?

9, 13, 15 1755 195, 135, 117