Positive and Negative Numbers

0 1 2 3 4 5 X

For a long time people had refused to believe in negative numbers. In our time, however, their existence is rarely questioned.

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Created by Inna Shapiro ©2007

Positive and Negative Numbers

0 1 2 3 4 5-1 -2 -3-4-5 X

Are you ready to take the Are you ready to take the challenge challenge of the negative of the negative numbers?numbers?

Problem 1• Find the rule and fill empty cells:

aa bb cc

55 -11-11 66

3232 -18-18 -14-14

-17-17 55

-14-14 1414

1111 5757

Answer

a b c

5 -11 6

32 -18 -14

-17 5 12

-14 14 0

11 57 -68

C = - ( a + b )

Problem 2

• Pete chose several positive and negative points on the coordinate line. Mary added all these numbers together and got 25.

Pete moved all points by 5 units to the left.

Mary added the new numbers together and got –35.

• How many numbers were chosen?

Answer• The sum was 25, and then it changed to -35.

• So the difference between the old sum and the new one is

25-(-35)=60

• That means that Peter chose

60/5=12 numbers

Problem 3

• Insert numbers into empty cells

so that each of them, starting from the third one, is equal to the sum of two previous numbers.

2 0

Answer

4 -2 2 0 2 2

-6 4 -2 2 0 2 2

10 -6 4 -2 2 0 2 2

-42 26 -16 10 -6 4 -2 2 0 2 2

-2 2 0 2

-16 10 -6 4 -2 2 0 2 2

26 -16 10 -6 4 -2 2 0 2 2

Filling of cells is shown on the schema, starting from the adjacent cells

Problem 4

• John has 9 cards with numbers

-6, -4, -2, -1, 1, 2, 3, 4, 6 on them.

• Can he choose some cards

so that the sum of chosen numbers

is equal to –8?

Answer

• He can choose different sets of cards:

• -6, -2

• -6, -2, -1, 1

• -6, -2, -4, 4

• -6, -2, -1, 1, -4, 4

• -6,-4, 2

• -6, -4, 2, -1, 1

• -6, -4, -2, -1, 2, 3

Problem 5

• Judy says that she can write 19 numbers in a row, so that the sum of each number with adjacent numbers from the left and from the right is positive, but the total sum of all numbers is negative.

•Is she right or wrong?

Answer• Judy is right. Her row is:

-7, 4, 4, -7, 4, 4, -7, 4, 4, -7, 4, 4, -7, 4, 4, -7, 4, 4, -7

Problem 6• Replace letters with numbers so that the sums

of the numbers in any row, any column, or main diagonals are all equal.

A 8 B -14

2 C -8 -4

-6 -2 D E

4 F -18 G

Answer• The diagonal sum is 4-2-8-14=-20, so

• A=-20-(4-6+2)=-20

• C=-20-(2-8-4)=-10

• B=-20-(-20=8-14)=6

• D =-20-(6-8-18)=0

• E =-20-(-6-2+0)=-12

• F =-20-(8-10-2)=-16

• G =-20-(4-16-18)=10

-20 8 6 -14

2 -10 -8 -4

-6 -2 0 -12

4 -16 -18 10

Problem 7

• Calculate:

-100-99-98-……-1+1+2+…+100+101

Answer• Let us change the order in the sum

-100-99-98-……-1+1+2+…+100+101=

= (- 100 + 100) + (- 99 + 99) + … + (-1 + 1) + + 0 + 101

= 0+101

• So the answer is 101.

Problem 8

• Calculate

1+2-3-4+5+6-7-8+…+301+302

Answer

• Let us group the numbers:

1+(2-3-4+5)+(6-7-8+9)+…(298-299-300+301)+302

(2-3-4+5)=0

(6-7-8+9=0) and so on.

• So the sum is 1+302=303 .

Problem 9

• Jill wrote a long sum:

17=17+16+15+14…+(X+1)+X

What was the number X in her sum?

Answer

• X=-16, because

17+16+15+14…+1+0+(-1)+…+(-16)=

= 17 + 0 = 17

Problem 10

• Fill in the cells with 1’s and –1’s so that the sum of the numbers in any 2x2 square is zero.

Answer

1 -1 1 -1

1 -1 1 -1

-1 1 -1 1

-1 1 -1 1