Promoting Creativity in Mathematics by Use of Non-standard Problems : Promoting Creativity in Mathematics by Use of Non-standard Problems Sara Hershkovitz & Pearla Nesher
Center for Educational Technology
&
The University of Haifa
Israel
Slide2 : Imagine a standard classroom of fifth-grade students, receiving the following nonstandard problems to solve:
Problem No. 1 : 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:
Slide4 :
What if not…
Problem No. 2: : 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:
Problem No. 3: : Problem No. 3:
Look at the following numbers: 23, 20, 15, 25, which number does not belong? Why?
Students' answers:
Problem No. 4: : 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:
Problem No. 5: (Elaborated after Paige, 1962) : 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:
Problem No. 6: : 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:
What if not…. : 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?
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Problem No. 7: : 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:
Problem No 8: : 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:
Slide13 : Discussion
A. The types of problems: : 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.
B. Promoting creativity: : 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.
Slide16 : Do such problems promote creativity?
Which additional types of problems can be employed for promoting creativity?
C. Using these problems in the classroom: : 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:
Slide18 : 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.
Slide19 : 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.
Slide20 : 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.
Slide21 : Thank you
Problem No. 1 - Avi : 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.
.
Slide23 : 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
Problem No. 1 - Ben: I’m going to find them in the “First-hundred table”: : Problem No. 1 - Ben: I’m going to find them in the “First-hundred table”:
Slide25 : Now I will count them: 1,2,3,…..45.
Problem No. 1 - Galit : 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
Slide27 : 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.
Problem No. 1 - Dana : 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.
Slide29 : 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.
Problem No. 1 - Hadar : Problem No. 1 - Hadar
Slide31 : Now I see that there are ten (while pointing to the arrows), twenty, thirty, forty, forty five …numbers.
Problem No. 1 - Vered : Problem No. 1 - Vered Vered wrote in the columns:
Slide33 : Now I can sum them; 8+7+6+5+4+3+2+1 are
15, 21, 26, 30, 33, 35, 36.
Actually, I see the same unit in each row.
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Problem No. 2 - Ziva : Problem No. 2 - Ziva This problem is symmetrical to the previous one, so the answer is the same:
45 numbers.
Problem No. 2 - Chedva : 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.
Problem No. 2 - Mick : Problem No. 2 - Mick Beginning with 10: 12, 13, 14. 15, 16, 17, 18, 19
Beginning with 20: 23, 24, 25, 26, 27, 28, 29
With 30: 34, 35, 36, 37, 38, 39
…………..
…………..
89 With 80: ---
with 90:
it’s
36= 8+7+6+5+4+3+2+1
Problem No. 2 - Yoni : 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.
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Problem No. 3 - Adi : Problem No. 3 - Adi 20 – It is the only even number.
Problem No. 3 - Bill : Problem No. 3 - Bill 15 - The only number that divides by 3.
20 - The units digit is 0; It doesn’t have units.
Problem No. 3 - Gur : 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.
Reasons for choosing a certain number as exceptional : 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
Slide42 : 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
Slide43 : 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
Slide44 : 25
A square number
It is the largest number
The sum of its digits is the largest
Is 30 in approximation
Distribution of Number Property Categories : Distribution of Number Property Categories
Unique Replies : Unique Replies
Slide49 :
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Problem No. 4 - Aluma : Problem No. 4 - Aluma It's impossible;
100 : 25 = 4 That means that each child gets 4 nuts, but it is an even number.
Problem No. 4 - Bat-Sheva : 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.
Problem No. 4 - Gidi : Problem No. 4 - Gidi It's impossible;
all the numbers I try are odd, and 100 is even.
Problem No. 4 - Dotan : 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.
Problem No. 4 - Hen : 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.
Problem No. 4 - Vivi : 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.
Problem No. 4 - Tzvi : 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.
Problem No. 4 - Yonit : Problem No. 4 - Yonit I tried many combinations, and did not succeed.
Problem No. 4 - Ofek : 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.
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Problem No. 5: Avigail : Problem No. 5: Avigail Me and my friend have tried and couldn’t do it, so it is impossible.
Problem No. 5: Bilha : Problem No. 5: Bilha 1 coin of 50 = 50
3 coins of 10 = 30
4 coins of 5 = 20
100
Problem No. 5: Gill : 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=70 and three times: 70+35=105 .
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Problem No. 6: Arava : 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.
Problem No. 6: Bill : 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.
Problem No. 6: Gila : 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.
Problem No. 6: Dalia : 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.
Problem No. 6: Hava : 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.
Problem No. 6: Viki : Problem No. 6: Viki We 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.
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Problem No. 7: : Problem No. 7: Adva
2142
Gad
3141 Ben
2143
Dalit
0141, 1140, 2142, 3141, 1143
There are many possibilities, but not endless.
Problem No. 7: Hilla : Problem No. 7: Hilla 1149
1140
1143
4140
1450
3141
2142
7140
2145
2448
3141
Problem No. 7: Osnat : Problem No. 7: Osnat
Problem No. 7: A teacher : Problem No. 7: A teacher Module1 – 1
Function hh (a As Integer) As Integer
Dim c As Integer
Dim I As Integer, J As Integer
b = 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 i
hh = c
End Function
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Problem No. 8: : Problem No. 8: Adam
25 dollars.
51-1=50
50:2=25 Ben
25X2+1=51
Problem No. 8: Galia : 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!!!
Problem No. 8: Dov : Problem No. 8: Dov
51 – 1 = 50, 50:2 = 25
25 – 1 = 24, 24:2 = 12
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