Appendix A: Questionnaire for determining the level of arithmetic thinking
development1
1. Below are groups of three numbers. Please circle the group whose numbers
cannot be used for composing any arithmetical exercise:
a. 3; 3; 9;
b. 15; 14; 1; c. 2; 4; 12; d. 100; 10; 10; e. None of the groups
can be used.
Explain your answer.
2.
Below are geometrical shapes
Please indicate the correct statement:
a. All the shapes are divided into 4 congruent parts, therefore each of the colored
parts is a 1/4;
b. The colored parts are not equal in their area, consequently none of the parts
is a 1/4;
c. Only in some shapes the colored part is a 1/4;
d. Only in a circle and a square we can indicate a part which is a 1/4;
e. None of the answers is correct.
Explain your answer.
3.
4.
Please circle the correct answers:
50 equals …
a. 5 digits of units; b. 50 digits of units; c. 0 digits of units;
tens; e. None of the answers is correct.
Explain your answer.
d. 50 digits of
1
2
5
Below is an exercise: 1 : (2  1 ) 
2
6
6
Please indicate what is the correct answer in your opinion:
𝟏
a. 0; b. 1𝟐; c. 1; d. 3; e. None of the answers is correct.
Explain your answer.
5.
Below is an exercise without the decimal point in the product2: 5.5 x 3.2 = 176
Please circle the correct answer:
a. 1.76; b. 0.176; c. 17.6; d. 176; e. None of the answers is correct.
Explain your answer.
6.
Below are two numbers in which some of the digits have been deleted.
Number A: * * * 3 * *
Number B: * * * 8 *
Which of the following claims is correct:
a. Without all the digits it is impossible to compare the numbers;
b. In Number A there are more places, therefore it is bigger;
1
The questionnaire is an adaptation of a questionnaire built by Usiskin, Z. (1982).
Taken from: Markovitch, Z. (1999). A sense for numbers. Tel Aviv: Yessod
Publishing. [Hebrew]
2
1
c. In Number A the digit of thousands is smaller than the digit of thousands in
Number B, therefore Number B is bigger;
d. 8<3, therefore Number B is bigger;
e. None of the answers is correct.
Explain your answer.
7.
Below is a table with the representative rate of exchange of a foreign currency
during one week:
Day of the week
Foreign currency rate of exchange
Monday
4.021
Tuesday
4.12
Wednesday
4.102
Thursday
4.012
Friday
4.201
Saturday
4.02
Please circle the line where the foreign currency rate of exchange is arranged in
an ascending order (direction from left to right):
a. 4.02; 4.012; 4.021; 4012; 4.201; 4.102;
b. 4.02; 4.12; 4.012; 4.201; 4.021; 4.102;
c. 4.021; 4.12; 4.102; 4.012; 4.201; 4.02;
d. 4.012; 4.02; 4.021; 4.102; 4.12; 4.201;
e. These numbers cannot be arranged in an ascending order.
Explain your answer.
8.
On Daisy Street the houses on both sides of the road are numbered from 1 to
2003.
While renovating the street, it was decided to order digits designed of ceramics, a
separate tile for each digit. What is the total amount of tiles which the craftsman
should prepare?
a. Approximately 200; b. Approximately 500; c. Approximately 1000;
d. A lot;
e. None of the answers is correct.
Explain your answer.
9.
Below is an exercise: 1116 + 704 + 258 + 884 + 296
In order to calculate quickly the total amount, we will use rules of mathematics in
the following order (please circle the correct answer in your opinion).
a. ( 1116 + 296 ) + ( 704 + 884 ) + 258;
b. ) 1112 + 227 ( + ) 892 + 407 ( + 852;
c. ) 1112 + 407 ( + ) 892 + 227 ( + 852;
d. ) 1112 + 407 ( + ) 852 + 227 ( + 892;
e. ) 1112 + 892 + 852 ( + ) 407 + 227 (;
Explain your answer.
3
This question is taken from a collection of irregular problems of the Centre of
Educational Technology, Tel Aviv, Israel.
2
1
10. The first addend in a sum of two addends is decreased by 1 .
2
2
What should be done with the second addend so that the sum increases by 3 4?
2
a. Increase by 3 4;
1
b. Decrease by 1 2;
c. Increase by 2;
d. Increase by 5;
e. None of the answers is correct.
Explain your answer.
11. If we add 1 to the numerator and denominator of a common fraction, a new
number is obtained.
Whatever the initial fraction is:
a. The obtained fraction will be bigger than the initial fraction;
b. The obtained fraction will be smaller than the initial fraction;
c. The initial fraction will not change;
d. It is impossible to draw an irrefutable conclusion;
e. None of the answers is correct.
Explain your answer.
12. How many times can we subtract the natural number a from the natural number b
so that the result is also a natural number?
The most effective rule for solving the question is:
a. Addition; b. Subtraction; c. Multiplication; d. Division
e. None of the answers is correct.
Explain your answer.
13. It is customary to define division of fraction in this way4:
a c a d
:  X when b,c,d ≠0.
b d b c
Is it possible to present the division differently?
a. No, there is only one way which is correct.
a c a:c
b. Yes, for example, : 
when b,c,d ≠0.
b d b:d
Explain your answer.
14. The total of five 2-digit numbers is smaller than 1005.
Please choose an assertion which is always true:
a. Each of the numbers is smaller than 20;
b. One of the numbers is bigger than 60;
c. If four numbers are bigger than 20, then one number is smaller than 20.
4
Tirosh, D. (1996). Mathematics and research. Tel Aviv: MOFET Institute. [Hebrew]
5
Markovits, Z., Hershkowitz, R., Bruckheimer, M. (1989). Research into Practice:
Number Sense and Nonsense. Arithmetic Teacher, 36 (6), pp. 53-55.
3
d. If two numbers are smaller than 20, then at least one number is bigger than 20.
e. No conclusion can be drawn without checking all the options.
Explain your answer.
15. Is it possible to divide by 7 the total of two numbers, none of which can be
divided by 7?
a. Yes, it is possible because _________________________
b. No, it is not possible because _______________________
16. The following claim is known to be correct:
For every three natural numbers a, b, c (a>c), the equality a+b = (a-c) + (b+c)
holds.
What follows from it?
a. Only after substitution we can be sure that this claim is correct for the certain
triplet numbers;
b. The claim is correct for many triplets of natural numbers but not
necessarily for all the natural numbers;
c. This claim is correct for all natural numbers.
e. None of the answers is correct.
Explain your answer.
17. A number is given.
Which of the following statements is correct:
a. If the number can be divided by 2, this means that it can be divided by 12
and, hence, it can be divided by 4;
b. If the number can be divided by 2, this means that it can be divided by 4
and, hence, it can be divided by 12;
c. If the number can be divided by 12, this means that it can be divided by 4
and, hence, it can be divided by 2;
d. If the number can be divided by 4, this means that it can be divided by 2
and, hence, it can be divided by 12;
Explain your answer.
18. Two theorems are given:
I. If a and b are two natural numbers, then m=
a
is a rational number.
b
a
is a rational number, then a and b are natural numbers.
b
Which of the following claims is correct:
a. In order to prove that I is correct, it is enough to prove that II is correct;
b. In order to prove that II is correct, it is enough to prove that I is correct;
c. In order to prove that II is correct, it is enough to find one pair of natural
numbers whose quotient is a rational number;
d. In order to prove that II is incorrect, it is enough to find one rational number
in which a or b are not natural numbers;
e. Claims a-d are incorrect.
Explain your answer.
II. If m=
19. Below is a claim:
"Division by 0 is a meaningless operation in mathematics".
4
This claim is correct because:
a. This is a definition;
b. This is an axiom;
c. 0 cannot be used in division;
d. Such a rule contradicts the 4 rules of mathematics;
e. Claims a-d are incorrect.
Explain your answer.
20. In the following theorems, a, b, c, d, n, k are natural numbers.
A given: a can be divided by c.
A claim: the sum (a+d) can be divided by c, if c+a=d.
In order to prove this claim one should rely on:
a. If each of two natural numbers can be divided by a natural number c, then their
sum can also be divided by c;
b. If one of the numbers can be divided by c and the sum of a+b can be divided
by c, then b can be divided by c;
c. If a can be divided by c with a remnant k,
and b can be divided by c with a remnant n,
and k+n = c, then the sum of a and b can be divided by c;
d. Either claim A or claim B;
e. Either claim B or Claim C.
Explain your answer.
5