problem solving non-standard problems

Problem SolvingNon-standard Problems

Sara Hershkovitz

Center for Educational Technology

Israel

Sara Hershkovitz, CET Israel, SEMT05 - Aug 2005

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Imagine a standard classroomof fifth-grade students, receiving the following nonstandard problems to solve:


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Problem No. 1

How many two-digit numbers, up to one hundred, have a tens digit that is

larger than the units digit?

Student’s answers:


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What if not…


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Problem No. 2:

How many two-digit numbers, up to one hundred, have a units digit that

is larger than the tens digit?

Student’s answers:


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Problem No. 3:

Look at the following numbers: 23, 20, 15, 25,which number does not belong?

Why?

Students' answers:


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Problem No. 4:

100 nuts are divided among 25 children, unnecessarily in equal portions. Each child receives an odd number of nuts.

How many nuts does each child receive?

Students' answers:


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Problem No. 5: (Elaborated after Paige, 1962)

We made change from one Shekel into smaller coins: 5, 10, and 50 Agorot (or cents) .

Make change so that you

hold three times as many 10-Agorot coins as 5-Agorot coins.

Students' answers:


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Problem No. 6:

A witch wants to prepare a frog drink.She can buy dried frogs only in packages of five or eight.

Note that the witch has to buy the exact number of dried frogs she needs .

How many frogs can she buy? What is the largest number of frogs she cannot buy?

Students' answers:


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What if not.…

In a similar witch problem,the dried frogs come in packages of four or eight. What is the largest number of

frogs that the witch cannot buy?

back


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Problem No. 7:

Insert different numbers in the blank spaces, so that the four-digit number received divides by 3.

__1 4__

Students' answers:

A teacher’s answer:


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Problem No 8:

I have a magic handbag. If I leave some money in it overnight, I find twice as much money in the morning, plus one unit.

Once, I forgot some money in the handbag for two nights, and on the third day I found 51 dollars.

How much money was in the handbag before the first night?

Students' answers :


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Discussion


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A. The types of problems:

How do the problems differ ?

What are the different types of problems? Can the types be characterized?

Write a few novel problems for each problem type .


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B. Promoting creativity:Creativity, as defined by some researchers (Guilford 1962, Haylock 1987, Silver 1994),

contains the following components :

Fluency: measured by the total number of replies.

Flexibility: measured by the variety of categories given .

Originality: measured by the uniqueness of a reply within a given sample of replies.


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Do such problems promote creativity?

Which additional types of problems can be employed for promoting creativity?


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C. Using these problems in the classroom:

We routinely use these problems in the classroom, approximately once a week, without connecting them to ongoing topics

studied in class .

Our goal is to allow students to experience the following:


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Extending and applying previous mathematical knowledge.

Constructing relationships among topics by integrating various topics within one problem.

Encouraging the use of different strategies: working systematically, while using naïve strategies.

Stimulating reflections on student experiences. Articulating the solution in various ways: words, or diagrams.


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We emphasize the following merits of class discussion :

An opportunity for students to explain how they think about problems; different problems provide different ways of analysis.

Individual ways of presentation may bring up disparate relations or different mathematical points of view.

Students are encouraged to understand and assimilate someone else’s strategies.


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A discussion emphasizes that mathematics entails active processes, such as investigating, looking for patterns, framing, testing, and generalizing, rather than just reaching a correct answer.

The discussion demonstrates that mathematical thinking involves more questions than answers.


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Thank you


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Problem No. 1 - Avi

T(Teacher): How many ……. (posing the question)?

A(Avi): Mmmm…. (thinking).

T: Do you understand the problem ?

A: Yes.

T: Can you find an example for such a number?

A: 52.

T: Ok; so how many two-digit numbers are there?

A: A lot .

.


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T: How many?

A: A lot.

T: How many? Give a number.

A: 8.

T: Tell me what they are.

A: 53, 42, 85, 31, 64, 97, 75, 61, 98; that’s all


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Problem No. 1 - Ben:I’m going to find them in the “First-hundred table:”

12345678910

11121314151517181920

21222324252627282930

31323334353637383940

414243444546474849 50

51525354555657585 960

61626364656667686 970

71727374757677787980

81828384858687888990

9192939495 96 979899100


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Now I will count them: 1,2,3,…..45.


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Problem No. 1 - Galit

21 ,30 ,31 ,32 ,Mmmm.

I see 10, 20, 21, 30, 31, 32 (thinking a bit before writing down each number.)

Then Galit began writing faster: 40, 41, 42, 43, 44, and deleted the 44.

T: Can you explain what you are doing?

Galit didn’t answer, and began writing quickly: 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74. 75, 76, 80, 81, 82, 83, 84, 85,

86, 87, 90, 91, 92,93, 94, 95, 96, 97, 98


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Galit raised her head and said: Now I have to count them: 1,2,3,4 ...

T: I saw that you began writing quickly. What happened at that point?

G: I saw that for each tens number in the list, I could write the numbers less one 1,2,3,4,5..,

T: Do you know how to sum the numbers by a shorter method?

G: 6,7,8,9,10,11,12,………43,44.45.

There are 45 numbers.


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Problem No. 1 - Dana

D: The smallest number is 20, Mmmm; no, sorry, 10,…. and the largest is …98

98 – 10 = 88 ;there are 88 numbers.

T: Mmmm.

D: Sorry, between the numbers there are also 55, and 34 and 28; I have to think. 20, 21 ……..30, 31, 32

T: You forgot 10.


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D: Yes, 10 , 20 ,21,

30 ,31 ,32 40 ,41 ,42 ,43

I see; In ten I have one number,in twenty there are

two numbers, in thirty there are three , in forty there are four; I can continue up to

ninety where there are nine .. So, 1+2+3+4+5+6+7+8+9=

1+2 is 3, plus 3 is 6, 10, 15, 21, 28, 36,….45. There are 45 numbers .


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Problem No. 1 - Hadar

110

22021

3303132

440414243

55051525354

6606162636465

770717273747576

88081828384858687

9909192939495969798


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Now I see that there are ten (while pointing to the arrows), twenty, thirty, forty,forty five …numbers.


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Problem No. 1 - Vered

Vered wrote in the columns:

2131415161718191

32425262728292

435363738393

5464748494

65758595

768696

8797

98


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Now I can sum them;8+7+6+5+4+3+2+1are

15 ,21 ,26 ,30 ,33 ,35 ,36.

Actually, I see the same unit in each row.

back


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Problem No. 2 - Ziva

This problem is symmetrical to the previous one, so the answer is the same:

45 numbers.


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Problem No. 2 - Chedva

If I think of the “First-hundred table ,”

100 has three digits, so 99 numbers are left;

subtract the previous 45, so there are 55 numbers.


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Problem No. 2 - Mick

Beginning with 10: 12, 13, 14. 15, 16, 17, 18, 19Beginning with 20: 23, 24, 25, 26, 27, 28, 29With 30: 34, 35, 36, 37, 38, 39..…………..…………

89 With 80--- :with 90 :

it’s 36 =8+7+6+5+4+3+2+1


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Problem No. 2 - Yoni

I’m thinking of the “First-hundred table .”

100 has three digits, and 99 numbers are left.

1-9 are one-digit numbers, so there are only 90 numbers left .

11-99 do not fit, as well, because they have the same digits; we have to remove 9 additional numbers, and then we are left with 81 numbers.

If we remove 45 numbers, from the previous

problem, we end up with 36 numbers.

back


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Problem No. 3 - Adi

20 – It is the only even number.


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Problem No. 3 - Bill

15 - The only number that divides by 3.

20 - The units digit is 0; It doesn’t have units.


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Problem No. 3 - Gur

15 - It is in the 2nd ten and the rest are in the 3rd ten.

20 - The only round number.

This number has more factors

23 - Not a multiple of 5.

25 - The sum of its digits is the largest.


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Reasons for choosing a certain

number as exceptional

15 It is under 20 Its tens digit is 1, and the rest have the digit 2 It is in the 2nd ten and the rest are in the 3rd ten It is the smallest number The only number that divides by 3


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20 The only even number A multiple of 2; divides by 2 The units digit is 0; it doesn’t have units The sum of its digits is doesn’t fit the series The number divides by 10 The only round number The only number divides by 4 The number that has more factors


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23 Not a multiple of 5 Doesn’t divide by 5 Not in the series that adds 5 to each number The only prime number Doesn’t appear in the multiplication table The only number that has the digit 3 in it


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25 A square number It is the largest number The sum of its digits is the largest Is 30 in approximation


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Distribution of

Number Property Categories Category15202325Total

Iconic properties75286109

Size consideration89124114

Additive series11718

Sums of digits25227

Multiples and divisors12341051152

Squares, Prime, Fractions, Negatives1739855

Evenness; Others 9613100

Total17719216838575


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Unique Replies

NumberThe given reason

15If you divide all numbers by 20, it is the only one that becomes a fraction

If you multiply each number by 100, it is the only one below 200


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20If you add 2 to this number, it remains

an even number.If you subtract the digits, it is the only negative numberIt is the only number that has the number 5 as its quarter (1/4)It is the average between 15 and 25


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23When subtracted from 30, it doesn’t divided by 5When multiplied by 2, it is not a round numberThe only one I cannot find an exercise for


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25The only number that didn’t have a reason to be exceptional

back


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Problem No. 4 - Aluma

It's impossible;

100 : 25 = 4That means that each child gets 4 nuts, but it is an even number.


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Problem No. 4 - Bat-Sheva

It's impossible ;

if we give an odd number of nuts to 24 children, together they have an even number of nuts ; odd + odd = even.

The 25th child can only get an even number of nuts.


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Problem No. 4 - Gidi

It's impossible ;

all the numbers I try are odd, and 100 is even.


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Problem No. 4 - Dotan

It's impossible.

Whenever we multiply an odd number by an odd number, the result is odd .

100 is always even, so it's impossible to divide it by 25 to get an odd number.


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Problem No. 4 - Hen

21 children get 3 nuts each .

One child gets 7 nuts.

Two children get 15 nuts each.

The last child gets 5 nuts.


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Problem No. 4 - Vivi

There are two possibilities:I 100 : 25 = 4; in this case each child gets an even number, and the answer

is wrong .

II Not all children get the same amounts of nuts; some get 3 nuts and the others 5 nuts.


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Problem No. 4 - Tzvi

Odd +odd = even.

There are 25 children; we can arrange them in pairs. So, 24 people have an even number of nuts together.. The 25th child doesn’t have a pair, so he must have an even number.

My conclusion: it is impossible to divide 100 nuts this way .


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Problem No. 4 - Yonit

I tried many combinations, and did not succeed.


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Problem No. 4 - Ofek

I tried to solve this problem with smaller numbers:

I divided 20 nuts.

I prepared a table, tried all the numbers, and did not succeed;

I think it is impossible.

back


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Problem No. 5: Avigail

Me and my friend have tried and couldn’t do it, so it is impossible.


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Problem No. 5: Bilha

1 coin of 50 = 50

3 coins of 10 = 30

4 coins of 5 = 20

100


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Problem No. 5: Gill

It is impossible ,because for each coin of 5 I have to take three coins of 10 (5+30=35)

If I multiply this sum I get 35+35=70and three times: 70+35=105.

back


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Problem No. 6: Arava

She can buy all the multiples of 8, of 5, and of (8+5).

She cannot buy the rest of the numbers.

There are endless numbers from both kinds.


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Problem No. 6: Bill

She can buy all the multiples of 5 and all the multiples of 8;

there are endless numbers she cannot buy.


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Problem No. 6: Gila

She cannot buy these numbers: 1,2,3,4,6,7,9,11,12,14,17,22,23,19,29,31.


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Problem No. 6: Dalia

I removed all the multiples of 5, 8, and 13 with their combinations.

The biggest number she cannot buy is 27.


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Problem No. 6: Hava

She cannot buy: 6,4,14,2,12,22,9,19,11,1,3,7,17,27

She can buy 8,16,24,32,40,48,56,64,72,80,88,96

All the unit digits: 5,0,8,6,4,2,9,3,1,7

The largest amount she cannot buy is 27.


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Problem No. 6: VikiWe began with the numbers she cannot buy, for

example, 1,2,3,4,6…We tried numbers up to 100.Then we tried to see how many frogs she can buy.

We figured out: she can buy all the multiples of 5,8,13,or common multiples as 26 (2*8+2*5).

We decided that she could buy endless amounts of frogs, as long as there were still frogs in the

shop and she had the money to buy them . back


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Problem No. 7 :

Adva2142

Gad

3141

Ben2143

Dalit

0141 ,1140 ,2142 ,3141 ,1143

There are many possibilities, but not endless.


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Problem No. 7: Hilla

1149

1140

1143

4140

1450

3141

2142

7140

2145

2448

3141


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Problem No. 7: Osnat

1140

1143

1146

1149

2142

2145

2148

3141

3144

3147

4140

4143

4146

4149

5142

5145

5148

6141

6144

6147

7140

7143

7146

7149

814281468148

9141

9144

9147

back


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Problem No. 7: A teacherModule1 – 1Function hh (a As Integer) As IntegerDim c As IntegerDim I As Integer, J As Integerb = a For i = 1 To 9 For j = 0 To 9 a = b a = i * 1000 + a + j*1

If a Mod 3 = 0 Then c = c + 1 Next j A = b Next ihh = cEnd Function

back


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Problem No. 8:

Adam

25 dollars.

51-1=50

50:2=25

Ben

25X2+1=51


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Problem No. 8: Galia

Each day, 1 + 2 X__= money in handbag

1 +2 X 8 = 17, 1 + 2 X 17 = 35

1 + 2 X 10 = 21, 1 + 2 X 21 = 43

1 + 2 X 11 = 23, 1 + 2 X 23 = 47

1 + 2 X 12 = 25, 1 + 2 X 25 = 51

that’s it!!!


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Problem No. 8: Dov

51 – 1 = 50 ,50:2 = 25

25 – 1 = 24 ,24:2 = 12

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problem solving non-standard problems

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